domain: spi phys

text:
Hco + Ch

Mehdi Ayouz

Jean-Michel Gilet

Pierre-Eymeric Janolin

Viatcheslav Kokoouline

Waste treatment BF 2 + ,PF 2 NO 2 , N 2 O

Keywords: Théorie du défaut quantique, méthode R-matrice, collisions électron-molécules, transformations de repère, excitation rovibronique, recombinaison dissociative, applications plasmas Theoretical description of electron-driven processes in astrophysical and technological plasmas: Applications for development of materials, waste treatment processes, and atmospheric depollution Quantum defect theory, R-matrix method, electron-molecules scattering, rovibronic frame transformation, rovibronic excitation, dissociative recombination, plasma applications

Description théorique des processus de collisions électroniques dans les plasmas astrophysiques et technologiques: applications pour le développement de matériaux, les processus de traitement des déchets et la dépollution atmosphérique

fundamental importance in a plethora of research fields ranging from engineering to plasma. The present habilitation compiles my achievements and contributions within each of the following research axis: (i) mechanisms of destruction of molecular ions in astrophysics and waste treatment, (ii) role of collisions involving electron-neutral molecules and electron-radicals for atmospheric depollution process and (iii) formation of fluorine based-molecules in the development of materials. We developed theoretical approaches based on first principles to compute cross sections and related rate coefficients for electron-impact rotational, vibrational and electronic excitations as well as dissociative recombina-tion of various molecules. These outcomes could be used later in plasma kinetics modeling in order to improve the performance of a process.

Another contribution within the physics education axis will be also outlined in this habilitation. We developed a complete course of undergraduate quantum mechanics based on numerical experiments, whose the beta version is available on http://prdmecaqu.centralesupelec.fr/index.html. This pedagogical tool is currently tested at Centralesupélec and at the University of Central Florida. Finally, I sketch the planned research work for the next 4-year period. This work is a continuation of the above research items.

Introduction

Over the last decades, low-temperature plasma science and technology has attracted the attention of many research groups due to a plethora of applications ranging from material development (semi-conductor and photovoltaic industries [Duchaine et al., 2012, Gonzatti et al., 2010, Bartschat and Kushner, 2016] for instance), to waste treatment ( [START_REF] Bundaleska | Air-water 'tornado'-type microwave plasmas applied for sugarcane biomass treatment[END_REF]), to air pollution control (atmospheric depollution [M andE, 1993, Baeva et al., 2002]) and to astrophysical environments (interstellar medium [Lepp et al., 2002, Millar et al., 2017]), nay to nuclei fusions reactors ( [START_REF] Hagelaar | Model of an inductively coupled negative ion source: I. general model description[END_REF], Boeuf et al., 2011]) and to the biomedical ( [START_REF] Boudaïffa | Resonant formation of dna strand breaks by low-energy (3 to 20 ev) electrons[END_REF], Shomali et al., 2015b]) or recently to food processing [START_REF] Misra | Cold plasma in food and agriculture: Fundamentals and applications[END_REF].

The above applications and advances were supported by significant efforts in understanding the mechanisms governing the plasma sources, especially those involving collisions between electron, atoms and molecules. Numerical plasma simulations require two main ingredients: (i) a physical model for describing the discharge plasma and (ii) a numerical algorithm for solving the set of equations for different species. In modeling plasma, there are two approaches: kinetic (atomistic) and fluid (hydrodynamic). In the former approach, one tracks the position and velocity of each particle of plasma, or more exactly a macroparticle representing a large number of particles of a given specie, by solving either the equations of motion (Newton's Law) or kinetic equation, viz. the Boltzmann or Vlasov-if collisions are ignored-equation, for a space-time distributions of the charged species energy or velocity distribution probabilities. As an example of kinetic models, we can cite the Particle In Cell with Monte-Carlo Collisions (PIC-MCC) which is commonly adopted to simulate the plasmas immersion ion implantation (PIII) for material development (see paper .7 dealing with the study of BF + 2 ).

On the other hand, in the fluid description, the plasma is considered as a continuous medium (macroscopic picture) where average quantities such density, temperature and mean velocity are of great interests. A set of equations for conservation of species mass, momentum and energy, derived form the Boltzmann equation, is then solved. Each approach has its own advantages and drawbacks. When some species are described with the kinetic approach and the others with the fluid one, we have the hybrid model. To describe the plasma chemistry, a 'reactive flow' model that describes the coupled phenomena of chemistry and diffusionconvection transport of chemical species is combined with the kinetic or continuum discharge model. For a self-consistent plasma simulation, it is ought to determine the self consistent electromagnetic field taking into account the plasma sheath space charges and/or the high frequency plasma oscillation using either Poisson's (electrostatic approximation) or Maxwell equations. As it is beyond the scope of this habilitation (Habilitation à Diriger des Recherches) to discuss details on plasma models and numerical algorithms, find here an interesting reviews vii viii Overview of research works [van Dijk et al., 2009], [START_REF] Samukawa | The 2012 plasma roadmap[END_REF] and [START_REF] Adamovich | The 2017 plasma roadmap: Low temperature plasma science and technology[END_REF], that give a detailed snapshot of the state of the art in this field.

As hinted above, characterizing the plasma chemistry, in order to improve the performance of a process, requires tremendous amount of data from atomic and molecular processes. Typically, atomic and molecular processes can be classified into surface reactions and gasphase reactions. The former are characterized by the adsorption process of a reactant, a chemical process between two adsorbed species or an adsorbed species and a gas phase species, and desorption of a product from the surfaces of small grains. It is generally acknowledged that surface reactions are responsible for formation of complex organic molecules in space [START_REF] Garrod | Complex chemistry in star-forming regions: An expanded gas-grain warm-up chemical model[END_REF]. As for gas-phase reactions, we can categorize them into one-, twoor three-body reactions. For example, an excitation/dissociation of molecules by electrons impact, e + M → M * , is a two-body reaction. The rate equations of species M that experience such reactions may be written

∂n M * ∂t + ∇ • (n M * v M * ) = Γ M * (.1)
where v M * is the velocity of specie M * and n M * its density. Above, Γ M * designates the rate production of M * per unit of time and volume. Note that solving the above equation, with the transport flux term (involving diffusion and convection) in the left-hand side (the second term), could be accomplished if the velocity v M * is well known. The latter can be obtained from solving the specie momentum equation.

The most striking feature of Eq.(.1) is the presence of Γ M * , which can take the following simplified form

Γ M * = αn e n M ( .2)
with α or k (in cm 3 •s -1 ) being the rate coefficient of the two-body reaction. As we will see in the next chapters, this coefficient is a macroscopic quantity depending on the cross section collision and the energy distribution probabilities of the colliding particles (see Equation I.31 in Chapter I). The challenge here lies in the determination of this rate coefficient which serves as input parameters in computer simulations of plasmas. Moreover, accuracy in the rate coefficients is very important since the uncertainty could affect the prediction of simulations. This HDR attempts to address these issues, at least, partially for certain fundamental elementary processes that are relevant at low-energy collision.

In this context, my main research interests are largely motivated by the lack of reliable data for atomic and molecular reactions especially those involving electron-molecule collisions at low-energy. In low-temperature and pressure plasmas, characterized by electron energies of the order of ionization potential of atoms/molecules (∼ 10 eV) and pressure of few Pa, electron-molecule collisions are dominant because electrons interact with electromagnetic fields and collide with the gas to produce radicals and excited neutral species in chemical reactions [Kolobov, 2009]. Availability of accurate cross sections and related rate coefficients for processes such as (i) rotational, vibrational and electronic excitations (RE,VE,EE), (ii) dissociative recombination (DR) and (iii) dissociative electron attachment (DEA) is crucial for plasmas modeling, as discussed above. Studies in this field can be conducted only in close ix collaboration with plasma experimentalists and modellers (see list of collaborations in CV). This report presents an overview of my research works as assistant professor at LGPM (from 2012 to 2020). Figure .1 resumes my achievements as well as the motivations regarding to applications in low-temperature plasma science. Over the last years, LGPM acquired a large expertise in chemical-based processes for environmental applications and in situ analysis of molecules of astrophysical interest (through the MSL-Curiosity mission in collaboration with NASA). My research fits into this context and is mainly focused on the development of theoretical approaches to study processes relevant for material science, waste treatment, and atmospheric depollution. This field of research is part of the LGPM project, proposed to the HCERES committee, in which the modelling of materials at different scales is one of the two target areas of excellence for the next five-year period.

The recent key achievements were motivated by the following:

need of understanding basic mechanisms in plasma kinetics in laboratory, in the interstellar medium (ISM), or in technological applications;

identify key elementary processes needed in plasma modeling;

x Overview of research works develop theoretical approaches for the identified processes, especially, those involving polyatomic molecules for which experimental and theoretical data are sparse;

in theoretical modelling of the processes, the need of taking into account that the presence of low-energy electronic resonances in electron-molecule collisions;

an importance of uncertainty estimations of the theoretical models when experiential data are not available, especially, for processes involving excited-state ions or such ions as radicals. These excited-state ions and radicals are unstable in collisions with other species present nearby;

the use of theoretical approaches as complement of experimental data;

improvement and validation of theoretical models by building collaborations with plasma experimentalists and modelers;

enrich KIDA, IAEA and LxCat databases for atomic and molecular reactions needed in various applications.

Within this wide context, the main goal is to develop and improve theoretical methods dictated not only by the lack of the fundamental understanding, but very practical interests in applications in plasma engineering, plasma-depollution technologies, space sciences and technologies (see Figure .1). Considering the cross-discipline importance of these applications (in physics, chemistry, astrophysics, plasma physics, materials), it is necessary to build an approach combining (i) the identification of the role of atomic and molecular species and key processes leading to their formation and destruction, (ii) development and improvement of theoretical models describing these processes, and (iii) experimental and theoretical validations with the help of collaborations.

The above lead to the following main research axes: I Mechanisms of destruction of molecular ions (in astrophysics and waste treatment). II Role of collisions involving electron-neutral molecules and electron-radicals (in atmospheric depollution).

III Formation of fluorine based-molecules and oxide particles (in the development of materials).

Besides the above research items, I would like to report in this manuscript my contribution in the field of physics education. I will further describe our initiative of transforming the traditional undergraduate and graduate quantum mechanics (QM) courses of Centralesupélec (CS) and the University of Central Florida (UCF) into actively engaging studio-mode styles. This pedagogical initiative led to the following achievement :

IV Toolkit for teaching quantum mechanics.

Over my experience of teaching QM and computational physics at CS, I have noticed the difficulties that students encounter in understanding the QM concepts. Typically, QM is taught as a pure math course or a descriptive phenomenological discipline, in which different xi core concepts of quantum physics appear to students to be disconnected from each other and from their real-life experience. Initially, the students are curious and participate in a course but very quickly, the more the course progresses, the more they lose their motivation and interest in studying QM. Nowadays, there is a broad consensus that the use of multimedia in QM classes helps to make teaching QM more efficient. This conclusion highlights the link between teaching and numerical tools as well as theoretical methods, developed in the above three main research axes for describing and understanding different fundamental elementary processes.

The present document compiles my achievements and contributions within each above items. Since details on the employed theoretical approaches and the obtained results have been published (see list of publication in Appendix V.2.C), I will restrict myself to underlining the major ideas, sometimes with the inclusion of a few illustrations or tests. This manuscript is organized as follows.

In Chapter I, I will present briefly my works on a variety of molecular ions, namely, HeH + , CH + , CH 2 NH + 2 and NH 2 CHOH + . My main contribution to this topic is the development of theoretical methods to compute DR and VE cross sections of molecular ions relevant in astrophysics and waste treatment. The presented works were carried out in the context of two thesis: Dr. Xianwu Jiang at CS and Dr. Chi Hong Yuen through a long term collaboration with UCF.

Chapter II is devoted to neutral molecules and radicals. Equipped with the theoretical methods developed on molecular ions, I extended them to neutral systems and computed VE cross sections of NO 2 and N 2 O that play an essential role in the atmospheric depollution processes. The outcomes were obtained in the context of Dr. Hainan Liu thesis at CS.

It is important to note that my past research also encompasses other applications such as metallic material and bio-material characterizations. For the sake of clarity, I will expose in Chapter III only the works on BF + 2 , playing a role in development of materials, since the study of metallic materials lies rather on describing experimentally/theoretically the formation of MnO oxides on a Fe-Mn alloy surface whereas the second one focuses on thermal and mass characterizations of bio-materials using the Lattice Boltzmann methods (based on the kinetic theory of gases). Both studies either require ab initio calculations (Li Gong's PhD on the determination of surface energy) or are based on methods similar to those developed in quantum physics (Mathilde Louërat and Huan Du PhDs on the calculation of thermal and mass diffusion coefficients). It should be stressed that the above research were carried out in close collaboration with experimentalists and modelers in plasma and material fields. Moreover, the BF + 2 study has enabled to develop and formulate ideas for further applications in particular for the HCO + and N 2 H + molecular ions in the context of Dr. Abdillah Abdoulanziz's thesis, co-supervised with Prof. Ioan Schneider at Université du Havre.

On the strength of my experience in numerical simulation and programming, I attempted to contribute to physics education by developing a set of PYTHON numerical codes (published in textbooks. See Appendix V.2.C) and designing a graphical interface allowing students to prepare and run numerically-simulated experiments demonstrating all basic concepts taught in QM course. An overview of the performed work, conducted in the context of the post-doc xii Overview of research works of Dr. Alexander Korovin and in collaboration with Prof. Viatcheslav Kokoouline at UCF, is given in Chapter IV.

Finally, Chapter V concludes this report and sketches the planned research work for the next 4-year period. Contributions within each of the above items and communications are also given. Most relevant published papers related to each above items are provided in Appendix.

I -Mechanisms of destruction of molecular ions

I.1 Context and Motivations

A number of process can occur when an electron collides with molecular ions. In lowtemperature ionized environments such as interstellar medium (ISM), planetary atmospheres and technological plasmas (see Figure .1) (where the pressure is low and the electron density is high), the main route for the ions destruction is the dissociative recombination (DR). This results in stable and metastable small species and/or radicals that are in turn the basis of chain of reactions leading to more complex species. Furthermore, DR competes with transitions between the rotational, vibrational and electronic states of the target ions. Hence, detailed knowledge of DR and competitive processes is necessary for modeling very cold and rarefied environments, often in non-equilibrium.

In this chapter, I will present my works on two types of ions: (i) those with low-energy electronic resonances appearing for geometries near the equilibrium of the target ion (in a fixed-nuclei picture) and (ii) the ions without such low-energy electronic resonances. Two theoretical treatments were developed in Section I.3 to be applied in Section I.5 to systems relevant in ISM as HeH + , example of ions of the first type, and in technological plasmas (as well as ISM) such as CH + , that have the two first excited electronic state at a relatively low energy, below 5 eV. With some additions, the theoretical method developed for ions of the first type was extended to more complicated systems with several vibrational degrees of freedom, such as NH 2 CHOH + or CH 2 NH + 2 (illustrated below) of planetary atmospheres interest. The theoretical method was initially introduced for BF + 2 with few vibrational degrees of freedom, viz. four-folded. This theory will be described in details in Chapter IV. The basics of theoretical approaches developed for both diatomics and polyatomics are similar and exposed in the next Section I.2.

Cross-sections and related rate coefficients of the electron-induced dissociation and/or excitation of molecular cations presented in this chapter were obtained in the context of collaboration with Prof. V. Kokoouline at the University of Central Florida (UCF) and two dissertations: Dr. Xianwu Jiang at Centralesupélec (CS) and Dr. Yuen Chi Hong at UCF. The following works have been published in different papers. Only relevant studies and related articles are included in Appendix V.2.C, with the inclusion of some illustrations or tests. The obtained results could serve in kinetic modeling of ionized media such as cold plasmas and 2 Chapter I. Mechanisms of destruction of molecular ions interstellar clouds.

I.2 The basics of the theoretical approaches

Generally, electron-molecule scattering calculations are performed either in the laboratoryframe (with respect to which the molecule rotates) in the framework of the Close-Coupling (CC) theory [START_REF] Arthurs | The theory of scattering by a rigid rotator[END_REF] or in the body-fixed frame making use of the Fixed-Nuclei (FN) theory [Chase, 1956].

In CC theory, a set of coupled second-order ordinary differential equations is solved for one-electron wave function where the total wave function is expanded on the unperturbed states of the isolated molecule, typically, via the products of functions of the internal states of the molecule (called a channel ) with angular functions (often spherical harmonics) describing the rotation of electron around the molecule. The scattering (or reaction) matrix can be deduced when matching the propagated one-electron wave function, from the internal region to a distance where the potential is negligible, to a well know asymptotic form [Lane, 1980]. This method was applied for studying electron-atom/molecule and atom/molecule-molecule collisions [mot, 1966, McGuire and Kouri, 1974, Green, 1975, Bohr et al., 2014, Mukherjee and Mukherjee, 2015, Zammit et al., 2017]. However, including the rovibrational states into CC equations introduces complications into the calculation, which is especially the case for polar molecules where the long-range nature of the dipole potential requires a large number of partial waves to be considered [START_REF] Sanz | Electron scattering cross sections from hcn over a broad energy range (0.1-10000 ev): Influence of the permanent dipole moment on the scattering process[END_REF]. This leads to a strong rotational coupling in the equations and, consequently, slow numerical convergence because the laboratory-frame, where the scattered electron is referred, is not appropriate at small electron-molecule separation, where the electron is close to the molecule. This discussion will be completed in Section II. 4 on the NO 2 study. More details can be found in Ref. [Lane, 1980].

On the other hand, FN theory adopts the philosophy of the Born-Oppenheimer approximation (BO) [START_REF] Born | Zur quantentheorie der molekeln[END_REF] in which the motion of nuclei can be separated from that of electrons because they are munch lighter (about 2000 times) and, hence, move faster than nuclei in a molecule. Performing bound states (electronic structure) calculations is thus possible and one can find the energy spacing of electronic, vibrational, and rotational levels of the order of 1000, 100, and 1 meV, respectively. The energy-time uncertainty principle enables to obtain the respective time scales 10 -17 , 10 -16 , and 10 -14 s. This result reveals that time scale for electronic processes is usually much shorter (about 10 or 100 times) than the nuclei motion, in agreement with the BO consideration. In a similar spirit, the motion of electrons in the continuum, in the framework of FN approximation, is fast in such a way that the nuclei can be viewed as fixed during the scattering process. This assumption, however, break downs when the scattering electron energies are close to excitation thresholds or to a narrow and long-lived resonance, where the collision time becomes long. As for CC, collision with a polar molecule, characterized by a strong long-range interaction potential, is also a circumstance for which FN approximation can not be readily applied.

In this context, the Frame Transformation (FT) approach of [Fano, 1970] takes the advantage of both representations by exploiting the features of electron-molecule interaction in accordance with the electron scattering distance. Indeed, close to the nuclei, the electron undergoes a strong Coulomb acceleration and acquires a high velocity such that nuclei can be viewed as being stationary, justifying the use of FN approach in the framework of BO approximation. The electron nearby the nuclei form together a new complex that should be treated as many-electron system. The rising exchange and electron-electron correlation interactions lead to a strong coupling between the angular momentum of the scattered electron and the internuclear axis (the angular momentum component is constant of motion in case of diatomic molecules). In such situation, the body-frame representation is more appropriate than the laboratory-frame. On the contrary, when the electron is far from nuclei, its collision energy becomes comparable to the energy of nuclei motion and their respective angular momenta remain uncoupled. Consequently, nuclear Hamiltonian has now to be introduced to take into account the effects of the rotational/vibrational molecule making thus the laboratory-frame more appropriate. In this representation, vibrational and rotational quantum numbers of the target molecule, in addition to the angular momentum of the scattered electron are utilized. The total angular momentum is here a collision constant.

The theoretical methods developed on diatomic (described in next section) and polyatomic molecules (described in next chapters) employs the above approach where once the scattering matrices are obtained in the framework of FN approximation, FT is performed for describing interactions between rovibronic channels of the target molecules induced by the incident electron. Nowadays, there are two types of computational approaches in the FN approximation that are the most employed; variational approaches and bound states approaches. The Kohn Complex Kohn Variational method [START_REF] Schneider | Complex kohn variational method: Application to low-energy electron-molecule collisions[END_REF]] is based on the variational principle, where the T -matrix is chosen as the variational quantities instead of the K-matrix [START_REF] Takatsuka | Extension of the Schwinger variational principle beyond the static-exchange approximation[END_REF]. Some recent applications making use of this method could be found in [Douguet et al., 2015a, Douguet et al., 2015c]. A second approach is the Schwinger Multichannel Method which is based on the Schwinger variational principle for obtaining the scattering amplitude [Schwinger, 1947]. This method utilizes the Green's function matrix element and the electron-molecule interaction method. Details on this approach can be found in Ref. [START_REF] Da | Recent advances in the application of the schwinger multichannel method with pseudopotentials to electron-molecule collisions[END_REF]] and references therein.

The ab initio R-matrix methods, to which we further direct our main attention, are other methods based on variational or on bound states approach [Burke, 2011]. They adopt the philosophy of the FT procedure by considering two regions of space in agreement with the interaction nature between an electron and a molecule. Compared to the above variational approaches, R-matrix method has a great advantage that the inner region problem is solved independently of the scattering electron energy. The dependence on that energy has only to be considered in the outer region where obtaining solutions is somewhat fast and straightforward. In practice, the employed inner region in FT approach is extended to a large distance and the whole problem can be thus solved in the framework of the body-fixed frame representation. This assumption called Adiabatic Nuclei Approximation is the central idea of R-matrix method. It is valid under certain circumstances, as stated above for the FN approximation, where electron energies are away from excitation thresholds or long-lived resonance and there are no significant long range interactions. The validity of these assumptions and, hence, of the theoretical models employed in the performed studies will be justified in the following section.

I.3.A

The ab initio R-matrix method

Initially introduced by Wigner [Wigner, 1946, Wigner andEisenbud, 1947] to treat nuclear reactions, R-matrix methods were early extended to study the scattering of electron-atom [START_REF] Burke | Electron scattering by complex atoms[END_REF]Robb, 1976] and electron-molecule [START_REF] Burke | R-matrix theory of electron-molecule scattering[END_REF], based on the developments of Schneider [Schneider, 1975, Schneider andHay, 1976]. A comprehensive reviews on the methodology and applications in atomic, molecular and optical processes can be found in [Burke andTennyson, 2005, Burke et al., 2007].

In the following studies, the open-access UKRmol suite [Carr et al., 2012] with Quantemol-N interface [Tennyson et al., 2007] will be used. Recent applications can be found in [START_REF] Loupas | Cross sections for electron scattering from thiophene for a broad energy range[END_REF], Gupta et al., 2019, Loupas and Gorfinkiel, 2019]. The implemented R-matrix approach is drawn in Figure I.2. This method aims in calculating, as its name suggests, the R-matrix (in its simplest form): R(r, ) = F(r) rF (r) , (I.1) that relates the radial wave function F(r) to its derivative F (r) at a radius r and for a specific scattering electron energy .

The above equation dictates the central idea of the R-matrix method: partition of configuration space into an inner and outer regions. The boundary is a sphere, with a radius noted r = a, centered at the center of mass of the molecule. This radius is chosen such as the electronic charge cloud of the target is negligible at the boundary. Usually in the inner part of configuration space, exchange and polarization interactions are dominant between N -electrons target and the scattering one. As a result, the collision problem within finite volume can be treated as a molecular bound state problem by constructing and diagonalizing an Hamiltonian matrix augmented with the Bloch operator (holding the Hermicity of the Hamiltonian) leading to solve a N + 1 eigenvalue problem (see Eq. (8) of Ref [Tennyson, 2010]). The scattering wave function is constructed using close-coupling expansion where the scatterer spin is taken into account and the target wave function is represented through a Configuration Interaction (CI) expansion. A standard quantum chemistry methods are employed to obtain the variational parameters in the CC expansion.

In the outer region, asymptotic expansion such as a long range multipolar interactions between the scattering electron and the target is employed. For target ions, the dominant term is Coulomb potential of the separate charges, viz. α = 0 in Eq.(II.12). Once the wave functions are generated in the inner region, the R-matrix is constructed at the boundary r = a for a given incident energy and it is propagated to the outer region (see Eq. (23) of Ref [Tennyson, 2010]) where the radial scattering electron wave function can be matched at the boundary r = a asy 70 bohr (such that coupling potentials to be negligible) to Coulomb functions

F Γ ij (r) ∼ 1 √ k i sin θ i (r)δ ij + cos θ i (r)K Γ ij (R) if i ≥ 0 0 otherwise (I.2)
where

θ i (r) = k i r - l i π 2 + ln(2k i r) k i + σ i , σ i = arg Γ l i + 1 - i π , (I.3)
and index i refers to the entrance channel before a collision and j refers to the outgoing channel after the collision. In the above equation, K Γ ij (R) are the elements of the reactance matrix [Aymar et al., 1996, Tennyson, 2010] depending on the nuclei geometry, here the internuclear distance R, parametrically and obtained for each scattering electron energy . Above, Γ designates the constants of motion corresponding to the irreducible representation of the molecular point group, in particular the total angular momentum projections Λ of the electron-molecule system.

In the cross section calculation, the scattering matrix

S Γ (R) is needed, which is obtained from K Γ (R) as S Γ (R) = (1 + ıK Γ (R))(1 -ıK Γ (R)) -1 . (I.4)
The above matrices relate the amplitudes of the incoming and outgoing waves functions. Another observable defined from these scattering quantities is the eigenphase sums (analogous to the phase shift of potential scattering)

δ Γ ( , R) = i arctan K Γ ii or δ Γ ( , R) = i log S Γ ii 2ı , (I.5)
where K Γ ii (resp. S Γ ii ) are the eigenvalues of the K-matrix (resp. S-matrix). We will see further that eigenphases are very helpful for testing the validity of the scattering models, identifying potential problems with calculations and studying the calculations convergence. Table I.1: Correlation tables for D ∞v → D 2h and C ∞v → C 2v point groups for diatomic molecules. In the same rows, irreducible representations Γ of the same symmetry are provided.

D ∞v D 2h C ∞v C 2v Σ + g A g Σ + A 1 Σ + u B 1u Σ - A 2 Σ - g B 1g Π B 1 ⊕ B 2 Σ - u A u ∆ A 1 ⊕ A 2 Π g B 2g ⊕ B 3g Π u B 2u ⊕ B 3u ∆ g A g ⊕ B 1g ∆ u A u ⊕ B 1u
In practice, our calculations are performed in the abelian subgroups D 2h , C 2v , C s or C 1 of the natural point groups of the molecules. Descent correlation tables for the most popular I.3 Theoretical methods for collisions between electron and diatomic molecular ions7 point groups encountered in our studies are given in Table I.1. We choose a large R-matrix sphere of radius ranging from a = 10 to 14 bohrs, according to the size of the studied systems, and continuum Gaussian-type orbitals (real spherical harmonics for the angular part) with partial waves l ≤ 4, viz. a total of 25 partial waves mapped to the irreducible representations of the abelian point groups in Table I.2. Thus, an element of the scattering matrix in Eq.(I.4) writes S Γ l λ i ,lλi (R) where l, λ (noted sometimes m or Λ, in case of Σ electronic states of the target) and i enumerate, respectively, the electron angular momentum and its projections on the molecular axis Z, and the electronic state of the target molecule. This element represents the scattering amplitude when the electron scatters from one channel (lλi) to another (l λ i ), while the nuclei do not have time to move.

I.3.B Frame transformation

As stated in Section I.2, the theory of frame transformation connects the laboratory frame and the body-frame representations that are each suited for different ranges. This is based on the fact that the couplings between the scattering electron and the target are different according to the r regions.

Lower panel of Figure I.2 depicts these regions, namely, the R-region, v-region and the j-region. In the first region, the strength of couplings, induced by the scattering electron, between different vibrational and rotational states (v,j) of the target and the scatterer are small in comparison with the interaction potential V (r), suffered by the incident electron. Therefore, the FN approximation is justified in this R-region (recall that R refers collectively to all nuclei coordinates) and the R-matrix calculations can then be performed where the scattering quantities (see above) depend on the nuclei geometry R parametrically. We will come back to this point in Section I. 4 Schematic diagrams for the R-matrix method (upper panel) as well as the R-/q-region, v-region and the j-region (lower panel) according to the local kinetic energy of the electron given in Eqs.(I.6) and (I.7). The frame transformation procedures (FT) connecting the molecular-frame to the laboratory-frame are also represented. Note that these transformations constitute the l-uncoupling transition from the Hund-coupling case b to case d [Chang and Fano, 1972].

Now, where the interaction potential V (r) becomes comparable to the strength of couplings between different v and/or j, these later can not be neglected and hence vibrational frame transformation and rotational frame transformation need to be accomplished in v-region and j-region, respectively. Transitions between each regions can be estimated by defining the local kinetic energy

vj = -V (r) -E vj , (I.6)
where is the scattering energy and V (r) is the attractive potential. E vj is the rovibrational energy of the target that can be expressed as

E vj = ω v + 1 2 + Bj(j + 1), (I.7)
white ω and B being respectively the vibrational frequency and the rotational constant of the target in the harmonic and rigid rotator approximations.

I.3 Theoretical methods for collisions between electron and diatomic molecular ions9

Making use of the above expressions, the values r v and r j in Figure I.2 can be estimated for -V (r v ) ω and -V (r j ) B. It follows that, when r < r v , one can apply a vibrational frame transformation (nay full frame transformation) assuming then vj ≈ v j ; while for r < r j , only rotational frame transformation can be applied with the assumption vj ≈ vj . In the region r > r j , FN approximation is no longer valid because the nuclei and electrons have to be treated on a more equal footing, justifying the laboratory-frame description. These considerations are the basis of the theoretical models described below.

Vibrational Frame Transformation

Once the matrix element S Γ l λ ,lλ (R) is obtained from the R-matrix calculations for a given geometry R and each symmetry (one of the irreducible representation Γ of the point group corresponding to a given projection Λ of the total angular momentum of the electron-molecule complex), one can employ the procedure of frame transformation

S Λ λ v l ,λvl = ψ v (R) S Λ λ l ,λl (R) ψ v (R) , S Λ λ l ,λl (R)δ(R -R) = R ; λ l Ŝ |R; λl , (I.8)
where the brackets above imply an integration over the vibrational coordinates which requires the vibrational wave functions.

Bound vibrational wave functions: We calculated vibrational wave functions ψ v (R) above and related energies E v by solving the Schrödinger equation for vibrational motion along R,

- 2 2µ ∂ 2 ∂R 2 + V (R) ψ v (R) = E v ψ v (R) , (I.9)
where µ the reduced mass and V (R) is the internal potential of the considered molecule. The above equation is solved using a DVR-type method, described in details in Section IV.2.B. In the numerical calculation, the distance R is discretized on a grid, on which the potential energy V (R) is calculated with the ab initio methods (plus a basis set) with the help of Molpro package [Werner et al., 2008].

Dissociative vibrational wave functions:

To compute dissociative recombination (DR) cross section of CH + , for instance, the vibrational continua states need to be considered. They could be discretized by the outgoing-wave basis functions defined by a complex absorbing potential (CAP). Similar to the technique of Siegert pseudostates calculations [START_REF] Tolstikhin | Siegert pseudostate formulation of scattering theory: One-channel case[END_REF]], a purely imaginary potential (CAP) is added to the above Schrödinger equation (I.9) in order to simulate an infinite grid. Different shapes of CAPs are described in the literature [START_REF] Vibok | Parametrization of complex absorbing potentials for time-dependent quantum dynamics[END_REF], we choose an exponential form

W (R) = A 5 exp -2L R-R CAP R ≥ R CAP 0 R < R CAP , (I.10)
where A 5 is the CAP strength and R CAP its starting point. L specifies the entire grid length of the internal potential V (R) of CH + . These parameters will be adjusted

Chapter I. Mechanisms of destruction of molecular ions

in the studies to reduce the uncertainty on the obtained DR cross sections.

Applying CAP, the Hamiltonian becomes non-Hermitian and eigenenergies turn to be complex (as eigenfunctions. See Figure I.14 for instance)

E v = E v -ı γ 2 , (I.11)
where E v and γ are the position and width (inversely proportional to half-time τ = γ ) of the dissociative state.

When plugging the dissociative vibrational wave functions into Eq.(I.8), the obtained scattering matrix appears to be subunitary in the bound and dissociative states basis. This defect of unitarity, i.e. loss of flux, can be identified with the dissociative flux.

Finally, summations over the partial waves (lλ) and (l λ ) yield the transition amplitude S v v from one vibrational level v to another v . This latter is used to compute the vibrational (de-)excitation (VE,VdE) cross sections.

Rovibrational Frame Transformation

As a second step, the rotational frame transformation is accomplished using the matrix elements of the above S-matrix, leading to the laboratory-frame scattering matrix

S J j µ l v ,jµlv = λλ (-1) l +λ +l+λ C j µ l -λ JΛ C jµ l-λJΛ S Λ l λ v ,lλv , (I.12)
where J is the total angular momentum of the e --ion system, j, µ and j , µ are the angular momenta with their projections on the molecular axis of the target before and after the rotational excitation of the projectile, and C j µ l -λ JΛ and C jµ l-λJΛ are Clebsch-Gordan coefficients. Note that J is specified in the above S-matrix because j is defined from the Clebsch-Gordan coefficients. Summing over l, l and J leads to the rovibrational (de-)excitation (RVE,RVdE) cross sections. A detailed derivation of Equation (I.12) and ensued cross section are given in Appendix of paper .4.

Note that when several electronic states of the target are involved in the calculations, rovibronic frame transformation has to be accomplished, with some careful attention for target states of µ = 0, where an extra indices (i , i) in Eq.(I.12) standing for the electronic state should be added. This development would a part of further works (see Section V).

The most striking features of Eqs.(I.8) and (I.12) is the presence of vibrational and rotational levels, initially absent in the nuclei-fixed S-matrix. The transition amplitude S Λ λ l ,λl (R) is obtained for a scattering electron energy computed with respect to the ground electronic state energy of the target, i.e. the entering channel, provided that the electron energy is not large enough to excite the target electronically. Consequently, performing frame transformation procedures give rise to uncertainty in the choice of the energy origin in the S-matrix which can resolved if S Λ λ l ,λl (R) is independent on the scattering energy within the spacing of rotational and vibrational levels. This is in fact consistent with the above approximation vj ≈ v j on I.3 Theoretical methods for collisions between electron and diatomic molecular ions11 which the R-matrix calculations are carried out. Further discussion is postponed to Section I.4.

As a result, one can apply frame transformations as long as S(R, vj ) ≈ S(R, v j ), where vj is given by Eq.(I.6). This assumption could be verified by a detailed analysis of eigenpahse sums of Eqs.(I.5), as we will see in the applications.

I.3.C

Elements of the Multi-channel quantum defect theory

Brief introduction

The quantum defect (QDT) theory was initially introduced by Ham [Ham, 1955] in solid physics and then reformulated by Seaton [Seaton, 1966, Seaton, 1983] for atomic physics. This theory is somewhat equivalent to the R-matrix or FT theories, described above, since the r-region space is divided in accordance with the interaction between an electron and a nuclei in an alkali atom. More specially, QDT deals with the properties of an excited electron of Rydberg surrounding an ionic core (nuclei+core electrons). The basic idea is to assume that beyond some distance from nuclei r > r 0 , the electron undergoes a purely Coulombic field created by a positive point charge. Eigenenergies of that excited (Rydberg) electron can be described by a phenomenological extension of the hydrogen expression, as formulated by Rydberg [Rydberg, 1890].

Exploiting the spherical symmetry of the atom, one needs only to solve the radial the Schrödinger equation whose the solution can be written as a linear combination of regular, f l (r) , and irregular, g l (r), Coulomb functions F l (r) = f l (r) cos πµ lg l (r) sin πµ l r > r 0 , (I. 13) where µ l denotes the quantum defect depending only on the angular momentum quantum number l and accounting for the corrections to the pure attractive Coulomb potential by the core electrons. Notice that for the diatomic molecules, one should add to the angular momentum l its projection along the Z axis of the molecule λ (sometimes noted m).

Making use of the asymptotic solutions of the above Coulomb functions, one can write Eq.(I.13), at any negative energy E = -1 2ν 2 , as follows (see Eq.(2.12) of Ref. [Aymar et al., 1996]):

F l (r) ∼ r→∞ 1 √ k sin (π(ν -l + µ l )) D -1 r -ν e kr -cos (π(ν -l + µ l )) Dr ν e -kr (I.14)
where ν is the effective quantum and D is a constant depending on the gamma function. As one is interested here by bound states, the exponential growth at infinity in Eq.(I.14) needs thus to be killed which leads to sin (π(νl + µ l )) = 0 or νl + µ l = n r , an integer. Putting the principal quantum number n ≡ n r + l and using the energy definition of the hydrogen atom, we fall back into the famous Rydberg formula of the allowed energy values (in atomic units)

E nl = I - 1 2(n -µ l ) 2 (I.15)
with I being the ionization energy (often set to zero). Equation (I.15) indicates that a series of resonant states can be formed with an excited ionic core. This series of resonances is called Rydberg series whose the energies are provided from the above formula. Therefore, the excited states of atoms with a large principle quantum number, leading to a very large electric dipole moments, are known as Rydberg atoms. In such atoms, the Rydberg electron is weakly bound to the ionic core which makes it sensitive to external perturbations resulting in a wide range of applications as in plasmas.

From Eq.(I.2) (by omitting here K-matrix), the radial wave function of the Rydberg electron can be expressed as follows

F l (r) ∼ r→∞ 1 √ k l sin [θ l (r) + πµ l ] , (I.16)
where θ l (r) is given by Eq.(I.3). πµ l appears as an additional phase shift with respect to Coulomb functions. Quantum defect µ l contains the information about short-range interaction between the outermost electron and the ionic core.

A key result above is the smooth manner in which the quantum defect can be extrapolated from below-threshold in Eq.(I.14) to above-threshold in Eq.(I.16) throughout a phase shifts

δ l = πµ l . (I.17)
This is usually expressed as the Seaton's theorem [Seaton, 1966]. Reshaping Eq.(I.15) yields

µ nl = n --1 2E
nl and hence µ nl depicts a weak dependence on n, in particular, on energy E nl . We fall back here on the circumstance mentioned in the preceding section: the use of smooth S-matrix in the frame transformation procedure since S l = e 2ıδ l = Eq.(I.17) e 2ıπµ l .

(I.18)

When several channels of electron excitation, distinguished by different core levels, or different orbital momenta are taken into account in the calculations, one has the multichannel QDT, noted MQDT. Generalizing QDT to include not specially Coulomb potentials, like molecular ones, arises with a more complicated relation between energies and quantum defects that can be described by the generalized quantum-defect theory GMQDT. More details can be found in [START_REF] Greene | General form of the quantum-defect theory[END_REF].

The quantum defect approach briefly discussed above has been widely used in the literature for studying different systems, from diatomic to polyatomic molecules, and processes (VE,RE,DR) [Mezei et al., 2015 et al., 2014b] using a theoretical approach based on the computation of the scattering matrix, via the complex Kohn variational technique, just above the ionization threshold and enables the explicit determination of all diabatic electronic couplings responsible for DR. In addition, authors used MQDT to demonstrate the precision of the scattering matrix by reproducing

I.3 Theoretical methods for collisions between electron and diatomic molecular ions13

accurately ab initio Rydberg state energies of the neutral molecules, obtained from the Rydberg formula of Eq.(I.15). As a result, complex Kohn variational technique reproduced quantum defects calculated from bound state calculations, verifying Eq.(I.17). A very good agreement with the experiments was found.

Closed-channel-elimination procedure

Going back to S-matrices, obtained previously after the frame transformation procedures, with the above considerations, one can notice that the scattering matrices S of Eqs.(I.8) or (I.12) can be used for cross sections calculations only if the total energy of the system is high enough for all collision channels to be open for ionization. When some channels are closed, the scattering matrix should be modified applying the "closed-channel-elimination" procedure and producing matrix S phys (E) according to

S phys (E) = S oo -S oc S cc -e -2iβ(E) -1 S co . (I.19)
Mathematically speaking, the above partition consists of eliminating exponential growth in closed channels to impose the correct physical behavior in solutions of Eq.(2.42) of Ref. [Aymar et al., 1996].

The matrix S phys (E) has N o × N o dimension, N o being the number of channels open for ionization for a given total energy E of the system. The total energy could be written as a sum of the energy of the entrance channel E i (rovibrational channel, E vj ) and the relative kinetic energy vj , see Eq.(I.6), of electron and the target ion: E = E i + vj . In the above equation, the matrices S oo , S oc , S cc and S co are submatrices of the original matrix S [Seaton, 1983, Aymar et al., 1996], partitioned as

S = S oo S oc S co S cc , (I.20)
where the partition of the matrix elements in the "o"-and "c"-parts is made on the basis whether the corresponding channel, i or i , is open or closed for ionization for a given total energy E. The quantity β(E) is a diagonal N c × N c matrix

β i i (E) = π 2(E i -E) δ i i , (I.21)
where E i refers to energy of the corresponding closed channel i (rotational, vibrational and/or electronic channels) and N c = N -N o is the number of closed channels. It relies on the effective quantum numbers, aside from π factor, of Eq.(I.15) with I being the closed ionization channel E i .

In summary, the first term in Eq.(I. 19) is restricted to the open channels and the second takes into account their mixing with the closed ones. The denominator is responsible for the resonant patterns in the shape of the cross section when where δ bg = log S oo /2ı is the background scattering phase shift contained in S oo and δ r is the resonant scattering phase shift given by the resonant S-matrix of the Breit-Wigner form

det S cc -e -2iβ
exp (2ıδ r ) = E -E r -∆ -ı γ 2 E -E r -∆ + ı γ 2 , (I.24)
with γ and ∆ being, respectively, the width and shift, due to the couplings between closed and open channels.

I.4 Validity of the theoretical models

In most theoretical studies, there are two main identifiable sources of uncertainty [START_REF] Chung | Uncertainty estimates for theoretical atomic and molecular data[END_REF]. The first one is related to the uncertainties of the theoretical model in which one often employs models with reduced dimensionality or neglecting competitive processes.

The second type of uncertainties arises from the choice of parameters of the model such as the use of limited basis set or ab initio/fitted data. Both uncertainties will be assessed for each studied system in Section I.5.

Besides the uncertainties stated above, the validity of the models is based on the frame transformation approach underlying the R-matrix method employed throughout this manuscript.

In the case of ionic target where attractive potential V (r) decaying slower than the centrifugal barrier l(l + 1)/(2r 2 ), the local kinetic energy vj of Eq.(I.6) depends on r at r > 100 bohr. At this distance, corresponding to the edge of the outer region in the R-matrix calculation, the kinetic energy of the scattering electron increases by about 0.27 eV which is larger than the spacing between vibrational levels. Hence, r 1 and r 2 in Figure I.2 are larger than 100 bohr and thus the whole region of r can be considered as the R-region with vj ≈ v j . This conclusion is consistent with the main hypothesis: adiabatic nuclei approximation can be used in the R-matrix calculations and frame transformation can afterwards be carried out, if S(R, vj ) is smooth with respect to vj , at the end of the calculation when K-matrix has been computed.

The energy dependence of the scattering matrix at threshold for attractive Coulomb potentials lies on the presence of a large range region, namely Coulomb zone, where the Coulomb potential dominates over the kinetic energy . The reactance matrix is defined as follows (from first-order perturbation theory) [Aymar et al., 1996]

K i i ≈ -π ψ i | V sr |ψ i , (I.25)
where ψ i and ψ i are respectively the initial and final wave functions of the scattering electron. One can see from the above equation that the matrix element of the short range potential V sr is proportional to a corresponding element of the reaction matrix. Therefore, the scattering matrix, deduced from Eq.(I.4), depends on the amplitude of the electron wave function which changes little (for a given normalization1 , i.e. usually a normalization per unit energy) with respect to the scattering energy at threshold since the local kinetic energy of (Eq.I.6) (thus the local wavelength) is governed by the Coulomb potential and not by the value of = 2 k 2 /2m. For r c > Z/ (obtained from = V (r c ) = Z/r c , in atomic units), the wave function oscillations are governed by the wavelength 1/k. When → 0, one has r c → ∞ and the Coulomb zone covers now the entire asymptotic range resulting to an energy independence of the electron wave function, thus the reactance matrix of Eq.(I.25) and the related scattering matrix.

To conclude, frame transformation procedures can be applied to ionic targets even at low scattering energy. This will be the subject of applications in the next section.

I.5 Applications

This section concentrates on a few studies using the theoretical approaches described above. Mechanisms of molecular ion destruction relevant to astrophysics are briefly presented and discussed. More details are provided in the published papers, the most relevant of which, are included in this manuscript.

I.5.A Rovibrational excitation of HeH + molecule by electron impact

To benchmark our theoretical approach, we started applying it to the simplest molecular ion formed in the early universe, namely, HeH + [ Lepp et al., 2002]. This cation is an important specie in the interstellar medium (ISM) [Roberge and Dalgarno, 1982] and in the chemistry of the planetary nebulae such as NGC7027 [Dabrowski and Herzberg, 1977, Black, 1978, Flower and Roueff, 1979]. In the ISM, it is mainly formed in the radiative association process of He and H + or of He + and H [Zygelman andDalgarno, 1990, Kraemer et al., 1995]. Collisions of HeH + with electrons play a significant role, in particular, leading to dissociation (dissociative recombination) and rovibrational (de-)excitation of HeH + [Jimenez- Serra et al., 2006], due to the relative abundance of electrons and HeH + in the ISM.

In this section, we focus on the rovibrational excitation and (de)-excitation (RVE,RVdE) process

e -+ HeH + (v, j) → e -+ HeH + (v , j ) , (I.26)
where v, j and v , j stand for the vibrational quantum number and angular momenta of the target before and after the rovibrational excitation of HeH + .

Since the first electronically excited state A 1 Σ + , correlated with H(1s) + He + (1s) is approximately 11 eV above the H + + He(1s 2 ) dissociation limit for the X 1 Σ + ground state, the ground state of electronic configuration 1σ 2 is isolated and non-adiabatic effects are expected to be small. Therefore, for low electron energy collisions (below 10 eV) only the lowest electronic state is open for ionization in e --HeH + collisions and the dimension of the geometry-fixed scattering matrix does not change with energy. Consequently, HeH + is Chapter I. Mechanisms of destruction of molecular ions an ion of the second type -having the first excited electronic state at a high energy-and we can thus directly apply our theoretical approach that combines the R-matrix code (the obtained S-matrix from Eq.(I.4)), the rovibrational frame transformation (Eq.(I.12)) and the elimination procedure in the sprite of MQDT (Eq.(I.19)). electron scattering energy (eV) In the R-matrix calculations (carried out in C 2v which is subgroup of C ∞v , natural point group of HeH + ), several basis sets, including 3-21G, DZP and 6-311G * as well as R-matrix sphere radii were tested to assess the uncertainties arising from the scattering model. Stability of the target properties such as dipole moment and ground state energy or positions of resonances have been investigated in order to select the adequate set of parameters for the final calculations. Moreover, in the calculations of the cross sections for reaction of Eq.(I.26), energies for vibrational and rotational transitions as well as vibrational wave functions of the target (in addition to its potential energy curve) are needed to perform the frame transformations of Eqs.(I.8) and (I.12). The obtained results are provided in papers .3 and .4 and compared with available data.

As widely discussed in the preceding sections, the application of frame transformations requires that K(R) and S(R) depend weakly on energy. A sharper energy-dependence is observed only at certain higher energies, corresponding to positions of Rydberg states attached to excited electronic states of the ion. A convenient way to represent a weak or a strong energy dependence of the matrices is the eigenphase sum of Eqs.I.5. Figure I.3 of paper .3 shows eigenphase sums for three symmetries ( 2 Σ + , 2 Π, and 2 ∆) of the e -+ HeH + system and for several internuclear distances in HeH + . The chosen interval of internuclear distances corresponds to the Franck-Condon region of the lowest vibrational level of HeH + . The variation of the eigenphase sums for 2 Π and 2 ∆ is smooth for energies below 10 eV and does not change significantly with R. The 2 Σ + eigenphase sums demonstrate a sharp energy dependence at certain energies, which corresponds to a resonant state of HeH with energy changing with the internuclear distance R.

Using the scattering matrix S phys of Eq.(I.8), the cross section for purely vibrational transition v → v is [Ayouz and Kokoouline, 2016]

σ v ←v ( ) = π 2 2m e λ l λl S phys λ l v ,λlv -δ λlv,λ l v 2 , (I.27)
where m e is the reduced mass of the electron-ion system. The inelastic cross section for the rotational excitation or de-excitation process j µ v ← jµv of a linear molecule by electron impact is obtained from the scattering matrix of Equation (I.12)

σ j µ v ←jµv ( ) = 1 2j + 1 π 2 2m e J,l,l
(2J + 1) e i(lπ/2+σ l ) S J,phys j µ l v ;jµlv e -i(l π/2+σ l ) 2 , (I. 28) where σ l is the Coulomb phase shift given in Eq.(I.3). In the above formula, the cross section is averaged over degeneracy of the initial rotational state j. Its derivation is given in paper .4. Note that in the ground electronic state of HeH + , the projection µ of the electronic angular momentum on the molecular axis of the target is zero. Therefore, for scattering energies below the first excited electronic state A 1 Σ + of HeH + , µ = µ = 0 in Eq.(I.28).

Cross sections for electron impact transitions between the lowest five j, j = 0 -4 rotational states of HeH + (without vibrational structure, i.e. a situation where the scattering electron is not energetic enough to excite the target vibrationally) are given in Figure I.5. Solid lines in the figure represent the results obtained using the complete theoretical approach, viz. the rotational frame transformation of Eq.(I.12) in combination with the closed-channels elimination procedure from Eq.(I. 19). The cross sections exhibit a strong resonant character. The resonances are produced by closed rotational states of the target. These resonances are washed out when thermally-averaged rate coefficients are computed. Therefore, in the calculation of the rate coefficients, one can use cross sections averaged over the resonances. Such averaged cross sections can be computed directly from the energy-independent scattering matrix S J , replacing in Equation (I.28) the matrix elements S J,phys j µ l ;jµl of the physical scattering matrix with the corresponding elements S J j µ l ;jµl of the energy-independent matrix. The cross sections calculated using the energy-independent scattering matrix are shown by dotted lines in Figure I.5.

Furthermore, data relevant to the other isotopologues 4 HeD + , 3 HeH + , and 3 HeD + could also be useful for plasma modeling. In this respect, cross sections as well as rate coefficients Chapter I. Mechanisms of destruction of molecular ions 4 HeH + molecule and its isotopologue 4 HeD + , for instance. The obtained results are in good agreement with the data available in literature. Due to the general 1/E el -dependence of cross sections of Eqs.(I.27) and (I.28), the calculated rate coefficients α v j ←vj (T ) behave as 1/ √ T for de-excitation and exp -∆ v j vj /T / √ T for excitation transitions, where ∆ v j v = E v j -E v is the excitation energy. Therefore, similar to papers .3 and .4, for convenience of use, the rate coefficients were fitted to an analytical interpolation formulas with up to four parameters. See Eqs.(4), ( 5) and (8) in Ref. [Ayouz and Kokoouline, 2019].

v'=4 v=3 → v'=0,1,2,4 v'=2 v'=0 v'=1
To summarize, we reported in Refs. [Ayouz and Kokoouline, 2016, Khamesian et al., 2018, Ayouz and Kokoouline, 2019] and briefly in this section the cross sections and rate coefficients for rovibrational excitation of HeH + and its isotopologues by electron impact. These results were obtained in context of my collaboration with Prof. Viatcheslav Kokoouline and, at least, partially during the dissertation of Marjan Khamesian (first author of paper .3) at University of Central Florida (UCF), in which I was involved. Our data were included in the IAEA database https://www-amdis.iaea.org/DATASETS/e_HeH/. They could be used in modeling the hydrogen/helium plasma experiments as well as for modeling interstellar clouds and planetary atmospheres, where the HeH + ion is present. Rot. exc. cross section (cm Rot. exc. cross section (cm

2 ) 0 → 1 1 → 0 0 → 1 1 → 0 0 → 2 2 → 0 0 → 2 2 → 0
2 ) 1 → 2 2 → 1 1 → 2 2 → 1
v'=3 v=0 → v' v'=1 v'=2 v'=4 rate constant (cm 3 /s) v=2 → v' v=1 → v' v=3 → v' v'=1 v'=0 v'=2 v'=2 v'=3 v'=3 v'=4 v'=4 v'=4 v'=1 v'=0 v'=0
j'=3 j=0 → j' j'=1 j'=2 j'=4 rate constant (cm 3 /s) j=2 → j' j=1 → j' j=3 → j' j'=1 j'=0 j'=2 j'=2 j'=3 j'=3 j'=4 j'=4 j'=4 j'=1 j'=0 j'=0

I.5.B Vibronic excitation of CH + molecule by electron impact

Another important specie relevant in ISM is CH + , first detected in cold ionized environments by Douglas and Herzberg [Douglas and Herzberg, 1941] in 1941 and later observed in many background stars. It is believed that its formation mechanism is governed by the collisions between C + and H 2 . The main route for its destruction is the dissociative recombination (DR), in competition with the rovibronic (de-)excitations processes.

This section aims to study the vibronic excitation and (de-)excitation process

e -+ CH + (i, v) → e -+ CH + (i , v ) , (I.29)
where v or v corresponds to the number of vibrational quanta in initial i or final i electronic state of the target ion. Rotational structure is neglected within this study. It will be considered in future work (see Chapter V).

Theoretical description of low-energy e --CH + collisions is complicated due to the presence of a low-energy electronic resonance and several low-energy excited electronic states of CH + which makes it an ion of the first type (see discussion in paper .5 and Ref. [Jiang et al., 2019]). The a 3 Π and A 1 Π curves above the ground state X 1 Σ + of electronic configuration 1σ 2 2σ 2 3σ 2 , correlate with the C + 2 P +H 2 S dissociation limit at large internuclear distances (see Figure I.12). In this situation, the standard frame transformation approaches, used above for HeH + and in many theoretical studies on electron-molecule collisions, is not well adapted due to the energy-dependence of the scattering matrix. For this purpose, we improved our model on HeH + by adapting the frame transformation procedures, described in Section I.3, to CH + . The central idea of the developed theoretical approach is that the e --CH + scattering physics below the A 1 Π state can be represented using an energy independent multichannel scattering matrix evaluated at a higher energy, above the A 1 Π ionization limit in a combination with the closed-channel elimination. Again, our approach employs the fixed-nuclear R-matrix (S-matrix in Eq.(I.4) is obtained from calculations performed in C 2v which is subgroup of C ∞v , natural point group of CH + ) with the vibronic frame transformation (in addition to electronic states) and the closed-channel elimination procedure in a spirit of MQDT (Eq.(I. 19)). To perform frame transformation of Eq.(I.8), with the inclusion of the electronic states i and i , we take the (almost) energyindependent 3×3 electronic scattering matrix, obtained at an energy above the A 1 Π ionization limit, to produce N × N matrix with N vibronic (or rovibronic) channels. Such a vibronic scattering matrix is essentially energy-independent and a closed-channel elimination procedure should be performed to obtain the physical energy-dependent matrix, which can be used to compute cross sections and related rate coefficients.

Several parameters of the scattering model were first tested to investigate the stability Chapter I. Mechanisms of destruction of molecular ions of the target properties. Figure I.8 depicts variation of the ground state energy and dipole moment of CH + regarding different basis sets and complete active spaces CAS(n e ,n ao ), viz. number of electrons n e distributed in n ao active orbitals. Considering the dipole moment 1.617 D obtained with cc-pVQZ is closer to the experimental value of 1.683 D, this basis with CAS(4, 12) and a R-matrix sphere of radius 13 bohrs were thus selected for the final calculations.

Another source of uncertainty is related to the model itself. The choice of the 3 × 3 scattering matrix used in the channel-elimination procedure is not unique, because the matrix depends on energy, even above the A 1 Π electronic state. To assess the result of uncertainty in the choice of the energy at which the 3 × 3 scattering matrix is taken, we plot in Figure I.9 eigenphase-sum derivatives obtained for 3 × 3 scattering matrices taken at three different energies above the A 1 Π state: at 3.413, 3.563, and 3.713 eV. Positions and the widths of the resonances are nearly the same in all three calculations.

To validate the developed model, we compare the fixed-nuclei electronic scattering matrices obtained (1) using the elimination procedure of the closed electronic states and by (2) a direct scattering R-matrix calculation at the same internuclear distance, viz. R e = 2.137 bohrs. To do this, the physical scattering matrix S phys ( , R e ) of Eq.(I. 19) describing electronic transitions at R e of CH + , was used to compute the fixed-nuclei cross section of the electronic excitations from the X 1 Σ + state to the a 3 Π state as Figure I.10 compares the cross sections for the X 1 Σ + → a 3 Π transition obtained in the two approaches. The general agreement between the two curves is good, even for the widths of the resonances. One noticeable difference is in the position of the minimum near 1.5 eV: In the MQDT calculations it is shifted slightly to the left. The agreement is better at energies approaching the a 3 Π ionization limit.

σ i ←i ( , R e ) = π 2 2m e l m ,lm S phys l m i ,lmi ( , R e ) -δ l m i ,lmi 2 , (I.30)
Once the model is validated, a vibronic frame transformation of Eq.(I.8) can be performed in order to compute the cross sections for vibronic excitation or (de-)excitation of CH + using Eq.(I.27, where S phys l m i ,lmi ( , R e ) is replaced with S phys l m v i ,lmvi ( ). We noted here m as the project of the angular momentum quantum number on the molecular axis instead of λ. Figure I.11 illustrates cross sections obtained for different combinations of initial and final vibronic states. Panel (a) shows results for pure vibrational excitations between levels of the ground electronic state X 1 Σ + . As expected, the cross section for the transition with ∆v = 1 is the largest one between inelastic processes. Panel (b) gives cross sections from the ground vibronic state X 1 Σ + , v = 0 to several vibrational levels of the a 3 Π state. Since the potential curves of the X 1 Σ + and a 3 Π states have similar shapes near equilibrium, the largest X 1 Σ + → a 3 Π cross section is expected to be for ∆v = 0, as the present calculation indeed demonstrated. Panel (c) gives cross sections for the de-excitation process

a 3 Π, v = 0 → X 1 Σ + , v = 0 -3.
The obtained cross sections differ significantly from those of Ref. [Chakrabarti et al., 2018b]. These disagreements can be attributed to approximations employed in their study:

(1) neglected differences in vibrational excitation threshold energies, (2) neglected dependence of e --CH + scattering parameters with the internuclear distance, and (3) the neglected resonances in closed vibronic channels. Thermally-averaged rate coefficients of the above vibronic (de-)excitations

ν'=1 ν'=2 ν'=3 ν=0, X 1 Σ + ν'=1,2,3 X 1 Σ + (a)
ν'=0 ν'=1 ν'=2 ν=0, a 3 Π ν'=0,1,2,3 X 1 Σ + ν'=3 (c)
Figure I.11: Cross sections for vibronic excitations of CH + from the ground vibrational level v = 0 of the X 1 Σ + state to v = 1, 2, 3 of the X 1 Σ + state (left panel) , to v = 0, 1, 2, 3 of the a 3 Π state (middle panel), and for vibronic de-excitations from the ground vibrational level v = 0 of a 3 Π to v = 0, 1, 2, 3 of the X 1 Σ + state (right panel).

α i v ←iv (T ) = 8π (2πk b T ) 3/2 ∞ 0 σ i v ←iv ( ) e -k b T d , (I.31)
are then computed and fitted to an analytical formula, analogously to HeH + . More details and discussions can be found in paper .5.

Finally, we report in Ref [Jiang et al., 2019] and briefly in this section the cross sections for vibronic excitation and de-excitation of CH + by electron impact using only first principles. These outcomes were obtained in the context of Xianwu Jiang's dissertation at Centralesupélec and in collaboration with UCF. The developed theoretical approach can be applied to ions with low-energy excited states, known to be difficult to treat theoretically using previous theoretical methods. Our approach is quite general and can be applied also for a number of different processes such as dissociative recombination, which is the subject of the next section.

I.5.C Dissociative recombination of CH +

In the continuation of the preceding study, we consider in this section the following dissociative recombination (DR) processes The state CH * * (2 2 Π) is formed when the electron collision energy is not transferred directly into kinetic motion of the nuclei, but rather to the electronic ionic clouds resulting in the excitation of the target bound electrons. The striking feature of CH * * (2 2 Π) is that it is situated in the ionization-continuum of CH + . The potential energy curve of the doublyexcited CH * * (2 2 Π) is repulsive and intersects the curve of CH + X 1 Σ + near the equilibrium geometry (see Figure I.12). The resonant state CH * * (2 2 Π) is unstable against autoionization, i.e. the re-emission of electron leading to vibronic excitations, because the excited electron has sufficient energy to escape from CH. Generally speaking, if the dissociative process happens fast enough so that the products of dissociation are already at a large distance from each other after a time equivalent to the order of the autoionizing lifetime, most of the collisional energy is already converted into nuclei kinetic energy and autionization becomes impossible. In such a case, the system irreversibly breakdowns in one of the dissociative channels. This is the direct dissociative recombination (DDR) process.

e -+ CH + (i, v) → CH * * (2 2 Π) → C 1 D + H 1 S direct DR CH * (Rydberg) → CH * * (2 2 Π) → C 1 D + H 1 S
In contrast to the above situation, the indirect dissociative recombination (IDR) mechanism can be considered as the result of two non-radiation transitions. When the kinetic energy of the electron is transferred to the motion of nuclei, the electron then moves around a vibrationally (or rovibrationally in general) excited core in a hydrogenic-like orbital of CH with high principle quantum number (see Eq.(I.15)). The excited Rydberg state CH * is in turn predissociated by the doubly-excited state CH * * (2 2 Π) leading to dissociation into C 1 D + H 1 S through non-adiabatic couplings, i.e. breaking the BO approximation. The IDR process occurs very often in a closed shell ions, such as HeH + or CH 2 NH + 2 for example. Both DDR and IDR in the reaction of Eq.(I.32) compete with the vibrionic excitation and (de-)excitation processes, investigated previously. Consequently, for a comprehensive study of mechanisms of destruction of CH + in ISM, for instance, one has to take into account the role of all these effects on the magnitude of the cross section at very low energy. As for the role of rotational effects, it has been postponed to a further study.

The theoretical approach employed in this section is similar to the one above for describing the VE and VdE processes. It combines the R-matrix code to compute electron-ion scattering matrices (Eq.(I.4)), the vibronic frame transformation (Eq.(I.8) with the inclusion of electron state indexes) and the closed-channel elimination procedure in a spirit of molecular quantum defect theory to construct an energy-dependent scattering matrix (Eq.(I. 19)). The main difference here is the inclusion of vibrational continua states in addition to vibrational bound states (see Figure I.12). This can be accomplished throughout the outgoing-wave basis functions defined by a complex absorbing potential as described in Section I.3. Since the present work is still in development, I will only provide the first results below.

In figure I.12 displaying the potential energy curves of CH + , the doubly-excited dissociative state CH * * (2 2 Π) crosses the ground electronic state X 1 Σ + of CH + near the left turning point of the first excited vibrational state. This gives rise to a DR process according to Eq.(I.32). In the figure, 35 vibrational levels for each electronic states are drawn and the CAP of Eq.(I.10) is also represented. The insight depicts, as an example, the long-range behavior of a vibrational wave function of the discretized continuum.

Accurate description of DR in CH + of Eq.(I.32) requires including the 2 2 Π resonant state in the MQDT treatment. So, performing the R-matrix calculations and selecting the 3 × 3 energy-independent scattering matrix (above the A 1 Π threshold) according to the procedure described in the preceding section, we can see in Figure I.13 the positions of the 2 2 Π state at different geometries R = 1.337, 1.437, and 1.537 bohrs. The figure compares the positions, i.e. the derivative of eigenphase sum of e --CH + system, given directly from R-matrix to those obtained from the closed-channel elimination procedure of Eq.(I. 19). Overall positions of the resonance are a little left shifted and the widths are slightly wider compared to that of the R-matrix calculation. However, we expect that these discrepancies do not affect the final thermally-averaged rates coefficients as seen in calculations of vibrionic (de-)excitation of CH + in paper .5. In the present treatment, the dissociation proceeds through a number of excited Rydberg states, which are bound with respect to dissociation. MQDT treats such a system as a set of coupled vibrational states. As hinted in section I.3, the introduction of the CAP leads to simulate an infinite grid in which the vibrational continuum becomes discretized. To test parameters of the CAP in Eq.(I.10), we used the Fourier transformation (FT) technique and analyzed the vibrational wave functions in the momentum space.

In left panels of Figure I.14, we display wave functions of dissociative vibrational states (vibrational continua states), for instance, v = 30, X 1 Σ + (black curve), v = 26, a 3 Π (red curve), and v = 20, A 1 Π (green curve) as functions of internuclear distance R. Right panels show the respective Fourier transform of that wave functions. The wave functions obtained with CAP calculations are given in solid lines whereas those without CAP are in dashed curves. The most striking feature in these figures is the absence of the second peak, corresponding to the incoming/reflected wave function, in the Fourier spectrum of the wave functions when the CAP is involved in the calculations. This indicates that the set of the CAP parameters, i.e. length and strength in Eq.(I.10 are appropriately chosen. (in black), v=26, a 3 Π (in red), and v=20, A 1 Π (in green) obtained with (solid curve) and without CAP (dashed curve). Right panels display the Fourier transform of the corresponding vibrational wave functions. The starting point of the CAP, R CAP , is drawn in the left lower panel.

After benchmarking the CAP parameters, the DR cross section can be computed from the following expression

σ vi ( ) = π 2 2m e i v l m ,lm 1 -S phys l m i v ,lmiv (E) S †phys l m i v ,lmiv (E) (I.33)
where m e is the reduced mass of e --CH + and vi = E -E vi ≈ v i ≈ is the incident energy of the electron. S phys l m i v ,lmiv and S †phys l m i v ,lmiv are calculated using the frame transformation of Eq.(I.8) with no complex conjugation for the vibrational states. In addition, performing the channel elimination procedure for S † : S † → S †phys (E), one has to use the complex conjugate of β(E) in Eq.(I.21). Recall that β(E) is the effective quantum number for energy E in closed channel multiplied by π (see Eq.(I.17)). It is complex valued because the energy of the closed channel has a non-zero imaginary part, resulting from the boundary condition imposed by the CAP, related to the dissociation lifetime of the ionic channel. More details of this method can be found in Ref. [Kokoouline et al., 2011].

e -+ CH 2 NH + 2 → CH 2 NH * 2 → CH 2 NH * * 2 →                CH 2 NH + H CH 2 + NH 2 HCN + H 2 + H CN + H 2 + H 2 H 2 CN + H + H HNC + H + H 2 , (I.34)
where a summary of the recommended rate coefficients and branching ratios for the above reactions is provided in Table 1 of Ref. [Yuen et al., 2019]. At low-energy, however, the electron is likely to be captured in a Rydberg level CH 2 NH * , strongly predissociated by CH 2 NH * * , associated with a vibrationally excited state of CH 2 NH + . Again, in environments of low temperatures, such as ISM, the indirect process becomes dominant with the direct DR retaining negligible contributions to the total DR process. We will thus shift the focus on the IDR which is in competition with vibrational transitions

e -+ CH 2 NH + 2 (v i ) → e -+ CH 2 NH + 2 (v i ). (I.35) CH 2 NH +
2 is a closed-shell ion belonging to the C 2v point group at its equilibrium, with the ground state electronic configuration

X 1 A : 1a 2 1 2a 2 1 3a 2 1 4a 2 1 1b 2 2 5a 2 1 2b 2 2 1b 2 1 . (I.36)
Although there are 12 degrees of freedom for CH 2 NH + 2 internal motion (see Figure I.16), the formalism for indirect DR developed for BF + 2 in Chapter III can be applied in a straightforward manner. Recall that the theoretical approach combines the normal modes approximation for the vibrational states of the target ion (described in Section II.2.A) and use of the R-matrix code to evaluate e --CH 2 NH + 2 scattering matrices, from Eq.(I.4), for fixed geometries of

CH 2 NH + 2 .
First, we computed the frequencies of the normal modes using a large active space CAS(12,10), i.e. 12 electrons distributed in 10 orbitals, and cc-pVQZ basis set. The obtained data such as frequencies and symmetries of the 12 vibrational modes are provided in [Yuen et al., 2019]. Secondly, we performed R-matrix calculations using identical parameters to the above and obtained the fixed-nuclei energy-independent scattering matrices, evaluated for two values of q i keeping the other normal coordinates q i fixed at q i = 0. Figure I.17 shows the eigenphases sums of the system e --CH 2 NH + 2 for different irreducible representations at the equilibrium position of the target ion. Eigenphases are smooth below 2.5 eV, where the first electronic resonance appears for the 2 A 1 state. The absence of electronic resonances at low collision energies justifies the indirect DR approach. With the formula of Eq.(III.12) and the summation extended to 12 normal modes, we calculated the DR cross section for CH 2 NH + 2 in its vibrationally ground state as a function of collision energy . Figure I.18 shows at low energies that the obtained DR cross section is featureless and behaves simply as 1/ . This can be viewed as a DR cross-section averaged over the vibrational autoionizing resonances. Such a result could be justified by the fact that the experimental/observation resolutions (storage-ring or merged-beam experiments) are too low to resolve individual Rydberg resonances in the DR spectra. For energies above 0.1 eV, the cross section drops in stepwise manner because the scattering electron excites the vibrational level of the ionic target by one quanta and leaves with a smaller kinetic energy.

Fitting the probabilities of Figure 2 in Ref. [Yuen et al., 2019], rate coefficients can be computed analytically, assuming the collision energy follows Maxwell-Boltzmann distribution, as

α V E v i (T ) = √ 2π(v i + 1) 2 m 3/2 e a i + ω i b i + ( ω i ) 2 c i / k b T + (b i + 2 ω i c i ) k b T + 2c i (k b T ) 3/2 exp - ω i k b T , (I.37) α V dE v i (T ) = √ 2πv i 2 m 3/2 e a i √ k b T + b i k b T + 2c i (k b T ) 3/2 , (I.38)
and

α DR v i (T ) = √ 2π 2 m 3/2 e 12 i (v i + 1) 2c i 1 -exp - ω i k b T (k b T ) 3/2 + b i -(b i + 2 ω i c i ) exp - ω i k b T k b T + a i -(a i + ω i b i + ( ω i ) 2 c i ) exp - ω i k b T / k b T , (I.39)
where a i , b i and c i are the fitted parameters from Table 3 in the same reference. Figure I.19 displays the thermally averaged VE and DR rate coefficients as functions of temperature for the initial ground vibrational state, v i = 0. For low temperatures, T < 400 K, the rate coefficients behave as 1/ √ T . Since the exponent factor exp (-ω i /k b T ) is much smaller than 1, the DR rate coefficient can be approximated as 4.65 × 10 -7 (300/T ) 0.5 cm 3 /s. As discussed previously, there are two main identifiable sources of uncertainty in the DR and VE rate coefficients obtained above. The most important source lies in the autoionization competitive process. Indeed, in our simplified model, once the electron is captured into a vibrational Rydberg resonance associated with a closed vibrationally-excited channel, the e --CH 2 NH + 2 system will dissociate rather than autoionize. In a previous studies dealing with smaller molecular ions such as H Chapter I. Mechanisms of destruction of molecular ions larger ions, the probability should be smaller and hence it is reasonable to assume that the corresponding uncertainty in the present DR calculations is below 20%.

To assess the second type of uncertainties related to the scattering model, we varied several parameters such as basis sets and complete active spaces. Table I.3 lists the three sets of parameters. The differences between results obtained in the three models are 1-2%. Therefore, the uncertainty due to the scattering model is negligible, with respect to the capture approximation and the overall uncertainty of the present theoretical rate coefficients, and is estimated to be below 20%. in the ground vibrational. To avoid overcrowded labeling in the figure, labels for three VE curves only are given. The lines for all VE rates cross the abscissa in the order of increasing number labeling the modes (and increasing energy) as in Table 2 of Ref. [Yuen et al., 2019]. For T < 400 K, the DR and VE rate coefficients behave as T -1/2 and T -1/2 exp (-ω i /k b T ) respectively. At higher temperatures, as vibrational excitation becomes more probable, the DR rate coefficient decreases faster than

T -1/2 .
To summarize, the first value for the total DR cross section and thermally averaged rate coefficient of CH 2 NH + 2 were briefly reported. The implications of the present calculations in the chemistry of extraterrestrial environments are important since the obtained value is consistent with the two most popular astrochemical data basis, KIDA, and UMIST but much smaller for T < 400 K than the value used in photochemical models of the upper atmosphere of Titan. This has in fact an impact on Titan ammonia abundance in the model. These results, however, provide an upper limit for the total rate coefficient of the CH 2 NH + 2 DR. See discussion in Ref. [Yuen et al., 2019].

I.6 Concluding remarks 35 Model C 1 (ω 3 ) C s (x = 0)(ω 2 , ω 5 , ω 10 , ω 12 ) C s (y = 0)(ω 1 , ω 4 ) 1 cc-pVTZ cc-pVQZ cc-pVQZ (5a|6a) (2a , 0a |8a , 2a ) (2a , 0a |7a , 3a ) 2 cc-pVTZ cc-pVTZ cc-pVTZ (5a|6a) (2a , 0a |8a , 2a ) (2a , 0a |7a , 3a ) 3 cc-pVTZ cc-pVQZ cc-pVQZ (5a|6a) (2a , 0a |8a , 2a ) (2a , 0a |7a , 3a ) Model C 2v (ω 6 , ω 7 , ω 8 , ω 9 , ω 11 ) 1 cc-pVQZ (4a 1 , 0b 1 , 1b 2 , 0a 2 |2a 1 , 2b 1 , 2b 2 , 0a 1 ) 2 cc-pVTZ (4a 1 , 0b 1 , 1b 2 , 0a 2 |2a 1 , 2b 1 , 2b 2 , 0a 1 ) 3 cc-pVQZ (2a 1 , 0b 1 , 0b 2 , 0a 2 |5a 1 , 2b 1 , 3b 2 , 0a 1 )
Table I.3: Parameters of three different scattering models. Different basis and electronic configurations are used for normal modes with different point group symmetries. The symbols in the parenthesis indicates the number of frozen orbitals and number of active orbitals.

I.6 Concluding remarks

This chapter outlines my contribution to the study of the molecular ions destruction mechanisms in astrophysical environments. Two main developments were achieved within this research axis. First, a theoretical approach for electron-impact rovibrational excitation of molecular ions was developed and applied to the HeH + benchmark system. Second, this approach was adapted to molecular ions with low-lying excited electronic states like CH + . With some additions whose details were deferred to Chapter III, these approaches were extended to more complicated systems such as prebiotic molecules NH 2 CHOH + and CH 2 NH + 2 that play an important role in the chemistry of ISM and the upper planetary atmospheres.

Our theoretical studies have allowed us to investigate the main mechanisms of destruction and/or formation of these ions in low collisional energy, namely, direct and indirect DR that are in competition with autoionization, viz. rovibronic excitation and (de-)excitation processes. These studies have also enabled, on the one hand, to provide an upper limit for the total rate coefficient of DR, where data are not available (CH 2 NH + 2 case), on the other hand, to confirm previous calculations as for HeH + and CH + . The obtained cross sections and related rate coefficients could be used in kinetic modeling of cold ionized media such as ISM.

The above works have been supported by different programs and funding listed in CV (see Appendix V.2.C). They have also been performed in the context of three thesis: Dr. Marjan Khamesian, Dr. Chi Hong Yuen and Dr. Xianwu Jiang. They result in at least height peer-reviewed articles. The following chapter pursues these studies to include neutral molecules.

II -Role of collisions involving electron-neutral molecules and electron-radicals

II.1 Context and Motivations

Neutral molecules and radicals play an important role in plasma discharges. They are mostly formed by surface processes on the walls and then involved in a volumetric chemical chain reactions with other species. Typically, neutral molecules are formed in excited states and radicals are well-known to be unstable in collisions with other species present nearby. Consequently, for both systems an experimental approach is difficult or impossible and data are needed for reactions taking place on the surfaces but also in the phase-gas [START_REF] Pelicon | Neutral molecules in tokamak edge plasma -role of vibrationally excited hydrogen molecules[END_REF]].

This chapter focuses on two type of neutral molecules NO 2 and N 2 O. These species are involved in a number of different chemical pathways in the atmosphere leading to a large variety of species. They also play a role in a non-equilibrium kinetics of plasmas containing N 2 and O 2 that are used for plasma based technologies and depollution purposes. Understanding/validation of the fundamental kinetic mechanisms for these molecules remains limited. This relies partially on the availability of the cross-section data of elementary processes, in particular to the rovibrational-dependent electron impact excitation/(de-)excitation data, which is very sparse.

Our following studies aim to fill, at least partially, this gap and provide reliable cross sections for electron-induced vibrational excitation of NO 2 and NO 2 . Cross sections and rate coefficients for transitions between different vibrational states of the targets are reported and compared to available experimental data. The rotational structure of each vibrational level of the targets was neglected which corresponds to the situation where rotational structure is not resolved in the initial and final states of the target molecule. A more comprehensive study will be considered in a near future (see Chapter V).

The main results of this chapter were obtained in the context of Dr. Hainan Liu's dissertation at Centralesupélec in collaboration with Prof. Viatcheslav Kokoouline at UCF and Dr. Samantha Fonseca Dos Santos at Rollins College. They have been published in two papers whose the one on N 2 O is included in this manuscript.

Simplified approach using derivatives of 𝑺

II.2 Theoretical method developed on neutral molecules

Similar to molecular ions discussed in the preceding chapter, theoretical models employed in the studies of Section II.4 are based on first-principles. They require three main ingredients summarized in Figure II.1. A brief description is given below.

II.2.A Normal modes approximation

In calculations of the cross section for vibration (de-)excitation of Section II.4, one needs the vibrational wave functions of the target in order to construct elements of the scattering matrix for transition from one vibrational level v to another v .

For polyatomic molecules, one can expand the potential energy V in a multi-dimensional Taylor expansion truncated at second order around equilibrium. The nuclear Schrödinger equation for an N atom system writes as

- 3N i 2 2m i ∂ 2 ∂x 2 i + 1 2 t (x -x eq ) d 2 V dx 2 (x -x eq ) ψ(x) = Eψ(x), (II.1)
where N is the atoms number, m i is the atomic mass and x = {x

1 x 2 x 3 • • • x 3N } is a vector containing the 3N components of coordinates of all nuclei.
x eq defines the equilibrium structure and ψ(x) is the nuclear wave function. Above, E stands for the total energy of the molecule, i.e. including translational, vibrational and rotational energies.

Defining the mass-weighted coordinates

xi = √ m i (x i -x eqi ) ⇒ ∂ 2 ∂ x2 i = 1 m i ∂ 2 ∂x 2 i , (II.2)
Equation (II.1) can be solved merely under the following form

  - 3N i 2 2 ∂ 2 ∂ x2 i + 3N i,j 1 2 H ji xi xj   ψ(x) = Eψ(x). (II.3)
where H ij defines the mass-weighted Hessian, also called matrix of force constants:

H ij = 1 √ m i √ m j ∂ 2 V ∂x i ∂x j . (II.4)
This matrix of the second derivatives of the potential energy plays a central role in energy minimization techniques.

Looking for a matrix D that diagonalizes the mass-weighted Hessian, the second term in the left-hand side of Equation (II.3) can be reshaped as

3N i,j 1 2 H ji xi xj = 1 2 t Q D -1 HD Q, (II.5)
where Q designates the set of 3N eigenvectors associated to 3N eigenvalues ω 2 i [START_REF] Janežič | Harmonic analysis of large systems. iii. comparison with molecular dynamics[END_REF]. D above is a transformation matrix from mass-weighted Cartesian coordinates to a set of 3N coordinates including the translation, rotation and vibration degrees of freedom.

In quantum chemistry softwares such as Molpro or Gaussian, the eigenvectors Q, i.e. the normal modes, are calculated after separating out the rotational and translational normal modes leaving 3N -6 or 3N -5 modes for vibrational analysis. The eigenvalues ω 2 i are the fundamental frequencies of the molecule. For rotation and translation modes they should be close to zero. More details can be found in Ref. [Frisch, 2004].

Generally, it is convenient to introduce a new set of coordinates called dimensionless normal coordinates that are related to the length-unit normal coordinates Q i as follows

q i = Q i µ i ω i . (II.6)
where ω i and µ i are the frequency and reduced mass of the ith mode, respectively. Hence, Equation (II.3) can be written for vibrational normal modes as

[T (q) + V (q)] ψ(q) = Eψ(q) 3N -6(5) i ω i 2 -∂ 2 ∂q 2 i + q 2 i ψ(q) = Eψ(q), (II.7)
where E denotes here the vibrational energy of the molecule in the uncoupled harmonicoscillator approximation

E = 3N -6(5) i v i + 1 2 ω i , (II.8)
associated to the following normalized wave function

ψ(q) = 3N -6(5) i η v i (q i ), (II.9)
with η v i (q i ) representing the harmonic vibrational function with v i quanta and corresponding vibrational frequency ω i , reduced mass µ i and dimensionless normal coordinate q i .

With the above expressions in hand, we can perform the vibrational transformation in Eq.(II.17) according to the well-known formula

η v i (q i ) = 1 π 1 4 1 √ 2 v i v i ! e -q i 2 2 H v i (q i ). (II.10)
where H v i is the Hermite polynomials. For example, the normalized wave function of the ground vibrational state is

η v i =0 (q i ) = 1 π 1 4
e -q i 2 2 , (II.11) associated to the zero-point energy ω i 2 .

In our studies, we employed Molpro [Werner et al., 2008] to obtain the frequencies ω i and the equilibrium geometry of the molecules throughout the Hessian matrix elements of Eq.(II.4). The matrix D enables to generate Cartesian displacements, X = Dq, along each normal coordinate that are used to perform scattering matrix calculations in the following section and then to compute the integral in Equation.(II.17). Permanent dipole moments of the molecules are also intermediate quantities useful to check the convergence of the final results, viz. cross sections and rate coefficients.

The normal mode approximation employed in this chapter is accurate for the lower vibrational levels. From the classical point of view, as v i increases, the nuclei spend more time in regions far from their equilibrium separation. For such regions the potential energy deviates substantially from that of a harmonic oscillator and the harmonic-oscillator approximation tends to become poor. Therefore, the developed theoretical approach can be applied for vibrational transitions involving few quanta in each mode.

II.2.B

The ab initio R-matrix method

Since a more detailed discussion on the R-matrix method is provided in Section I.3.A, I sketch below only the main difference.

Again, the R-matrix method (see upper panel of Figure I.2) involves the division of the configuration space into an inner region where a molecular (N -electrons targets+the scatterer) bound state is solved, and an outer region with an long range multipolar expansion of the form

V ij = α=0 C (α) ij r α+1 , (II.12)
where C

(α) ij is the asymptotic potential coefficient of order α between i and j channels. This coefficient is given by [START_REF] Burke | R-matrix theory of electron-molecule scattering[END_REF]:

C (α) ij = 2l i + 1 2l j + 1 C ( l i , α, l j ; λ i , λ α , λ j )C(l i , α, l j ; 000)Q (α) ij , (II.13)
with l j being the orbital angular momentum of the scattering electron for the ith channel and

C(l i , α, l j ; λ i , λ α , λ j ) the Clebsch-Gordon coefficient. Q (α)
ij is the target moment between the channels i and j such as if i = j, it corresponds to a permanent moment, and for i = j it is a transition moment. In general, only the first few terms in the expansion over α play a significant role in the collision in the outer region. For neutral molecules such as NO 2 and N 2 O, α is 1 which corresponds to the interaction between the charge of electron and the dipole moment of the target molecules.

Once the wave functions are generated in the inner region, the R-matrix is constructed at the boundary r = a for a given scattering electron energy and it is propagated toward the outer region (see Eq.( 23) of Ref [Tennyson, 2010]) where the radial scattering electron wave function can be matched at the boundary r = a asy 70 bohr (such that coupling potentials to be negligible) to Bessel functions

F Γ ij (r) ∼ 1 √ k i sin θ i (r)δ ij + cos θ i (r)K Γ ij (q) if i ≥ 0 0 otherwise (II.14)
where

θ i (r) = k i r - l i π 2 , (II.15)
and index i refers to the entrance channel before a collision and j refers to the outgoing channel after the collision. K Γ ij (q) above are the elements of the reactance matrix [Aymar et al., 1996, Tennyson, 2010] depending on the dimensionless normal coordinates q = {q 1 , q 2 , • • • } parametrically and obtained for each scattering electron energy . Recall, Γ denotes the constants of motion corresponding to the irreducible representation of the molecular point group, in particular the total angular momentum projections Λ of the anion system (electronmolecule).

Finally, in the cross section calculation, the scattering matrix S Γ (q) is needed and can be obtained from

S Γ (q) = (1 + ıK Γ (q))(1 -ıK Γ (q)) -1 , (II.16)
at a given geometry q, similar to Eq.(I.4). The eigenphase sums is another observable, defined from these matrices through Eqs.(I.5), which is very useful for testing the validity of our theoretical approach. These key points will be discussed in Section II.3.

II.2.C Vibrational frame transformation

The application of the frame transformation procedures was discussed in Sections I.2 and I.3. It relies on the assumption that the strength of couplings between the scattering electron and vibrational states v of the target are small compared to the interaction potential V (r) experienced by the incident electron. As we have seen, this assumption remains valid for ions even at low collisional energy because of Coulomb asymptotic behavior in V (r). In case of neutral molecules, the situation is less obvious. More remarks are differed to the next section.

Assuming v ≈ v and a smooth scattering matrices along each dimensionless normal coordinates, one can employ the procedure of frame transformation

S v v = λ l ,lλ ψ v (q) S Λ λ l ,λl (q) ψ v (q) , (II.17)
where the matrix element S Λ l λ ,lλ (q) is obtained from the R-matrix calculations above, and v and v denote the collection of initial and final vibrational quantum numbers of all the normal modes. The brackets in Eq.(II.17) imply an integration over the vibrational coordinates where the vibrational wave functions are given in the harmonic approximation picture by Eq.(II.9).

In the studied cases of Section II.4, the integral above is computed numerically using a Gaussian-Legendre quadrature with 10 points.

II.3 Validity of the theoretical model

43

II.3 Validity of the theoretical model

As hinted in Section I.4, there are two main identifiable sources of uncertainty [START_REF] Chung | Uncertainty estimates for theoretical atomic and molecular data[END_REF]. One is related to the uncertainties of the theoretical model often employing reduced dimensionality or neglecting competitive processes. A second type of uncertainties relies on the choice of parameters of the model such as the use of limited basis set or ab initio/fitted data. Both uncertainties will be assessed in NO 2 and N 2 O studies of Section II.4.

In addition to the above uncertainties, the validity of the model developed for neutral molecules is mainly based on the circumstance of applying the frame transformation procedure in Eq.(II.17). Indeed, the attractive potential V (r) decays faster than the centrifugal barrier, l(l+1) 2r 2 (in bohrs), in case of neutral targets, according to Eq.(II.15). In a such situation, the local kinetic energy of electron in Eq.(I.6) is independent on r at the edge of the R-matrix simulation box, i.e. vj ∼ 0.003 eV at r = 100 bohrs for V (r) behaving as 1/r 2 for instance. Thus, vj ≈ v j can be fulfilled outside r = 100 bohrs only if the scattering energy is large enough, compared to the spacing between vibrational levels, such that the entire region of r can be considered here as the q-region (see lower panel of Figure I.2).

The above condition on vj is consistent with an energy-independent scattering matrix. In general, S(q) depends on the scattering energy especially for neutral targets. To overcome this difficulty, (1) one can use a smooth S(q), if possible, as done in the preceding studies for target ions (HeH + or CH 2 NH + 2 ) or ( 2) choose a different normalization factor for the scattering wave function to remove the energy-dependence of S(q) [START_REF] Greene | General form of the quantum-defect theory[END_REF], Greene et al., 1982, Greene and Jungen, 1985]. This later suggestion is reserved for future study (see Chapter V).

In our following studies, the scattering electron energy is (i) chosen so as to not be so large to excite the target electronically and (ii) also below of electronic resonances appearing when the potential energy surface (PES) of the anionic molecule (electron-molecule system) crosses the neutral PES near the equilibrium geometry. Moreover, as we will see, a detailed analysis of enigenphase sums of e -NO 2 and e --N 2 O collisions reveals that S(q, v ) ≈ S(q, v ). Hence, the vibrational frame transformation can be accomplished here directly without further development (refereed above as point (2)).

II.4 Applications

As a demonstration, we applied the theoretical approach described above to neutral species such as NO 2 and N 2 O of atmospheric interest. Details on these studies, performed by Hainan Liu in her thesis at Centralesupélec, can be found in Refs. [Liu et al., 2019, Liu et al., 2020]. Below, I will underline only the major ideas and results in particular for the N 2 O study whose the article is included to this manuscript as paper .6.

II.4.A Electron-induced vibrational excitation of NO 2

The nitrogen dioxide (NO 2 ) molecule has a number of applications in engineering and science.

For instance, it is used in the sterilization of medical instruments [Shomali et al., 2015a]. It plays a major role in the middle and upper atmosphere by the chemiluminescent emission process [Clough and Thrush, 1967]. NO 2 is also well-known to be an undesirable pollutant in the troposphere such as recent plasma technologies are used for its removal. Knowledge of physical and chemical processes taking place in NO 2 plasma especially collisions electrons and NO 2 molecules are of great importance in improving process efficiency in these technologies.

Electron collisions with the NO 2 molecule have been extensively investigated in the past, with initial studies focusing on the NO 2 ionization [Collin, 1959, Kiser and Hisatsune, 1961, Stephan et al., 1980]. In more recent studies, other processes in the e --NO 2 collisions were reported. We can cite, for example, the measurements of absolute total cross sections for electron-NO 2 collisions by Ref. [Szmytkowski and Krzysztofowicz, 1992] or those of dissociative electron attachment (DEA) to NO 2 as a mechanism on negative-ion formation from the groups [Fox, 1960] and [Rangwala et al., 2003]. Munjal et al. [Munjal et al., 2009] reported theoretical data on elastic integral, differential, momentum transfer cross sections, as well as electronic-excitation cross sections from the ground electronic state to the five lowest electronically excited states of NO 2 .

As for electron-impact vibrational excitation (VE) of NO 2 , only little information is available while this process plays an important role in plasma depollution because vibrationally excited NO 2 react differently, compared to the ground-state NO 2 , with other species present in the plasma. To the best of our knowledge, there exist only one measurement on VE cross sections for energies 0.3 ∼ 2.5 eV by Benoit and Abouaf [Benoit and Abouaf, 1991]. No theoretical VE study of NO 2 has been reported until now. As excitation of different modes having almost the same fundamental frequencies cannot be separated experimentally, at least by the conventional technique, the theoretical treatment for the VE of NO 2 is indeed necessary.

Combining the normal mode approximation for vibrational states of the target molecule (Eq.(II.7)), fixed-nuclei electron-NO 2 scattering matrices (Eq.(II.16)) and the vibrational frame transformation (Eq.(II.17)) employed to evaluate the scattering matrix for vibrational transitions, we provided cross sections for VE,

e -+ NO 2 (υ i ) → e -+ NO 2 (υ i ), (II.18)
and corresponding thermally-averaged rate coefficients. Above, the quantum number υ i and υ i stand for the initial and final vibrational state of the target for normal mode i. The reaction channel for which υ i = υ i corresponds to elastic scattering. Systematic calculations were performed for the excitation of the 3 lowest excited vibrational levels in the ground electronic state of NO 2 . For higher vibrational states, processes such as DEA become more predominant than vibrational excitation.

The ab initio calculations

NO 2 is an open-shell molecule belonging to the C 2v point group at its equilibrium, with the ground state electronic configuration

X 2 A 1 : 1a 2 1 2a 2 1 1b 2 2 3a 2 1 2b 2 2 4a 2 1 5a 2 1 3b 2 2 1b 2 1 4b 2 2 1a 2 2 6a 1 1 .
It is characterized by three normal modes of vibration: bending, symmetric stretching, and asymmetric stretching, displayed in Figure II.2. Displacements along the bending and symmetric stretching modes do not break the C 2v symmetry of the molecule, while the asymmetric stretching mode reduces the symmetry to the C s group. The frequencies of the normal modes were computed with a complete active space selfconsistent field method (CASSCF) of CAS (17,15), i.e. 17 electrons are distributed over 17 orbitals, and cc-pVTZ basis set using the ab initio quantum chemistry package Molpro [START_REF] Molek | A remeasurement of the products for electron recombination of N 2 H + using a new technique: No signficant NH+N production[END_REF]. Table II.1 lists the optimized geometry and vibrational frequencies obtained in the present calculation and compares the results with the available experimental data.

Table II.1: Structure and vibrational frequencies of NO 2 obtained in this study and compared with experimental data from Ref. [Johnson III, 2010]. ω 1 , ω 2 and ω 3 are the frequencies of bending mode, symmetric stretching mode and asymmetric stretching mode, respectively. As the table demonstrates, the present results match excellently with the experiment. From the results of the frequencies calculation, we also obtained the transformation matrix, D of Eq.(II.5), between the normal and Cartesian coordinates. The matrix is needed to generate geometries input data from X = Dq for electron-NO 2 scattering calculations.

Bond (Å) Angle(Degrees) ω 1 (cm -1 ) ω 2 (cm -1 ) ω 3 (cm -1 )

Chapter II. Role of neutral and radicals molecules

Scattering calculations

In the scattering calculation, we used for several values of normal coordinates q i an R-matrix sphere radius of 14 bohrs, the cc-pVTZ basis set and the complete active space configuration interaction (CAS-CI) method built on orbitals obtained from the same CASSCF calculation in Molpro. An accurate description of the target molecule is crucial in this model. In each panel, only one mode q i is varied, while the other modes are kept fixed at their equilibrium positions q i = 0. Red solid curves are the actual potential energies obtained from the UKRmol suite, while black dashed curves represent energies calculated in the harmonic approximation. Horizontal dashed lines denote energies of vibrational states.

Cross sections and rate coefficients

Taking an average over initial rotational states and a sum over final rotational states in the process, i.e. neglecting the rotational structure of the molecule, the cross section for VE of the mode i is expressed as

σ υ i ←υ i ( ) = π 2 2m |S υ i υ i -δ υ i υ i | 2 , (II.19)
where again is the energy of the incident electron and m is the mass of electron. The computation of the scattering matrix S v i v i for VE relies on the vibrational frame transformation of Eq.(II.17). with respect to all possible allowed combinations of lλ, l λ as a function of the difference ∆l = ll for all four singlet irreps of the e --NO 2 complex. The lowest value of l is 0 for 1 A 1 , 1 for 1 B 1 and 1 B 2 , 2 for 1 A 2 . For 1 A 1 (solid red line with circles), the couplings between the s and f, s and g partial waves are very small. For 1 B 1 (solid green line with triangles) and 1 B 2 (solid blue line with squares), since s-wave scattering is forbidden by symmetry, only couplings between p, d, f and g partial waves is shown. For 1 A 2 (solid pink line with stars), s and p partial waves scattering is not allowed. As discussed in the preceding section, vibrational frame transformation treatment becomes feasible, at least in principle, only if the elements of the fixed-nuclei S-matrix for the e --NO 2 system are smooth with respect to the incident energy. Therefore, the treatment is not appropriate if there are low-energy resonances in the e --NO 2 spectrum at low energies. In an attempt to analyze the behavior of S-matrix elements, we computed the absolute value squared S l λ ,lλ 2 (Λ is omitted hereafter for the sake of simplicity) of the matrix elements as a function of the electron scattering energy at the equilibrium geometry. Figure II. 4 gives an idea about couplings between different partial waves in the scattering process. Here, singlet states are chosen as an example. Although five partial waves is included in the computation, only couplings between channels with ∆l < 2 are not negligible for inelastic scattering. The contribution from the ∆l ≥ 2 couplings is modest.

A 1 3 A 1 3 B 2 3 B 1 3 A 2 1 B 1 1 B 2 1 A 2 |S l'λ',lλ | 2 1 A 1 1 B 1 1 B 2 1 A 2 3 A 1 3 B 1 3 B 2 3 A 2
Notably, the S l λ ,lλ 2 coupling producing the dominant contribution to the inelastic process for each symmetry depends only weakly on the scattering energy, as shown in Figure II.5. Therefore, it is reasonable to employ the vibrational frame transformation for the VE calculations. Vib.exc.cross section (cm Figure II.6 displays the VE cross sections for the singlet and triplet states of the e --NO 2 complex with the target molecule being initially in the ground vibrational level. Not surprisingly, the 1 ← 0 cross sections of the e --NO 2 singlet complex (solid red curves) are the largest compared to the triplet and 2 ← 0 transitions. The 1 ← 0 VE cross section for triplet (dashed red curve) bending mode has the same shape as that of the singlet. Its magnitude is smaller than that of the singlet by more than a factor of 4. For the symmetric stretching mode (see Figure 4 (b)), the 1 ← 0 VE cross sections for both singlet and triplet depend very weakly on the scattering energy up to 0.6 eV. Note that the 1 ← 0 VE cross sections for the asymmetric stretching mode are zero due to the symmetry of the scattering matrix with respect to positive and negative values of displacements along the mode. Furthermore, there is no significant difference in the magnitude of the cross sections for the symmetric stretching and asymmetric stretching modes due to the close fundamental frequencies (see Table II.1).

2 ) triplet singlet triplet υ=0 υ'=1,2 υ'=1 υ'=1 υ'=2 υ'=2
It should be stressed that the ∆v = 1 transitions for asymmetric stretching mode are forbidden by symmetry. Indeed, Figure II.8 shows the dominant elements of S-matrix as function of asymmetric stretching mode at a given energy. As we can see, S l λ ,lλ 2 is symmetric with respect to q 3 yielding a zero integral in Eq.(II.17) for one quanta transition.

( 1 A') 1,1 1,1 ( 3 A') 1,1 1,1 ( 1 A') 1,1 0,0 ( 3 A') 1,1 0,0 ( 1 A") 2,1 1,0 ( 3 A") 2,1 1,0 ( 1,3 A") 1,0 1,0 ( 1,3 A") 2,1 2,1
Figure II.8: Dominant (absolute value squared) elements S l λ ,lλ 2 of the scattering matrix as a function of normal coordinate q 3 (asymmetric stretching mode) at 0.3 eV scattering energy. Solid curve: singlet for NO 2 + e -system. Dashed curve: triplet for NO 2 + e -system. The corresponding symmetry is indicated in parentheses. The mumbers indicate different partial waves l λ ← lλ.

Chapter II. Role of neutral and radicals molecules

The thermally averaged rate coefficient α υ i ←υ i (T ) is obtained from the energy-dependent cross sections of Equation (I.28) according to Eq.(I.31). The rate coefficient is first calculated separately for singlet and triplet transitions and then the final rate coefficient is obtained by taking into account the corresponding statistical weights: 1/4 for singlet and 3/4 for triplet. Figure II.7 displays the spin-and thermally-averaged rate coefficients for (de-)excitation transitions between the three lowest vibrational states of the bending mode as an example. Similar to the previous studies for conveniently using the computed rate constants, we fitted the numerical spin-and thermally-averaged rate coefficients to an analytical formula. The fitted coefficients for each individual transitions are listed in Table 2 of Ref. [Liu et al., 2019].

Cross section obtained with a simplified approach Making use of the simplified approach described in Chapter III, we computed the VE cross section for the change only one quanta form the analytical formula (III.9) and compared the result to the one obtained using the complete numerical treatment of Eq.(II. 19).

Figure II.9 displays, as an example, the compared VE cross sections for the bending mode where the derivative of the scattering matrix in Eq.(III.6), employed in Eq.(III.9), was evaluated with two points q 1 = 0.2 and q 1 = 1. In the case of singlet, the VE cross section calculated with the simplified model (green solid curve) is larger than the presently calculated one (red solid curve) by about 30% while this deviation is less than 10% for triplet. To get an idea about these discrepancies, we plotted in Figures II. 10

ν 1 =0 ν 1 '=1

Bending

Figure II.9: Comparison between the presently calculated v 1 = 0 → v 1 = 1 cross sections (red curves) for bending mode and that one calculated by Equation (III.9) (green curves) with two normal coordinates q 1 at 0.2 and 1 respectively. In the upper subfigure of Figure II.10, we can see small oscillations in the S-matrix elements of triplet state with a maximum amplitude of about 2%, for the middle panels (S 00,00 versus q 1 ). This implies that the S-matrix elements are in fact linear and hence the first derivative approximation for triplet is reasonable. As for singlet in the bottom subfigure of Figure II.10, the S-matrix elements seems smooth with respect to q 1 but a detailed analysis of the derivatives, displayed in Figure II.11, reveals that they depend on q 1 . Consequently, the slop in Eq.(III.6) should be different according to the chosen points to evaluate it which can explain the overestimation observed in the obtained VE cross section. However, the simplified approach (III.9) apparently can provide an estimation of the VE probabilities used to compute the VE cross section for one quanta transition. In a better treatment higher orders in the Taylor expansion of the scattering matrix elements of Eq.(III.1) should be left.

q 1 -0,2 -0,15 -0,1 -0,05 0 0,0 0,0 0,0 0,0 1,0 0,0 1,0 0,0 2,-2 2,-2 1,1 1,1 1,-1 1,-1 2,-2 2,-2 1,1 1,1 1,-1 1,-1 3,-2 2,-2 2,-1 1,-1 2,1 1,1 2,1 1,1 2,-1 1,-1 3,- 2
1,0 0,0 1,0 0,0 2,-2 2,-2 1,1 1,1 1,-1 1,-1 2,-2 2,-2 1,1 1,1 1,-1 1,-1 3,-2 2,-2 2,-1 1,-1 2,1 1,1 2,1 1,1 2,-1 1,-1 3,-2 2,-2
1,0 0,0 1,0 0,0 2,-2 2,-2 1,1 1,1 1,-1 1,-1 2,-2 2,-2 1,1 1,1 1,-1 1,-1 3,-2 2,-2 2,-1 1,-1 2,1 1,1 2,1 1,1 2,-1 1,-1 3,- 2

Uncertainty estimations

In a similar manner to previous studies, there are sources of uncertainties in our calculations that need to be assessed. To our best knowledge, there exists no experimental or theoretical VE cross sections or rate coefficients data available in the literature. Hence, the uncertainties of the theoretical model can not be estimated since there is not other more accurate model to benchmark the present results. Nevertheless, for an ab initio study like this one, uncertainties could and should be provided by intermediate quantities as well as for final calculated observables, such as excitation and ionization energies, potential energy surfaces, dipole moments, polarizabilities, etc, that need to be stable regarding to parameters of the model. From a point of view of an electrostatic model potential for the electron-NO 2 collisions, the major contribution to the scattering amplitude for vibration excitation is expected to be caused by variations of the permanent dipole moment and the polarizabilities of NO 2 along the normal mode coordinates. We don't use the model potential method in the study: The accuracy of the final cross sections depends on the accuracy of wave functions of the target and the scattering electron. The accuracy of computed wave functions cannot be directly compared with previous results. However, comparing the dipole moment, evaluated from the wave functions of the target molecule, with the available accurate data can give an idea about the accuracy of computed wave functions used in the R-matrix calculations and, correspondingly, on the accuracy of the final cross sections.

Here, we investigate the stability of the dipole moment of target molecule as an example by performing a complete R-matrix calculation with different active spaces and basis sets, comparing the results with experimental data. We carried out two set of computations: (1) using the CAS (referred here as CAS4) mentioned above and increasing the size of basis sets;

(2) increasing the complete active space (CAS) with the cc-pVTZ basis set. The obtained 54 Chapter II. Role of neutral and radicals molecules results at the equilibrium geometry using the various parameters are illustrated in Figure II.12.

As one can see in Figure II.12(a), the dipole moment approaches the experimental value 0.316 D when we increase the basis set. Evidently, the dipole moment obtained by cc-pVTZ basis set used in this study agrees with the experimental data very well. Augmented (aug-) basis sets are not used as they would significantly extend outside the R-matrix sphere. Figure II.13: The figure compares selected values of the 1 A 1 scattering matrix elements (absolute value squared) before (solid color lines) and after (dashed lines of the same color) the unitary transformation that eliminates the long-range dipolar coupling between asymptotic channels in the e --NO 2 scattering matrix in the body frame. See the detailed discussion in the text. Each curve is labeled at the left of the figure with the pair of indexes (l λ ← lλ), corresponding to the final channels and initial channel. The results for other irreps are not displayed but the situation is very similar to the 1 A 1 symmetry.

Another source of uncertainty is the accuracy of the S-matrix derived from the R-matrix code, when the dipole term is included in the interaction. For the electron scattering by a nonpolar molecule, the electronic angular momenta l are decoupled at large distances from the target, such that l is a good quantum number for large separations between the electron and the molecule. However, for a dipolar molecule the electronic angular momenta l are coupled at long, as at short distances. The dipolar interaction exhibits the same long-range behavior as the centrifugal potential, thus, it is possible to combine the centrifugal and dipole interactions and obtain effective angular momentum. The Schrödinger equation for an electron in a dipole field is given by (in atomic units)

- 1 2 d 2 dr 2 + L2 2r 2 - D cos θ r 2 -E ψ = 0 (II.20)
where L2 is the square of the orbital angular momentum, D is the dipole moment of NO 2 , r and θ (polar angle) are the relevant spherical coordinates of the incident electron with respect to the center of mass of the target, E is the total (electronic) energy of the system. Note that the term D cos θ r 2

in the above equation arises form Eq.(II.12) where D is given by Eq.(II.13).

In practice, expanding the angular part of the electron wave function ψ (the radial part is F (r) whose the asymptotic form is given by Eqs.(II.14) and (II.15)) in the spherical harmonics basis and noting that D cos θ = D 4π 3 Y 0 1 , Eq.(II.20) becomes an eigenvalue problem in which the elements

l λ Y 0 1 |lλ = (2l + 1)(2l + 1) 4π l 1 l 0 0 0 l 1 l λ 0 λ , (II.21)
should be evaluated acknowledging that l λ | L2 |lλ = 2 l(l + 1)δ ll δ λλ . This can be accomplished using the Clebsch-Gordan coefficients or alliteratively the Wigner 3j-symbol calculator.

Once all elements in Eq.(II.21) are obtained for l ≤ 4 (recall that in the R-matrix calculations, continuum Gaussian-type orbitals was included up to g-wave), we find eigenvalues of the operator L2 2 -D cos θ corresponding to effective values of l. The eigenvectors of this operator are used to build the unitary matrix U to transform the scattering matrix in Eq.(II.16) into the effective angular momentum representation according to S = U † SU, where S designates a scattering matrix without long-range dipolar coupling. Therefore, the unitary transformation U allows eliminating the long-range dipolar coupling between asymptotic channels in the e --NO 2 scattering matrix in the body frame. The values of the effective angular momentum l could be noninteger and even complex. In our case, l are real such that

U † = U -1 and U -1 is orthogonal resulting in U -1 = t U).
In the next step, we fitted the obtained S-matrix elements (absolute values squared) before S l λ ,lλ 2 and after Sl λ ,lλ 2 the unitary transformation with power law: S l λ ,lλ 2 = βE α and Sl λ ,lλ

2

= β E α , respectively, where α, β and α , β are the fitted parameters. The obtained parameters are given in Table 3 of Ref [Liu et al., 2019]. Figure II.13 shows a few examples of scattering matrix elements of dominant channels (∆l ≤ 2 and λ is zero) of the 1 A 1 symmetry. The largest matrix element for the 00 ← 00 transition remains unchanged after the unitary transformation. It is the variation of this matrix element with respect to the normal coordinates that yields the largest contribution to the cross sections. Therefore, the uncoupling of the partial-wave channels at large distances would not produce a significant change in the final cross sections. This also means that the coupling between partial waves induced by the permanent dipole moment of the target has a minor effect on the final cross sections.

Assuming that the order of magnitude of the vibrational excitation cross sections is determined by the square of the derivatives of the permanent dipole moment and polarizabilities of the target with respect to the normal coordinates, and also assuming that the relative uncertainties of the derivatives are of the same order as the relative uncertainties of the dipole moment, we can estimate the uncertainty of the obtained cross sections with respect to the accuracy of the wave functions of the target. From panels (a) and (b) in Fig. II.12, we estimate that the uncertainty in the dipole moment of NO 2 is less than 2%, producing the uncertainty in cross sections less than 4%. Another source of uncertainty in the final cross sections is due to the variation of the geometry-fixed scattering matrix with energy. The choice of the energy at which the scattering matrix S l λ ,lλ (q) in Eq. (II.17) is computed produces the corresponding uncertainty. Figure II.5 gives an idea about the energy variation of S l λ ,lλ (q). The largest components with ∆l = 0 vary for about 3% over the energy interval of 0.4 eV. It gives an uncertainty in the cross sections of the order of 6%. No other significant uncertainty sources were identified. Therefore, the overall uncertainty of the present calculations seems to be below 10%.

Summary

We reported in this section the first theoretical results on vibrational excitation and deexcitation of the NO 2 X 2 A 1 molecule in collisions with a low-energy electron using first principles. Extensive uncertainty estimations were performed by changing parameters of the scattering model. Converged results for the target properties and eigenphase sums demonstrated the validity of the obtained results. We expect that the data reported in the present study could be valuable in kinetic studies of low-temperature NO 2 -containing plasma.

II.4.B Vibrational excitation of N 2 O by electron impact

As is the case with NO In this section, we present the first theoretical vibrational (de-)excitation (VE) study of N 2 O by electron impact. The studied process can be depicted by

e -+ N 2 O(v i ) → e -+ N 2 O(v i ), (II.22)
in which v i and and v i denoted the initial and final vibrational states of N 2 O, respectively.

In a similar manner to the NO 2 study described above, we computed cross sections and rate coefficients for transitions between ground and first vibrational states of N 2 O using an approach that employs the normal mode approximation for vibrational states of the target molecule (Eq.(II.7)), fixed-nuclei electron-NO 2 scattering matrices (Eq.(II.16)) and the vibrational frame transformation (Eq.(II.17)).

II.4 Applications

57

At its equilibrium geometry, N 2 O (with 22 electrons) has a linear asymmetric "N-N-O" molecular structure, described by the C ∞v symmetry point group where a closed-shell electronic ground state configuration is given by As a second step, we performed the scattering calculations in C 2v (a subgroup of C ∞v according to Table I.1) with the same basis set and CAS employed for the structure calculations. However, Quantemol-N does not have CASSCF built into it, a series of convergence tests using the available complete active space configuration interaction (CAS-CI) model with the Hartree-Fock orbitals built with Molpro were performed to investigate the stability of the target properties. We obtained with a R-matrix sphere radius of 11 bohrs a permanent electric dipole moment of the target molecule of 0.1 D, which is considered to be in satisfactory agreement with the experimental value, 0. Red solid curves are the actual potential energies obtained from the R-matrix code, while black dashed curves represent energies calculated in the harmonic approximation, i.e. simply ∼ ω i 2 q 2 i . Horizontal dashed lines denote the energies of vibrational states.

1 Σ + : 1σ 2 2σ 2 3σ 2 4σ 2 5σ 2 6σ 2 1π 4 7σ 2
Figure II.15 depicts the ground-state electronic potential energy curves of N 2 O for each normal mode obtained with the R-matrix method and compared to the potential energy curves of harmonic oscillators generated with the frequencies provided by Molpro. As we can see the curves match reasonably well with each other but with small discrepancies that can be attributed to the anharmonicity of the actual N 2 O potential. q 2 =-0.8668 q 1 =-0.8668 q 1 =+0.8668

q 3 =-0.8668 q 3 =+0.8668 2 Π 2 Σ - 2 Σ + 2 A" 2 A' Figure II.16:
The sum of eigenphases as a function of the electron scattering energy for equilibrium geometry and displacements q i = ±0.8668 (dimensionless) along each normal mode. Due to the symmetry of bending mode, the eigenphase sums of q 2 = +0.8668 and q 2 = -0.8668 are the same. The curves are color coded according the different symmetries of the e -+ N 2 O system.

II.4 Applications

59

Again, the applicability of the theoretical approach described throughout this chapter is based on the fact that the scattering matrix should be smooth with respect to the collision energy. For this purpose, we plotted in Figure II.16 the eigenphase sums (from Eq.(I.5)) of different irreducible representations at equilibrium and at displacements away from the equilibrium along each normal mode coordinate. As we can see, the variation of the eigenphase sums is smooth for energies below 2.3 eV. Above this value, a sharp energy dependence at certain energies is observed due to the presence of electronic resonances 2 Π of the complex e --N 2 O.

-1,5 -1 -0,5 0 0,5

(q 1 )S 00,00 (q 1 )ψ 0 (q 1 )| -0,8 -0,7 -0,6 -0,5 -0,4 0,24 0,26 0,28 0,3 0,32

E el =0.3 eV E el =0.4 eV E el =1.6 eV -1 0 1 q 1 0 0,1 0,2 0,3 0,4 |ψ 1 *
(q 1 )S 11,11

(q 1 )ψ 0 (q 1 )| -0,8 -0,7 -0,6 -0,5 -0,4 0,24 0,26 0,28 0,3 0,32

E el =0.3 eV E el =0.4 eV E el =1.6 eV -2 -1 0 1 2 q 1 0 0,1 0,2 0,3 0,4 |ψ 1 *
(q 1 )S 2-2,2-2 (q 1 )ψ 0 (q 1 )| -0,8 -0,7 -0,6 -0,5 -0,4 0,24 0,26 0,28 0,3 0,32

E el =0.3 eV E el =0.4 eV E el =1.6 eV
Figure II.17: The integrands of Equation (II.17) for the NO stretching mode for three among the largest matrix elements 00 ← 00, 11 ← 11 and 2 -2 ← 2 -2 as an example. Each panel shows the integrand for one matrix element for three energies 0.3eV, 0.4eV and 1.6eV.

A second demonstration of the energy dependence is the analysis of the integrand in Eq.(II.17). Figure II.17 shows the integrands for the NO stretching mode for three among the largest matrix elements 00 ← 00, 11 ← 11 and 2 -2 ← 2 -2. Each panel depicts the integrand for one matrix element for three energies 0.3eV, 0.4eV (which are just above the vibrational excitation threshold), and 1.6eV (the energy is well above the threshold). As one can see, the energy dependence is weak. For the 11 ← 11 and 2 -2 ← 2 -2 transitions drawn in the two lower panels, the curves for different energies are indistinguishable. With the above considerations, we calculated vibrational (de-)excitation cross sections for transitions between the ground and the first excited vibrational states for each of the normal modes using the formula of Eq.(II. 19). Figure II.18 displays a comparison of the theoretical 1 ← 0 VE cross sections with the available experimental data [Hayashi and Akashi, 1992, Allan and Skalický, 2003, Nakamura, 2007]. All three experimental data do not resolve the v 1 = 1/v 2 = 2 and v 3 = 1/v 1 = 2 thresholds, i.e. the experimental cross section for the excitation of the NO mode (v 1 = 0 → 1) includes also a contribution for the transition v 2 = 0 → 2, and the cross section for the excitation of the NN mode includes a contribution for the v 1 = 0 → 2 transition. But these additional contributions are expected to be significantly smaller due to the vibrational propensity rule: The transitions with a change of only one vibrational quanta are the largest. There is a significant disagreement between the experimental data, up to a factor of 20-50 for certain energies. On the other hand, the theoretical results also don't agree better with one experiment or another: For the NO stretching mode (panel a), the theory agrees better with the experiment by Hayashi [Hayashi and Akashi, 1992]. For the bending mode (panel b), the theory agrees better with the two other experiments, although the agreement is quite poor. Note that the presented integral cross sections by Allan and Skalický [Allan and Skalický, 2003] were obtained from the measured differential cross sections. Finally, for the NN stretching mode (panel c), the theory agrees better again with the data by Hayashi [Hayashi and Akashi, 1992].

The present theoretical cross sections are expected to be valid only for energies below the energy of the 2 Π resonance mentioned above, i.e. below 2.3 eV.

Thermally averaged rate coefficients α v i ←v i for vibrational excitation are obtained from the cross sections using the standard formula of Eq.(I.31). The computed rate coefficients were also fitted using an analytical formula employed in the preceding studies. The obtained coefficients are shown in Figure 6 and fitted parameters are listed in Table II of paper .6.

As in previous studies, we performed a number of calculations to assess the uncertainty of the obtained theoretical results. Once again, two main sources of uncertainty in the present theoretical approach can be identified. The first is the accuracy of the fixed-nuclei S-matrix elements computed for the polar molecule (with a small dipole moment) in the limited basis of spherical harmonics. This point has been discussed in the NO 2 study where the uncertainty associated with this approximation was estimated to be of the order of 6%. This uncertainty should not be larger for N 2 O because it has a smaller dipole moment so that the couplings between partial waves induced by the permanent dipole moment of this molecule have a weaker effect on the final cross sections.

The second source of uncertainties is related to the particular scattering model used in the calculation. Performing a complete calculation with different parameters such as basis sets and CASs, uncertainty was assessed in paper .6. We found that the difference in the rate coefficients produced in by three models is about 6%. Consequently, the overall uncertainty of the present theoretical result is estimated below 12%.

One of the most striking features of this performed study is the role of Renner-Teller coupling in N 2 O vibrational excitation by electron impact. The Renner-Teller coupling is a particular type of the non-Born-Oppenheimer (non-adiabatic) coupling leading to a coupling between the vibrational and the electronic motions. It has been discussed in several previous studies [Mikhailov et For N 2 O with a ground electronic state of 1 Σ + symmetry, the Renner-Teller effect couples σ and π partial waves of the incident electron with degenerate bending mode of N 2 O. It manifests itself by a linear dependence of the coupling between partial wave components for small displacements along the bending coordinate as displayed in Figure 6 of paper .6. Renner-Teller effect results in a bending configuration of the equilibrium geometry of the N 2 O -anion. More discussions can be found in [Hopper et al., 1976, Tschumper andSchaefer III, 1997] and [Liu et al., 2020] for N 2 O.

Summarizing this section, we briefly reported the cross sections for vibrational (de-Chapter II. Role of neutral and radicals molecules )excitations of N 2 O by a low-energy electron using first principles. The computed results show a reasonable agreement with experimental data for the NO and NN stretching modes while for the bending mode the agreement is rather poor at energies above 0.4 eV. It was found that Renner-Teller coupling is responsible for the excitation of the bending mode, as was expected from general theoretical considerations. These outcomes suggest that more measurement of vibrational excitation in N 2 O, at least for a few energies, are needed. Current data could, however, be used in kinetic studies of low-temperature N 2 O-containing plasma.

II.5 Concluding remarks

This chapter gives an overview of my contribution within the research axis on the role of collisions involving electron-neutral molecules in low-temperature plasma technology, used for atmospheric depollution purpose. A theoretical approach for the electron-impact vibrational excitation of neutral molecules was developed and applied to NO 2 and N 2 O. Availability of accurate data for such process is believed to be crucial for plasma modeling. The study of electron-radicals collisions as well as taking into account for the rotational structure of each vibrational level of the targets were postponed to future works (see outlooks in Chapter V).

The present theoretical studies have allowed, on the one hand, to provide the first theoretical results on vibrational (de-)excitation of NO 2 , on the other hand, to benchmark the obtained results with experimental data in case of N 2 O. These studies have been carried out in the context of Hainan Liu dissertation at Centralesupélec and in close collaboration with UCF and Rollins College. These collaborations benefit from supports of NSF and the Thomas Jefferson Foundation, in addition to other programs listed in Appendix V.2.C.

Finally, it has to be stressed that the theoretical approach employed throughout this section handles a smooth scattering matrix with respect to the collision energy. This circumstance of energy-independent S-matrix was fortunately fulfilled for the performed studied. In general, S-matrix depends on the scattering energy for neutral molecules which requires a further development as proposed in Chapter V.

III -Formation of fluorine based-molecules

III.1 Context and Motivations

As stated in the introduction, this chapter investigates the formation/destruction mechanisms of fluorine based-molecules that play a vital role in plasma technologies used for the development of materials. For the sake of clarity, studies performed in the context of other projects such as the formation of MnO oxides on a Fe-Mn alloy surface or thermal/mass characterizations of bio-materials will not be reported here.

Currently, the non-equilibrium plasmas produced with electrical discharges in BF 3containing feed gas have increasing been of interest for a large number of applications. In particular, BF 3 is frequently the boron carrier when plasmas are used for material processing [Sennikov et Central Florida (UCF), has also allowed for laying the theory for further applications in particular for the HCO + and N 2 H + molecular ions in the context of Dr. Abdillah Abdoulanziz's thesis, co-supervised with Prof. Ioan Schneider at Université du Havre.

In the following, only main results are provided. More details can be found in paper .7 included to this manuscript.

III.2 Simplified theoretical approach

Similar to previous models of Chapters I and II, our theoretical approach combines the normal modes approximation for the vibrational states of the target ion and use of the UK R-matrix code to evaluate electron-ion scattering matrices for fixed geometries of the ion (see Figure II.1). The main difference here is that the vibrational frame transformation is evaluated analytically. Indeed, at low-energy collisions, the scattering matrix frequently depends, in a linear fashion, on the normal mode coordinate near the equilibrium geometry. Hence, from the mathematical point of view, one can expand S Λ λ l ,λl (q) of Eq.(II.17) to first order in the normal coordinates q = {q 1 , q 2 , • • • } yielding

S Λ λ l ,λl (q) ≈ S Λ λ l ,λl (q 0 ) + i ∂S Λ λ l ,λl ∂q i dq i , (III.1)
where i is the index of different normal modes and S Λ l λ ,lλ is the scattering amplitude when the electron scatters from one channel (λl) to another (λ l ), l being still the electron angular momentum and λ its projections on the molecular axis. The partial derivative in Eq.(III.1) is taken at the equilibrium configuration q 0 = {0} of the ion, below BF + 2 for instance.

Plugging the above expansion in Eq.(II.17), we arrive at

S v i v i = λ l ,lλ S Λ λ l ,λl (q 0 )δ v i v i + 1 √ 2 λ l ,lλ ∂S Λ λ l ,λl ∂q i √ v i δ v i v i -1 + √ v i + 1δ v i v i +1 , (III.2)
for the mode i where we used the expressions of the vibrational wave functions in the harmonic approximation picture from Eq.(II.10) as well as the following identities

dq i = 1 √ 2 âi + â † i (III.3) dq i ψ v i (q i )q i ψ v i (q i ) = v i 2 δ v i v i -1 + v i + 1 2 δ v i v i +1 (III.4) dq i ψ v i (q i )ψ v i (q i ) = δ v i v i . (III.5)
The second term on the right-hand side of Equation (III.2) implies that each normal mode is most likely (de-)excited by one quanta. As a result, performing the integral in Eq.(II.17) becomes an evaluation of the derivatives in Eq.(III.2) that can obtained using the finite difference method calculated from two values of q i . The computational effort is thus significantly reduced.

III.2 Simplified theoretical approach 65

For convenience, we introduce the quantities

P i = g i 2 λ l ,lλ ∂S Λ λ l ,λl ∂q i 2 , (III.6)
which could be interpreted as the probability of excitation of the vibrational mode i with g i being its degeneracy.

Again, applying the vibrational frame transformation (FT) of Eq.(III.2) requires a smooth scattering matrices with respect to the incident electron energy and v i ≈ v i . Recall that FT theory extends the fixed-nuclei (FN) approximation by including the vibrational motion of the target adiabatically. As widely discussed in Chapter I (see Section I.4), this theory is powerful for ions without low-energy electronic resonances (doubly-excited dissociative states) appearing for geometries near their equilibrium and if the collision energy is not enough to excite the target ion electronically. The theory could also be applied to neutral targets if these assumptions are fulfilled (see Chapter II).

With the above considerations, if the doubly-excited dissociative states are not energetically accessible, one can only consider the indirect dissociative recombination (IDR) process in competition with rovibrational (de-)excitation. Based on the expansion of Eq.(III.2), the present theoretical model employs the following assumptions.

• The rotation of the molecule is neglected leading to a cross section without rotational structure. This is justified since these rotational resonances are washed out in the experimental cross sections.

• The cross section is averaged over the autoionizing resonances leading to a featureless curve. This procedure is justified because these resonances are not individually resolved in the experiment.

• For the indirect DR route, the autoionization lifetime is assumed to be much longer than the predissociation lifetime. The kinetic energy of the electron is quickly converted to vibrational motion of the ion just after the electron capture in a Rydberg resonant state.

This simplified approach allows to describe the vibrational (de-)excitation (VE,VdE) process in which the electron energy is enough to excite the ion in each normal mode q i and then leave it. In contrast, if that energy is not sufficient to excite the ion, the present model suggests that the probability of excitation of the ion by the electron is described by the same physics, but instead of leaving the vibrationally excited ion, the electron is captured in a Rydberg resonance attached to that vibrational state, excited by the electron. If the electron is captured by the ion, the system will most likely dissociate, rather then autoionize, leading to the IDR process.

It should be finally stressed that the method implemented is based on ideas developed in previous studies [Mikhailov et al., 2006b, Douguet et al., 2008b, Douguet et al., 2012b]. The expansion of Eq.(III.2) was performed on the quantum defect matrix that is used afterwards Chapter III. Formation of fluorine based-molecules in Eq.(I.15) with a set of ab initio potential surfaces of the neutral molecule and of the corresponding molecular ion obtained numerically near the equilibrium position of the ion. In a later study [Fonseca dos Santos et al., 2014a], a scattering matrix approach was used through the complex Kohn variational method and has allowed reproducing accurately ab initio Rydberg state energies of the neutral molecule.

In the next section, the scattering matrix approach for indirect DR using this time around the R-matrix method will be employed to obtain the DR and VE/VdE rate coefficients for BF + 2 , as a benchmark system. This formalism was also applied in a straightforward manner to more complex molecules such as CH

-+ BF + 2 (v i ) → BF * 2 , BF * * 2 →    BF + F F 2 + B e -+ BF + 2 (v i ) (III.7)
BF + 2 is a linear molecule in its equilibrium geometry belonging to D ∞h point group. Its main electronic ground state configuration 1 Σ + g is (with 22 electrons)

1σ + u 2 1σ + g 2 2σ + g 2 2π + u 2 3σ + g 2 4σ + g 2 3π + u 2 1π - u 2 1σ + u 2 1π + g 2 1σ - g 2 .(III.8)
The structure calculations were performed using a coupled pair approximation CEPA(2) method and the cc-pVTZ basis via the Molpro suite of codes [Werner et al., 2008]. Two stretching modes v 1 and v 3 (asymmetric and symmetric) with respective frequencies ω 1 and ω 3 and corresponding coordinates q 1 and q 3 , and a doubly degenerate transverse mode v 2 with a lower frequency ω 2 and coordinates (q 2x , q 2y ) were found. Figure III.1 shows the vibrational frequencies listed in Table 1 of Ref. [Kokoouline et al., 2018].

The electron-scattering calculations were carried out using the UK R-Matrix [Tennyson, 2010] code in the abelian subgroup D 2h with a configuration of interaction (CI) model; a basis cc-pVTZ and a complete active space CAS(12,10), i.e. 12 electron are distributed in 10 orbitals. We choose a R-matrix sphere of radius 10 bohrs. It should be noted that bending and asymmetric stretching mode calculations were performed in the C 2v abelian subgroup, while for the symmetric mode the group D 2h was used in the calculations.

As discussed above and widely in the preceding chapters, at low collision energies the fixed-nuclei scattering matrix depends weakly on energy whereas a sharper energy-dependence is observed at certain relatively high energies, corresponding to positions of Rydberg states attached to the excited electronic states of the ion. To analyze the energy dependence of the Plotting the excitation probabilities defined from Eq.(III.6) for each normal mode, we found in Figure 2 of paper .7 that these quantities have a weak dependence on energy and, therefore, they could be used as constants in calculations of the cross sections and thermallyaveraged rate coefficients.

Therefore, combining Eqs.(III.6) and (III.2), the cross sections1 for VE and VdE of one quanta for normal mode i take the respective simplified forms

σ v i +1←v i ( ) = π 2 2m (v i + 1)P i θ( -ω i ) , (III.9)
σ v i -1←v i ( ) = π 2 2m v i P i , (III.10)
where m is the reduced mass of the electron-ion system, is the incident energy of the electron, ω i and v i are respectively the energy and the vibrational quantum of mode i. θ(ω i ) denotes the Heaviside step function that opens the vibrational excitation channel when the collision energy is larger than one quanta of the vibrational energy for the ith normal mode.

68

Chapter III. Formation of fluorine based-molecules

The formulas of Eqs.(III.9) and III.10 describe the situation where the electron excites or de-excites the ion and then leaves it.

With the assumptions stated in the preceding section, the DR cross section for capture of the electron by an ion initially on its vibrational level v i via the temporary capture into a Rydberg state associated to the vibrational ionic level v i writes

σ DR v i ( ) = π 2 2m (v i + 1)P i θ( ω i -), (III.11)
where the brackets symbolically refer to a cross section averaged over the Rydberg series of resonances associated with the same vibrational state v i = v i + 1 (one quantum transitions) of the ion as well as over the rotational structure of the molecule [Douguet et al., 2012b].

Adding contributions from all normal modes q i , yields the total DR cross section (accounting for only indirect route)

σ DR {v} ( ) = π 2 2m i (v i + 1)P i θ( ω i -), (III.12)
where {v} denotes the collection of initial vibrational quantum numbers of all the normal modes. Above, the sum runs over the three modes of BF + 2 ; two stretching modes and doubly degenerate transverse mode. The Heaviside step function in σ DR {v} ( ) subtracts the contribution from the normal modes which are energetically allowed to excite. It results in a sharp drops in the cross section. 2 B 1 eigenphase (rad) 2 is the ground vibrational level and thus the electron can only be captured in a Rydberg resonances attached to the first excited vibrational state of each normal mode, excited by the electron. At very low scattering energies, i.e. below 0.02 eV, the DR cross-section is a smooth function inversely proportional to the incident energy of the electron, as predicted by the Wigner threshold law, whereas at higher energies, it exhibits a drop at each vibrational threshold, as expected from Eq.(III.12). In the calculations, the derivatives of P i in Eq.(III.12) were obtained using the finite difference method calculated from two values, q i = 0.01 and 0.1, for the displacements in each mode. Due to the simple analytical form of the cross-sections, the corresponding rate coefficients are readily evaluated from the general expression of Eq.(I.31) yielding

2 Σ g + 2 B 2 2 Σ + 2 Π asymmetric str. bending 2 Σ - 2 Π u - 2 Π u + 2 Σ g - 2 Σ u + 2 Π g - 2 Π g + 2 Σ u - 2 A 1
α v i (T ) = 2π k b T 2 m 3/2 (v i + 1)P i exp -ω i k b T VE v i P i VdE , (III.13) α DR v i (T ) = 2π k b T 2 m 3/2 i (v i + 1)P i 1 -exp - ω i k b T , (III.14)
where k b is the Boltzmann coefficient and T is the temperature. The thermally-averaged rate coefficients for DR and VE are shown in figure 4 of paper .7.

To check convergence of the obtained DR and VE/VdE rate coefficients, we have performed calculations for three sets of parameters (1) the CAS 1 (12,10) with the cc-pVTZ basis set; (2) a calculation (referred here as cc-pVTZ CAS 2 ) similar to (1) but with a smaller orbital space, Chapter III. Formation of fluorine based-molecules where 8 electrons are kept free in the active space; and (3) a calculation (referred here as cc-pVQZ CAS 1 ) similar to (1) but with the larger basis cc-pVQZ. The results are shown in Figure III.4. The difference between the rate coefficients produced in the three calculations for the DR process and the VE of the asymmetric stretching and bending modes is about 4%. The uncertainty of the rate coefficient for the VE of the symmetric stretching mode is larger, varying in the interval 10-40% for different temperatures. To give an idea about the uncertainty of the present results, we also plotted results of two calculations with the cc-pVTZ CAS 2 (dashed line) and the cc-pVQZ CAS 1 (dotted line) sets of parameters of the model. For the vibrational excitation of the symmetric stretching mode, the cc-pVTZ CAS 2 and cc-pVQZ CAS 1 curves are indistinguishable and slightly below the cc-pVTZ CAS 1 curve. For the remaining three processes, the three calculations produce the curves almost indistinguishable in the figure .   In addition to uncertainties related to the choice of the model parameters estimated above, the second source of uncertainty deals with the approximation (the capture model) that once the electron is captured into a vibrational Rydberg resonance associated with a closed vibrationally-excited channel, the e --BF + 2 system will dissociate rather than autoionize. For small and medium-size polyatomic closed-shell ions such as HCO + , NH it has been estimated that the probability of autoionization of vibrational Rydberg resonances is of the order of 10-20% compared to the probability of dissociation. Extending this result to BF + 2 , which is also a closed-shell ion, the overall uncertainty of the present theoretical rate coefficients is below 20-40%.

To test our simplified approach, we compared the cross sections obtained for the bending mode of BF + 2 to those computed by the complete treatment described in Chapter II, i.e. cross section of Eq.(II. 19) evaluated numerically with and/or without the inclusion of the III.4 Concluding remarks 71 closed-channel elimination procedure of Eq.(I.19). As an example, Figure III.5 shows this comparison for transitions between the two lowest vibrational levels of bending mode of BF + 2 . The cross sections (solid lines) obtained using the complete theoretical approach exhibit a strong resonant character, absent in a treatment without elimination procedure (dashed lines). The resonances are produced by closed vibrational states of the target. These resonances are washed out when thermally-averaged rate coefficients are computed. Therefore, in the calculation of the rate coefficients, one can use cross sections averaged over the resonances obtained using the simplified approach (dotted lines). Simplified approach (VE): σ 1<-0 (ε) Figure III.5: Vibrational (de-)excitation cross sections for transitions between the two lowest vibrational levels of the bending mode of BF + 2 . Solid lines represent the results obtained using the complete treatment of Equation (II. 19) in addition to the closed-channel elimination procedure of Equation (I. 19) in dashed lines. Dotted lines show the results from the simplified approach of Equations (III.9) or (III.10).

III.4 Concluding remarks

Contribution within the research axis on the formation of fluorine based-molecules, in particular BF + 2 , was reported briefly in this chapter. A simplified theoretical approach using the normal modes approximation, the vibrational frame transformation and the UK R-matrix code was developed to compute cross-sections and rate coefficients for DR, VE and VdE of BF + 2 by electron-impact. The thermally-averaged rate coefficients have a simple analytical form. Uncertainty estimations were performed by changing parameters of the scattering model. After benchmarking our model with the complete theoretical approach, we applied it to more complex systems in Chapter I. 2 . Solid lines represent the results obtained using the complete treatment in addition to the closed-channel elimination procedure in dashed lines. Dotted lines show the results from the simplified approach using Eq.(III.6).

The obtained thermally-averaged rate coefficients are relevant for the kinetic modeling of molecule based cold non-equilibrium plasmas, in the context of a complete lack of other theoretical or experimental data on these processes for this cation, and are ready to be used in the modeling of fluorine/boron plasma for etching or implantation processes.

The rotational structure of the target ion and of the neutral molecule was neglected in the present approach to be deferred in a further study (see outlooks in Chapter V). This implies that the obtained cross-sections and rate coefficients should be viewed as averaged over initial rotational states and summed over final rotational states of the corresponding initial and final vibrational levels (for VE) or dissociative states (for DR). Purely rotational transitions, i.e. without changing the vibrational state, might be useful to model very cold environments, below 40 K, which is not the case for the presently investigated BF 3 plasma. BF + 2 has no permanent dipole, so the rotational transitions are likely to have very small cross-sections.

At low collision energies, if the doubly-excited states of the electron-ion system are not energetically accessible, the indirect DR is usually dominant. Therefore, our simplified approach can provide an ab inito estimation of the total DR cross section when other more accurate approaches are computationally expensive or not available for polyatomic ions. An implementation of the developed model as module in Quantemol-N is currently performed with Dr. Maria Tudorovskaya, scientific consultant form Quantemol team. The module could be used to compute DR cross sections of molecular ions relevant in the interstellar medium and planetary atmospheres or low-temperature plasma technologies. Finally, the above work was published in Ref. [Kokoouline et al., 2018]. It has received supports from different programs listed in CV (see Appendix V.2.C). This study has also allowed to develop and formulate ideas for further applications in particular for the HCO + and N 2 H + molecular ions in the context of Dr. Abdillah Abdoulanziz's thesis, co-supervised with Prof. Ioan Schneider at Université du Havre.

IV -Toolkit for teaching quantum mechanics

IV.1 Context and motivation

Since about 70 years, quantum physics is responsible for the technological advances that make modern life possible. For example, without quantum mechanics there would be no transistors, and hence no personal computers and smartphones; no lasers, and hence no Blu-ray players or modern TV; no atomic clocks, and hence no GPS navigation. A new and promising area of research in recent years, at the intersection of mathematics, computer science, and quantum physics, is quantum computers and quantum information [Shor, 1995, Britt andHumble, 2017]. The developed processes in this area, based on a novel way of information coding using the quantum-bit (qbit), are going to transform many sectors such as artificial intelligence [START_REF] Cai | Entanglement-based machine learning on a quantum computer[END_REF], online security [START_REF] Lucamarini | Robust unconditionally secure quantum key distribution with two nonorthogonal and uninformative states[END_REF], or drug development [START_REF] Zhou | Quantum mechanical methods for drug design[END_REF].

To support these advances and further new technologies, based also on principles governing the microworld, knowledges of quantum mechanics are essential for future scientists, engineers, medical doctors, and many other professions.

Typically, quantum physics/mechanics (QM) is taught either as a pure math course or as a descriptive phenomenological discipline, in which different core concepts of quantum physics appear to students to be disconnected from each other and from their real-life experience. Moreover, designing laboratory experiments for a course of QM, as commonly made for the course topics in Newtonian mechanics and classical electromagnetism, is expensive or/and not available for the most of the core QM concepts [START_REF] Singh | Review of student difficulties in upper-level quantum mechanics[END_REF]]. These difficulties make quantum mechanics to new students counter-intuitive, in contrast to other areas of physics that allow describing phenomena objectively observed in the real world, and hence uneasy to apprehend. Over the last two decades, several excellent physics-education studies reported this issue [Carr et There is a general consensus that the use of multimedia in QM classes helps to make teaching QM more efficient. This conclusion consolidated me in the idea of integrating experiential learning in my QM classes (since 2016) at Centralesupélec (CS), taking the form of computer-simulated experiments.

For this purpose, with my colleague Prof. Viatcheslav Kokoouline from the University of Central Florida (UCF), we have formulated and developed an initial version of a complete course of undergraduate quantum mechanics based on numerical experiments. The framework combines a set of PYTHON numerical codes and a graphical interface allowing students to prepare and run numerically-simulated experiments demonstrating all topics of a standard QM course, usually covered in a formal mathematical approach. The most of the codes are simplified versions of previous codes developed in my research activities in the area of atomic, molecular, and optical physics. These codes are published in two textbooks [START_REF] Ayouz | Les fondamentaux de la mécanique quantique sous Python[END_REF], Ayouz et al., 2022]. As for the graphical interface, it was designed in the context of a post-doc, Dr. Alexander Korovin, funded by IDEX Paris-Saclay.

In the following, I will provide the basic ingredients of the numerical experiments, i.e. programming language, mathematical techniques and the theoretical topics covered by simulations, as well as the two components of the developed digital learning environment.

IV.2 Basic ingredients

IV.2.A Programming language

For the purpose of the suggested transformation of our QM courses, PYTHON language was found convenient because it offers a wide range of operations and functions allowing to manipulate easily with vectors and matrices (product of matrices, diagonalization,...) as well as special mathematical functions (Legendre, Laguerre, Chebyshev, Hermite, Bessel and other functions). In addition, graphical representations can be produced almost directly-using the plot () function, unlike some languages, which require additional graphical software (C or FORTRAN for example). The plot can also be displayed at each iteration, which, for example, makes it easy to visualize the time-evolution of wave functions. The simplicity of the syntax, the great flexibility in the utilization and execution of the programs as well as the free access are also among the advantages.

The codes were developed in PYTHON under Spyder (via ANACONDA), which is a powerful interactive development environment for the Python language with advanced editing, interactive testing, debugging and introspection features.

IV.2.B Mathematical techniques

For computer modeling several numerical techniques will be employed. As hinted above, they are chosen with two objectives that (1) they provide an intuitive and visual picture of the QM concepts and processes and (2) they could be used in an actual research work for simple projects. The methods may not be extremely accurate for idealized situations (such as a purely theoretical δ-potential or the infinite square wall) due to discontinuities in idealized potential, but they are very accurate for realistic potentials, which usually don't have any discontinuities. Below, we will describe briefly the employed techniques.

Discrete variable representation method for solving the stationary Schrödinger equation

In the described numerical method, wave functions are mainly represented using the Fourier grid Hamiltonian method (FGH), which is widely used in different research areas, in particular, to describe quantum molecular dynamics [Kosloff, 1988] and to solve the stationary Schrödinger equation [START_REF] Dulieu | Quasibound states in long-range alkali dimers: Grid method calculations[END_REF], Kokoouline et al., 1999, Ayouz and Kokoouline, 2016].

FGH is one of the variety of the Discrete Variable Representation (DVR) methods that uses a basis of complex exponent functions. The DVR methods were initially introduced by Harris et al. [START_REF] Harris | Calculation of matrix elements for one-dimensional quantum-mechanical problems and the application to anharmonic oscillators[END_REF] for calculating matrix elements of complicated potential functions in a truncated basis set. Later, Dickinson and Certain [START_REF] Dickinson | Calculation of matrix elements for one-dimensional quantum-mechanical problems[END_REF]] derived the transformation matrix with Gaussian quadrature for standard orthogonal polynomials. The DVR method is sometimes referred to as the Lagrange method [Baye, 2006], where the functions are usually represented on a coordinate grid or lattice. Analytical expressions for the matrix elements of operators in the DVR basis is an important advantage for the approach adopted here for numerical experiments: An explicit form of the kinetic energy operator allows one to avoid computing derivatives numerically using finite-difference approximations, as is the case for iterative methods such as Euler or Runge-Kutta. It therefore simplifies the structure of the codes, making them simple to handle by an inexperienced student. The main elements of the method used here are given below.

One of the core concepts of the non-relativistic QM is the equation of state: the stationary Schrödinger equation. Typically (in research or in QM courses), one needs to find solutions of the equation in one dimension

[T(x) + V(x)] Ψ(x) = - 2 2m d 2 dx 2 + V(x) Ψ(x) = EΨ(x), (IV.1)
where m is the mass of the particle. In general, FGH starts by defining a grid of N points over coordinate x such as

x 0 = x 0 + i∆x i = 0, 1, • • • N -1, (IV.2) where ∆x = x L -x 0 N , (IV.3)
is the constant grid step and x 0 and x L are respectively the initial and final grid points of the space interval where the equation is solved. It is assumed that outside of the interval the wave functions of interest are negligible within the required numerical accuracy.

The grid step ∆x and the interval [x 0 , x L ] are the key parameters determining the accuracy of obtained numerical results, which are the energies E and wave functions Ψ(x). The grid step ∆x determines the largest possible momentum p grid , which could be represented by the grid. If the largest and smallest values of the potential V (x) are V max and V min , respectively, the step should satisfy [Kokoouline et al., 1999]

∆x = π p grid ≤ π 2m(V max -V min ) . (IV.4)

Chapter IV. Toolkit for teaching quantum mechanics

The wave function Ψ(x) in this method is expanded in terms of the orthonormal DVR/FGH basis set functions as

Ψ(x) = i c i ϕ i (x), ϕ i |ϕ i = δ ii . (IV.5)
The functions ϕ i (x) form an orthonormal set [START_REF] Bulgac | Use of the discrete variable representation basis in nuclear physics[END_REF]] In the basis of functions ϕ i (x), both operators T and V are represented by N × N matrices. The potential operator V is diagonal in this representation

ϕ i (x) = N/2-1 k=-N/2 1 L exp {ıp k (x -x i )} = 1 L sin π(x-x i ) ∆x cot π(x-x i ) L , (IV.6) with p k = 2πk L and k ∈ -N 2 + 1, -N 2 + 1, • • • , N 2
V ii = dxϕ * i (x)V (x)ϕ i (x) = V (x i )δ ii . (IV.7)
The kinetic energy operator T is diagonal in the momentum representation, which enables one to evaluate the analytical form of matrix elements of T in the coordinate representation:

T ii = dxϕ * i (x)T (x)ϕ i (x) = π 2 (N 2 +2) 6m(N ∆x) 2 i = i (-1) i-i π 2 m(N ∆x) 2 1 sin 2 [(i-i )π/N ] i = i .
(IV.8)

A convenient property of the FGH representation is that the coefficients c i of the expansion in Eq. (IV.5) give also the values of Ψ(x) at the grid points Ψ(x i ) = c i / √ ∆x.

Once the matrix elements of the potential and kinetic energy operators are obtained, the matrix of the Hamiltonian operator T+V is written in the explicit form as

H ii = π 2 (N 2 +2) 6m(N ∆x) 2 i = i (-1) i-i π 2 m(N ∆x) 2 1 sin 2 [(i-i )π/N ] i = i + V (x i )δ ii , (IV.9)
where N is assumed here to be an even integer.

Eigenvalues E n and eigenfunctions Ψ n (x) of the Hamiltonian, H = T + V, are obtained using numerical PYTHON routines of N × N matrix diagonalization. The diagonalization produces the eigenfunctions in terms of the coefficients c n i of expansion (IV.5). The normalization of the Ψ n and c n i is such that

x L x 0 |Ψ n (x)| 2 dx ≈ N -1 i=0 |Ψ n (x i )| 2 ∆x = N -1 i=0 |c n i | 2 = 1 for all n. (IV.10)
It should be stressed that for a few case studies, we also use the DVR with basis functions constructed from sin, Legendre, Laguerre, or Chebyshev polynomials. Details on the form that should take the Hamiltonian (IV.9) in such basis can be found in Refs. [Szalay, 1993, Ayouz et al., 2020] and therein.

The Chebyshev polynomial expansion method for solving the time-dependent Schrödinger equation

When solving the one-dimensional time-dependent Schrödinger equation ı ∂Ψ(x, t) ∂t = HΨ(x, t), (IV.11) in most cases one employs the second-order differences, split operator propagation, Chebyshev polynomial expansion or Lanczos method to represent the time variable. Among these methods, the propagation approach using the FGH method (introduced above) and the Chebyshev expansion (FGH-C) is one of numerous pseudo-spectral techniques for the time evolution problem. Details on this method and comparison with others have been reported in [START_REF] Leforestier | A comparison of different propagation schemes for the time dependent schrödinger equation[END_REF], Monnerville and Robbe, 1994, Kosloff, 1994].

For the sake of consistency with the DVR/FGH method employed in the stationary case, we choose FGH-C method in which the solution of Eq.(IV.11) is given by the operator exp -ı H∆t :

Ψ(x, t + ∆t) = e -ı H∆t Ψ(x, t) ≡ U(∆t)Ψ(x, t) (IV.12)
Thus, the problem of the time propagation can be separated in two essential parts.

1. A choice of a basis of the representation of the wave function Ψ(x, t); exponential function in FGH.

Chapter IV. Toolkit for teaching quantum mechanics 2. The procedure of evaluation of the propagation operator e -ı H∆t on the wave function Ψ(x, t): U(∆t)Ψ(x, t).

The propagation (IV.12) is carried out using an expansion of the propagation operator exp -ı H∆t as a function of t in a series of Chebyshev polynomials:

U(∆t) ≈ Nc k=0 a k g k -ı H∆t r , a k = e ı(r+q) C k ג k (r), (IV.13)
where ג k (r) designates the Bessel function of order k and N c is the number of expansion coefficients. The following abbreviations were also used

r ∆t = E max -E min 2 and q = V min ∆t, (IV.14)
where r ∆t denotes the half of Hamiltonian energy spectrum with E min = V min and E max = V max + p 2 max 2m such as p max = π/∆x. Coefficients a k are defined from the Bessel function ג k (r) with C k = 1 for k = 1 and C k = 2 when k > 1. Finally, g k functions are obtained making use of the recurrence relation

g k (ẑ) = 2ẑg k-1 (ẑ) + g k-2 (ẑ), (IV.15)
where

ẑ = - ı H∆t, ∀ ẑ ∈ [-1, 1] and g 0 (ẑ) = 1 g 1 (ẑ) = z .
Note that the refinement of the scheme is based on increasing N c and not the time step ∆t, as usual. It is well known that the expansion by complex Chebyshev polynomials gives the best approximation among other possible expansions [Tal-Ezer and Kosloff, 1984]. Above, H∆t is N × N matrix operator where N being the number of grid points, defined above in Eq.(IV.3). Elements of the matrix H = T + V are given by Eq.(IV.9).

IV.2.C Theoretical topics

The main topics of the developed course are the same as in a standard undergraduate course of QM for physics and chemistry majors. The PYTHON codes solve elementary and more complex problems covering topics such as the piecewise potentials, measurements in QM, uncertainty and sperposition principles, WKB-approximation, α-decay, harmonic oscillator, hydrogen atom, diatomic molecules, propagation of wave packets, methods of approximation, Hartree-Fock method, Spin, etc. Demonstrating all topics listed above requires some additions to methods introduced in Section IV.2.B. The main elements are given below.

• To treat α-decay in polonium, resonant states of the atom have to be determined. This can be accomplished making use of the complex absorbing potential (CAP) formalism, combined to the DVR methods. Again, the CAP formalism was developed in research work and employed, for instance, in computing the DR cross section of CH + . Details on that method are given in Chapter I. Recall that by adding the CAP, the Hamiltonian matrix becomes non-Hermitian, requiring a special library for diagonalization. This latter, namely, LAPACK, is available in PYTHON without extra efforts.

IV.3 Achievements: the digital learning environment

81

• Extending Eq.(IV.9) to two (or three) dimensions is straightforward and reads

H i,j = T ii δ jj + T jj δ ii + V (x i , y j )δ ii δ jj , (IV.16)
where j-index stands for y-coordinate.

• The rotational structure of a molecule can be taken into account using the spherical harmonics for representing the angular wave functions. It follows that a centrifugal barrier should be added into the radial Schrödinger equation. The potential in Eq.(IV.9) reshapes as

V (r i )δ ii → V (r i ) + 2 l(l + 1) 2mr 2 i δ ii , (IV.17)
with l being the eigenvalues of the angular momentum operator.

• To obtain reflection and transmission coefficients of barriers, a projection method is employed in which the wave packet, obtained at the end of the propagation using Eq.(IV.12), is projected onto reflected or transmitted wave functions. Similar method will be used to compute branching ratios in DR of small polyatomic ions in Section V.2.C (planned research work).

• When a non-orthonormal basis sets (in contrast to those employed in the DVR methods) are used to represent the kinetic operator, such as atomic orbitals in Hartree-Fock methods, a generalized eigenvalue problem has to be solved: HΨ = ESΨ where S is the overlap matrix. This latter rises from the non-orthogonality of the basis. In practice, an orthogonalization procedure (canonical, symmetric or Gram-Schmid) is performed to save time spent on calculation.

• Take into account the spin, of an electron for instance, is also straightforward in the DVR methods since the equation to be solved will be somewhat similar to Eq.(IV. 16), where j-index now stands for the two possible projections of the spin ±1/2.

With the above additions, all major concepts of QM are covered by the developed digital learning environment and this is made possible due to the adopted DVR method. Indeed, this latter is easy to implement in computer modeling (differential operators have analytical expressions according to Eq.(IV.8)) and intuitive because it fully reconciles mathematical tools of QM and digital programming. Thus, mathematics that are employed, in general, heavily in explanations of postulates could be demonstrated with codes since students will be able to manipulate with matrices representing QM operators; inverting them, finding and analyzing their eigenvalues, analyzing eigenvectors and eigenfunctions, calculating expectation values of observables using dot/matrix products.

IV.3 Achievements: the digital learning environment

IV.3.A Python numerical codes

A standard code starts its execution by the variable "__name__" having the value "__main__" which represents the main program. In this program, we declare variables of the problem, Chapter IV. Toolkit for teaching quantum mechanics tables, constants, and the required libraries to be used during the execution. The called functions from the main program will return an operation or calculations such as the desired wave functions. Figure IV.2 displays the general structure of the majority of the codes dealing with the above topics. These codes are simplified versions of those developed in my research activities. Hence, they are use-friendly allowing a quick handling for further utilization: as tool, for instance, to teach the course of Computational Physics and test the approximation methods commonly thought in QM course for solving the Schrödinger equation. Note that atomic units for masses, energies, distances and times are adopted in the codes.

IV.3.B Graphical interface

In order to make the codes accessible to a large public, we have designed, in the context of post-doc of Dr. Alexander Korovin, a preliminary version of the graphical interface, that can be accessed at this address http://prd-mecaqu.centralesupelec.fr/. Currently, the interface works and contains draft versions of 10 chapters including most of the preceding discussed topics.

Figures IV. NGINX: a web server used as a reverse mail proxy and HTTP cache. AJAX: Asynchronous JavaScript + XML programming languages used to create web applications. JSON: is an open-standard file and data interchange format that uses human-readable text to transmit data objects consisting of attribute-value pairs and array data types. js -JavaScript: a high-level programming language used for web applications. • Upload potential energy data.

• Choose one of the "standard" potentials (step-like, square well, etc.).

• Enter an analytical formula for a potential energy or a physical disturbance, in studying methods of approximation, especially, for the perturbation theory.

• Modify the input parameters, • Download PYTHON codes or data such as the wave functions.

• Save figures and videos (wave packet propagation) for further utilizations.

The current interface can be used by instructors from other schools and universities IV.4 Concluding remarks 85 through our servers (CS or UCF) or in their own servers (online version). A stand-alone distribution (as an installation file) that can be installed on their own computers, as offline local use, was also developed for Linux and Windows operating systems. All these tools could be employed as course support.

IV.4 Concluding remarks

Based on codes carried out in my research activities, we developed an original tool to make teaching QM more efficient and accessible to undergraduate/graduate students (via online and blended modes). The idea of this project came from our experience of teaching at CS and UCF. The digital learning environment combines a set of PYTHON codes and a graphical interface gathering up a numerically-simulated experiments that demonstrate all topics of a standard QM course. The PYTHON codes are user-friendly, and the employed methods, namely, DVRs, permit to reconcile mathematical concepts of QM and programming. As for the web-interface, it allow users to manipulate with the codes, the wave functions, and observables. This pedagogical tool will be made available to the community either using our own servers (CS and UCF) or via a stand-alone distribution (as an installation file) that can be installed on other instructor's servers (online)/computers (offline) for their local use.

Further developments are planned in near future and would be a part of the NSF project on Improving Undergraduate STEM Education. Such developments will include preparation of (a) short lectures covering theoretical aspects of the QM concepts, (b) in-class activities and (c) laboratory assignments with and without the learning environment, (d) homework assignments. Some missing codes (in the list of QM topics above) should still be developed. Some other features that could be also considered are to integrate the developed tool either with (i) LMS (Learning Management System as Daskit that is available at CS) giving the possibility for the students to submit directly results of their experiental work (lab reports) or with (ii) a quantum chemistry packages (Molpro/Gaussian) for visualizing atomic/molecular orbitals and understanding the nature of the bonds (covalent, ionic) in a molecule.

Finally, we started testing the proposed teaching approach at CS in my quantum and statistical mechanics and research program classes. First feedback from students are positive which encourage us to accomplish tasks mentioned above. More tests are planned for the upcoming academic year. In parallel, this pedagogical tool will be tested in the Fall 2020 semester at the UCF Physics department in the class of undergraduate quantum mechanics taught by Prof. Kokoouline. Studies about the efficiency of the teaching strategies will be performed and published. V -Conclusions and perspectives

User interface input data

V.1 Conclusions

The most relevant studies of my projects and supervisions (see CV in Appendix V.2.C) were described in the preceding chapters. I have made a significant contribution into the determination of reliable data for atomic and molecular reactions especially those involving electron-molecule collisions at low energy (see Figure .1). My students and I first focused on studying the simplest diatomic systems to benchmark our developed theoretical models, after which we worked on extending these models to large molecular ions and to the neutral species.

Our theoretical models (depicted in Figures I.1 and II.1) are based on first principles. They employ the following: (i) the R-matrix method to compute ab initio electronic bound and continuum states, (ii) the vibrational frame transformation combined to, in case of ions, (iii) the closed-channel elimination procedure, in a spirit of molecular quantum defect theory, in order to construct an energy-dependent scattering matrix. This matrix describes interactions between vibronic channels of the target induced by the incident electron. For polyatomic molecules, an additional approximation of the theoretical methods was introduced: the use of the normal mode approximation for vibrational states of the target.

The proposed methods have allowed for computing ro-vibronic excitation (RVE) and dissociative recombination (DR) cross sections, as well as related rate coefficients, of various polyatomic systems of astrophysical (HeH + , CH 2 NH + 2 , NH 2 CHOH + ), material development (BF + 2 ) and atmospheric depollution (NO 2 , N 2 O) interest. Applications to ions with low-energy excited states were successful, as with CH + . Such ions were known to be difficult to treat theoretically using previous theoretical methods. Certain precautions, nevertheless, were taken to extend the models to neutral molecules, especially on the use of frame transformation procedures. Our studies have afforded us, on one hand, to provide an upper limit for the total rate coefficient of DR (where data are not available) and on the other hand, to confirm previous calculations. These outcomes could be used for plasma models in planetary science, interstellar medium, and for technologies.

It has to be stressed that the developed theoretical approaches are general since they require only a method to determine the scattering or reaction matrices. This can be accomplished from the electron-scattering codes such as the U.K. R-matrix, used over this manuscript through the Quantemol-N suite, the complex Kohn method or the Schwinger variational principles. We are currently implementing our models as a module in Quantemol-N, to compute cross sections and related rate coefficients of VE and DR of polyatomic molecules by electron impact; conducted in collaboration with Prof. Kokoouline from UCF and Dr Maria Tudorovskaya, scientific consultant form Quantemol team. We have also developed a complete course of undergraduate quantum mechanics, whose framework combines a set of PYTHON numerical codes and a graphical interface allowing students to prepare and run numerically-simulated experiments, demonstrating all topics of a standard QM course, usually covered in a formal mathematical approach. This project aims to address difficulties for undergraduate students in learning QM concepts. Further developments planned for the near future in both educational and research terms are stated thereafter.

V.2 Perspectives

My planned research program is a continuation of the previously-conducted studies.

V.2.A Rotational excitations of molecular ions in their electronically-excited states

In many low-temperature ionized environments such as ISM, planetary atmospheres and technological plasmas, cross sections for vibronic excitations are important for understanding and modeling these environments, especially, when including the rotational structure of molecules in collisions with electrons. Therefore, I plan to extend the study of HeH + , where we developed models for RE+VE of linear molecules, to CH + and include its rotational structure, in addition to its vibronic structure.

For this purpose, a complete rovibronic frame transformation (RVFT) is needed in combination with CCE-MQDT procedure to construct an energy-dependent scattering matrix in the laboratory frame. This matrix describes the interactions between rovibronic channels of CH + induced by the incident electron. Numerically, we obtain the body-frame scattering matrix (using the UK R-matrix code) in the subgroup C 2v , included in the natural C ∞v symmetry group of the ion. When various electronic states with different symmetries are included in the calculations, one needs to transform ŜC 2v into ŜC∞v . This can be accomplished through the spherical harmonics transformation according to Tables I.1 and I.2.

The second step will generalize the developed model for linear molecules to asymmetric top polyatomic targets. This step is straightforward and should be carried out in the context of a Ph.D. research work funded by CSC or ministry (INTERFACES).

V.2.B Electronically-excited states of neutral molecules

As emphasized in the preceding chapters, the developed models are based on the approximation that the S-matrix for electron-molecules collisions is energy-independent, so far. This holds for ions when (i) the scattering electron energy is not large enough to excite the target ion electronically but also (ii) in absence of electronic resonances appearing when the potential energy surface (PES) of the doubly-excited neutral molecule (electron-ion system) crosses the ionic PES near the equilibrium geometry (CH + case for instance).

Fortunately, this assumption was fulfilled for e -NO 2 and e --N 2 O collisions as revealed by a detailed analysis of eigenphase sums, so that the vibrational VFT could be accomplished. The S-matrix, however, does depend on the scattering electron energy for neutral molecules, in general.

Due to a variety of applications in plasma technology, the EE process including VE should be investigated. I am thus planning to extend the previous study on NO 2 making use of the method developed for CH + . I identity three main tasks:

-Choosing a different normalization factor for the scattering electron wave function, the energy dependence of Ŝ can be removed. Then, the renormalized Ŝ is energy-independent so that VFT can be employed. After VFT, the renormalization factor is multiplied back, and the energy dependence is recovered to compute the cross sections.

-Another approach would be needed were there to exist electronic resonances or a sufficiently high total energy of the system for all collision channels to be open for ionization. Similar to the "closed-channel-elimination" procedure used for ions, I am planning to adapt this procedure in combination with the Breigt-Wigner formula so that the e --NO 2 scattering physics below the first electronic state can be represented using an energy-independent multichannel scattering matrix evaluated above.

-To accomplish VFT in Eq.(II.17) within the preceding procedure, one has to evaluate the matrix elements ψ 1 (q 1 )| Ŝ |ψ 2 (q 2 ) , where ψ 1 (q 1 ) and ψ 2 (q 2 ) represent, respectively, the harmonic vibrational functions depending on dimensionless normal coordinates of the ground and first electronically excited states of the target. Here also, a new development is required since the integration mentioned above runs over different coordinates.

Chapter V. Conclusions and perspectives

To achieve these tasks, support will be asked from CSC/Ministry for hiring a PhD student.

V.2.C Branching ratios

Besides the DR process mentioned in the preceding chapters, branching ratios (BR) in dissociation of polyatomic molecules and negative ions after a collision of a molecular ion or a neutral molecule with an electron are also important data. Experimental determination of BR for certain systems is possible but expensive. From the theoretical point of view, a similar situation is observed: There are studies in which the DR branching ratios were determined, albeit with a highly unsatisfactory theoretical description. Hence, there is a clear need for a reliable theoretical approach towards determining of BR with respect to electronic and vibrational excitation of products. My long term project will address this issue.

In collaboration with Prof. Kokoouline (UCF), we plan to develop a simple theoretical method to determine the BR in DR of small polyatomic ions. We benefit from the study of DR in H + 3 (of astrophysical interest) in which e --H + 3 wave functions are explicitly computed, available, and could be analyzed in the asymptotic region, where the branching ratios should be evaluated. Thus, all the information about the scattering is accessible, particularly how much of the outgoing flux goes into dissociative channels within a finite region of space. Projecting this wave function on each rovibrational level of the H 2 +H of H+H+H fragmentation yields the current density associated to each channel leading to BR. The development of a detailed theoretical approach for BR in a benchmark system as H + 3 will help us to understand the main pathways that produce different products H 2 +H/H+H+H. We expect that this approach could be further applied to other ions.

Analogously, the electron-neutral molecules collisions may lead to DEA process, in competition with VE, RE or EE. Also, the DEA branching ratios can be as important as the overall DEA cross sections for plasma modeling. Treating the complete DEA of triatomic molecules, which is the work being focused on this part, is theoretically and computationally challenging. To do this, we will follow another strategy. After computing the PESs of the neutral and ion molecules, NO 2 and NO - 2 for instance using MOLPRO, we will employ the MCTDH (Multi-channel Configuration Time-Dependent Hartree) method in order to propagate the wavepacket on these PESs. At the end of the propagation, the wavepacket is projected onto vibrational and electronic states of the final products and DEA cross sections and BR can be evaluated.

These proposed works will be carried out with the help of collaborations: Profs. Ioan Schneider and Khalid Hassouni who have a large expertise on the plasma modelling. New collaborations may be built with Dr. Asa Larson from Stockholm university for MCTDH method through Dr. Samantha Douguet. This could be a part of projects such as ANRs, NSF and NASA.

D

Conference presentations

Introduction

The hydrohelium (helium hydride) cation HeH + is an important species in the interstellar medium (ISM) [1] and in the chemistry of the planetary nebulae such as NGC 7027 [2][3][4]. It is one of the simplest molecular ions and the first one formed in the early universe [5]. In the ISM it is formed mainly in the process of radiative association of He and H + or of He + and H [6,7]. Due to the relatively large abundance of electrons and HeH + in the ISM, collisions of HeH + with electrons play a significant role, in particular, leading to dissociation (dissociative recombination) and rovibrational (de-)excitation of HeH + . Other than in the ISM, the HeH + ion is also present in fusion reactors. Because it is one of the most abundant molecular ions near walls of the reactors, its contribution to the complete network of processes taking place in the reactors should be accounted for in fusion plasma modeling. In particular, cross sections for electron-HeH + collisions are needed.

A number of processes can occur when an electron collides with HeH + . One of the most important processes is the dissociative recombination. Cross sections for this process were measured and calculated in several previous studies [8][9][10][11][12][13]. Another process, relevant for the ISM [14] and fusion reactors, in the vibrational excitation and (de)-excitation,

e -+ HeH + (v, j) → e -+ HeH + (v , j ) , (1)
for which cross sections were calculated in a previous study [15] for the transitions between the three lowest vibrational levels v = 0, 1, 2. In that study, an R-matrix approach was employed to determine scattering matrices for fixed geometries. Rydberg series of vibrational and rotational resonances present in the spectrum of the e --HeH + system were neglected. In this work we determine cross sections and rate coefficients for the process using an improved treatment, in which the Rydberg series of vibrational resonances in HeH are accounted for using the multichannel quantum defect theory (MQDT). Electron-impact transitions between the five lowest vibrational levels, v = 0 -4, are considered.

The rest of the article is organized in the following way. The next section of the article discusses the theoretical approach used in the present calculation. In Section 3, the obtained rate coefficients for vibrational (de-)excitation are discussed and compared with the data available in literature. Section 4 concludes the study.

Theoretical Approach

Scattering Matrix for Fixed Geometries Of HeH +

The theoretical model employed in the present study combines the UK R -matrix method [16,17] and the multi-channel quantum defect theory (MQDT) [18][19][20]. The electron scattering calculations were performed employing the Quantemol-N interface [21] to run the UK R-Matrix code [16,17]. The target HeH + ion was assumed to be in the ground electronic state X 1 Σ + .

In the R-matrix calculations, the target and the scattering wave functions were represented within a full CI treatment, i.e., by allowing all electrons to occupy all orbitals. The calculations were performed using the Gaussian basis set 6-311G * . The continuum orbitals of Faure et al. [22], representing the scattering electron, was included up to the g-wave. The R-matrix radius was set to a 0 = 10 bohrs. All generated states up to 25 eV were retained in the final close-coupling calculation. As the first electronically excited state, A 1 Σ + correlated with H(1s) + He + (1s) is approximately 11 eV above the H + + He(1s 2 ) dissociation limit for the X 1 Σ + ground state, the ground state is essentially isolated and non-adiabatic effects are expected to be small. Therefore, for low electron energy collisions (bellow 10 eV) only the lowest electronic state is open for ionization in e --HeH + collisions and the dimension of the geometry-fixed scattering matrix does not change with energy.

Wave functions of e --HeH + continuum states obtained in the R-matrix calculations have the following asymptotic behavior at large distances r between the electron and the target ion in channels open for ionization [16]

F ij (r) ∼ 1 √ k i sin θ i (r)δ ij + cos θ i (r)K ij (R) , ( 2
)
θ i (r) = k i r - l i π 2 + ln(2k i r) k i + arg Γ l + 1 - i π , (3)
where index i refers to the entrance channel before a collision and j refers to the outgoing channel after the collision. In the above equation, K ij (R) are the elements of the reactance matrix [16,20], which depend on the internuclear distance R. In the cross section calculation, the scattering matrix S(R) is needed, which is obtained from K(R) as

S(R) = (1 + iK(R))(1 -iK(R)) -1 . (4)
Generally, for fixed geometries of the target and low collisional velocities, the matrices S(R) and K(R) depend weakly on energy. A sharper energy-dependence is observed only at certain higher energies, corresponding to positions of Rydberg states attached to excited electronic states of the ion. A convenient way to represent a weak or a strong energy dependence of the matrices is the eigenphase sum. Figure 1 shows eigenphase sums for three symmetries ( 2 Σ + , 2 Π, and 2 ∆) of the e -+ HeH + system and for several internuclear distances in HeH + . The chosen interval of internuclear distances corresponds to the Franck-Condon region of the lowest vibrational level of HeH + . The variation of the eigenphase sums for 2 Π and 2 ∆ is smooth for energies below 10 eV and does not change significantly with R. The 2 Σ + eigenphase sums demonstrate a sharp energy dependence at certain energies, which corresponds to a resonant state of HeH with energy changing with the internuclear distance R. electron scattering energy (eV)

HeH + Vibrational Dynamics

In calculations of the cross section for vibration (de-)excitations, vibrational wave functions of the target ion are needed in order to construct elements of the scattering matrix for transitions from one state v to another v. We calculated vibrational wave functions ψ v (R) and energies v by solving the Schrödinger equation for vibrational motion along R,

- 1 2µ
∂ 2 ∂R 2 + V(R) ψ v (R) = v ψ v (R) , (5)
where µ the reduced mass and V(R) is the internal potential of HeH + . The above equation was solved using a DVR-type method [23]. In the numerical calculation, the distance R was varied from 0.5 to 20 bohrs. The potential energy curve V(R) was calculated with the ab initio CCSD(T) method and the cc-pVQZ basis of Gaussian-type orbitals using the Molpro package [24]. Figure 2 shows the potential energy curve V(R) and wave functions for four vibrational levels v = 0, 1, 2, 8. Energy differences for transitions v → v + 1 and rotational constants of the HeH + vibrational levels are listed in Table 1 and compared with previous benchmark calculations [25,26]. Once vibrational energies, wave functions, and geometry-fixed S-matrix are calculated, the scattering matrix describing the transition amplitude S i ,i from one vibrational level v to another v is obtained using the method of vibrational frame transformation [19,27]. The indexes i and i enumerate vibrational states v, states of different electron angular momentum l, and its projections Λ on the molecular axis Z. Therefore, for the process

e -(l, Λ) + HeH + (v) -→ e -(l , Λ ) + HeH + (v ) , (6)
the scattering matrix in the molecular frame coordinate system can be written as

S Λ v l ,Λvl = ψ v (R) S Λ l ,Λl (R) ψ v (R) , (7)
where the brackets imply an integration over the vibrational coordinates. Many elements among S Λ v l ,Λvl are zero. In particular, the symmetry (one of the irreducible representations Γ of the C ∞v group) of the total e -+ HeH + system stays the same before and after a collision. Also, for low energies when there is only one electronic state X 1 Σ + open for ionization, the S Λ v l ,Λvl matrix is diagonal over Λ. Note that the matrix element S Λ l ,Λl (R) in the above equation is obtained from R-matrix calculations for a given value of R. It represents the scattering amplitude when the electron scatters from one channel (Λl) to another (Λ l ), while the nuclei do not have time to move. To evaluate the integral in Equation ( 7) the R-matrix calculations were performed for thirty values of R in the interval R = 1.0-3.9 bohrs with a step of 0.1 bohrs.

Elimination of Channels Closed for Ionization From the Scattering Matrix

The scattering matrix S of Equation ( 7) can be used for cross sections calculations only if the total energy of the system is high enough for all collision channels to be open for ionization. When some channels are closed, the scattering matrix should be modified applying the "closed-channel-elimination" procedure [20] and producing matrix S phys (E) according to

S phys (E) = S oo -S oc S cc -e -2iβ(E) -1 S co . (8)
The matrix S phys (E) has N o × N o dimension, N o being the number of channels open for ionization for a given total energy E of the system. The total energy could be written as a sum of the energy of the entrance channel E i and the relative kinetic energy E el of electron and the target ion: E = E i + E el . In the above equation, the matrices S oo , S oc , S cc and S co are submatrices of the original matrix S [18,20], partitioned as

S = S oo S oc S co S cc , (9)
where the partition of the matrix elements in the "o"-and "c"-parts is made on the basis whether the corresponding channel, i or i , is open or closed for ionization for a given total energy E. The quantity

β(E) is a diagonal N c × N c matrix β i i (E) = π 2(E i -E) δ i i , (10)
where E i refers to energy of the corresponding closed channel i and N c = N -N o is the number of closed channels. For a given vibrational quantum number v, channel energies are degenerate with respect to l and Λ. In principle, the channel energy E i depends also on the electronic excitation of the target ion, but in this study the initial electronic state is X 1 Σ + and collisional energies are too low to excite other electronic states.

Rate Coefficients and Cross Sections for Vibrational (De-)Excitation

Using the scattering matrix S phys , the cross section for electron-impact vibrational transition v → v of the ion is [28]

σ v v (E el ) = πh 2 2m e E el ∑ Λ l Λl S phys Λ l v ,Λlv -δ Λlv,Λ l v 2 , (11)
where m e is the reduced mass of the electron and the HeH + ion. It is also convenient to represent the cross section in the form

σ v v (E el ) = π k 2 el P v v (E el ) , (12)
where k el is the wave vector of the incident electron and P v v (E el ) could be viewed as the probability for vibrational (de-)excitation at collision energy E el . Figures 3 and4 demonstrate, as examples, the cross sections of Equation ( 11) and the corresponding probabilities from Equation (12) for the v = 3 → v = 0, 1, 2, 4 transitions. At very low scattering energies, below 0.02 eV, the de-excitation cross sections are smooth functions inversely proportional to the incident energy of the electron as predicted by the Wigner threshold law. But at higher energies, especially, just below the energy of the each excited threshold, v = 4, 5, • • • , the (de-)excitation cross sections and probabilities demonstrate series of Rydberg resonances, where they vary significantly.

v'=4 v=3 → v'=0,1,2,4 v'=2 v'=0 v'=1
v=3 → v'=2 v=3 → v'=4 v=3 → v'=1 v=3 → v'=0 Figure 4. Probabilities P v v (E el
) for vibrational (de-)excitation from the vibrational level v = 3 to the v = 0, 1, 2, 4 vibrational levels. Notice that the energy scale is linear in contrast to the logarithmic scale in Figure 3.

The thermally averaged rate coefficient α v ←v (T) (in atomic units) is obtained from the energy-dependent cross section of Equation ( 11),

α v v (T) = 8π (2πk b T) 3/2 ∞ 0 σ v v (E el )e - E el k b T E el dE el , (13)
where k b is the Boltzmann coefficient and T is the temperature. Examples of obtained rate coefficients α v ←v (T) are shown in Figure 5.

v'=3 v=0 → v' v'=1 v'=2 v'=4 rate constant (cm 3 /s) v=2 → v' v=1 → v' v=3 → v' v'=1 v'=0 v'=2 v'=2 v'=3 v'=3 v'=4 v'=4 v'=4 v'=1 v'=0 v'=0 Figure 5.
Thermally averaged rate coefficients for several (de-)excitation transitions obtained by direct integration using Equation (13). Vibrational transition v → v labels are shown in each panel. Results of a previous calculation [15] are shown by dashed lines.

In order to simplify eventual applications of the calculated numerical coefficients α v v (T), the obtained numerical rate coefficients were fitted using the following analytical interpolation formula

α f it v ←v (T) = 1 √ T e - ∆ v v T P f it v v (x), (14)
where

P f it v v (x) = a 0 + a 1 x + a 2 x 2 + a 3 x 3 and x = ln(T). (15)
with

P f it v v (x) ≈ P f it vv (x).
In equation ( 14), ∆ v v is the threshold energy having the following expression:

∆ v v = E v -E v > 0 for excitation , 0 for (de-)excitation . (16)
The coefficients a i (i = 0, 1, 2, 3) are obtained for each pair of transitions v ↔ v from a numerical fit. The quantity P f it v v (x) in the above equation could be viewed as an averaged (de-)excitation probability that varies weakly with energy.

Figure 6 illustrates the rates coefficients for the v = 3 → v = 0, 1, 2, 4 transitions obtained numerically from Equation ( 13) and by the fit of Equation ( 14). Overall, the fitting curves agree well with the curves obtained numerically. The numerical values of a i listed in the Table 2 are such that, when plugged into Equation ( 14), they give rate coefficients in units of cm 3 •s -1 . The temperature in the fitting formulas of Equation ( 15) is in kelvin. 14) for several pairs of initial and final vibrational level. The upper line in the header of the table specifies the pairs of initial and final vibrational levels for which the parameters are fitted. For convenience, we also specify (the second line of the header) the threshold energy ∆ v v for excitation process of the corresponding pair. For all excitation and de-excitation processes, same parameters a i are used in Equations ( 14) and (15) where the threshold for de-excitation is zero, ∆ v v = 0, as in Equation (16). a 0 0.23e-5 0.20e-6 0.75e-7 0.11e-7 0.21e-5 0.47e-6 0.80e-7 0.24e-5 0.44e-6 0.28e-5 a 1 -0.11e-6 -0.20e-8 0.88e-8 0.31e-9 0.95e-7 0.62e-7 -0.71e-9 0.34e-6 -0.53e-8 0.16e-8 a 2 0.25e-8 0.10e-8 -0.28e-8 -0.24e-10 -0.18e-7 -0.16e-7 0.76e-9 -0.56e-7 0.40e-8 0.13e-7 a 3 0.85e-10 -0.11e-9 0.15e-9 -0.56e-12 0.79e-9 0.85e-9 -0.76e-10 0.22e-8 -0.36e-9 -0.92e-9

/s) v'=4 v'=0 v'=1 v'=2 v=3 → v'=0,1,2,4
f it v v (x) = P f it vv (x) of Equation (
v -v 0-1 0-2 0-3 0-4 1-2 1-3 1-4 2-3 2-4 3-4 ∆ v v (

Conclusions and Discussion

In this study, cross sections and rate coefficients for electron-impact vibrational transitions in HeH + were computed for different combinations of the five lowest vibrational levels of HeH + . The calculations were performed using the UK R-matrix package combined to the quantum-defect approach. The obtained thermally averaged rate coefficients were fitted with a simple analytical formula with four parameters. The numerical values of the fitting parameters are provided in Table 2. The obtained cross sections and rate coefficients could be used in modeling the hydrogen/helium plasma experiments as well as for modeling interstellar clouds and planetary atmospheres, where the HeH + ion is present.

The rotational structure of each vibrational level was neglected in the present approach such that the obtained cross sections and rate coefficients should be viewed as averaged over initial rotational states and summed over final rotational states of corresponding vibrational levels. Neglecting the rotational structure would correspond to an experiment for which the energy resolution is worse than a typical energy splitting between rotational levels. The rotational constants for the lowest vibrational levels of HeH + are in the interval 20-33 cm -1 (see Table 1). The energy resolution of existing experimental data on e -+ HeH + collisions is much worse [9]. Detailed rotational state-to-state thermally-averaged rate coefficients for rotational transitions without changing the vibrational state might be useful to model very cold environment, below 40 K. Due to this reason the rate coefficients obtained in the present study should not be viewed as accurate below 40 K. Theoretical rate coefficients for rotational excitation at low temperatures will be presented in a separate study.

It is worth stressing that narrow electronic resonances present in the e -HeH + spectrum (see Figure 1), around 1 eV and above, are taken into account only in a very crude manner. Namely, they produce scattering matrix S(R) of Equation ( 4), which depends significantly on energy. Because the resonances are situated at relatively high energies and their widths are small (see, for example, the resonance at 4 eV for R = 1.455 bohrs in Figure 1), their effect is expected to be small on the obtained cross sections at low energies. The reason why the effect of such resonances is accounted for in a crude manner is because after the vibrational frame transformation of Equation ( 7), the resonances will be smeared out and will no appear in the excitation cross section. In a better treatment such resonances associated with HeH * potential curves should be represented by one or several additional vibronic channels and would produce a few more additional resonances in the cross section for vibrational excitation. In the present treatment, the energy-dependence of S(R), increases the excitation cross section near the corresponding energies but only on average, in a washed-out manner, without producing a few additional resonances. In any case, the effect of these resonances on the thermally-averaged rate coefficient is averaged out. A wider resonance appearing at large internuclear distances (for example, at 5 eV for R = 1.7 bohrs) could produce an important contribution to the vibrational excitation cross section at energies above 5 eV. A better treatment of the vibrational excitation process accounting for electronic resonances is possible and will be developed in a further publication.

Introduction

Cross sections for the electronic, rotational, and vibrational excitation of molecules in collisions with electrons are important for understanding and modeling various plasma environments, such as the interstellar medium (ISM), planetary ionospheres and exospheres, in plasma processing and de-pollution technologies, and others. Measuring cross sections experimentally for such processes is usually difficult and expensive. However, theoretical methods for electron-molecule scattering together with abundant computational resources have made it possible to obtain reliable cross sections numerically, at least for diatomic and small polyatomic molecules.

The excitation of rotational and vibrational states of molecular ions has been studied theoretically for several decades. In particular, in one of the earliest studies of this kind, Boikova and Ob'edkov [1] considered the process using the Coulomb-Born approximation for the low-energy region. The first-order perturbation theory was applied, and a general analytical formula was derived in which dipole and quadrupole moments of the target ion determine the cross section for rotational excitation, while derivatives of the moments with respect to nuclear distances determine the cross section for vibrational excitation. The direct non-resonant excitation mechanism of the molecules was assumed to maintain the validity of the Born-Oppenheimer approximation for such processes. Later, Chu and Dalgarno [2] applied the same Coulomb-Born approximation to compute rate coefficients for the rotational excitation j = 0 → j = 1 of the CH + ion.

Flower [3] applied the semi-classical approximation and the time-dependent perturbation theory to the j = 0 → j = 1 transition for CH + and HeH + . The rotation of the target molecule was quantised, while the motion of the incident electron was treated classically. The applicability of method is restricted to incident electron energies E el

2(E j -E j ), where E j and E j are the energies of the rotational states of the target ion.

In a series of publications, Rabadan et al. [4][5][6] modified the method developed by Chu and Dalgarno [2] for diatomic molecules. In their approach, the scattering matrix is obtained from first principles using the R-matrix approach [7], rather than from the Coulomb-Born approximation. Similarly to Flower [3], cross sections for rotational excitation of CH + and HeH + , as well as for NO + [4,5] are computed. However, as in the other studies mentioned above, the method does not account for near-threshold effects and assumes that different rotational levels are degenerate. In particular, it does not account for Rydberg resonances associated with closed rotational states of the neutral molecule, such as rotational Rydberg resonances of the HeH molecule in the case of e --HeH + collisions. As it was shown later [8,9], this assumption is not appropriate if the incident electron couples strongly different rotational states of the target ion. In such a situation, the near-threshold effects should be accounted for to produce accurate rate coefficients at temperatures below 150 K. To address this problem, in Reference [10] the authors extrapolated the numerically-obtained cross sections down to the threshold using Wigner's threshold law, producing more accurate results compared to those obtained from a kinetic scaling of the previous -matrix study [6] on HeH + and CH + . However, closed-channel effects associated with rotational resonances were still neglected in this study [10].

The theoretical method accounts for near-threshold effects, including rovibrational Rydberg resonances, and makes use of first-principle calculations (or experimental spectroscopic data if necessary), and is based on (1) the electron-molecule scattering matrix computed for fixed positions of nuclei (molecular-frame scattering matrix), (2) the idea of the rotational frame transformation [11], and (3) the molecular quantum defect theory (QDT) [12,13], which makes it possible to evaluate the scattering matrix in the laboratory frame (with respect to which the molecule rotates) and excitation cross sections. This method will be referred to below as the QDT method. It has been used in slightly different implementations in theoretical studies of rotational excitation for several molecular ions: H + 2 [14,15], H + 3 [16], HeH + [17], and CH + [18]. The molecular-frame scattering matrix in the QDT method can be evaluated in different ways. For example, in Reference [16], in calculations of the rotational excitation of H + 3 , the matrix was obtained by extrapolating quantum defects extracted from ab initio calculations of excited electronic states of H 3 for several internuclear geometries of the molecule. A similar method for the evaluation of the scattering matrix was used by Takagi et al. [19][20][21][22] in the study of the dissociative recombination of HeH + . Another way to obtain the molecular-frame scattering matrix is to perform electron scattering calculations directly, using first principles. In a recent study [10], Čurík and Greene employed the molecular scattering matrix computed directly using the UK R-matrix method [7] in the calculation of the rotational excitation cross sections in e --HeH + collisions.

In the present study, we discuss a general theoretical approach for the determination of rotational excitation cross sections for collisions of electrons with molecular ions at low scattering energies. A detailed derivation of the theory is presented for symmetric-top and linear target ions. A generalization to asymmetric top polyatomic targets is straightforward. We apply the method to the benchmark e --HeH + system and compute cross sections and rate coefficients for excitation and de-excitation of the ion from the five lowest rotational states. The HeH + ion is one of the simplest molecular ions. It is present in helium-containing plasma, such as in fusion devices. It is thought to be the first molecule formed in the early Universe. It has also been suggested [23][24][25][26][27] that HeH + could be detected in planetary atmospheres, white dwarfs, and the interstellar medium. At low energies, e --HeH + collisions can lead to dissociative recombination, rotational excitation, or dissociation recombination. The dissociative recombination of HeH + has been extensively studied both experimentally and theoretically [19][20][21][22]28]. There are no experimental measurements of cross sections for the rotational excitation of HeH + , but there are a few previous theoretical studies [1,3,10,17]. Therefore, the present results can be compared with the previous calculations.

In the present study, essentially the same theoretical method as in Reference [17] is used to represent the rotational excitation of HeH + , but the results for a larger number of rotational transitions in HeH + are obtained and an analytical fit of the thermally-averaged rate coefficients is performed, allowing the use of the data in plasma modeling.

The next section of the article presents the theoretical approach. In Section 3 we discuss the cross sections and rate coefficients obtained for the e --HeH + collisions. Section 4 summarizes the obtained results. Finally, in Appendix A, details of the theoretical derivation of the main formulas of Section 2 are provided.

Theoretical Approach

A detailed derivation of the present theory is given in Appendix A. Here, we discuss only the main steps in the implementation of the approach.

As the first step in the calculation, the body-frame e --HeH + reactance matrix KΛ is determined. For rotational-excitation transitions without changing the vibrational state of the target, the reactance matrix can be evaluated using just one internuclear distance corresponding to the equilibrium position of the target ion, R eq = 1.445 bohr for HeH + . The ground electronic state of the HeH + ion is X 1 Σ + with the Hartree-Fock electronic configuration of 1σ 2 . The matrix is obtained numerically using the UK R-matrix code [7,29]. Performing the scattering calculations, the R-matrix sphere of radius 10 bohr is used. Initially, several basis sets, including 6-311G*, DZP, and cc-pVTZ, were tested to investigate the stability of target properties such as dipole moment and ground state energy. Finally, the cc-pVTZ basis set was selected to perform the final calculations. A multicentered configuration interaction wave function expansion was used in the inner region, including two target states. The e --HeH + reactance matrix is smooth at low electron energies, and the lowest electronic resonance appears at about 4 eV at geometries near the equilibrium.

At electron energies below the first excited electronic state of the target ion, different channels of the body-frame reactance matrix are labeled with partial wave labels, including the angular momentum quantum number l and its projection λ of the molecular axis. The matrix is block-diagonal, where each block corresponds to a given projection Λ of the total angular momentum (ion+electron) of the system. Thus, in practice, the body-frame reactance matrix is calculated separately for each Λ. At low scattering energies, below 10 eV, the target ion can only be in the ground electronic state 1 Σ + . Therefore, in the present calculations, the projection of the electronic angular momentum in a given channel is equal to Λ.

The reactance matrix is used to compute the body-frame scattering matrix

ŜΛ = 1 + i KΛ 1 -i KΛ , ( 1
)
where 1 is the identity matrix. The laboratory-frame scattering matrix is then obtained by the transformation [30]

S J j µ l ;jµl = ∑ λλ (-1) l +λ +l+λ C j µ l -λ JΛ C jµ l-λJΛ S Λ l λ ;lλ , (2)
where J is the total angular momentum of the e --HeH + system, j, µ and j , µ are the angular momenta with their projections on the molecular axis of the target before and after the rotational excitation of HeH + , and C j µ l -λ JΛ and C jµ l-λJΛ are Clebsch-Gordan coefficients. The total energy E of the system is the sum E = E el + E jµ of the relative kinetic energy E el and the energy E jµ of the initial state of the target. At a given energy E, the size N o of the scattering matrix should be equal to the number of open channels with energies below E. Therefore, N o varies with the energy depending on how many rotational states are open for a given E. However, the scattering matrix S J in Equation (2) does not contain information about which channels are open or closed. The actual scattering matrix S J with the correct energy dependence is obtained from S J using the procedure of the closed-channel elimination [31,32] according to Equation (A15) of Appendix A. The resulting "physical" scattering matrix S J represents properly rotational resonances associated with the closed rotational levels of the target.

Having the "physical" scattering matrix S J , the cross section for the rotational excitation or de-excitation of the linear molecule by an electron impact is given as

σ j µ ←jµ (E el ) = 1 2j + 1 π k 2 j ∑ J,l,l (2J + 1) e i(lπ/2+σ l ) S J j µ l ;jµl e -i(l π/2+σ l ) 2 , (3)
assuming that the initial jµ and final j µ rotational states are different (an inelastic process) and that the vibrational state is not changed during the process. In the above formula, σ l is the Coulomb phase shift (see Equation (A5) in Appendix A).

Cross Sections and Rate Coefficients

Cross sections for electron impact transitions between the lowest five j, j = 0-4 rotational states of HeH + were computed. Note that the only allowed projection µ in Equation ( 3) for HeH + in its ground electronic state is zero. Therefore, for scattering energies below the first excited ionization threshold of HeH + , µ = µ = 0 in Equation (3). Some examples are given in Figure 1. Solid lines in the figure represent the results obtained using the complete theoretical approach described in the previous section. The cross sections exhibit a strong resonant character. The resonances are produced by closed rotational states of the target. These resonances are washed out when thermally-averaged rate coefficients are computed. Therefore, in the calculation of the rate coefficients, one can use cross sections averaged over the resonances. Such averaged cross sections can be computed directly from the energy-independent scattering matrix S J , replacing in Equation ( 3) the matrix elements S J j µ l ;jµl of the physical scattering matrix with the corresponding elements S J j µ l ;jµl of the energy-independent matrix. The cross sections calculated using the energy-independent scattering matrix are shown by dotted lines in Figure 1. Due to the overall 1/E el dependence of the cross sections as a function of the collision energy, it is convenient to see the products k 2 j σ j µ ←jµ , which could be viewed as excitation probabilities. They are shown in Figure 2. It is evident that the excitation probabilities obtained in the full treatment, including the closed-channel elimination, oscillate near the averaged value obtained in the treatment without considering the closed channels (i.e., from the energy-independent scattering matrix S J ).

The obtained averaged rate coefficients are shown in Figure 3. They are compared with recently published data for the j = 0 ↔ j transitions: dotted lines are the calculations by Hamilton et al. [10] and the dashed lines are those of Čurík and Greene [17]. The agreement between the three sets of calculations are perfect to the 0 ↔ 2 transitions, while for the 0 ↔ 1 transitions the rate coefficients of Reference [10] are somewhat larger than the present result and the one from Reference [17]. For the 0 ↔ 3 transitions, the coefficients of Reference [17] are somewhat larger than the present result and the one from Reference [10]. For the 0 ↔ 4 transitions, the coefficients from the two other calculations agree with each other and are slightly larger than the present result.

The thermally averaged coefficients at low temperatures are sensitive to exact positions and widths of the lowest resonances, because the averaging integral over thermal velocities at low temperatures T is determined only by the small collision energies, E el ∼ k B T. For example, the actual value of the cross sections for the j = 1 → j = 0 and j = 2 → j = 1 transitions in Figure 1 depends strongly on the position and the widths of the lowest resonances: at very low energies (below 2 meV), the cross sections are very different from the averaged ones shown with blue dotted lines. Therefore, the closed-channel elimination procedure is essential at low temperatures. Computationally, the procedure is not expensive if the number of channels is not very large (e.g., less than a thousand), and therefore, one can use the cross sections with all resonances in calculation of the rate coefficients for all temperatures. Rot. exc. cross section (cm Electron energy (eV)

2 ) 0 → 1 1 → 0 0 → 1 1 → 0 0 → 2 2 → 0 0 → 2 2 → 0
0 → 3 3 → 0 0 → 3 3 → 0 Figure 1.
Rotational (de-)excitation cross sections for transitions between the four lowest rotational states of HeH + . Solid lines represent the results obtained with the applied closed-channel elimination procedure of Equation (A14), while the dotted lines show the results for which the procedure was not applied.

0 → 1 1 → 0 0→ 1 1 → 0 0 → 2 → 0 → 2 → 0 0.05 0.1 Electron energy (eV) 10 -3 10 -2 10 -1 10 0 10 1 Excitation probabilities 1 → 2 2 → 1 1 → 2 2 → 1 0 0.05 0.1 Electron energy (eV) 0 → 3 → 0 → 3 → Figure 2.
Rotational (de-)excitation probabilities for transitions between the four lowest rotational states of HeH + . Solid lines represent the results obtained with the applied closed-channel elimination procedure, while the dotted lines show the results for which the procedure was not applied. [10], and the dashed lines are those of Čurík and Greene [17]. For the 0 → 1 transition, the dashed and solid lines overlap. For the 0 → 2 transition, the curves for all three calculations overlap. For the 0 → 3 transitions, the dotted and solid lines overlap. For the 0 → 4 transitions, the dotted and dashed lines overlap.

j'=3 j=0 → j' j'=1 j'=2 j'=4 rate constant (cm 3 /s) j=2 → j' j=1 → j' j=3 → j' j'=1 j'=0 j'=2 j'=2 j'=3 j'=3 j'=4 j'=4 j'=4 j'=1 j'=0 j'=0 temperature (K)
Due to the general 1/E el -dependence of cross sections σ j ←j , the calculated rate coefficients behave as 1/ √ T for de-excitation and exp -∆ j j /T / √ T for excitation transitions, where ∆ j j = E j µ -E jµ (with µ = µ = 0 for the present case) is the excitation energy. Therefore, similarly to References [16,33], for convenience of use, the rate coefficients are fitted to the formula

α f it j ←j (T) = 1 √ T e - ∆ j j T P f it j j (x) , (4)
where P j j (T) are smooth functions of temperature and represented by the quadratic polynomial

P f it j j (x) = a 0 + a 1 x + a 2 x 2 and x = ln(T) , (5)
where ∆ j j is the threshold energy defined as

∆ j j = E j -E j > 0 for excitation , 0 for (de-)excitation . ( 6
)
The numerical parameters given in Table 1 are such that when used in Equations ( 4) and ( 5) along with the temperature T expressed in kelvin, the obtained numerical value of the rate coefficient in Equation (4) will be in units of cm 3 /s. 4) and ( 5) for several pairs of initial and final rotational states for de-excitation j ← j of HeH + , with j < j . The probabilities P f it j j (x) for the opposite (excitation) process, j → j , are obtained from P f it jj (x) multiplying them with the factor (2j + 1)/(2j + 1) (see Equation ( 7)). For convenience, we also specify (the second column) the threshold energy ∆ j j in units of temperature (K) for the excitation process of the corresponding pair. For the de-excitation processes, ∆ j j = 0. Due to the detailed balance principle, the probabilities for the direct P f it j j (x) (j ← j) and the inverse P f it jj (x) (j ← j ) processes are related to each other by the relative degeneracy factor

j ← j ∆ j j (K) a 0 a 1 a 2 0 ←
P f it j j (x) = 2j + 1 2j + 1 P f it jj (x) . (7)
The coefficients a i (i = 0, 1, 2) are obtained numerically for each pair of transitions j ↔ j and are given in Table 1.

Conclusions

In this study, cross sections and thermally-averaged rate coefficients for electron impact rotational transitions in HeH + are computed for the five lowest rotational levels of HeH + using the UK R-matrix method combined with the multichannel quantum defect theory (MQDT). Our improved channel elimination procedure removes this ambiguity to evaluate accurate results at low energy (<0.01 eV). This and our previous study [33] make us believe that once the collisional excitation cross section data is available, the analysis of the intensities of infra-red and microwave regions can provide information for the diagnostics of tokamak, as well as the study of planetary atmospheres and of the interstellar medium.

With certain modifications accounting for a different threshold behavior of the cross section at low collision energies, the present theoretical approach can be extended for collisions between an electron and a neutral molecule. These developments will be published later.

Moving the exponent factor to the right-hand side, we obtain

∑ m j f j µ m j ←jµm j ( k j )φ j µ m j = 2πi √ k j ∑ J M ∑ lm l Y lm l ( kj ) ∑ l C J M lm l jm j k j Φ J M
j µ l i l e iσ l δ j j δ µ µ δ l l -S J j µ l ;jµl e iσ l i -l .

(A8)

Multiplying both sides of the above equation with φ j µ m j , integrating over electronic coordinates r 2 of the target and the angles Ω, we obtain the scattering amplitude for the transition jµm j → j µ m j :

f j µ m j ←jµm j ( k j ) = 2πi √ k j ∑ J M ∑ lm l Y lm l ( kj ) ∑ l m l C J M lm l jm j k j C J M l m l j m j
Y l m l (θ ϕ)i l e iσ l δ j j δ µ µ δ l l -S J j µ l ;jµl e iσ l i -l .

(A9)

To make notations slightly more uniform, in the equations below, we use symbol m j instead of m j . The cross section for rotational excitation σ j µ ←jµ (E el ) averaged over initial projections m j and summed over final projections m j = m j is obtained as follows:

σ j µ ←jµ (E el ) = sin θdθdϕ 1 2j + 1 k j k j ∑ m j ,m j f j µ m j ←jµm j ( k j ) 2 , ( A10
)
where E el = (hk j ) 2 /(2m) is the energy of the incident electron, and θ and φ are spherical angles of the wave vector k j of the scattered electron in the LF. Below, we assume that the incident plane wave propagates along the z-axis of the LF (i.e., Y lm l ( kj ) = δ m l ,0 (2l + 1)/(4π) in Equation (A9)). Therefore, the number of summation indexes in Equation (A9) is reduced from six to five, over J, M, l, l , and m l . The square of the amplitude in Equation (A10) doubles the number of summation indexes. We will refer to the additional indexes as J, M, l, l , and ml . With the sums over m j and m j in Equation (A10), the number of summation indexes becomes twelve. Due to the orthogonality of Y l m l (θ ϕ) and Y l ml (θ ϕ), the integral over θ and ϕ reduces the number of summation indexes to ten with l = l and m l = ml . In the remaining ten-fold sum, the double sum over m j and m l is

∑ m l ,m j C J M l m l j m j C J M l m l j m j = δ J, J δ M, M , (A11)
which reduces the number of indexes to J, M, l, l , l, and m j . Again, in the remaining sum (because

m l = ml = 0): ∑ m j ,M C J M l0jm j C J M l0jm j = 2J + 1 2l + 1 δ l, l . (A12)
With the above simplifications, the cross section of Equation (A10) becomes:

σ j µ ←jµ (E el ) = 1 2j + 1 π k 2 j ∑ J,l,l
(2J + 1) e i(σ l +lπ/2) S J j µ l ;jµl e i(σ l -l π/2) 2 , (A13) assuming that the initial jµ and final j µ states are different (i.e., an inelastic process).

For a given total energy E = E el + E jµ of the e --HeH + system, the size N o of the matrix S J in the formula above is equal to the number of open scattering channels with energies E j µ < E. However, the closed channels with E j µ > E, which are not included explicitly in Equation (A13), usually significantly influence the S-matrix and the cross section. Such closed channels are taken into account using the "closed channel elimination" procedure [31,32]. The S-matrix in Equation (A13) is obtained from another matrix, having a larger number of channels, including the channels that are closed at given E. Namely, S J in Equation (A13) is given by: S J = e i ηc S oo -S oc S cce -2iβ(E) -1 S co e i ηc , (A14)

where ηc is a N o × N o [32] diagonal matrix with diagonal elements equal to the Coulomb phase shift in the corresponding channel, [ ηc ] i,i = -lπ 2η ln(2k j r) + σ l . The matrices S oo , S oc , S cc , and S co are submatrices of the larger N × N S-matrix, which includes open and closed channels (N ≥ N o ). The larger S-matrix, partitioned as:

S J = S oo S oc S co S cc , ( A15
)
where the partition of the matrix elements in the "o"-and "c"-parts is made on the basis whether the corresponding channel, jµ or j µ , is open or closed for ionization at the total energy E. The quantity β(E) is a diagonal N c × N c matrix:

β j µ ;jµ (E) = π 2(E jµ -E) δ j j δ µ µ , ( A16
)
and N c = N -N o is the number of closed channels. We assume that the initial and final vibrational states of the target ion are the same. In this situation, it is a good approximation to consider that the averaged internuclear distance in the target ion is unchanged during the rotational excitation process and is equal to the equilibrium distance R e . The scattering matrix S J in Equation (A15) is therefore obtained for a fixed geometry R e in the basis of channel functions of Equation (A3). In the R-matrix calculation, the reactance matrix K is obtained in a different basis of functions X J M lλ [30], in which the molecule is fixed in space (i.e., the body-fixed (BF) basis). The channel functions X J M lλ transform into Φ J M jµl in the following way:

Φ J M jµl = ∑ λ X J M lλ (-1) l+λ C jµ l-λJΛ , ( A17
)
where λ is the projection of the orbital momentum l of the incident electron on the molecular axis, and Λ is the projection of the total orbital momentum of all electrons on the molecular axis. For the HeH + ion in the ground electronic state, λ = Λ and µ = 0. The S-matrix obtained in the BF is diagonal over quantum numbers J and Λ. Therefore, the transformation between the S-matrices obtained in the two bases is given by:

S J j µ l ;jµl = ∑ λλ (-1) l +λ +l+λ C j µ l -λ JΛ C jµ l-λJΛ S Λ l λ ;lλ , ( A18
)
where S Λ l λ ;lλ is an element of the the BF S-matrix obtained from the reactance matrix K as:

ŜΛ = 1 + i KΛ 1 -i KΛ , (A19)
with 1 being the identity matrix and KΛ is the reactance matrix obtained numerically using the UK R-matrix code.

I. INTRODUCTION

In many fields of research and applications, it is essential to have accurate cross sections for different processes taking place in collisions between molecular ions and electrons. Among such processes are electron-impact rotational (RE), vibrational (VE), and electronic (EE) excitation of the ions; dissociative recombination (DR); and photoionization and its inverse process, radiative recombination. Some cross sections could be obtained in experiments. However, for many processes, especially for the processes involving excited-state ions (ions here and below are assumed to be molecular ions, not atomic) or such ions as radicals, which are unstable in collisions with other species present nearby, an experimental approach is difficult or impossible. Even for stable ions in their ground quantum state, an experimental approach is often very expensive.

On the other hand, for theoretical approaches a significant complication in computation of the cross sections is the presence of vibrational and rotational degrees of freedom that have to be accounted for to obtain an accurate description of the processes. Electronic excitation and ionization of molecules can be treated theoretically, at least to some extent in the Born-Oppenheimer approximation or by taking into account the Franck-Condon factor. For other processes, such as rovibrational excitation or dissociative recombination, non-Born-Oppenheimer effects should be accounted for explicitly.

With modern development of electron-scattering methods and abundant computational resources, it became possible * slavako@ucf.edu to compute, with an acceptable uncertainty, cross sections for many processes in electron-ion collisions. Significant progress was made for processes in diatomic ions formed by light elements: H with its isotopologs [13][14][15][16][17], where non-Born-Oppenheimer effects in electron-ion collisions were accurately accounted for, typically by using a quantum-defect approach combined with rotational and vibrational frame transformations. With some additional simplifications, such processes as rovibrational excitation and dissociative recombination were also successfully described theoretically for larger molecular ions: CH Theoretically, non-Born-Oppenheimer couplings in electron-ion collisions are treated differently for the ions with low-energy electronic resonances appearing for geometries near the equilibrium of the target ion (in a fixed-nuclei picture) and for the ions without such low-energy electronic resonances. In the former case, usually the potential energy surface (PES) of the doubly excited neutral molecule crosses the ionic PES near the equilibrium geometry; in the latter case, there is no such a resonance PES. The ions of the first type usually (not always) have the first excited electronic state at a relatively low energy, below 5 eV; the ions of the second type have the first excited electronic state at a higher energy.

The presence of low-energy electronic resonances in the first type of the ions increases significantly compared to the ions of the second type, the DR, EE, VE, and RE cross sections at low collision energies. Because of the significant difference in the physics of couplings in electron-ion collisions in the two types of the ions, one developed two types of approaches. The first approach, developed for DR, VE, and RE processes and originated from studies by O'Malley [35] and , takes into account explicitly the PES crossing. The second approach, based mainly on studies by Lee [39], Jungen et al. [40,41], and Giusti [42] and employed when there is no PES crossing, accounts for the coupling between the incident electron and the rovibronic Rydberg resonances of the neutral molecule. In the absence of a PES crossing, such resonances are responsible for the major contribution to the DR cross section at low energies [13,42,43]. This is especially important for polyatomic ions, listed above. All these ions have a closed electronic shell, the first excited electronic state at a high energy, and no PES crossing near the equilibrium geometry of the ions.

There are situations where there is a PES crossing near the ion equilibrium geometry and, in addition, there are one or several low-energy electronic resonances in the collisional spectrum. Many open-shell ions are of this type, for example. The two approaches mentioned above are not able to describe satisfactory the DR and excitation processes. On the basis of an earlier theory suggested by Giusti [42,44], Jungen et al. have developed an efficient approach that can deal with such a situation. The approach was applied to several diatomic ions for which the dissociative electronic PES of the neutral molecule crosses the ionic PES near the ion equilibrium [6,[45][46][47]. The approach is based on the quantum defect theory (QDT), where, in addition to one or several electronic states of the ion, the dissociative state is explicitly included into the coupling scheme [42,44]. Couplings between different electronic states of the target ion are derived from ab initio calculations of electronic (Rydberg) bound states of the neutral molecule. Couplings between the ionic and dissociative states are obtained from the autoionization widths of dissociative states of the neutral molecule (where autoionization is allowed). The widths are typically obtained in electron-scattering calculations.

The above theoretical approach is the only one able to describe non-Born-Oppenheimer effects on electron-ion collisions in the presence of coupled electronic channels of the target. One significant limitation of the approach is the difficulty in obtaining couplings between the electronic states. The procedure of diabatization of coupled Rydberg states obtained in ab initio calculations, used in the approach, is laborious, not unique, and sometimes not accurate. It becomes even more ambiguous and very complicated for polyatomic ions, such that an extension of the approach to polyatomic ions becomes impractical.

In this study, we propose another approach, which combines some of the original ideas from the molecular quantum defect theory [40,42,48], more recent DR and VE studies in polyatomic ions [14,21,25,27,43], and recent progress in electron-scattering calculations. The approach can be applied to determine EE, VE, RE, and DR cross sections for a wide range of small polyatomic ions, including the ions with one or several low-energy excited ionic and/or resonant states of the system. In this article, we focus on the VE process and, for the simplicity of discussion, on a particular case of a diatomic ion CH + . However, the treatment can easily be applied to small polyatomic ions and, with some additions similar to Refs. [42,44], for the DR process.

II. ELECTRON-CH + COLLISIONS

Collisions of the CH + ion with electrons have been studied theoretically since, at least, 1951 [38,[49][50][51]. The interest was motivated by the detection of the ion in diffuse interstellar clouds, made initially by Douglas and Herzberg [52] and by Adams [53] in 1941. These and later detections confirmed that CH + is ubiquitous as a major constituent of interstellar clouds. The ion is also an important intermediate in combustion and in the formation of large hydrocarbons in the interstellar medium (ISM). Reactive collisions of CH + with a low-energy electron determine the energy balance and evolution of lowtemperature hydrocarbon plasmas such as in the ISM. The theoretical study of the e --CH + collision system is thus of considerable astrophysical interest. Processes taking place in e --CH + collisions are also of interest for technological plasmas: For example, they play an important role in plasma processing of diamond films [54] and at the edge plasma of fusion reactors [55], where graphite is used as plasma-facing material.

A theoretical description of low-energy e --CH + collisions is complicated due to the presence of a low-energy electronic 2 2 resonance and several low-energy excited electronic states of CH + [4,51,56] (see Fig. 1). The excited ionic states produce series of Rydberg resonances that influence all collisional processes. In this situation, the standard vibrationalframe-transformation approach by Chang and Fano [57], used in many theoretical studies on electron-molecule collisions [48,58], is not well adapted: The approach requires that the scattering matrix or, alternatively, the matrix of quantum defects, obtained for fixed internuclear positions (in the Born-Oppenheimer approximation), to be a smooth function of the collision energy-ideally, to be energy independent. However, the presence of the 2 2 resonance and the low-energy excited electronic states makes the fixed-nuclei scattering matrix to be strongly energy dependent.

The PES of the 2 2 resonance crosses the PES of the ground electronic state X 1 + of the CH + ion slightly to the left of the CH + equilibrium geometry. As Giusti has pointed out [51], the electronic configuration of the resonance is mainly due to the coupling between the 3σ 1π ( 1 )4σ and 3σ 1π ( 3 )4σ orbitals of the CH + + e -system. The resonance at small internuclear distances has a Rydberg character and is produced by the a 3 and A 1 parent states of the CH + ion. These states are the first and second electronically excited states of the ion. Near and to the left of the crossing of the resonance and ionic X 1 + PES, the character of the resonance is mainly 3σ 1π ( 1 )4σ (see Fig. 1 of Ref. [51]). Therefore, the resonance could be included in the QDT description of the CH + + e -scattering if the a 3 and A 1 excited ionic states are accounted for in the complete scattering matrix. Of course, at low scattering energies at fixed geometries to the left of the crossing, these two electronic channels are closed for ionization and should be accounted for using the closed-channel elimination procedure [59], often employed in the QDT studies. Below, we describe in detail the developed theoretical approach.

III. QDT DESCRIPTION OF ELECTRONIC RESONANCES

In applications of the theoretical method presented below, one needs scattering matrices obtained numerically for fixed geometries of the target ion. The scattering matrices could be obtained in different ways. We used the UK R-matrix code [60]. The details of the numerical calculations using the R-matrix code for e --CH + collisions are given in Sec. V.

As mentioned above, the geometry-fixed scattering matrix is strongly energy dependent for e --CH + collisions. This is demonstrated in Fig. 2, showing derivatives of the eigenphase sums for the three symmetries 2 + , 2 , and 2 -of the e --CH + system computed at the equilibrium with internuclear distance R e = 2.137 bohrs. Several series of Rydberg resonances converge to the electronic states a 3 and A 1 as marked by the blue vertical lines in Fig. 2.

To describe low-energy electronic resonances in different e --CH + scattering processes, we use the QDT approach and need an energy-independent scattering matrix, which includes not only the ground electronic state of CH + but a few more states that can produce resonances at scattering energies of the interest. In the e --CH + case, low-energy resonances are well reproduced if one takes into account only three electronic states of the ion. Figure 2 shows derivatives of eigenphase sums in two calculations. In one calculation (black solid curves), only the three lowest X 1 + , a 3 , and A 1 states are included. In the second calculation (red dashed lines), 14 lowest states were included. As one can see, at low energies, below the A 1 ionization limit, the two calculations agree quite well with each other. In the second calculation with a larger number of ionic states, there are a few narrow resonances at low scattering energies that are not reproduced in the first, smaller calculation. These resonances are attached to very excited electronic states of the ion and do not influence significantly the low-energy spectrum.

Therefore, the electronic scattering matrix at low energies could well be represented by the three states X 1 + , a 3 , and A 1 of the ion. With this set of electronic states, the abovementioned 2 2 resonance is included in the scattering model.

In order to account for vibrational and rotational excitation of the target, the standard QDT approach is to use vibrational and rotational frame transformations [40,57]. The approach is applicable only if the electronic scattering matrix, obtained for a number of different geometries of the ion, is energy independent. As Fig. 2 shows, the e --CH + scattering matrix depends strongly on energy below the A 1 ionization limit and cannot be immediately used in the frame transformation. A possible solution is to take the (almost) energy-independent scattering matrix, obtained at an energy above the A 1 ionization limit, and use it at energies below the limit. Therefore, the vibrational (and rotational) frame transformation is performed on a 3 × 3 electronic scattering matrix, which produces a N × N matrix with N vibronic (rovibronic) channels. Such a rovibronic scattering matrix is essentially energy independent and a QDT closed-channel elimination procedure [59,61] should be performed to obtain the physical energy-dependent matrix, which can be used to compute cross sections for various processes.

Before discussing the vibronic frame transformation applied to the e --CH + collisions, we compare the fixed-nuclei electronic scattering matrices obtained (1) using the elimination procedure of the closed electronic states and by (2) a direct scattering R-matrix calculation at the same internuclear distance. 1) the direct Rmatrix calculations (red dashed curve) and using (2) the energyindependent 3 × 3 scattering matrix and the procedure of elimination of closed electronic channels (black solid curve). The inset shows an enlarged view for the 2.6-to 3.0-eV interval of energies.

The elimination of closed electronic channels at a geometry R is given by [59,61]

S phys (E el ) = S oo -S oc [S cc -e -2iβ(E el ) ] -1 S co , (1)
where E el is the scattering energy and S oo , S oc , S cc , and S co are submatrices of the weakly dependent electronic scattering matrix (3 × 3 in the present case of the e --CH + system),

S(E el ) = S oo S oc S co S cc .

(

) 2
Partition of matrix elements in the o and c parts is made on the basis of whether the corresponding channels are open or closed for excitation for the particular scattering energy E el .

The quantity of β(E el ) in Eq. ( 1) is a diagonal N c × N c matrix

β(E el , R) = π √ 2[E i (R) -E el ] δ i ,i , (3)
where E i (R) denotes the energy values of the ith electronic states at internuclear distance R.

Figure 3 shows derivatives of eigenphase sums obtained from the scattering matrices computed at the equilibrium distance R e . The red dashed curve is the result from the R-matrix calculation; the black solid curve is the calculation using the energy-independent 3 × 3 electronic scattering matrix and the closed-channel elimination procedure. Overall, positions of the resonances in the two calculations are the same but widths in the R-matrix calculation are wider. This means that diagonal elements of the scattering matrices in the two calculations are very similar but the nondiagonal elements, responsible for channel couplings and widths of the resonances, are slightly different, suggesting that highly excited electronic states, neglected in the 3 × 3 channel elimination procedure, have non-negligible contributions to the coupling between the lowest channels.

The choice of the 3 × 3 scattering matrix used in the channel-elimination procedure is not unique, because the matrix depends on energy, even above the A 1 electronic state. To assess the result of uncertainty in the choice of the energy at which the 3 × 3 scattering matrix is taken, we plot in Fig. 4 eigenphase-sum derivatives obtained for 3 × 3 scattering matrices taken at three different energies above the A 1 state: at 3.413, 3.563, and 3.713 eV. Positions and the widths of the resonances are nearly the same in the three calculations.

An important conclusion from the results discussed above is that the e --CH + scattering physics below the A 1 state can be represented using an energy-independent multichannel scattering matrix evaluated at a higher energy, above the A 1 ionization limit in a combination with the closed-channel elimination.

A rough idea about the magnitude of cross sections for electron-impact electronic excitation of a molecule is obtained from a fixed-geometry calculation. Here, for a comparison between the QDT and direct R-matrix approaches, we present such excitation cross sections. The vibrational dynamics during the process is discussed in the next section.

Using the physical scattering matrix S phys (E el , R e ) of Eq. ( 1) describing electronic transitions at the equilibrium geometry R e of CH + , the fixed-nuclei cross section of the electronic excitations from the X 1 + state to the a 3 state is computed in the QDT approach as [14]

σ i ,i (E el , R e ) = π h2 2m e E el × l m ,lm S phys l m i ,lmi (E el , R e ) -δ l m i ,lmi 2 , ( 4
)
where m e is the reduced mass of electron and i and i refer to the initial (X 1 + in this case) and final (a 3 here) electronic states. Indexes lm and l m numerate initial and final angular momenta and their projections in the molecular reference frame (where ab initio calculations are performed).

The cross section in the R-matrix approach is obtained by the same formula, except that the scattering matrix in the above equation is replaced with the one obtained directly in the R-matrix calculations at the corresponding energy E el . Figure 5 compares the cross sections for the X 1 + → a 3 transition obtained in the two approaches. The general agreement between the two curves is good, even for the widths of the resonances. One noticeable difference is in the position of the minimum near 1.5 eV: In the QDT calculations, it is shifted slightly to the left. The agreement is better at energies approaching the a 3 ionization limit.

Differences observed in the cross sections obtained by the two methods are smeared out in the thermally averaged rate coefficient (5) computed from the cross sections. In the above equation, k b is the Boltzmann coefficient and T is the temperature. The obtained rate coefficients, shown in Fig. 6, are in very good agreement with each other. This confirms that major couplings between electronic channels are accurately represented in the QDT approach and validates the approach.

k i ,i (T, R e ) = 8π (2π k b T ) 3/2 ∞ 0 σ i ,i (E el , R e )e -E el k b T E el dE el ,

IV. VIBRONIC EXCITATION

The energy-dependent physical scattering matrix for vibronic transitions is obtained in two steps. First, one computes the energy-independent vibronic scattering matrix assuming that all vibronic channels are open. In the second step, an

ν'=1 ν'=2 ν'=3 ν=0, X 1 Σ + ν'=1,2,3 X 1 Σ + (a)
ν'=0 ν'=1 ν'=2 ν=0, a 3 Π ν'=0,1,2,3 X 1 Σ + ν'=3 (c)
FIG. 7. Cross sections for vibronic excitations of CH + from the ground vibrational level v = 0 of the X 1 + state to v = 1, 2, 3 of the X 1 + state (left panel), to v = 0, 1, 2, 3 of the a 3 state (middle panel), and for vibronic de-excitations from the ground vibrational level v = 0 of a 3 to v = 0, 1, 2, 3 of the X 1 + state (right panel).

/s) ν'=1 ν'=2 ν'=3 X 1 Σ + ,ν=0 → ν'=0-3 (a)
ν'=0 ν'=1 ν'=2 a 3 Π,ν=0 → X 1 Σ + ,ν'=0-3 ν'=3 (c)
FIG. 8. Rate coefficients for same vibronic transitions as shown in Fig. 7.

elimination of closed vibronic channels is applied, producing the required energy-dependent vibronic scattering matrix. The first step is performed by the vibronic frame transformation (6) where ϕ vi (R) and ϕ v i (R) are wave functions of the initial and final vibrational states. Index v or v corresponds to the number of vibrational quanta in initial i or final i electronic states. The brackets imply an integration over the vibrational coordinate R.

S l m v i ,lmvi (E el ) = ϕ v i (R)|S l m i ,lmi (E el , R)|ϕ vi (R) ,
In the second step, the energy-dependent physical scattering matrix S phys (E el ) is obtained by the QDT vibronic closedchannel elimination procedure, described by the same Eqs. ( 1) and ( 3), except that the energies E i (R) of closed channels are replaced with energies of vibronic channels E vi , i.e.,

β(E el ) = π √ 2(E vi -E el ) δ v i ,vi . (7)
The cross sections σ v i ,vi (E el ) for vibronic excitation or de-excitation of CH + are computed using Eq. ( 4), where S phys l m i ,lmi (E el , R e ) is replaced with S phys l m v i ,lmvi (E el ). Figure 7 illustrates cross sections obtained for different combinations of initial and final vibronic states. Figure 7(a) shows results for pure vibrational excitations between levels of the ground electronic state X 1 + . As expected, the cross section for the transition with v = 1 is the largest one between inelastic processes. Figure 7(b) gives cross sections from the ground vibronic state X 1 + , v = 0 to several vibrational levels of the a 3 state. Since the potential curves of the X 1 + and a 3 states have similar shapes near the equilibrium, the largest X 1 + → a 3 cross section is expected to be for v = 0, as the present calculation indeed demonstrated. Figure 7(c) gives cross sections for the deexcitation process a

3 , v = 0 → X 1 + , v = 0 -3.
Cross sections for vibronic excitations were recently estimated by Chakrabarti et al. [5] using a rough theoretical approach, in which cross sections for electronic excitations computed at the CH + equilibrium geometry were multiplied with Franck-Condon overlaps for various combinations of initial and final vibrational levels to obtain the cross sections for vibronic transitions. In that study, vibronic Feshbach resonances as well as differences in vibrational excitation energies were neglected. The cross sections obtained in Ref. [5] differ significantly-more than an order of magnitude for several transitions-from the present results. We attribute the disagreement to the mentioned approximations employed in Ref. [5]: (1) neglected differences in vibrational excitation threshold energies, (2) neglected dependence of e --CH + scattering parameters with the internuclear distance, and (3) the neglected resonances in closed vibronic channels.

Thermally averaged rate coefficients k v i ,vi (T ) for these vibronic (de-)excitations from 10 to 10 000 K are then computed using Eq. ( 5), where σ i ,i (E el , R e ) is substituted with σ v i ,vi (E el ). Figure 8 shows computed rate coefficients for the same transitions as the cross sections in Fig. 7.

As in previous studies [62][63][64][65] and for convenience of use, the computed thermally averaged rate coefficients k v i ←vi were fitted using the following analytical formula,

k fit v i ←vi (T ) = 1 √ T e -v i ,vi T P fit v i ,vi (x), (8)
where P fit v i ,vi (x) is a quadratic polynomial

P fit v i ,vi (x) = a 0 + a 1 x + a 2 x 2 and x = ln(T ) (9)
with P fit v i ,vi (x) ≈ P fit vi,v i (x). This quantity could be viewed as the (de-)excitation probability. v i ,vi in Eq. ( 8) is the threshold energy defined as

v i ,vi = E v i -E vi > 0 for excitation, 0 for de-excitation. (10)
Numerically fitted parameters for vibronic transitions are given in Tables I-VI. When the parameters given in the tables are used in the fitting formulas of Eqs. ( 8) and ( 9) with T in K, obtained numerical values of rate coefficients will be in units 8) and ( 9) for several pairs of initial and final vibrational levels of the ground electronic state X 1 + of CH + . We specify the threshold energy v i ,vi for the excitation process in the pair v i , vi in the second column of each table. For the de-excitation process, of cm 3 /s. In the tables, the electronic states are numerated with index i (or i ) with i = 0 corresponding to X 1 + , i = 1 to a 3 , and i = 2 to A 1 .

vi,v i = 0. v i ↔ vi v i ,vi (K) a 0 a 1 a 2

V. CALCULATIONS OF ELECTRONIC STRUCTURE, SCATTERING, AND VIBRATIONAL DYNAMICS OF THE PROCESS

In this section, we provide details about numerical calculations of the vibrational wave functions of the ion and ab initio calculations performed for the bound electronic states of the ion and the e --CH + scattering.

The configuration of the ground electronic state X 1 + of CH + is 1σ 2 2σ 2 3σ 2 in the C ∞v symmetry group of the ion. The potential energy curves V (R) of CH + were calculated using the C 2v symmetry group with a multireference configuration interaction (MRCI) method and the cc-pV5Z basis set using the MOLPRO code [66]. We kept the 1σ orbital of carbon doubly occupied and used 14 orbitals, i.e., 2σ -7σ , 1π -3π , and 1δ as the complete active space (CAS). The calculated potential energy curves of the X 1 + , a 3 , A 1 , b 3 -, and c 3 + electronic states are shown in Fig. 1. The X 1 + , a 3 , A 1 , and c 3 + curves correlate with the C + ( 2 P) + H( 2 S) dissociation limit at large internuclear distances.

In order to determine the vibrational energies E v and the corresponding vibrational wave functions ϕ v (R) within these electronic states of CH + , we solved the Schrödinger equation for vibrational motion along R -h2 2μ

d 2 dR 2 + V (R) ϕ v (R) = E v ϕ v (R), (11)
using a discrete variable representation (DVR) method [67].

In the above equation, μ denotes the reduced mass of CH + . The lowest four vibrational energy levels v = 0, 1, 2, 3 of the X

↔ v a 3 . v i ↔ vi v i ,vi (K) a 0 a 1 a 2
The e --CH + scattering calculations were carried out using the UK R-matrix code [60,69] with the help of the Quantemol-N interface [70]. The cc-pVQZ basis set and CAS configuration interaction (CI) method in the C 2v Abelian subgroup were used in the calculations. The inner orbital 1a 1 2 of CH + was frozen, and four external electrons were distributed in the space of the [2a 1 , 3a 1 , 4a 1 , 5a 1 , 6a 1 , 7a 1 , 8a 1 , 1b 1 , 2b 1 , 3b 1 , 1b 2 , 2b 2 , 3b 2 , 1a 2 ] orbitals (2σ, 3σ, 4σ, 5σ, 6σ, 7σ, 1π, 2π, 3π, 1δ in C ∞v symmetry group). We chose an R-matrix sphere of radius 13 bohrs and continuum Gaussian-type orbitals with partial waves l 4. The two different R-matrix calculations described in Sec. III closed-coupling expansions with 3 and 14 lowest electronic states of CH + were used for constructing the total wave functions for the e --CH + system. In the

↔ v A 1 . v i ↔ vi v i ,vi (K) a 0 a 1 a 2
↔ v A 1 . v i ↔ vi v i ,vi (K) a 0 a 1 a 2
vibrational frame transformation of Eq. ( 6), the electron scattering calculations were performed in the interval between 1.537 and 3.937 bohrs with a step of 0.1 bohrs along the internuclear coordinate R.

VI. CONCLUSIONS

In conclusion, cross sections and rate coefficients for vibronic excitation and de-excitation of CH + by electron impact were computed in a framework using first principles only. The theoretical approach combines fixed-nuclei scattering matrices obtained for a number of internuclear distances using the UK R-matrix code, the vibronic frame transformation, and the QDT closed-channel elimination procedure. The approach is validated in model calculations for electronic excitation, performed for a single fixed internuclear distance of the target, comparing the results from a direct R-matrix calculation and the QDT channel elimination procedure. The main advantage of this method compared with the previous state of theory is that it can be applied to collisions of electrons with molecular

. v = 0 v = 1 v = 2 v = 3
Biglari et al. [68] In this study, we took into account only the electronic and vibrational structure of the target ion. The rotational structure of each vibrational level was neglected. The description of the rotational structure and couplings could be included into the treatment in the same way as was made in many previous treatments (see, for example, Ref. [65] and references therein). Therefore, the obtained cross sections and rate coefficients should be viewed as averaged over initial rotational states and summed over final rotational states of the corresponding vibrational levels. The inclusion of the rotational structure and couplings is important if one needs rotationally resolved cross sections or thermally averaged rate coefficients at temperatures T comparable or smaller than the CH + rotational constant, i.e., at T 20 K. An extension of this method to include nuclear rotation will be discussed in a later publication.

I. INTRODUCTION

Vibrational (de-)excitation by electron impact is a process in which an electron scatters off a molecule and exchanges energy with it in a way that leaves the molecular target in a different vibrational state. For N 2 O, vibrational (de-)excitation can be depicted by

e -+ N 2 O(ν i ) → e -+ N 2 O(ν i ), (1)
where ν i and and ν i denote the initial and final vibrational states of N 2 O, respectively. Due to the importance of nitrous oxide (N 2 O) in a plethora of research fields ranging from astrochemistry [1][2][3] to lowtemperature plasma technology [4] and medicine [5], different electron-N 2 O collisional processes have been experimentally and theoretically explored over the years. Differential and integrated cross sections for elastic and certain inelastic processes have been measured by several groups . Although the experimental investigations generally agree on the position of an observed resonance near 2.3-2.5 eV, they disagree on the assignment for the symmetry of the resonant state. Furthermore, there is also disagreement with respect to a second resonance observed around or above 8.0 eV. On the theoretical side, the earlier studies by Morgan [27], Sarpal et al. [28], and Bettega et al. [29] aimed at clearly specifying the nature of the two resonances observed in the experiments. Morgan and Sarpal et al. employed the R-matrix method with different models to study electron scattering by N 2 O in its equilibrium geometry. They obtained a resonance near 2.3 eV * mehdi.ayouz@ecp.fr with 2 symmetry. Later, using a slight modification of the Schwinger multichannel method of incorporating polarization effects, Bettega [29] was able to reproduce the experimental features between the two resonances.

To our knowledge, no theoretical vibrational cross sections have been reported to date, while several experimental cross sections have: by Hayashi and Akashi [30], Kitajima et al. [18], Allan and Skalický [21], and Nakamura [22]. A compilation of their work can be found in a recent review [31]. Hayashi and Akashi presented cross sections for electron-induced vibrational excitations from electron swarm parameters in pure N 2 O. Kitajima et al. as well as Allan and Skalický measured absolute differential cross sections for vibrationally inelastic electron scattering with a range of electron scattering energies from the threshold region up to 20 eV. Allan and Skalický reported measurements for only one scattering angle, 135 • , and multiplied each of the measured differential cross sections by the factor 4π to estimate the integral cross section. Nakamura [22] derived cross sections for vibrational excitation from swarm parameters.

The present work represents the first theoretical vibrational excitation (VE) study of N 2 O by electron impact. We present cross sections and rate coefficients for transitions between ground and first vibrational states of N 2 O using the theoretical and computational formalism recently employed to study the VE of NO 2 [32]. The rotational structure is neglected in the present study.

The paper is organized as follows. The next section, Sec. II, describes the theoretical approach and computational details employed in our calculations. In Sec. III, the results obtained numerically are analyzed using the model of Renner-Teller coupling for linear molecules. In Sec. IV, the obtained VE cross sections and corresponding rate coefficients are shown and discussed. Section V presents uncertainty estimations of the present approach, and the last section, Sec. VI, is devoted to our conclusions.

II. THEORETICAL APPROACH

Our approach can be summarized as follows. We start by characterizing the molecular target according to its equilibrium geometry, vibrational frequencies, and dipole moment value at equilibrium: features that can be obtained by performing ab initio electronic bound molecular states calculations. We proceed by performing ab initio electronic continuum molecular states calculations to obtain a scattering matrix at different molecular geometries along the vibrational normalmode coordinates. We then transform the scattering matrix into the basis of vibrational states of the target molecule. Finally, we compute the vibrational (de-)excitation cross sections from the transformed scattering matrix.

The approach used in our study has been previously described and applied to the NO 2 molecule [32], and a more detailed narrative of the simplified model on which our approach is based can be found in Refs. [33,34]. Therefore, we limit the description presented in this section to the main ingredients of the theoretical formalism-the normal-mode approximation and the vibrational frame transformation-and to the computational details of our calculations.

At low energies around the equilibrium position, the potential energy curve of the most rigid molecules is fairly well described by the quadratic potential of a harmonic oscillator. In our approach, we describe vibrational wave functions of the molecular target using the normal-mode approximation. The approximation allows us to perform a significant part of the calculations analytically. For molecules of astrophysical and low-temperature plasma interest, like N 2 O, only the lowest vibrational levels are significantly populated at low temperatures and the range of scattering energies needed to study vibrational excitation is within the validity of the normalmode model.

After computing the scattering matrix, we perform a vibrational frame transformation [35] to change the scattering matrix obtained for clumped nuclei for a number of molecular geometries to the vibrating-molecule picture that the electron sees when it is at large electronic distances.

The clumped nuclei basis of asymptotic channels is denoted by the channel quantum numbers {l, λ} that label the angular momentum of the incoming and outgoing electrons and their respective projections on the z axis in the molecular frame coordinate system. The three axes of the molecular coordinate system are chosen along the principal axes on the inertia of the molecule, such that the quantization axis (the z axis) is directed along the molecular axis in calculations for linear geometries. For bent geometries of the molecule, the z axis is perpendicular to the plane of the molecule, with the x axis aligned along the axis of the smallest moment of inertia. In Sec. III we introduce another set of quantum numbers {l, λ} and corresponding channel functions, which replace the spherical harmonic Y λ l with their real-valued combinations of Y ±λ l . The target vibrational wave functions are labeled by the {ν i , ν i } set of quantum numbers.

The vibrational frame transformation of the scattering matrix elements is given by

S ν i ν i l λ ,lλ = χ ν i (q)|S l λ ,lλ (q)|χ ν i (q) , (2)
where q collectively represents the normal-mode coordinates and the index i denotes the vibrational mode. N 2 O has three normal modes of vibration, namely, NO stretching, the doubly degenerate bending mode, and NN stretching, represented by ν 1 , ν 2 , and ν 3 , respectively. The physical meaning of an element of the transformed scattering matrix is the scattering amplitude from one vibrational state, χ ν i (q), of the target molecule to another, χ ν i (q). The vibrational frame transformation of Eq. ( 2) can only be performed if the fixed-nuclei S-matrix element, S l λ ,lλ , is a smooth function of the incident electronic energy [36]. This means, in particular, that for this approach to be applicable, the fixed-nuclei S matrix should not have low-energy electronic resonances. As discussed above, the lowest electronic resonance in e --NO 2 collisions occurs at collision energies of about 2.5 eV. The cross section σ ν i ν i for vibrational (de-)excitation can be obtained from the corresponding matrix element S ν i ν i l λ ,lλ by the expression

σ ν i ν i (E el ) = π h2 2m e E el l λ ,lλ S ν i ν i l λ ,lλ -δ ν i ν i l λ ,lλ 2 , (3)
where m e and E el are, respectively, the reduced mass of the electron-N 2 O system and the energy of the incident electron. Although the fixed-nuclei scattering matrix S l λ ,lλ (q) is weakly dependent on the energy, the remaining energy dependence introduces an ambiguity in the choice of the matrix in the integrand of Eq. ( 2). In the present calculation, we choose the following procedure: Integrating over the normal mode q in Eq. ( 2) for a given energy E el of the electron in the incident channel (see the above equation) and at each integration point q, the scattering matrix S l λ ,lλ (q) is taken from the R-matrix calculations performed at this particular fixed-nuclei geometry q and the electron scattering energy E el .

Because the energy dependence of the fixed-nuclei scattering matrix is weak below 2.5 eV, the corresponding uncertainty of the final cross section is much smaller than the uncertainty related to the choice of the ab initio model (discussed below). The cross section of Eq. ( 3) for vibrational excitation does not account for the rotational structure and can be compared with experiments or used in applications where the rotational structure of the initial and final vibrational levels is not important or not resolved. This is, generally, the case for most current experiments (including swarm measurements) and plasma applications at room or higher temperatures: With the rotational N 2 O constant of 0.419 01 cm -1 = 5.195×10 -5 eV [37] at 300 K, at least 25 rotational states are significantly populated.

Computational details

At its equilibrium geometry, N 2 O has a linear asymmetric "N-N-O" molecular structure, described by the C ∞v symmetry point group with the group electronic state of the 1 + symmetry. The equilibrium geometry and the normal-mode coordinates with corresponding frequencies were computed with the MOLPRO suite [39] using the complete active space self-consistent field (CASSCF) method and the cc-pVTZ basis set [40] centered on each atom. N 2 O has 22 electrons in a closed-shell electronic ground-state configuration given by 1 + : 1σ 2 2σ 2 3σ 2 4σ 2 5σ 2 6σ 2 1π 4 7σ 2 2π 4 .

In the calculations preserving the C ∞v symmetry group, the 10 electrons which occupy the lowest five σ molecular orbitals were kept frozen and the remaining 12 electrons were allowed to distribute themselves according to symmetry and spin restrictions in the complete active space (CAS) formed by the remaining 6σ 1π 7σ 2π ground-configuration orbitals and the next three molecular orbitals, 8σ , 9σ , and 3π , which are empty in the ground configuration. Because available quantum chemistry codes cannot handle continuous groups like C ∞v , the calculations were performed in the C 2v group for the geometries describing NO and NN stretching displacements. For geometries breaking the C ∞v symmetry group-the bending-mode displacements-the same 10 electrons were kept frozen in the lowest 5 a orbitals and the remaining 12 electrons were distributed in the 6-12 a and 1-3 a orbitals of the corresponding C s symmetry group. Upon optimization of the equilibrium geometry, the N-N and N-O bond lengths were found to be 1.131 and 1.186 Å, respectively, in good agreement with the experimental values, 1.128 and 1.184 Å [38]. Table I reports a comparison between the obtained normal-mode frequencies and the available experimental data [38]. Our frequencies agree with the experimental references with a difference of less than 3%.

After characterizing the equilibrium geometry and normalmode frequencies with MOLPRO [39], we carried out calculations of the potential energy of the ground electronic state of N 2 O and calculations of continuum states using the U.K. R-matrix code [41] with the Quantemol-N suite [42]. For consistency with the MOLPRO calculations, we have used the same basis set and CAS. However, Quantemol-N does not have CASSCF built into it, and a series of convergence tests showed that the available complete active space configuration interaction (CAS-CI) model with the Hartree-Fock orbitals built with MOLPRO gave the best results. Figure 1 displays the ground-state electronic potential energy curves of N 2 O for each normal mode obtained with Quantemol-N. For comparison, we also show the potential energy curves of harmonic oscillators generated with the frequencies obtained from MOLPRO. The Quantemol-N potential energy curves agree reasonably well with the potential energies calculated in the harmonic approximation. Small discrepancies are attributed to the anharmonicity of the actual N 2 O potential. The permanent electric dipole moment of the target molecule obtained from the R-matrix calculation is 0.1 D, which is considered to be in satisfactory agreement with the experimental value, 0.16 D [38].

Using the molecular orbitals obtained from the structure calculations and the continuum Gaussian-type orbitals with partial waves up to l 4, we performed the electronic continuum molecular states calculations with Quantemol-N. The radius of the R-matrix sphere was set to be 11 bohr. All the electronic states of the target below the cutoff energy, 16 eV, have been included in the close-coupling expansion. From the scattering calculations we can obtain the eigenphase sums and the reactance matrix (K matrix) at clumped nuclei.

Figure 2 displays the eigenphase sum of different irreducible representations at equilibrium and at displacements q 2 =-0.8668 q 1 =-0.8668 q 1 =+0.8668 q 3 =-0.8668

q 3 =+0.8668 2 Π 2 Σ - 2 Σ + 2 A" 2 A' FIG.
2. Sum of eigenphases as a function of the electron scattering energy for equilibrium geometry and displacements q i = ±0.8668 (dimensionless) along each normal mode. Due to the symmetry of the bending mode, the eigenphase sums of q 2 = +0.8668 and q 2 = -0.8668 are the same. The curves are color-coded according to the different symmetries of the e -+ N 2 O system. away from the equilibrium along each normal-mode coordinate. We obtained the position and width of calculated resonances by fitting the eigenphase sum to a Breit-Wigner form. At equilibrium, the lowest resonance is found at 3.0 eV and has 2 symmetry. To compare with the available experimental data (the resonance around 2.3-2.5 eV [21,43]), the zero-point energy h(ω 1 + 2ω 2 + ω 3 )/2 = 0.3 eV of the ground vibrational level should be accounted for. Therefore, in the present calculation, the energy of the resonance is 2.7 eV above the ground vibrational level. The difference from the experimental position of the resonance is attributed to the large uncertainty associated with the Born-Oppenheimer approximation used to identify the energy of the resonance in the theoretical calculation: The position of the resonance depends strongly on the choice of the fixed geometry near the N 2 O equilibrium at which the scattering calculations were performed. In addition, the width (about 1 eV) of this shape resonance is larger than the difference between the experimental and the theoretical results.

The K matrix obtained from the scattering calculations was used to compute the clumped-nuclei scattering matrix (S matrix). Figure 3 displays selected dominant elements (the absolute value squared) of the S matrix at equilibrium geometry. In the figure (as well as in Fig. 4), the indices λ refer to real-valued combinations of spherical harmonics Y ±λ l with positive and negative projections λ. The real-valued harmonics Y l λ with positive λ transform as cosine-type functions with respect to the rotational angle φ about the axis z perpendicular to the plane of the molecule, while the harmonics with negative λ transform as sine-type functions. Except for the S 10,00 element, all other elements behave smoothly with the electronic energy below the first resonance. The minimum is observed near 0.4 eV for the 10 ← 00 transition. Although the S 10,00 element has a strong energy dependence, which breaks the condition of the applicability of the vibrational frame transformation (the energy dependence should be smooth), its contribution to the VE cross section in Eq. ( 3) is negligible compared to that of the dominant terms (diagonal over l λ), and therefore, it does not compromise the present theoretical approach. The vibrationally transformed S matrix is calculated according to Eq. ( 2), where the integration over vibrational coordinates is performed numerically using a Gaussian-Legendre quadrature with 10 points. It is instructive to analyze the dependence of major coupling elements of the scattering matrix as a function of the normal coordinates, especially for the bending mode. For the NO and NN stretching modes, the main contribution to the VE cross section is due to variation of the diagonal elements of the scattering matrix with low l.

In constrast, for the bending mode, the major contribution is due to the q 2 dependence of nondiagonal elements between the 2 + and the 2 states of the e -+ N 2 O system near the linear geometry. This is the Renner-Teller coupling, whose effect on electron-molecule collisions has been discussed in several previous studies [33,[44][45][46][47][48]. For a linear triatomic (and larger) molecule with a ground electronic state of 1 symmetry, the Renner-Teller effect couples σ and π partial waves of the incident electron with vibrational bending motion of the target molecule.

Due to the symmetry of the bending mode, all matrix elements S l λ ,l λ are symmetric or antisymmetric with respect to the change in the sign q 2 → -q 2 of the displacement along the bending mode. The elements which are symmetric, such as diagonal and some nondiagonal elements, do not contribute to the vibrational excitation by one quantum of the bending mode. For an element S l λ ,l λ to be antisymmetric with respect to the q 2 → -q 2 operation, one of λ and λ should be negative, with the other positive or 0. In addition, there is a selection rule regarding the elements that do not vanish: For displacements along q 2 , both spherical harmonics in S l λ ,l λ should be of a or a irreducible representations of the C s symmetry group (of the bent molecule). Figure 4 shows the largest (in magnitude) antisymmetric S-matrix elements as a function of the bending coordinate.

In Fig. 4, we see that most of the elements are linear with the q 2 coordinate, with the notable exception of S 1-1,00 , which has a strong cubic dependence q 3 2 . We attribute the significant cubic contribution to the coupling to the fact that the sσ partial wave penetrates closer to the N 2 O core electrons such that the linear approximation for the coupling between the Y 0,0 and the Y 1,-1 harmonics is no longer valid and higher terms are needed if a Taylor expansion is used to represent the coupling.

The linear dependence of the coupling between partial wave components in a linear molecule for small displacements along the bending coordinate is one of the main characteristics of the Renner-Teller effect. The effect cannot be easily observed in e --N 2 O scattering experiments, but it manifests itself in the bound states of the e --N 2 O system: Due to the degeneracy of the 2 electronic state of the e --N 2 O complex and the degenerate bending mode of N 2 O, the relatively strong Renner-Teller coupling results in a bending configuration of the equilibrium geometry of the N 2 O -anion [49,50].

IV. CROSS SECTIONS AND RATE COEFFICIENTS

We calculated the vibrational (de-)excitation cross sections for transitions between the ground and the first excited vibrational states for each of the normal modes. Figure 5 displays a comparison of the theoretical 1 ← 0 VE cross sections with the available experimental data [21,22,30], mentioned in [30,31] (solid line with circles), Allan and Skalický [21] (solid line with triangles), and Nakamura [22] (dashed-dotted line).

Sec. I. None of the three experimental data resolve the ν 1 = 1/ν 2 = 2 and ν 3 = 1/ν 1 = 2 thresholds, i.e., the experimental cross section for the excitation of the NO mode (ν 1 = 0 → 1) includes also a contribution for the transition ν 2 = 0 → 2, and the cross section for the excitation of the NN mode includes a contribution for the ν 1 = 0 → 2 transition. But these additional contributions are expected to be significantly smaller due to the vibrational propensity rule: The transitions with a change of only one vibrational quantum are the largest. There is a significant disagreement between the experimental data, up to a factor of 20-50 for certain energies. On the other hand, the theoretical results also do not agree better with one or another experiment: For the NO stretching mode [Fig. 5(a)], the theory agrees better with the experiment by Hayashi [30]. For the bending mode [Fig. 5(b)], the theory agrees better with the other two experiments, although the agreement is quite poor. Note that the integral cross sections presented by Allan and Skalický [21] were obtained from the measured differential cross sections. Finally, for the NN stretching mode [Fig. 5(c)], the theory again agrees better with the data of Hayashi [30].

ν 2 '=0 ν 1 '=0 ν 2 =1 ν 3 '=1 ν 1 '=0 ν 3 =0 ν 1 =1 ν 3 '=0 ν 3 =1 ν 1 =1 ν 2 '=1 ν 2 =0
FIG. 6. Thermally averaged rate coefficients for excitation and de-excitation transitions between the ground and the first excited vibrational states of the NO stretching mode (black curves), bending mode (blue curves), and NN stretching mode (red curves). Vibrational (de-)excitations are labeled ν i ← ν i . To provide an idea of the uncertainty of the present results, we also plotted the results of calculations with model 2 (dotted lines) and model 3 (thin lines). The three calculations produce curves that are almost indistinguishable.

In a recent review article [31], the swarm data of Nakamura [22], shown by the dashed-dotted lines in Fig. 5, were recommended as the most accurate among the available experimental cross sections. However, it should be stressed that the recommended swarm data may not be very accurate because of an ambiguity in their interpretation [22] (see the discussion in Secs. 3.3 and 3.5 in Ref. [31]). Therefore, the recommended experimental data should have a relatively large uncertainty and could be improved in a more accurate future experiment.

The present theoretical cross sections are expected to be valid only for energies below the energy of the 2 resonance mentioned above, i.e., below 2.3 eV. They are included in the Supplemental Material to this paper [51].

Thermally averaged rate coefficients α ν i ν i (T ) for vibrational excitation are obtained from the cross sections of Eq. (3) using the standard formula

α ν i ν i (T ) = 8π (2π k B T ) 3 2 ∞ 0 σ ν i ν i (E el )e -E el k B T E el dE el , ( 4
)
where k B is the Boltzmann constant and T is the temperature. The computed rate coefficients are shown in Fig. 6 in different colors.

For convenient use in plasma models, the computed coefficients were also fitted using the analytical formula employed in our previous studies [36,52,53],

α fit ν i ν i (T ) = 1 √ T e -ν i ν i T P fit ν i ν i (x) , (5)
where P fit ν i ν i (x) is a quadratic polynomial, P fit ν i ν i (x) = a 0 + a 1 x + a 2 x 2 and x = ln(T ),

TABLE II. Parameters a 0 , a 1 , and a 2 of the polynomial P fit ν i ν i (x) in Eqs. ( 5) and (6) for transitions between the ground and the first vibrational states in each normal mode. We specify the excitation threshold energies ν i ν i of Eq. (7) in the second column. The threshold ν i ν i = 0 for the de-excitation process. where P fit ν i ν i (x) ≈ P fit ν i ν i (x) can be viewed as the excitation and de-excitation probabilities, with v i ,vi being the threshold energy, defined as

ν i ν i = E ν i -E ν i > 0 for excitation, 0 for de-excitation. ( 7
)
The numerically fitted parameters for vibrational excitation are listed in Table II. When the parameters listed in the table are used in the fitting formulas of Eqs. ( 5) and ( 6) with the temperature in kelvins, the obtained numerical values of the rate coefficients will be in units of cm 3 /s.

V. UNCERTAINTY ESTIMATIONS

We have performed a number of calculations to assess the uncertainty of the obtained theoretical results. There are two main sources of uncertainty in the present theoretical approach. The first is the accuracy of the fixed-nuclei S-matrix elements computed for the polar molecule (with a small dipole moment) in the limited basis of spherical harmonics. It has been discussed by Liu [32]. The uncertainty associated with this approximation was estimated to be of the order of 6% for NO 2 [32]. It should not be larger for N 2 O because it has a smaller dipole moment so that the couplings between partial waves induced by the permanent dipole moment of this molecule have a weaker effect on the final cross sections. Therefore, it is reasonable to assume that the corresponding uncertainty in the present case is below 6%.

The second source of uncertainty derives from the particular scattering model used in the calculation. The uncertainty can be assessed by performing a complete calculation with different parameters of the model. With the parameters discussed in Sec. II, referred to as model 1, we obtained the results shown above. In the second calculation, with model 2, the CAS in the configuration calculation was the same as in model 1 but a larger basis set, cc-pVQZ, was used. Finally, in the model 3 calculation the electronic basis set remained cc-pVTZ, but the CAS was reduced compared to that in model 1: 12 electrons were placed in frozen orbitals and the remaining 10 electrons in the lowest orbitals were allowed to be freely distributed in the active space. The rate coefficients obtained in the three models are shown in Fig. 6. The difference in the rate coefficients produced in the three models is about 6%. Consequently, the overall uncertainty of the present theoretical result is estimated to be below 12%.

VI. CONCLUSIONS

In this study, we computed cross sections for vibrational (de-)excitations of N 2 O by a low-energy electron using (a) the normal-mode approximation to describe the vibrational states of the target molecule, (b) the R-matrix method to evaluate the fixed-nuclei electron-N 2 O scattering matrices, and (c) the vibrational frame transformation to evaluate the amplitudes for vibrational transitions. In this approach, we neglected the rotational structure of each vibrational level, which corresponds to the situation where the rotational structure is not resolved in the initial and final states of the target molecule.

The computed results show a reasonable agreement with experimental data for the NO and NN stretching modes. For the bending mode the agreement is rather poor at energies above 0.4 eV. It was found that Renner-Teller coupling is responsible for the excitation of the bending mode, as was expected from general theoretical considerations. We are quite confident about the present theoretical cross sections for the bending mode because the numerical calculations of the fixed-nuclei scattering matrix fit well to the theory of Renner-Teller coupling. It should be stressed here that the most reliable experimental cross section for the bending mode by Nakamura [22,31] was obtained from swarm data and a direct measurement of the differential cross section at a single scattering angle by Allan and Skalický [21] and, therefore, may have a large uncertainty. This suggests that a better direct measurement of vibrational excitation in N 2 O, at least for a few energies, is needed.

Introduction

Non-equilibrium plasmas produced by electrical discharges in BF 3 -containing feed gas are of continually increasing interest for a large number of applications. In particular, BF 3 is very often the boron carrier when plasmas are used for material processing [1][2][3]. Basically, BF 3 plasmas are used either for (i) the synthesis of ultra-hard boron compounds, e.g. boron carbides [1], (ii) the deposition of boron nitride, an advanced material with a large number of functionalities [2], and (iii) p-type doping by boron in the semi-conductor and photovoltaic industries [3,4]. As far as doping applications are concerned, plasma immersion ion implantation (PIII) processes are probably one of the most promising in terms of cost and technical performance [4,5]. These processes make use of very low pressure, very high density magnetized plasmas generated in a BF 3 -containing feed gas. Depending on the level of the power coupled to the plasma, several cations-BF 3 + , BF 2 + , BF + , B + -and anions-e.g. F --may be produced [6,7]. The positive ions are then extracted from the source to an implantation chamber where the processed silicon substrate is submitted to very high negative voltage pulses. These pulses result in a large acceleration of the positive ions that are implanted in the substrate, which results in the doping of the latter. The implantation depth depends on the nature and the energy distribution of the ions impinging the substrate, while the doping level depends on the ion flux and, consequently, on the plasma density. The plasma density and the relative predominance of the different ions are determined by the ionization kinetics in the source region. The plasma sources used in PIII processes are usually magnetized [8]. The ambipolar diffusion in the radial direction that is perpendicular to the magnetic field is strongly reduced with respect to the parallel diffusion. Under typical plasma conditions used in PIII processes, i.e., B≈100-500 G and p≈1 Pa, electron cyclotron frequency in the GHz range, while the electron-neutral momentum transfer collision frequency is in the MHz range, according to equation (5.4.5) of [9] the diffusion coefficient in the radial direction is reduced by approximately 6 orders of magnitude with respect to the parallel diffusion coefficient. The characteristic time estimated for perpendicular diffusion for typical PIII reactor geometries is at the order of 100 s, while the dissociative recombination (DR) characteristic time is around 0.1 s. Following a similar procedure, one can easily show that mutual neutralization is also likely to dominate diffusion losses provided the negative ion density is of the same order of magnitude as the electron density, i.e. moderately electronegative plasmas. It appears, therefore, that under PIII discharge conditions, positive ion losses at the reactor wall are dominated by their DR with electrons and by their neutralization through collisions with negative ions. The investigation of DR of molecular ions is therefore of major interest for these processes. This is especially the case with BF 2 + , which is often the major ion in BF 3 -containing plasma in discharge conditions corresponding to PIII process [5]. This study is a continuation of a previous work performed in the theoretical framework of the multichannel quantum defect theory on the DR and competitive processes of BF + [10].

The article is organized as follows. After the Introduction, section 2 describes the theoretical approach used in the present calculation. In section 3, the obtained cross-sections and the corresponding rate coefficients are displayed and discussed. Section 4 concludes the study.

Theoretical approach

Dissociative recombination and vibrational excitation cross-section formulas

Since the basic formalism used in our model is presented in detail in [11,12], we restrict ourselves here to underlining its major ideas. The molecular cation under study, BF + 2 , is linear in its equilibrium geometry.

The theoretical model starts with the following assumptions (see for example [12]): (i) the rotation of the molecule is neglected, (ii) the cross-section is averaged over the autoionizing resonances, (iii) the autoionization lifetime is assumed to be much longer than the predissociation lifetime and (iv) the harmonic approximation is used to describe the vibrational state of the core ion. Using (i)-(iv) and applying the frame transformation, the DR cross-section is given by equation (13) of [12], in which the scattering matrix elements were expanded to first order in the normal coordinates. The cross-section for vibrational excitation (VE) of the mode i is written as

E mE g S q E 4
. 1 Here q i , i w and g i (i=1, 2,3) are respectively the dimen- sionless coordinate, the energy and the degeneracy of the mode i. The degeneracy is two for the bending mode 2 and one for the symmetric and asymmetric stretching modes. S l l , l l ¢ ¢ is an element of the fixed-nuclei scattering matrix for electron-BF 2 + collisions with initial channel (λl) and exit channel ( l l¢ ¢), l being the electron angular momentum and λ its projection on the molecular axis. Finally, m is the reduced mass of the electron-ion system, E el the incident energy of the electron and θ the Heaviside step function. The present theoretical approach can describe the (de-)excitation process changing only one quantum in each normal mode of the target ion. (De-)excitation cross-sections for changing two or more quanta in a mode are much smaller (the propensity rule) and are neglected in this study. In the present case, the initial state of the ion is the ground vibrational level, so the electron can only be captured into the first excited vibrational state of each normal mode.

The situation is similar if the electron energy is not sufficient to excite the ion and then to leave it. In such a situation, the present model suggests that the probability of excitation of the ion by the electron is described by the same physics, but instead of leaving the vibrationally excited ion, the electron is captured in a Rydberg resonance attached to that vibrational state, excited by the electron. If the electron is captured by the ion, the system will most likely dissociate, rather then autoionize. Correspondingly, the cross-section for DR is then obtained [12] as

E mE g S q E 4
. 2 Here i runs over all three modes: two stretching modes ν 1 and ν 3 (symmetric and asymmetric) with respective frequencies ω 1 and ω 3 and corresponding coordinates q 1 and q 3 , and a doubly degenerate transverse mode ν 2 with a lower frequency ω 2 and coordinates q q , x y 2 2 ( ).

To calculate the derivative of the scattering matrix S q l l i , ¶ ¶ l l ¢ ¢ with respect to the normal coordinate q i , the scattering matrix is evaluated for two values of q i keeping the other normal coordinates q i¢ fixed at q 0 i = ¢ . The elements S q l l , l l ¢ ¢  ( ) of the scattering matrix S q  ˆ( ) at a given geometry q  specified by four normal coordinates q  = q q q q , , ,

{ } = q q q q , , , where 1 ˆis the identity matrix.

The properties of the BF + 2 ion and the scattering calculations

The main electronic ground state configuration g 1 S + of the ion in its natural point group symmetry D h ¥ is 1 u and including s, p and d orbitals. Performing configuration interaction (CI) calculations in the D 2h symmetry group, using the MOLPRO suite of codes [13], we found that the equilibrium geometry of the ion has a B-F distance of 1.2215 Å. Table 1 gives the vibrational frequencies obtained from CI calculations using the cc-pVTZ basis set, and compares with the data available in literature.

The electron-scattering calculations were performed using the UK R-Matrix code [17,18] with the Quantemol-N interface [19]. The calculations were performed in the abelian subgroup D 2h and the target BF 2 + ion was assumed to be in its ground electronic state. In our CI model, we freeze 10 electrons in the core 1a g , 2a g , 3a g , 1b 2u , 2b 2u , while the remaining 12 electrons are kept free in the active space of the 4a g , 5a g , 1b 3u , 2b 3u , 3b 2u , 4b 2u , 1b 1g , 1b 1u , 2b 1u , 1b 3g molecular orbitals. A total of eight electronic excited target states are represented by 1844 configuration state functions (CSFs) for the ground state. All the generated states up to 16 eV were retained in the final close-coupling calculation. We used an R-matrix sphere of radius 10 bohrs and continuum Gaussiantype orbitals with partial waves up to l<4. In the following, this calculation will be referred to CAS 1 .

Initially, several basis sets-including DZP and cc-pVTZ -were tested to investigate the stability of the target properties such as polarizability and ground state energy and, finally, we chose the cc-pVTZ basis set in order to perform the scattering calculations. Since the first electronically excited state g 3 S -is approximately 13 eV above the dissociation limit for the ground state, this latter state is essentially isolated and the non-adiabatic effects are expected to be small. Therefore, for low electron energy collisions, i.e. below 10 eV, only the ground electronic state is open for ionization in e-BF 2 + collisions, and the dimension of the geometry-fixed scattering matrix remains the same at low collision energies. At low collision energies the fixed-nuclei scattering matrix depends only weakly on energy. A sharper energydependence is observed at certain relatively high energies, corresponding to positions of Rydberg states attached to the excited electronic states of the ion. A quantity convenient for the analysis of the energy dependence of the scattering matrix is the eigenphase sum. Figure 1 displays the eigenphase sum for three different geometries corresponding to a small displacement from equilibrium along each normal mode of the BF 2 + ion. Bending and asymmetric stretching mode calculations were performed in the C 2v abelian subgroup, while for the symmetric mode the group D 2h was used in the calculations. The variation of the eigenphase sums is smooth for energies below 6 eV. Above this value, a sharp energy dependence at certain energies is observed due to the presence of electronic Rydberg resonances attached to closed ionization limits.

Cross-sections and rate coefficients

For convenience, we introduce the quantities

P g S q 2
, 4 which could be interpreted as the probability of excitation of the vibrational mode i, and list them in table 2. Figure 2 shows the weak dependence of those quantities on energy and, therefore, they could be used as constants in calculations of the thermally-averaged rate coefficients. Using P i , the cross-sections of equations (1) and (2) are written as In order to access the uncertainty of the present theoretical model, we have performed a complete calculation of the DR and VE rate coefficients using different basis sets and orbital spaces in the electron-scattering calculations. The calculations were performed for three sets of parameters (1) the CAS 1 with the cc-pVTZ basis set, mentioned above; (2) a calculation (referred here as cc-pVTZ CAS 2 ) similar to (1) but with a smaller orbital space, where 8 electrons are kept free in the active space; and (3) a calculation (referred here as cc-pVQZ CAS 1 ) similar to (1) but with the larger basis cc-pVQZ. The results are shown in figure 4. The difference between the rate coefficients produced in the three calculations for the DR process and the VE of the asymmetric stretching and bending modes is about 4%. The uncertainty of the rate coefficient for the VE of the symmetric stretching mode is larger, varying in the interval 10%-40% for different temperatures. Notice that the overall probability for the symmetric stretching excitation is much smaller than the probabilities for other modes and DR.

E

Conclusions and discussions

In this study, cross-sections and rate coefficients for DR and VE of BF 2 + by electron-impact were obtained using a theoretical approach that combines the normal modes approximation for the vibrational states of the target ion, the vibrational frame transformation, and the UK R-matrix code. The thermally-averaged rate coefficients have a simple analytical form.

The obtained thermally-averaged rate coefficients are relevant for the kinetic modeling of molecule based cold nonequilibrium plasmas, in the context of a complete lack of other theoretical or experimental data on these processes for this cation, and are ready to be used in the modeling of fluorine/boron plasma for etching or implantation processes. Indeed, we are presently able to make important statements on the relative importance of BF 2 + with respect to BF + on the population and excitation balance. In particular, as shown in figure 5, the DR of BF 2 + strongly dominates that of BF + below 7000 K, and the VE displays the same feature below 5000 K. The rotational structure of the target ion and of the neutral molecule was neglected in the present approach, which implies that the obtained cross-sections and rate coefficients should be viewed as averaged over initial rotational states and summed over final rotational states of the corresponding initial and final vibrational levels (for VE) or dissociative states (for DR). Purely rotational transitions, i.e. without changing the vibrational state, might be useful to model very cold environments, below 40 K, which is not the case for the presently investigated BF 3 plasma. Moreover BF 2

+ has no permanent dipole, so the rotational transitions are likely to have very small cross-sections.

Figure . 1 :

 1 Figure .1: Schematic diagram summarizing the research activities performed over the previous years. Note that the molecules drawn in bold have already been studied while the others are in progress to date. It should be stressed that certain molecules could play different role in various applications.

Figure I. 1 :

 1 Figure I.1: The flow chart of different ingredients employed in the theoretical model developed for the description of rotational, vibrational and electronic (de-)excitation (RE,VE,EE) as well as dissociative recombination (DR) of diatomic molecular ions such as HeH + and CH + . The following abbreviations are used. FT: frame transformation. PECs: potential energy curves. DVR: discrete variable representation. CAP: complex absorbing potential. Tasks or methods in non-bold text are not performed in this HDR yet.

  Figure I.2: Schematic diagrams for the R-matrix method (upper panel) as well as the R-/q-region, v-region and the j-region (lower panel) according to the local kinetic energy of the electron given in Eqs.(I.6) and (I.7). The frame transformation procedures (FT) connecting the molecular-frame to the laboratory-frame are also represented. Note that these transformations constitute the l-uncoupling transition from the Hund-coupling case b to case d[Chang and Fano, 1972].

Figure I. 3 :

 3 Figure I.3: Eigenphase sum as a function of the electron scattering energy E el (in eV) for several internuclear distances R (in units of bohr) for the 2 Σ + (left panel), 2 Π (right upper panel), and 2 ∆ (right bottom panel) symmetries. The color scheme used to label different values of R is the same for all three panels. The dashed line represents the equilibrium internuclear distance R e = 1.445 bohrs [Rabadán et al., 1998] of HeH + .

  Figure I.4 demonstrates, as examples, the cross sections of Equation (I.27) for the v = 3 → v = 0, 1, 2, 4 transitions. At very low scattering energies, below 0.02 eV, the de-excitation cross sections are smooth functions inversely proportional to the incident energy of the electron as predicted by the Wigner threshold law. But at higher energies, especially, just below the energy of the each excited threshold, v = 4, 5, • • • , the (de-)excitation cross sections and probabilities demonstrate series of Rydberg resonances, where they vary significantly.

Figure I. 4 :

 4 Figure I.4: Cross sections of vibrational (de-)excitation from the vibrational level v = 3 to several other levels v . The inset enlarges the region of Rydberg series of resonances close to the v = 4 and v = 5 ionization thresholds, situated at energy of 0.2465 and 0.4533 eV, respectively, above the v = 3 threshold.

Figure I. 5 :

 5 Figure I.5: Rotational (de-)excitation cross sections for transitions between the four lowest rotational states of HeH + . Solid lines represent the results obtained with the applied closedchannel elimination procedure of Equation (I.19), while the dotted lines show the results for which the procedure was not applied.

Figure I. 6 :

 6 Figure I.6:Thermally averaged rate coefficients for several vibrational (de-)excitation transitions of4 HeH + (solid lines) and4 HeD + (dashed lines). Vibrational transition v → v labels are shown in each panel. Results of a previous calculation[Rabadán et al., 1998] are shown in dotted lines with circles.

Figure I. 7 :

 7 Figure I.7: Thermally averaged rate coefficients for several rotational (de-)excitation transitions of 4 HeH + (solid lines) and 4 HeD + (dashed lines). Dotted lines in the upper left panel are the calculations by Hamilton et al. [Hamilton et al., 2016], and lines with circles are those of Čurík and Greene [Čurík and Greene, 2017] for 4 HeH + .

Figure I. 9 :

 9 Figure I.9: Comparison of eigenphase-sum derivatives computed for a fixed CH + geometry using three different energy-independent 3 × 3 scattering matrices and the procedure of elimination of closed electronic channels. The three matrices are taken at energies 3.413 eV (black curve), 3.563 eV (red curve) and 3.713 eV (green curve). The inset shows an enlarged view for 1.2-2.4 eV energies.

Figure I. 10 :

 10 Figure I.10: Cross sections for the X 1 Σ + → a 3 Π electronic excitation of CH + at a fixed geometry R e obtained in the direct R-matrix calculations (red curve) and using the QDT channel elimination procedure (black curve).

Figure I. 13 :

 13 Figure I.13: Low-energy resonances corresponding to the doubly-excited CH * * (2 2 Π) obtained from the derivative of eigenphase sum of the 2 Π symmetry in R-matrix calculation (solid curves) and MQDT channel elimination procedure (dashed curves). Each color refer to different resonances obtained at R=1.337 (black), 1.437 (red), and 1.537 bohrs (green). Details of the R-matrix calculations could be found in paper .5.

3 E

 3 Figure I.14: Left panels show the vibrational wave function as function of R for v=30, X 1 Σ + (in black), v=26, a 3 Π (in red), and v=20, A 1 Π (in green) obtained with (solid curve) and without CAP (dashed curve). Right panels display the Fourier transform of the corresponding vibrational wave functions. The starting point of the CAP, R CAP , is drawn in the left lower panel.

Figure I. 15

 15 Figure I.15 shows the first result of the computed DR (including direct dn indirect routes) cross section of the ground vibronic state of CH + , compared with different ab initio studies and experimental measurements available in the literature. To compare these results with

Figure I. 17 :

 17 Figure I.17: Energy-dependence of eigenphases of different irreducible representations for the CH 2 NH + 2 + e -collisions.

1 Figure I. 18 :

 118 Figure I.18: The DR cross section as a function of the collision energy. For energies above 0.1 eV one or several vibrationally-excited channels are open. As a result, the DR cross section decreases in a stepwise manner.

+ 3 ,

 3 HCO + , NH + 4 , H 3 O + , CH + [Mikhailov et al., 2006b, Douguet et al., 2012b, Douguet et al., 2012a, Fonseca dos Santos et al., 2014], it was estimated that the probability of autoionization of vibrational Rydberg resonances in these closed-shell ions is of the order of 10-20% compared to the probability of dissociation. For

8 Figure I. 19 :

 819 Figure I.19: The DR (dashed line) and VE (solid lines) rate coefficients for target ion CH 2 NH + 2

Figure II. 1 :

 1 Figure II.1: The flow chart of different ingredients employed in the theoretical model developed for the description of vibrational (de-)excitation (VE) as well as dissociative recombination (DR) of polyatomic molecular ions such as NO 2 , N 2 O, CH 2 NH + 2 (described in Chapter I) and BF + 2 (described in Chapter III). The following abbreviations are used. FT: frame transformation. DVR: discrete variable representation. Tasks or methods in non-bold text are not performed in this HDR yet.

𝜔 1

 1 Figure II.2: Normal modes of NO 2 . The arrows indicate the direction and magnitude of displacements for each mode. Note that arrowheads are not shown for the displacements of relatively large magnitudes. Vibrational frequencies and symmetries are listed in Table II.1.

  Figure II.3 displays the potential energy surface of the ground state of NO 2 computed with the R-matrix code, where the surface is plotted along the three normal modes and compared with the harmonic potential energy. Almost no anharmonic contribution is observed for the bending (panel (a) in the figure) and asymmetric stretching (panel (c)) modes. The potential energy of the symmetric stretching mode is, however, slightly anharmonic as evident from panel (b) in the figure. The dipole moment of NO 2 obtained in this model is 0.317 D, which is in remarkably good agreement with the experimental value of 0.316 D [Leonardi et al., 1996].

Figure II. 3 :

 3 Figure II.3: Potential energy curves for the ground electronic state of NO 2 as a function of the (a) bending; (b) symmetric stretching; (c) asymmetric stretching modes. The abscissa axes in the figure represent dimensionless normal coordinates.In each panel, only one mode q i is varied, while the other modes are kept fixed at their equilibrium positions q i = 0. Red solid curves are the actual potential energies obtained from the UKRmol suite, while black dashed curves represent energies calculated in the harmonic approximation. Horizontal dashed lines denote energies of vibrational states.

Figure II. 4 :

 4 Figure II.4:The largest value of S l λ ,lλ2 with respect to all possible allowed combinations of lλ, l λ as a function of the difference ∆l = ll for all four singlet irreps of the e --NO 2 complex. The lowest value of l is 0 for 1 A 1 , 1 for 1 B 1 and 1 B 2 , 2 for 1 A 2 . For 1 A 1 (solid red line with circles), the couplings between the s and f, s and g partial waves are very small. For 1 B 1 (solid green line with triangles) and 1 B 2 (solid blue line with squares), since s-wave scattering is forbidden by symmetry, only couplings between p, d, f and g partial waves is shown. For 1 A 2 (solid pink line with stars), s and p partial waves scattering is not allowed.

Figure II. 5 :

 5 Figure II.5: The figures shows dominant (absolute value squared) elements S l λ ,lλ 2 of the scattering matrix as a function of the electron scattering energy at the NO 2 equilibrium. Black lines: coupling between channels with ∆l = 0. Color lines: coupling between channels with ∆l = 1

Figure II. 6 :

 6 Figure II.6: Calculated cross sections as functions of the electron scattering energy for the vibrational excitation of NO 2 being initially in the lowest vibrational state v = 0 for the three normal modes (see the text for detailed discussion): (a) cross sections for v = 1, 2 ← v = 0 transitions for bending mode; (b) for symmetric stretching mode; (c) for asymmetric stretching mode.

Figure

  Figure II.7: Spin-and thermally-averaged rate coefficients for (de-)excitation transitions between the three lowest vibrational states of the bending mode. Vibrational excitations are labeled by υ ← υ. Corresponding de-excitations are shown by dashed lines of the same color. The figure also demonstrates the quality of the fit of Equation (6) in Ref.[Liu et al., 2019]: Fitted curves are shown by stars with the same color for each transition.

  Figure II.9 displays, as an example, the compared VE cross sections for the bending mode where the derivative of the scattering matrix in Eq.(III.6), employed in Eq.(III.9), was evaluated with two points q 1 = 0.2 and q 1 = 1. In the case of singlet, the VE cross section calculated with the simplified model (green solid curve) is larger than the presently calculated one (red solid curve) by about 30% while this deviation is less than 10% for triplet. To get an idea about these discrepancies, we plotted in Figures II.10 and II.11 the dominant fixed-nuclei S-matrix element S l λ ,lλ and the derivatives (for singlet only) as function of q 1 .

Figure II. 10 :

 10 Figure II.10: Dominant elements S l λ ,lλ of the scattering matrix as a function of normal coordinate q 1 (bending mode) at 0.3 eV scattering energy. Upper figure: S-matrix elements for triplet . Bottom figure: S-matrix elements for singlet. The real (resp. imaginary) parts of the scattering matrix are given in solid curves in left panels (resp. dashed curves in right panels). The numbers indicate different partial waves l λ ← lλ.

2 ,- 2 Figure II. 11 :

 2211 Figure II.11: Derivatives of dominant elements S l λ ,lλ as function of normal coordinate q 1 at 0.3 eV scattering energy, displayed in the lower Figure II.10. Recall that the real parts of the scattering matrix are given in left panels (solid curves) while the imaginary ones are in the right panels (dashed curves).

Figure II. 12 :

 12 Figure II.12: Variation of the computed permanent dipole moment of NO 2 versus different basis sets (panel (a)) and CAS (panel (b)). CAS1: 7 electrons are kept free in the active space including 7 orbitals; CAS2: 9 electrons are kept free in the active space including 8 orbitals; CAS3: 11 electrons are kept free in the active space including 9 orbitals; CAS4 (CAS used in this paper): 13 electrons are kept free in the active space including 10 orbitals; CAS5: 15 electrons are kept free in the active space including 11 orbitals. The blue dashed line indicates the experimental value of the dipole moment of Ref[Leonardi et al., 1996]. (c) The eigenphase sums of scattering of the 3 B 1 symmetry of the e --NO 2 complex as a function of the electron scattering energy for different CAS's. The inset enlarges the region where a sharper energy-dependence is observed around 1.6 eV corresponding to a resonance.

  Figure II.12(b) displays the variation of the dipole moment as a function of different CASs for the cc-pVTZ basis set. Obviously, the current CAS, i.e. CAS4 corresponds to the dipole moment closest to the experimental data. Therefore, we concluded that the target properties obtained by Quantemol-N are well converged and accurately represented. One might also provide uncertainties for other intermediate quantities computed in collisional studies, such as eigenphase sums. These are also shown in Figure II.12, panel (c). As shown in the figure, the small shift observed in calculations with different CASs indicates the convergence of the scattering data.

  2π 4 . (II.23) N 2 O has three normal modes of vibration, namely: NO stretching, the doubly degenerate bending mode, and NN stretching represented by v 1 , v 2 , and v 3 , respectively. Figure I displays the normal modes of N 2O computed with Molpro using the complete active space self-consistent field (CASSCF) method with CAS(12,7), i.e. 12 electrons are distributed in 7 orbitals, and the cc-pVTZ basis set. Computational details and the obtained data on frequencies and equilibrium geometry are provided in paper .6 (see Table1

Figure II. 15 :

 15 Figure II.15: Potential energy curves for the ground electronic state of N 2 O as a function of the (a) NO stretching, (b) bending and (c) NN stretching normal-mode coordinates. The abscissa axes in the figures represent dimensionless normal mode coordinates. In each panel, only one mode is varied, while the other modes are kept fixed at their equilibrium positions.Red solid curves are the actual potential energies obtained from the R-matrix code, while black dashed curves represent energies calculated in the harmonic approximation, i.e. simply ∼ ω i 2 q 2 i . Horizontal dashed lines denote the energies of vibrational states.

Figure II. 18 :

 18 Figure II.18: Comparison of the present theoretical results with available experimental cross sections for the vibrational v = 0 → v = 1 excitation of the (a) NO stretching, (b) bending, and (c) NN stretching modes. The experimental results are taken from Hayashi [Hayashi and Akashi, 1992, Song et al., 2019] (solid line with circles), Allan and Skalický [Allan and Skalický, 2003] (solid line with triangles), and Nakamura[Nakamura, 2007] (dashed-dotted line).

  al., 2006b, Douguet et al., 2008b, Douguet et al., 2009b, Douguet et al., 2011, Douguet et al., 2015b, Fonseca dos Santos et al., 2014a].

  al., 2017,Torigoe et al., 2016, Gonzatti et al., 2010]. Basically, BF 3 plasmas are used either for (i) the synthesis of ultra-hard boron compounds, e.g. boron carbides[Sennikov et al., 2017], (ii) the deposition of boron nitride, an advanced material with a large number of functionalities[Torigoe et al., 2016], and (iii) p-type doping by boron in the semi-conductor and photovoltaic industries[START_REF] Gonzatti | Plasma immersion ion implantation applied to n + p junction realisation in 4h-sic[END_REF], Duchaine et al., 2012]. As far as doping applications are concerned, plasma immersion ion implantation (PIII) processes are probably among the most promising in terms of cost and technical performance[Duchaine et al., 2012, Young et al., 2016]. These processes make use of very low pressure, very high density magnetized plasmas generated in a BF 3 -containing feed gas.Under such conditions, several ions as BF + 3 , BF + , B + or F -may be produced [M. Farber and Srivastava, 1984, Yong-Ki andIrikura, 2000] where the major one is BF + 2[Young et al., 2016]. In a typical PIII reactor, the positive ions losses are dominated by their dissociative recombination with electrons. Consequently, the investigation of DR of the major ion BF + 2 and the competitive processes such as the vibrational (de-)excitation are of great importance for monitoring these plasmas.The present study is motivated by the lack of data on e --BF + 2 collisions. The theoretical model described in next Section III.2 is a simplified approach of the model employed for neutral molecules in Chapter II. With certain assumptions, we computed cross-sections for DR and electron-impact vibrational excitation of the BF + 2 molecular ion in Section III.3. This study, performed in close collaboration with Prof. Viatcheslav Kokoouline at the University of 64 Chapter III. Formation of fluorine based-molecules

III. 3

 3 Figure III.1: Normal modes of BF + 2 . The arrows indicate the direction and magnitude of displacements for each mode. Note that arrowheads are not shown for the displacements of relatively large magnitudes. Vibrational frequencies and symmetries are listed inTable 1 of Ref.[Kokoouline et al., 2018].

Figure III. 2 : 2 69Figure

 22 Figure III.2: Eigenphase sums as a function of the electron-scattering energy for q i = 0.01 (dimensionless) for the symmetric stretching (left panel), bending (right upper panel), and asymmetric stretching (right bottom panel) modes. The curves of different color correspond to different symmetries of the e --BF + 2 system.

ω 1 Figure III. 3 :

 13 Figure III.3: Cross-section for the DR of BF + 2 . The lowest vibrational threshold of each normal mode is indicated by arrows.

Figure III. 4 :

 4 Figure III.4: Dissociative recombination and vibrational excitation of BF + 2 : rate coefficients -Eqs. (III.13) and (III.14).To give an idea about the uncertainty of the present results, we also plotted results of two calculations with the cc-pVTZ CAS 2 (dashed line) and the cc-pVQZ CAS 1 (dotted line) sets of parameters of the model. For the vibrational excitation of the symmetric stretching mode, the cc-pVTZ CAS 2 and cc-pVQZ CAS 1 curves are indistinguishable and slightly below the cc-pVTZ CAS 1 curve. For the remaining three processes, the three calculations produce the curves almost indistinguishable in the figure.

+ 4 or H 3 O

 3 + [Mikhailov et al., 2006b, Douguet et al., 2012b, Douguet et al., 2012a, Fonseca dos Santos et al., 2014a],

  Figure III.6 displays the employed probabilities in Fig.III.5. The averaged probabilities obtained from a complete treatment are in good agreement with the one of the simplified approach given by Eq.(III.6).

  elimination procedure (VdE): σ 0 <-1 (ε) Complete treatment without elimination procedure (VdE): σ 0 <-1 (ε) Complete treatment without elimination procedure (VE): σ 1<-0 (ε) Complete treatment with elimination procedure (VE): σ 1 <-0 (ε)

Figure III. 6 :

 6 Figure III.6: Vibrational (de-)excitation probabilities, used in Fig.III.5, for transitions between the two lowest vibrational levels of the bending mode of BF +2 . Solid lines represent the results obtained using the complete treatment in addition to the closed-channel elimination procedure in dashed lines. Dotted lines show the results from the simplified approach using Eq.(III.6).

- 1 .

 1 Note that p N/2 = p grid . Examples of functions ϕ i (x) with ∆x = 1 are shown in Figure IV.1 by solid lines.

Figure IV. 1 :

 1 Figure IV.1: FGH basis functions ϕ i (x) for N = 5. Solid and dashed lines represent the basis functions of Eq.(IV.6) for ∆x = 1 and ∆x = 1, respectively.

Figure IV. 2 :

 2 Figure IV.2: The general structure of the majority of the codes for modeling QM phenomena and demonstration of main QM concepts.

3

 3

  and IV.4 display two snapshots of the draft version of the web-interface. The snapshot (Fig.IV.3) of the front page of the preliminary version of the learning environment looks like a table of content of a book (in the spirit of published textbooks). Links to different chapters lead to web-pages of corresponding topics of the class. Figure IV.4 shows the logical structure of the interface of the learning environment as well as an example of in-class activities on the topic of the harmonic oscillator. In the figure, the following abbreviations are used. CGI: Computer-generated imagery. It is a computer graphics application to create images.

Figure IV. 3 :

 3 Figure IV.3: The front page of the draft version of the QM learning environment under development.

Figure IV. 5

 5 Figure IV.5 demonstrates a draft version of the interface for material on the time-dependent Schrödinger equation, for instance. Various functionalities are available in the web-interface for the other topics:

  Figure IV.4: Logical structure of the interface of the learning environment in the upper panel. Lower panel shows the preliminary graphical environment for in-class activities on the topic of the harmonic oscillator (one of the topics listed non-exhaustively in Section IV.2.C). The figure shows the potential and several boxes to modify parameters of the problem and observables to watch ("measure") in the computer-modeled experiment. The lower part of the interface, allowing to visualize wave functions and different expectation values of observables, is not shown here to save space (see the online version

Figure IV. 5 :

 5 Figure IV.5: The draft version of the interface for wave packet propagation and the timedependent Schrödinger equation.The students can choose one of the "standard" potentials (step-like, square well, etc.) to monitor the time-dependent evolution, or to use their own potential uploadable as a data file with values of the potential on a grid of points. Several observables can be monitored during the wave-packet propagation. Laboratory and homework assignments on the topic will be added later.

Figure V. 1 :

 1 Figure V.1: Schematic diagram summarizing the planned research program for the next 4-year period. Main tasks are listed with respect to their difficulties of realization.

Figure 2 .

 2 Figure 2.Wave functions ψ v (R) of the v = 0, 1, 2, and 8 vibrational states of the HeH + ion. The inset shows the potential energy curve V(R) of the X 2 Σ + electronic state of HeH + used in the calculation.

Figure 3 .

 3 Figure 3. Cross sections of vibrational (de-)excitation from the vibrational level v = 3 to several other levels v . The inset enlarges the region of Rydberg series of resonances close to the v = 4 and v = 5 ionization thresholds, situated at energy of 0.2465 and 0.4533 eV, respectively, above the v = 3 threshold.

Figure 6 .

 6 Figure 6. The figure demonstrates the quality of the fit of Equation (14): Numerical and fitted values are shown by solid and dashed lines respectively.

Figure 3 .

 3 Figure 3. Rate coefficients (solid lines) for transitions from the four lowest rotational states of HeH + , j = 0, • • • , 3 rotational states. Dotted lines in the upper left panel are the calculations by Hamilton et al.[10], and the dashed lines are those of Čurík and Greene[17]. For the 0 → 1 transition, the dashed and solid lines overlap. For the 0 → 2 transition, the curves for all three calculations overlap. For the 0 → 3 transitions, the dotted and solid lines overlap. For the 0 → 4 transitions, the dotted and dashed lines overlap.

FIG. 1 .

 1 FIG.1. Potential energy curves for the X 1 + (black curve), a 3 (red curve), A 1 (blue curve), b 3 -(green curve), and c 3 + (purple curve) electronic states of CH + . Four lowest vibrational levels for the four lowest electronic states are shown by horizontal thin lines in potential wells of the states. The inset displays the four vibrational states v = 0-3 of the X 1 + state. The CH 2 2 resonance state is plotted as thin dashed (violet) line. The CH 2 2 resonance energies are obtained in fixed-nuclei R-matrix calculations at R varying from 1.137 to 1.737 bohrs with an interval of 0.1 bohrs.

FIG. 2 .

 2 FIG.2. Derivatives of the eigenphase sum for three symmetries 2 + , 2 , and 2 -of e --CH + obtained for the equilibrium internuclear distance R e = 2.137 bohrs in two different calculations: Black solid curves show the results obtained taking into account only three lowest electronic states of CH + . Red dashed curves are obtained with 14 states. The differences between the curves are subtle and can hardly be seen in the figure.

FIG. 3 .

 3 FIG. 3. Comparison of eigenphase-sum derivatives below theA 1 state obtained at a fixed CH + geometry in (1) the direct Rmatrix calculations (red dashed curve) and using (2) the energyindependent 3 × 3 scattering matrix and the procedure of elimination of closed electronic channels (black solid curve). The inset shows an enlarged view for the 2.6-to 3.0-eV interval of energies.

FIG. 4 .

 4 FIG.4. Comparison of eigenphase-sum derivatives computed for a fixed CH + geometry using three different energy-independent 3 × 3 scattering matrices and the procedure of elimination of closed electronic channels. The three matrices are taken at energies 3.413 eV (black solid curve), 3.563 eV (red dashed curve), and 3.713 eV (green dotted curve). The inset shows an enlarged view for 2.1-to 2.4-eV energies.

3 ΠFIG. 5 .

 35 FIG.5. Cross sections for the X 1 + → a 3 electronic excitation of CH + at fixed geometry R e obtained in the direct R-matrix calculations (red dashed curve) and using the QDT channel elimination procedure (black solid curve).

10

 10

3 ΠFIG. 6 .

 36 FIG.6. The rate coefficients for the X 1 + → a 3 electronic excitation of CH + at R e obtained in the direct R-matrix calculations (red dashed curve) and the QDT channel elimination procedure (black solid curve).

10

 10

FIG. 1 .

 1 FIG.1. Potential energy curves for the ground electronic state of N 2 O as a function of the (a) NO stretching, (b) bending, and (c) NN stretching normal-mode coordinates. The abscissa axes represent dimensionless normal-mode coordinates. In each panel, only one mode is varied, while the other modes are kept fixed at their equilibrium positions. Solid red curves show the actual potential energies obtained from the R-matrix code, while dashed black curves represent energies calculated in the harmonic approximation, i.e., simply ∼ hω i 2 q 2 i . Dashed horizontal lines denote the energies of vibrational states.

FIG. 3 .

 3 FIG.3. Largest S-matrix elements (absolute values squared, |S l λ ,l λ| 2 ) computed at the N 2 O equilibrium geometry as a function of the scattering energy. Top: Couplings between channels with l = 0 in dashed curves. Bottom: Couplings between channels with l = 1 in solid curves.

FIG. 4 .

 4 FIG.4. Largest fixed-nuclei S-matrix elements as a function of the bending coordinate q 2 , computed for scattering energy 0.26 eV. The upper (lower) panel shows the real (imaginary) part of the S-matrix elements. Couplings l λ ← l λ between different partial waves, represented by real-valued harmonics Y l, λ, are labeled by curves of different colors.

FIG. 5 .

 5 FIG. 5. Comparison of the present theoretical results with the available experimental cross sections for the vibrational v = 0 → v = 1 excitation of the (a) NO stretching, (b) bending, and (c) NN stretching modes. The experimental results are taken from Hayashi[30,31] (solid line with circles), Allan and Skalický[21] (solid line with triangles), and Nakamura[22] (dashed-dotted line).

  from the reactance matrix K ˆ, obtained numerically as discussed below:

Figure 3 /

 3 Figure 3 illustrates the DR cross-section E DR el s ( ) computed based on equation (5). At very low scattering energies, i.e. below 0.02 eV, the DR cross-section is a smooth function inversely proportional to the incident energy of the electron, as predicted by the Wigner threshold law, whereas at higher energies, it exhibits a drop at each vibrational threshold. Due to the simple analytical form of the cross-sections, the corresponding rate coefficients are easily evaluated from the general expression T k T E E k T E E 8 2 exp d , 7 b b 3 2 0 el el el el

Figure 1 .

 1 Figure 1. Eigenphase sums as a function of the electron-scattering energy E el for q i =0.01 (dimensionless) for the symmetric stretching (left panel), bending (right upper panel), and asymmetric stretching (right bottom panel) modes. The curves of different colors correspond to different symmetries of the e --BF 2 + system.

Figure 2 .

 2 Figure 2. VE of BF 2 + : probabilities corresponding to the normal vibrational modes of the target ion.Figure 3. Cross-section for the DR of BF 2 + . The lowest vibrational

Figure 3 .

 3 Cross-section for the DR of BF 2 + . The lowest vibrational threshold of each normal mode is indicated by arrows.

Table I .

 I 2: Mapping of spherical harmonics of partial waves (lλ) in each irreducible representations Γ of the abelian point groups that are encountered in our studies.

		C 1 A
		00,1-1,10,11 2-2,2-1,20,21,22 3-3,3-2,3-1,30,31,32,33 4-4,4-3,4-2,4-1,40,41,42,43,44
	C s A			A
	00,1-1,11,2-2,20,22 3-3,3-1,31,33,4-4,4-2,40,42,44 3-2,30,32,4-3,4-1,41,43 10,2-1,21
	C 2v A 1		B 1	B 2		A 2
	00,10,20,22 30,32,40,42,44 31,33,41,43 3-3,3-1,4-3,4-1 3-2,4-4,4-2 11,21 1-1,2-1 2-2
	D 2h A g	B 1g	B 2g	B 3g	A u B 1u	B 2u	B 3u
	00,20,22 2-2 40,42,44 4-4,4-2 41,43 4-3,4-1 3-2 30,32 3-3,3-1 31,33 21 2-1 10 1-1 11

  , Moulane, Y. et al., 2018, Chakrabarti et al., 2018b, Douguet et al., 2012b, Mikhailov et al., 2006a, Douguet et al., 2008a, Douguet et al., 2009a, Kokoouline and Greene, 2001, Novotný et al., 2009, J. B. A. Mitchell and A. I. Florescu-Mitchell, 2005]. As an example, DR cross sections of NH + 2 and HCO + were computed by [Fonseca dos Santos

  This yields the position of Rydberg resonances (at first order if couplings between closed and open channels are neglected). According to the partition (I.19), the eignephase can be written as follows δ(E) = δ bg + δ r , (I.23)

(Er)

has a local minimum. (I.22) Chapter I. Mechanisms of destruction of molecular ions

  CHOH + that is largely influenced by the rate of dissociative recombination. Indeed, it has been shown that such molecules have a significant impact on chemical evolution in different astrophysical Chapter I. Mechanisms of destruction of molecular ions environments. In the following, I will briefly report our work on CH 2 NH + 2 carried out in the context of the dissertation of Chi Hong Yuen at UCF. Details on NH 2 CHOH + can be found in Ref.[Ayouz et al., 2019].Figure I.16: Normal modes of CH 2 NH + 2 . The arrows indicate the direction and magnitude of displacements for each mode. Note that arrowheads are not shown for the displacements of relatively large magnitudes. Vibrational frequencies and symmetries are listed in Table 2 of Ref.[Yuen et al., 2019]. Methanimine and protonated methanimine are also important in the upper atmosphere of Titan, the massive moon of Saturn. It was detected by the Ion and Neutral Mass Spectrometer (INMS) on-board Cassini orbiter [Vuitton et al., 2006]. Once formed, CH 2 NH + 2 can undergo DR and, indeed, this is the only destruction pathway of CH 2 NH + 2 considered in the two most popular databases of astrochemical networks: KIDA [Wakelam et al., 2012] and UMIST [McElroy et al., 2013]. We are yet to obtain experimental or theoretical data about the DR of CH 2 NH + 2 .The rate coefficient of this process was generally estimated using data on complex hydrocarbon ions. Consequently, a reliable value for the rate coefficient of the CH 2 NH + 2 DR is needed for chemical modeling.

					01
	00	10	20 30	40
	pπ
	sσ, dπ	pπ	dδ	pσ, dσ, dπ
	Figure I.15: Comparison of the DR cross section of the ground vibronic state of CH + . The computed total DR cross section in the present work (black thin dashed curve) and the convoluted results (black solid curve) are compared with the experimental measurements of Ref.[Amitay et al., 1996] (red solid curve), theoretical results Ref.[Takagi et al., 1991] (green solid curve), Carata Ref.[Carata et al., 2000] (blue solid curve) and Ref.[Chakrabarti et al., 2018a] (magenta solid curve), and rotationally resolved DR cross section obtained by Ref.[Mezei et al., 2019] (violet solid curve). Partial waves of the incident electron are assigned to the prominent resonances by arrows. Energies of ground vibrational threshold in each electronic state are labeled with arrows and the corresponding vibrionic quantum numbers vi. experimental data, the DR cross section, shown as a dashed curve, was averaged over the non-isotropic distribution throughout the formula of Eq.(2) of Ref.[Kokoouline and Greene, 2005], shown as a solid line. The parameter of the experimental energy distribution were picked from Ref.[Amitay et al., 1996]. This first result demonstrates a good agreement at high energy in contrast to the low-energy range. We attribute this discrepancy to the fact that the rotational structure in each vibrational level of CH + was neglected in the present study. Tests and improvements are ongoing to date. I.5.D Dissociative recombination of polyatomic molecular ions: CH 2 NH + 2 This section presents an application of the simplified approach, described in Chapter III, on a population of some prebiotic molecules such as CH 2 NH + 𝜔 2 𝜔 3 𝜔 4 𝜔 5 𝜔 6 𝜔 7 𝜔 8 𝜔 9 𝜔 10 𝜔 11 𝜔 12 2 and NH 2 𝜔 1 Protonated methanimine CH 2 NH + 2 acts as a precursor of simple amino acids. It is formed in ISM after proton transfer reaction with methanimine CH 2 NH 2 and HCO + , H + 3 or H 3 O + . Other CH 2 NH + 2 formation routes exist, such as the rapid reaction CH 2 with NH + 3 giving [Anicich, 1993].

Analogous to BF + 2 , CH 2 NH + 2 requires a highly-energetic incoming electron to form a doubly-excited resonant dissociative state of the neutral CH 2 NH 2 throughout the process

  2 NH + 2 , described in Chapter I, or NH 2 CHOH + investigated in Ref.[Ayouz et al., 2019]. Below, a brief description of the BF + 2 study is given. The ion BF + 2 requires a highly-energetic incoming electron to form a doubly-excited resonant dissociative state of the neutral BF 2 . Low energy electrons are more likely to be captured in a Rydberg level BF * 2 associated with a vibrationally excited state of BF + 2 . This Rydberg state BF * 2 is predissociated by BF * * 2 . In this section, we aim to study the IDR and VE/VdE depicted by

	III.3 Application: DR cross section of BF + 2
	e

  al., 2009, McKagan et al., 2010, Deslauriers and Wieman, 2011, Zhu and Singh, 2012a, Zhu and Singh, 2012b, Sadaghiani and Pollock, 2015] and proposed solutions on how to improve teaching QM [Müller and Wiesner, 2002, McKagan et al., 2008, Kohnle et al., 2015, Krijtenburg-Lewerissa et al., 2017].

  Cross sections for vibronic excitation and dissociative recombination of CH + by low-energy. Virtual RSVP Challenges for Plasma and Catalysis, October 17-18, Le Havre, France. Dissociative recombination and vibrational excitation of large molecules by electron impact. Invited The 50th Annual Meeting of the APS Division of Atomic, Molecular and Optical Physics APS Meeting, May 27-31, Milwaukee, Wisconsin, US. Cross sections for vibronic excitations of CH + by low-energy. Hainan LIU, Samantha Fonseca dos Santos, Chi Hong Yuen, Pietro Cortona, Viatcheslav Kokoouline and Mehdi Ayouz Journée Initiatives Pédagogiques de l'Université Paris-Saclay, JIP 2019, November 28, Institut d'Optique Graduate School (IOGS), France. Enseigner la physique quantique (EnsPhyQ) autrement: approche numérique Opensource International Conference of Assistant professors, October 25, Russian cultural center, Paris, France. Teaching quantum physics differently: an open source numerical approach. Invited +D -→ HD+H -H. C. H. Yuen, M. Ayouz, E. S. +D -→ HD+H -reaction at low energies. Chi Hong Yuen, Mehdi Ayouz, Viatcheslav Kokoouline The 11th International Conference on Zinc and Zinc Alloy Coated, November 12-18, Tokyo, Japan. Selective oxidation of Fe-Mn (1wt.%) binary alloy. N. Ruscassier, Li Gong, M. Ayouz, T. Reiss, P. Haghi-Ashtiani and M-L. Girogi GDR, Nancy, November 13, France. De l?expérimentation à l?échelle anatomique à la modélisation multi échelle des propriétés physiques du bois. Mehdi Ayouz and Giana Perré. Eurodrying'2013, October 4th, Paris, France. An original method for the determination of mass diffusion of hygroscopic materials in unsteady regime. F. Pierre, M. Olivier Dulieu, Mehdi Ayouz, Romain Guérout, Jacques Robert, and Viatcheslav Kokoouline Spectroscopy of Molecular Ions in the Laboratory and in (Extraterrestrial) Space, October 3-6 Kos, Greece. Theoretical study of the H - 3 anion; proposal for its observation in the interstellar medium. Mehdi Ayouz, Olivier Dulieu, Jacques Robert and Viatcheslav Kokoouline DPG Spring Meeting of the Section AMOP, "Deutsche Physikalische Gesellschaft", March 8-12, Hanover, Germany. http://hannover10.dpg-tagungen.de/index.html. Calculation of the structure of negative ion, and of its formation rate in the interstellar medium. Mehdi Ayouz, Olivier Dulieu, Jacques Robert and Viatcheslav Kokoouline The 10th European Conference on Atoms, Molecules and Photons, July 4-9, Salamanca, Spain. Formation of H - 3 by radiative association of H 2 and H -. Could H - Theoritical model for vibronic excitation and dissociative recombination of CH + by electron impact. Mehdi Ayouz, Xianwu Jiang, Chi Hong Yuen and Viatcheslav Kokoouline Processes in ATmospheric and AStrochemical environments, January 18-21, PATAS 2021 Virtual WG3 meeting. Electron induced reactivity of molecule cations: from mechanisms to new stateto-state cross sections and rate coefficients. J.-Zs. Mezei, E. Djuissi, A. Abdoulanziz, C. Argentin, F. Iacob, N. Pop, A. Orban, K. Chacrabarti, D. Talbi, M. Ayouz, V. Kokoouline and I.-F. Schneider. The 51st Annual Meeting of the APS Division of Atomic, Molecular and Optical Physics, June 1-5, Portland, Oregon. US. Theoretical study of vibrational (de-)excitation of NO 2 and N 2 O by low-energy electron impact. Hainan Liu, Mehdi Ayouz, Pietro Cortona, Samantha Fonseca dos Santos, Chi Hong Yuen and Viatcheslav Kokoouline Physique et Chimie du Milieu Interstellaire (PCMI) biennal conference, July 6-9, Le Havre, France. Cross sections for vibronic excitation of CH + by low-energy electron impact. Xianwu Jiang, C. H. Yuen, P. Cortona, V. Kokoouline, M. Ayouz The 50th Annual Meeting of the APS Division of Atomic, Molecular and Optical Physics, May 27-31, Milwaukee, Wisconsin, US. Dissociative recombination of CH 2 NH + 2 and NH 2 CH 2 O + ions. Viatcheslav Kokoouline, Chi Hong Yuen, Ioan Schneider, Cecilia Ceccarelli, Nadia Balucani and Mehdi Ayouz The 31st International Conference on Photonic, Electronic and Atomic Collisions, 23-30 July, Deauville, France. Theoretical study of vibrational excitation and dissociative electron attachment of NO 2 by an electron impact. Hainan Liu, Samantha Fonseca dos Santos, Chi Hong Yuen, Pietro Cortona, Viatcheslav Kokoouline and Mehdi Ayouz The 31st International Conference on Photonic, Electronic and Atomic Collisions, 23-30 July, Deauville, France. Model for vibronic excitation of CH + by resonant scattering of electron. Xianwu Jiang, Chi Hong Yuen, Pietro Cortona, Viatcheslav Kokoouline and Mehdi Ayouz Intl'Conf. on Processing and Manufacturing of Advanced Materials, July 9-13, Paris, France. Nucleation and Growth of MnO on Fe-Mn (1 wt.%) Binary Alloy. Li Gong, Ruscassier Nathalie, Ayouz Mehdi and Giorgi Marie-Laurence Colloque de Recherche Inter Écoles Centrale, June 8-12, Lyon, France. Modélisation et expérimentation à l?échelle microscopique de biomatériaux. Huan Du, Mehdi Adrien Ayouz, Pin Lv and Patrick Perré Colloque de Recherche Inter Écoles Centrale, June 8-12, Lyon, France. Calcul de propriétés des matériaux par méthodes discrète MLattice Boltzmann. M. Louërat, M. Ayouz and P. Perré October 4th, Paris, France. Numerical simulation of diffusion in porous media using the Lattice Boltzmann Method. Mehdi Ayouz and Patrick Perré The 42nd Annual Meeting of the APS Division of Atomic, Molecular and Optical Physics, June 13-17, Atlanta, Georgia, US. Bound states, resonances, and formation of the H - 3 anion. Viatcheslav Kokoouline, Mehdi Ayouz, Jacques Robert and Olivier Dulieu La Société Française de Physique, July 6-10, École polytechnique, Palaiseau, France Faraday Discussion, Cold and Ultracold Molecules, April 15-17, Durham University, U K.

	1 Publications -Scientific outreach The 42nd Annual Meeting of the APS Division of Atomic, Molec-5 ular and Optical Physics, June 13-17, Atlanta, Georgia, US. Processes in ATmospheric and AStrochemical environments, January 18-21, PATAS 7 2021 Virtual WG3 meeting. Publications -Scientific outreach Eurodrying'2013,
		http://meetings.aps.org/Meeting/DAMOP11/Event/147334.	Radiative associ-
		ation of H 2 and H -at low temperature : can we observe H -3 in the interstellar
		medium ?. Mehdi Ayouz, Olivier Dulieu, Maurice Raoult and Viatcheslav
		Kokoouline
	2010	Meeting of division of atomic, molecular and optical physics of the French Physics
		Society, (PAMO), June 29-July 2, Orsay, France. http://www.pamojsm2010.fr/. Étude théorique de la formation de l'ion moléculaire H -3 : proposition d'observation de H -dans le milieu interstellaire, V. Kokoouline, M. Ayouz, M. Raoult, J. Robert,
		O.Dulieu
		The 41st Annual Meeting of the APS Division of Atomic, Molec-
		ular	and	Optical	Physics,	May	25-29,	Houston,	Texas,	US.
		http://meetings.aps.org/Meeting/DAMOP10/Session/C2.5.	Potential energy
	2021 2009 2008 2020	Journée Initiatives Pédagogiques de l'Université Paris-Saclay, JIP 2020-2021, Febru-The 49th Annual APS Division of Atomic, Molecular, and Optical Physics, May 28-June 1, Fort Lauderdale, US. http://meetings.aps.org/Meeting/DAMOP18/Content/3507. Quantum tunnel-ing isotope exchange reaction H 2 Endres, O. Lakhamanskaya, R. Wester and V. Kokoouline The 48th Annual APS Division of Atomic, Molecular, and Optical Physics, June 5-9, Sacramento, California, US. http://meetings.aps.org/Meeting/DAMOP17/Content/3310. Theoretical study and dipole moment surfaces of H -3 molecule. Robert and O. Dulieu European Graduate College (EGC) on Cold atoms and molecules, June 29-July 02. Berlin, Germany. http://www.egc.uni-hannover.de/. Low energy dynamics of H -3 : a prototype for collisions between molecules and anions of the H 2 Ayouz and P. Perré European Graduate College (EGC) on Cold atoms and molecules, June 30-July ary 10, Institut d'Optique Graduate School (IOGS), France. Enseigner la physique quantique (EnsPhyQ) autrement: approche numérique Opensource ular and Optical Physics June 1-5, Portland, Oregon. US. polynomial function. Invited The 51st Annual Meeting of the APS Division of Atomic, Molec-International conference on ozone. October 3-11, Reims, France. Improved potential energy surface of ozone constructed using the fitting by permutationally invariant 03, Gif Sur Yvette, France. http://n.saquet.free.fr/EGC2008/Home.html. N -body systems dynamic: application at cold collision between atoms and molecules
	E	http://meetings.aps.org/Meeting/DAMOP20/Session/M05.8. Posters

3

be observed in the interstellar medium ?. V. Kokoouline, M. Ayouz, M. Raoult, J.

  Figure 1. Eigenphase sum as a function of the electron scattering energy E el (in eV) for several internuclear distances R (in units of bohr) for the 2 Σ + (left panel), 2 Π (right upper panel), and 2 ∆ (right bottom panel) symmetries. The color scheme used to label different values of R is the same for all three panels. The dashed line represents the equilibrium internuclear distance R e = 1.445 bohrs [15] of HeH + .

		2.1
			R=1.6
				R=1.7		R=1.6			0.08
		2	R=1.5	R e =1.455					0.06
	eigenphase (rad)	1.8 1.9	R=1.3 R=1.1 R=1.4 R=1.2	2 Σ	+	2 Π			R=1.1 R=1.5	0.02 0.04 -0.04
									R=1.2
		1.7							R=1.3	-0.06
									R=1.4
		1.6				2 ∆		R=1.7 R=1.5	-0.08
					0	1 2	3 4	5 6	7 8 9	-0.1
						electron scattering energy (eV)

Table 1 .

 1 The vibrational frequencies ∆ v = E v+1 -E v for transitions v → v + 1 and rotational constants B v obtained in the present study and compared with previous calculations. All values are in cm -1 .

	Level v	∆ v	∆ v [26]	B v	B v [25]
	0 1 2 3 4 5 6 7 8	2913 2911.0007 33.527 2607 2604.1676 30.816 2300 2295.5787 28.088 1988 1982.0562 25.305 1668 1660.3559 22.425 1339 1327.7860 19.394 1000 984.3599 16.151 660 639.1959 12.645 328 327.3615 8.813	33.558 30.839 28.090 25.301 22.402 19.344 16.058 12.479 8.621
	2.3. The Scattering Matrix for an Electron Colliding with HeH +

Table 2 .

 2 Parameters a 0 , a 1 , a 2 and a 3 of the polynomial P

Table 1 .

 1 Parameters a 0 , a 1 , and a 2 of the polynomial P

	f it jj (x) of Equations (

TABLE I .

 I Parameters a 0 , a 1 , and a 2 of the polynomial P fit vi,v i (x) of Eqs. (

  10 -6 1.60 × 10 -8 -6.60 × 10 -9 30 ↔ 00 11 299 2.90 × 10 -7 5.00 × 10 -8 -5.10 × 10 -9

	10 ↔ 00	3 934	2.90 × 10 -6 -1.20 × 10 -7 2.30 × 10 -9
	20 ↔ 00 1.10 × 20 ↔ 10 7 700 3 766 7.85 × 10 -7 2.17 × 10 -7 -1.22 × 10 -8
	30 ↔ 10	7 365	4.57 × 10 -7 1.02 × 10 -7 -8.64 × 10 -9
	30 ↔ 20	3 599	2.39 × 10

-6

2.51 × 10 -8 -5.26 × 10 -9

TABLE II .

 II Same as TableIfor the electronic state a 3 .

	v i ↔ vi v i ,vi (K)	a 0	a 1	a 2
	11 ↔ 01	3 633	5.43 × 10 -6 4.68 × 10 -7 -4.41 × 10 -8
	21 ↔ 01	7 039	2.01 × 10 -6 5.14 × 10 -9 -3.18 × 10 -9
	31 ↔ 01 10 216 1.17 × 10 -6 -3.37 × 10 -8 -2.65 × 10 -9
	21 ↔ 11	3 405	7.53 × 10 -6 6.95 × 10 -8 -2.30 × 10 -8
	31 ↔ 11	6 583	2.99 × 10 -6 3.58 × 10 -8 -2.48 × 10 -9
	31 ↔ 21	3 178	6.55 × 10 -6 2.04 × 10 -7 -8.40 × 10 -9

  1 + state are listed in Table VII. As one can see, the present computed energies agree well with the theoretical calculations by Biglari et al. [68].

TABLE III .

 III Same as Table I for the electronic state A 1 .

	v i ↔ vi v i ,vi (K)	a 0	a 1	a 2
	12 ↔ 02	2 290 5.44 × 10 -6 1.74 × 10 -7 -1.09 × 10 -8
	22 ↔ 02	4 271 2.16 × 10

-6 -1.24 × 10 -8 -6.08 × 10 -10 32 ↔ 02 5 971 1.49 × 10 -6 -8.82 × 10 -9 5.33 × 10 -10 22 ↔ 12 1 981 1.33 × 10 -5 -1.13 × 10 -7 -3.47 × 10 -10 32 ↔ 12 3 680 7.74 × 10 -6 8.39 × 10 -9 -8.35 × 10 -10 32 ↔ 22 1 699 3.48 × 10 -6 -1.17 × 10 -8 6.50 × 10 -10

TABLE IV .

 IV Same as Table I for vibronic transitions vX 1 +

TABLE V .

 V Same as TableIfor vibronic transitions vX1 +

TABLE VI .

 VI Same as TableIfor vibronic transitions va3

TABLE VII .

 VII Comparison of the four lowest vibrational energy levels (in eV) of the X 1 + state obtained in this study with the calculations by Biglari et al.[68].

	Ref

TABLE I .

 I Energies hω i (in eV) of N 2 O normal modes obtained in the present study and compared with experimental data from Ref.[38].

				hω i
	Mode (ν i )	Degeneracies	Symmetry	Experimental	Calculated
	NO stretching (ν 1 )	1	+	0.1610	0.1622
	Bending (ν 2 )	2		0.0739	0.0761
	NN stretching (ν 3 )	1	+	0.2830	0.2849

Table 1 .

 1 Vibrational frequencies (in cm -1 ) obtained in this study and compared with previous data available in literature.

	Mode Symmetry	Symmetric stretching g S +	Bending u P + and u P -	Asymmetric stretching u S +
		ω 1	ω 2	ω 3	Method	References
		1062.8	469.2	2146.1	CI/cc-pVTZ This work
		1023	443	2088	MP2	[14]
		1030	450		CI	[15]
				2 026.1±0.2	Exp.	[16]

Table 2 .

 2 Parameters of equations (8) and (9) calculated at E=0.1 eV collision energy.

	Mode i	P i
	symmetric stretching	0.053
	bending	0.20
	asymmetric stretching	0.05

The normalization should eliminate any energy dependence of the electron wave function amplitude arising from the normalization factor in Eq.(I.16).

Definition of the cross section is σ υ i ←υ i ( ) = π

2m |S υ i υ iδ υ i υ i | 2 .

Atoms 2016, 4, 30; doi:10.3390/atoms4040030 www.mdpi.com/journal/atoms

Atoms 2018, 6, 49; doi:10.3390/atoms6030049 www.mdpi.com/journal/atoms

Acknowledgments: This work was supported by the National Science Foundation, Grant No. PHY-15-06391. V.K. acknowledges also a support from the Austrian-American Educational Commission.

ACKNOWLEDGMENTS

This work acknowledges support from the National Science Foundation, Grant No. PHY-1806915, China Scholarship Council, and the Thomas Jefferson Fund of the Office for Science and Technology of the Embassy of France in the United States. It has also received funding from the program "Accueil des chercheurs étrangers" of CentraleSupélec and "Séjour à l'étranger 2019" of École doctorale INTERFACES of Université Paris-Saclay. ACKNOWLEDGMENTS This work was supported by the China Scholarship Council, the Thomas Jefferson Fund of the Office for Science and Technology of the Embassy of France in the United States, and National Science Foundation Grant No. PHY-1806915. It also received funding from the programs "Accueil des chercheurs étrangers" of CentraleSupélec and "Séjour à l'étranger 2019" of école doctorale INTERFACES of Université Paris-Saclay. Acknowledgments This work was supported by the National Science Foundation, Grant No PHY-1806915. Ioan F. Schneider and János Zsolt Mezei are grateful for generous financial support from La Région Haute-Normandie, via the GRR Electronique, Energie et Matériaux and the project BIOENGINE, from the Fédération de Recherche 'Energie, Propulsion, Environnement', and from the LabEx EMC 3 and FEDER via the projects PicoLIBS (ANR-10-LABEX-09-01), EMoPlaF and CO 2 -VIRIDIS. They acknowledge support from the CNRS via the GdR THEMS, IAEA (Vienna) via the Coordinated Research Project 'Light Element Atom, Molecule and Radical Behaviour in the Divertor and Edge Plasma Regions', the Programme National 'Physique et Chimie du Milieu Interstellaire' (PCMI) of CNRS/INSU with INC/INP co-funded by CEA and CNES, and Fédération de Recherche 'Fusion par Confinement Magnétique' (CNRS and CEA).

János Zsolt Mezei and Khalid Hassouni acknowledge support from USPC via ENUMPP and Labex SEAM.

Ioan F Schneider thanks Laboratoire Aimé Cotton for its outstanding hospitality.

Funding: This work was supported by the National Science Foundation, Grant No. PHY-15-06391 and the University Grant Commission (UGC), Government of India under Raman Post Doctoral Fellowship No. 5-159/2016 (IC).

Acknowledgments: V.K. acknowledges a support from the Austrian-American Educational Commission.

A

Author Contributions: Authors contributed equally to this work.

Conflicts of Interest:

The authors declare no conflict of interest.

Author Contributions: Methodology, M.A., M.K. and V.K.; Software, M.A., M.K. and J.S.; Validation, M.A., M.K., J.S. and V.K.; Investigation, M.A., M.K. and V.K.; Resources, V.K.; Writing-Original Draft Preparation, M.K., J.S. and V.K.; Writing-Review & Editing, M.A. and V.K.; Supervision, V.K.

Conflicts of Interest:

The authors declare no conflict of interest.

Appendix A

This section provides details on the formulas used for the cross section calculations. First, rotational and electronic states of the target ion are introduced as:

where j, m j , and µ are the angular momentum of HeH + and its projections in the laboratory frame (LF) and the molecular frame (MF), respectively. The function ω(r 2 ) specifies the two-electron wave function of HeH + , which depends on µ. D j m j µ (Ω) is the Wigner function depending on three Euler angles, which are called collectively by symbol Ω. The vibrational state of the target ion is not specified (i.e., pure rotational transitions are considered). We assume that the incident electron plane wave propagates along the z-axis in the LF with a wave vector of magnitude k j . The complete scattering wave function Ψ jµm j of the system in the asymptotic region is given by the sum of the contributions due to the pure Coulomb field ψ C and the short-range potential V sr , representing the difference between the actual e -+HeH + interaction and the Coulomb potential, as follows [34]:

where η = -1/(k j h2 ) is the Sommerfeld parameter. In the above expression, energetically open channels are labeled by the quantum numbers j and µ ; f j µ m j ←jµm j ( k j ) denotes the differential amplitude for scattering from state {j, µ, m j } to {j , µ, m j }. The amplitude includes only the contributions due to V sr . We assume that the incident wave in ψ C is a plane wave for large r, such that the incident current density is k j . Similar to Reference [30], channel functions with a definite total angular momentum J = l + j and its projection M = m l + m j in the LF are introduced:

where l and m l are the incident electron angular momentum and its projection in the LF. The scattering state of Equation (A1) takes the form

The pure Coulomb scattering wave function can be written in a similar form:

The difference between Equations (A4) and (A6) gives the last term in Equation (A2):

j µ l δ j j δ µ µ δ l l -S J j µ l ;jµl e iθ j (r) . (A7)

PAPER

Theoretical study of dissociative recombination and vibrational excitation of the ion by an electron impact