domain: chim phys

text:
Y Belkhodja

J Loreau

A Van Der Avoird

Y Berger

P Asselin

Y Belkhodja

Intermolecular dynamics of NH 3 -rare

come

Introduction

Over the three last decades, a large number of spectroscopic and theoretical studies of NH 3 -containing van der Waals (vdW) complexes have been undertaken, leading to a better understanding of weak interactions involving NH 3 . [1][2][3][4][5][6][7][8][9][10] Among them, binary complexes consisting of a rare gas (Rg) atom bound to NH 3 offer the possibility to build advanced theoretical models to describe such interactions since the Rg atom behaves as a structureless probe. However, past experimental and theoretical results highlighted that the internal rotation and inversion degrees of freedom of the NH 3 subunit within the complex make it quite challenging to characterize such systems. Nevertheless, NH 3 -Rg complexes are good prototypes to investigate in detail inter-and intramolecular dynamics. In particular, the inversiontunneling splitting in NH 3 represents a valuable probe of the asymmetry in the intermolecular potential energy surface (IPES) and in the vibration-rotation-tunneling (VRT) states of NH 3 complexes along the inversion coordinate. In previous high resolution spectroscopic studies of complexes with NH 3 binding to various other molecules [11][12][13] , it was shown that the complexation leads to a large asymmetry in the NH 3 inversion potential which quenches the tunneling. The NH 3 -Rg complexes form an exception to this case. NH 3 -Ar was the first vdW complex of the series to be extensively studied both in the microwave 3,4 and far-infrared 7 regions. Besides assigned pure rotational transitions, many other transitions due to the non-rigidity of the complex were observed and attributed to inversion-tunneling transitions, which showed that ammonia behaves as a nearly free rotor in the complex. [5][6][7][8][9] This large collection of results provided further evidence for the free-rotor picture from the precise knowledge of the bound states of the complex with approximate quantum numbers j and k that correlate to the j k = 0 0 ,1 0 and 1 1 levels of free NH 3 . Also, vibrational excitations in the vdW  s stretch coordinate were identified in these states. Early ab initio studies combined with these in-depth far-infrared measurements and considering the interplay of anisotropy in the well depth, Coriolis interactions, NH 3 inversion tunneling and degenerate NH 3 states led to a semi-empirical IPES. 10 Recently, more sophisticated IPESs have been developed to improve the understanding of spectroscopic data, such as the four-dimensional one computed by Loreau et al. 14 that includes explicitly the  2 umbrella motion of NH 3 and ND 3 , which will be used in the present study (see Section I). Further insight into the internal-rotation dynamics of the complex could be obtained from a direct comparison between the bound states of both NH 3 -Ar and ND 3 -Ar calculated for total angular momentum values up to J = 10 with the inclusion of Coriolis interactions, and all spectroscopic data collected in both the microwave and far-infrared regions for both complexes.

Advanced microwave studies of three other members of the NH 3 -Rg family were reported by Jäger et al [15][16][17] , including the study of several isotopologues (up to ten for the NH 3 -Kr complex) and the support of ab initio calculations. Ground state spectroscopic constants were used to derive molecular parameters from a pseudodiatomic model. The observation of two inversion components for each deuterated isotopologue provided evidence of inversion tunneling in the ground internal rotor state. As expected, the ab initio potential energy surfaces display deeper wells for Xe compared to Ne-, Ar-and Kr-, while the observed rotational spectra indicate nearly free internal rotation of NH 3 unit within the complex whatever the Rg atom.

Mid- 18,19 and near-infrared 20 data on NH 3 -Rg remain scarce due to the lack of broadband laser sources in both regions. Using a microwave-sideband CO 2 laser optothermal spectrometer, Fraser et al. 18 observed in the  2 inversion mode region an ortho  e/f (j=1,k=0) f (j=1,k=0) -type transition (where the subscripts 'e' and 'f' designate symmetric and antisymmetric NH 3 inversion wave functions, respectively) and derived spectroscopic constants containing information about the  2 dependence of the NH 3 interaction. Previously, Fraser et al. 3 had detected a laser coincidence at 938.6893 cm -1 assigned to the P(5) line of the para (j=1, k=1) (j=1, k=1) transition while Bizzari et al. 19 found two more located at about 32 and 36 cm -1 above this frequency. These results were taken as further evidence that the NH 3 inversion frequency is essentially unchanged upon complexation, since the tunneling splitting in the v 2 =1 state of free NH 3 is about 36 cm -1 .

More recently, high resolution infrared laser jet-cooled measurements were realized in the  2 umbrella mode region of NH 3 -Ar using a pulsed supersonic jet coupled to a laser spectrometer (named SPIRALES) implemented at the MONARIS laboratory. 21 Two bands in ortho-NH 3 -Ar and three bands in para-NH 3 -Ar were unambiguously assigned in the 936-975 cm -1 region based on a rovibrational analysis for the ortho ones, and by comparison with ab initio VRT levels from the 4D IPES of Loreau et al. 14 for the para ones. The ab initio results for both the ortho and para species turned out to be very predictive in terms of band centers, upper state rotational constants, and relative weight of Coriolis couplings in the ground and excited  2 states.

The present study aims to investigate more in detail the intense ortho bands of the NH 3 -Rg family with Rg = Ne, Ar, Kr, Xe using the SPIRALES set-up taking advantage of recent improvements of the set-up's performances in terms of line widths and absorption. The upper vibrational states investigated, above 900 cm -1 , are well above the dissociation limits inferred from quantum dynamics calculations on ab initio potentials. This raises the interesting issue of investigating vibrational predissociation dynamics within the NH 3 -Rg series to prove the existence of ultralong lived vibrational Feshbach resonances in atom-molecule systems. Unfortunately, our experimental line widths are dominated by residual static and dynamic Doppler contributions at the slit jet temperature, and are on the order of about 60 MHz, which precludes a vibrational dynamics study. In the following, we will treat the excited vibrational states of NH 3 -Rg as quasi-bound states. This assumption is justified by the study of Fraser et al. 18 who measured linewidths up to 3 MHz (compared to 0.5 MHz for free NH 3 ) for many transitions of the  2 inversion mode of NH 3 -Ar, which establishes a lower bound to the vibrational predissociation lifetime of about 50 ns. This result might be compared to measurements on ground state NH 3 -Ar in the farinfrared range, where linewidths as low as 600 KHz were obtained from hyperfine analyses, 6 which confirms the predissociation broadening of the  2 excited levels.

On the other hand, the substantial gain in absorption in our experiments makes it possible to derive accurate excited state rotational and quartic distortion parameters from rovibrational analyses of the ortho  e/f (j=1,k=0) f (j=1,k=0) transition as well as the doubling constant q which contains indirect information about the relative positions of the  e/f states of different parity, and about the position of the neighboring  f state perturbing the  f state. The precise position of the  f state could be directly determined from the analysis of the ortho  f (j=1,k=0) f (j=0,k=0) transition. Using a pseudodiatomic model, molecular parameters of the NH 3 -Rg family are estimated from the set of experimental spectroscopic constants, which enabled us to follow their evolution with the polarizability of the Rg atom. These experimental molecular parameters are compared with those derived from ab initio VRT levels, transition frequencies and line strengths based on 4D IPESs that explicitly include the NH 3 umbrella inversion motion. The predictive character of the ab initio VRT levels enabled us to explain the unexpected molecular parameter values for NH 3 -Xe, which do not show the good agreement between spectroscopic constants and derived molecular parameters obtained for the three other NH 3 -Rg complexes. Based on the energy difference between both the  e/f (j=1,k=0) states and the  f (j=1,k=0) states of NH 3 -Rg complexes, the possibility of a perturbation has been examined.

I Theory: rovibrational state calculations

The bound states of the NH 3 -Rg complexes have been computed on the global 4D IPES reported in Refs. [14,22], for the most abundant isotope of neon and argon and for the standard atomic weight in the case of krypton and xenon. The IPES depends on four coordinates: the length R of the vector R pointing from the center of mass of the ammonia molecule to the rare gas atom, two spherical angles  and  that define the orientation of this vector R, and the NH 3 umbrella inversion coordinate .The N-H bond length r was fixed at its vibrationally averaged value, r 0 = 1.9204 a 0 derived from a submillimeter and far-infrared spectroscopic study of the ground state of NH 3 23 . The IPESs were constructed through calculations with the coupled-cluster method with single, double, and perturbative triple excitations [CCSD(T)] with the aug-cc-pVQZ basis set for all atoms, supplemented by a set of mid-bond functions. In addition, the 10 and 28 core electrons of Kr and Xe, respectively, were described by relativistic pseudopotentials. 24 The IPESs were fitted based on 10472 unique geometries for NH 3 -Ne, 6820 points for NH 3 -Ar and NH 3 -Kr, and 6160 points for NH 3 -Xe. The IPESs show similar characteristics along the rare gas series; further details of the calculation of the IPESs and their features can be found in Ref. [22]. The minimum of the IPES always occurs for  = 60°, i.e., when the rare gas atom is located in the plane that contains the C 3 axis and passes midway between the two hydrogen atoms.

The equilibrium value of  decreases from 87.5° for NH 3 -Ne to 79.9° for NH 3 -Xe, while the equilibrium distance R e increases from 6.23 a 0 for NH 3 -Ne to 7.35 a 0 for NH 3 -Xe. There are three saddle points at  = 0° (with values of  decreasing from 77.2° for Ne to 46.3° for Xe),  = 0°, and  = 180°; the corresponding energy barriers hinder the rotation of NH 3 in the complex. The accuracy of the IPES has already been established for NH 3 -Ar by comparison with spectroscopic 14,21 and scattering experiments 25,26 , while for Ne, Kr, and Xe the present work represents the first test of the IPESs. The calculation of the bound states of the complexes with NH 3 either in the  2 = 0 state or in the  2 = 1 state has been performed following the approach detailed in our previous work. 21 Briefly, the Hamiltonian of the complex is (1) where is the reduced mass of the system and is the Hamiltonian of the NH 3 monomer that depends explicitly on the umbrella coordinate . We obtained the bound states energies and wave functions of the complex by means of a variational approach 27 with the basis (2) where and are the rotational quantum numbers of NH 3 , is the total angular momentum, is its projection on the intermolecular bond axis, is its projection on the space-fixed z-axis, denotes the Wigner functions, and are the overall rotation angles of the complex.

are eigenfunctions of the umbrella motion in free NH 3 , with the superscript labeling the symmetric and antisymmetric umbrella tunneling components of each v. In this work we considered the first four wave functions corresponding to  2 = 0 ( and ) and  2 = 1 ( and ), which are obtained by solving the one-dimensional Schrödinger equation for the umbrella motion. 28 Finally, is a set of radial basis functions. These functions are obtained using an effective radial potential of the form V eff (R) = [V 0 (R) + R], where V 0 (R) is a radial cut through the minimum of the 4D PES. The parameters and  were optimized by minimizing the energy of the lowest rovibrational state of the complexes. The optimized values are given in Table 1. The basis contained angular functions with j max = 10 and 20 radial functions calculated on an equidistant radial grid of n R points between R min and R max . The grid parameters are also given in Table 1. For each irreducible representation of the full molecular symmetry group D 3h (M), we calculated the lowest 20 eigenvalues of the Hamiltonian for J ranging from 0 to 10 by means of the iterative Davidson algorithm. 29 As in Ref. [21], the intensities of all allowed transitions between the VRT states were calculated for each complex in order to generate a theoretical spectrum corresponding to the  2 = 0 → 1 transition. The line strengths of the transitions were computed from the wave functions of the bound states and the dipole moment function corresponding to the  2 = 0 → 1 transition in the complex. The spectra are obtained from the line strengths assuming a Boltzmann distribution of the lower states.

The ground state of all ortho-NH 3 -Rg complexes is a  state with the approximate quantum number = 0, with dissociation energies D 0 that range from 30.35 cm -1 for NH 3 -Ne to 144.36 cm -1 for NH 3 -Xe. In Table 2 these values of D 0 are compared with the binding energies D e . We observe that the zero-point energy accounts for about 55% of the binding energy for NH 3 -Ne, while this decreases along the rare gas series to 27% for NH 3 -Xe. By comparison with the results presented in Table II of Ref. [22] we observe that the energy of the ground state of all the complexes is above the two saddle points of the PES that appear respectively at = 0° (and values of that depend on the Rg atom) and = 0°, while it is above the saddle point at = 180° only for NH 3 -Ne.

II-Experimental

The rovibrational spectra recorded in the region of the  2 umbrella mode of NH 3 -Rg (Rg = Ne, Ar, Kr, Xe) have been obtained using the SPIRALES set-up, a jetcooled laser spectrometer which couples an external-cavity quantum cascade laser (EC-QCL) and a pulsed free supersonic jet to probe by direct absorption the molecular complexes cooled down in the adiabatic expansion. SPIRALES has already been described in detail in recent papers 30,31 and only the main characteristics and most recent developments will be presented hereafter.

The light source is a continuous-wave room-temperature mode hop-free EC-QCL with a spectral width of about 10 MHz, which covers the 930-990 cm -1 range (Daylight Solutions). Less than 10% of the total light is sent through two additional laser channels for relative and absolute frequency calibrations reaching a precision of about 0.0005 cm -1 , while the main part is sent through a multipass optical cavity mounted in the jet chamber and composed of two 1.5" astigmatic mirrors (R = 99.2%, AMAC-36, Aerodyne Research) aligned according to a 182-pass configuration.

The molecular jet is produced from a pulsed nozzle (General Valve Series 9) by fitting a 0.9 mm diameter pin hole nozzle with two modified industrial blades, forming a 30 mm length x 150 µm width slit opening, and serving as the molecular expansion source. With respect to the first version of pulsed slit nozzle 21 two major changes were made: (i) Due to the very short distance (~1 mm) between the pin hole and the blades forming the slit, a strongly inhomogeneous distribution of the gas between the center and both ends of the slit is expected leading to a nearly axially symmetric behavior and consequently to residual Doppler line broadenings and reduced absorption pass. To overcome this a six-way distribution gas channel already used by the Calgary group 32 is implemented between the pinhole and the slit zones. Experimental full-width-at-half-maximum (FWHM) line widths of rovibrational lines measured in the present work reach typically 40 MHz, (ii) Owing to the delicate parallelism adjustment of blades initially spaced by 50 µm, the slit width is enlarged by a factor 3 to minimize width change effects. With this new slit version, the gain in intensity in similar experimental conditions is a factor of 5.

To reach a better precision of spectroscopic parameters derived from the rovibrational analyses for all NH 3 -Rg complexes, the dilution conditions of the ternary mixture NH 3 /Rg/He have been systematically explored to get the highest absorption signal. Compared to the previous NH 3 -Ar study, larger Rg/NH 3 ratios and higher total backing pressures up to 8 bar have been used. Rovibrationally cooled complexes were probed by the infrared laser light at axial distances not larger than 3 mm from the nozzle exit to keep a sufficient density of van der Waals complexes to be probed in the slit jet expansion.

III-Jet-cooled laser spectra results

In the following, we present two series of rovibrational bands of ortho NH 3 -Rg observed in the region of the  2 = 1  0 NH 3 monomer band. Both transitions involve the same initial level in the ground state, i.e.  2 = 0, =0 ( state) and two different levels in the  2 = 1 excited state, = 0 ( state) and = 1 ( state). In the following, these transitions will be labeled  e/f (j=1,k=0) f (j=0,k=0) and  f (j=1,k=0)  f (j=0,k=0). Guided by the results of previous infrared experiments on NH 3 -Ar, gas mixture and backing pressure conditions could be optimized on the intense ortho  e/f (j=1,k=0) f (j=0,k=0) transition of this complex before recording the much weaker bands such as for NH 3 -Ne and NH 3 -Xe. In spite of the use of large helium backing pressures to observe the ortho bands of NH 3 -Rg, no absorption signal of NH 3 -He (expected around 950 cm -1 for the  e/f (j=1,k=0) f (j=0,k=0) transition) could be detected, probably due to the very small binding energy of this complex.

Fig. 1 displays the four EC-QCL jet-cooled spectra of NH 3 -Rg for Rg = Ne, Ar, Kr, and Xe observed in the 947-957 cm -1 range. The spectra are locally blurred around 948.222 and 951.778 cm -1 by intense absorptions of the NH 3 monomer but no overlap has been observed with rovibrational lines of the vdW complexes. The rovibrational band contour is typical of a -type vibrational band of a linear molecule with characteristic features such as a strong Q branch, the presence of a R(0) line and the absence of a P(1) line evidenced by the simulations shown below. As seen in Fig. 1, the Q branch is fully resolved in J for each of the NH 3 -Rg complexes, as a consequence of the improved gas homogeneity provided by the multichannel distribution along the slit nozzle which strongly reduces the Doppler broadening. The spectrum of NH 3 -Kr in natural abundance, as used in the present study, displays a characteristic P(J) and R(J) line structure which evidences the presence of four main isotopes: 82 Kr (11.3%), 83 Kr (11.5%), 84 Kr(57.0%) and 86 Kr (17.3%) (see a zoom of the P(7) line of the NH 3 -Kr spectrum in Fig. 1). In the following, our rovibrational analyses will be focused on the dominant isotope of each rare gas while those concerning the different Kr isotopes are presented in the ESI.

It is noteworthy that P(J), Q(J) and R(J) lines with high values of J (up to 15 for Ne, larger than 24 for Ar, Kr, Xe) are still sufficiently intense to be observed, which means that the rotational temperature of the vdW complexes is higher than in our previous study. Finally, Fig. 1 (left) clearly illustrates that the asymmetry of the Q branch changes notably with the Rg atom, first degraded to the high-frequency side for Ne and then more and more degraded to the low-frequency side, from Ar to Xe.

For the ortho  e/f (j=1,k=0) f (j=0,k=0) transition, we should consider the Coriolis coupling between  f (j=1,k=0) and  f (j=1,k=0), two states spaced by about 10 cm -1 with the same j and J and the same parity f, but with different K values. The doubling constant q, which represents the energy splitting

 
 between the unperturbed e and perturbed f components of the  state, occurs in the energy expression of the P, Q and R transitions

 (3)
where  , , , , and are fit parameters for the excited state. The + sign is taken for the upper state in the Q-branch transition and thesign is taken for the upper state in the P-and R-branch transitions of the K-doubled pair in the excited (j=1,k=0) state. This expression has been applied in the rovibrational analysis of all transitions in ortho NH 3 -Rg, but it is not applicable to transitions in para NH 3 -Rg, due to the larger density of interacting states. 21 We use a model dealing with the interacting  f and  f excited states in which the energy spacing   enabled us to evaluate the strength of the Coriolis coupling. When   the density of interacting states is considered to be negligible.

To check the reliability of our jet-cooled EC-QCL spectra we start the rovibrational analysis with the ortho  e/f (j=1,k=0) f (j=0,k=0) transition of NH 3 -Ar which can be compared with the study of Fraser et al. 18 who used a sub-Doppler Lamb-dip technique and the recent study by our group with the same but a less advanced SPIRALES setup as in the present study. 21 All the simulations of the  e:f (j=1,k=0) and  f (j=1,k=0) states were performed with the PGOPHER software which contains the spectroscopic parameters derived from the experimental analyses. For all NH 3 -Rg complexes we estimate that the accuracy of the ground state parameters ( , ) derived from jet-cooled microwave experiments [15][16][17] is sufficiently high to keep them fixed in the simulations, which enabled us to better determine the eight adjustable excited state parameters. Table 3 lists the excited state molecular parameters derived for the  e/f (j=1,k=0) state from the simulation of the rovibrational band contours of four NH 3 -Rg complexes, as well as the parameter values extracted from a fit to the VRT levels that we calculated for total angular momentum values up to J = 8. Table S1 (ESI) lists the experimental and ab initio calculated (up to J = 10) transitions for the two bands of four ortho NH 3 -Rg complexes. Table S2 lists the experimental transitions for the three less abundant NH 3 -Kr isotopologues with 82 Kr, 83 Kr and 86 Kr. Table S3 contains the corresponding excited state molecular parameters for the  e/f (j=1,k=0) state. Fig. 2 displays the comparison between jet-cooled laser and simulated spectra of the  e/f (j=1,k=0) f (j=0,k=0) transition for the series of NH 3 -Rg complexes. In all cases, the asymmetry of the Q branch is fairly reproduced to the low frequency side for Ar, Kr and Xe, indicating that the effective rotational constant in the excited state, , is smaller than in the ground state, while the opposite holds for the Ne complex.

The rotational temperatures (T R ) of NH 3 -Rg derived from the simulations suggest several comments: first, the evolution of T R (from 6.5 K for Ne to 15 K for Xe) as a function of the seeded rare gas agrees well with the thermodynamic properties of the supersonic expansion, namely that the translational temperature varies inversely with the average mass of the gas mixture. This dependence is transferable to the rotational temperature because of similar translational and rotational relaxation achieved in the high collision regime of pulsed jets with large backing pressures. Secondly, the much higher T R value obtained from the NH 3 -Ar spectrum with respect to our first study (10 K versus 3 K) is mainly due to the planar expansion behavior of the multichannel slit nozzle. The increase of the slit width probably also affects the warming of the pulsed jet, but we did not investigate further to what extent.

The jet-cooled spectra of the ortho  f (j=1,k=0)  f (j=0,k=0) transition of NH 3 -Ne, -Ar and -Kr provide reliable information about the location of the  f (j=1,k=0) neighboring state perturbing the  e/f (j=1,k=0) state. Fig. 3 displays the comparison between jet-cooled laser and simulated spectra of the ortho  f (j=1,k=0) f (j=0,k=0) transition for the three lightest NH 3 -Rg complexes. The excited state parameters of the  f (j=1,k=0) state are collected in Table 3. The band center of NH 3 -Xe that could not be detected is estimated from the available frequency shifts between observed and ab initio calculated frequencies. From the list of observed and ab initio frequencies reported in Table S1, frequency shifts equal to -0.283(3), +0.044(8), +0.460(8) and +1.364(21) cm -1 from Ne to Xe should be applied to the ab initio frequencies to match the observed values for the  e/f (j=1,k=0) f (j=0,k=0) transition. These shifts amount to -0.323(20), -0.723( 14) and -0.226(12) cm -1 from Ne to Kr for the  f (j=1,k=0)  f (j=0,k=0) transition. Taking into account that the variation of the frequency shifts of the three lightest NH 3 -Rg complexes between both transitions ranged between -0.04 (Ne) and -0.769 cm -1 (Kr) , we assumed that the experimental band center of the ortho  f (j=1,k=0)  f (j=0,k=0) transition of NH 3 -Xe is shifted from the ab initio one by a value ranging from 0.60 to 1.32 cm -1 (Table 3). These small shifts probably originate from the 4D PESs: in the case of NH 3 -Ar in the ground state, a difference of 0.01-0.5 cm -1 between the origins of experimental and theoretical bands was observed. 14 Also the lines corresponding to the transition frequencies and line strengths obtained from the ab initio calculated energy levels and wave functions up to J = 8 are shown in Figs. 2 and3. The highly predictive level of the ab initio VRT calculations is illustrated in Table 4 which collects measured and ab initio band origins and e/f splittings of the  e/f (j=1,k=0) levels for both ortho bands of NH 3 -Rg complexes. The agreement between experiment and theory is excellent with average band origins shifts of 0.5(4) cm -1 and very small deviations of the values between 3 and 7 % for NH 3 -Ne, -Ar and -Kr.

IV-Discussion

Validity of the pure-precession model

By assuming that  f (j=1,k=0) is the only state Coriolis perturbing the  e/f (j=1,k=0) state we can use the pure-precession model of van Vleck 33 to estimate the energy difference   to be . Its positive sign confirms that the  f (j=1,k=0) level is above the  e/f (j=1,k=0) level. The NH 3 -Ar complex already investigated in our previous study has demonstrated that the pure-precession model overestimates the value of q by about 17 %. 21 Spectroscopic constants derived from rovibrational analyses of the NH 3 -Rg series (Table 3) have been exploited to evaluate the error level of the pureprecession model from the comparison between the energy difference of interacting  f (j=1,k=0) and  f (j=1,k=0) states measured experimentally and calculated from the relation of van Vleck. 33 The pure-precession model systematically overestimates the energy difference by 8%, 17% and 25 % for NH 3 -Ne, NH 3 -Ar and NH 3 -Kr, and in the 6-16 % range for NH 3 -Xe by extrapolating the ab initio band center of the  f (j=1,k=0)  f (j=0,k=0) transition as explained in Section III.

Comparison between experimental and theoretical structural parameters

From the spectroscopic constants that we obtained from fits of the spectra, as well as from fits of the ab initio calculated VRT levels, we extracted a set of molecular parameters with the use of a pseudodiatomic model. First, we defined effective rotational and distortion constants for the upper state, namely and -. Then we used the following pseudodiatomic relationships [START_REF] Gordy | Microwave Molecular spectra[END_REF] ,  and  , where is the reduced mass, to obtain an effective R-value , a force constant and a vdW stretching frequency  for each isotopologue. Table 5 compares the molecular parameters of the NH 3 -Rg complexes, all derived from the pseudodiatomic model : in the ground state from microwave measurements 15,16,17 , in both the  e/f (j=1,k=0) and  f (j=1,k=0) states from the present experimental study and from ab initio calculated VRT levels. 14 Table S3 displays the very small variations of and  (about 0.1 %) between the different NH 3 -Kr isotopologues.

In the ortho v 2 =1,  e/f (j=1,k=0) state, the same tendencies are observed for the NH 3 -Rg complexes as in the lowest v 2 =0, f (j=0,k=0) state: from Ne to Xe, the experimental vdW bond length increases by 9% in the  f (j=0,k=0) ground state and by 11% in the  e/f (j=1,k=0) state, the force constant increases by 400% and 200%, while the experimental vdW stretching frequency  increases by 57% and 21 % (87% and 61% from the ab initio levels) in both ground and  2 states. These results clearly show that the vdW bond is more rigid when the polarizability of the Rg atom increases. However, the values of  and derived experimentally for the  e/f (j=1,k=0) state of NH 3 -Xe are surprisingly lower than for the NH 3 -Ar and -Kr complexes. Moreover, the experimental value of  is lower than the ab initio value, while very good agreement (~2.5-7 %) between experiment and theory is obtained for the three other NH 3 -Rg complexes. Lastly, the ab initio calculations display the same frequency variation in the v 2 =1 state as in the ground state, which suggests an anomaly in the results derived from the NH 3 -Xe spectra.

Two possible causes are examined hereafter: -the small number of P(J) lines (only 2) assigned in the band contour fitting of the  e/f (j=1,k=0) spectrum. To evaluate the consequences of a rovibrational analysis with only two rotational branches, the NH 3 -Ne, NH 3 -Ar and NH 3 -Kr jet-cooled spectra have been reanalyzed without the P(J) lines. The set of molecular parameters thus obtained is very close to the values reported in Table 3, which discards this hypothesis.

-an unreliable determination of the energy splitting  . The RMS error obtained for the NH 3 -Xe spectrum analysis (two times larger compared to other NH 3 -Rg spectra) supports this assumption. Table S4 reports, for the four NH 3 -Rg complexes the energy splitting  extracted for each rotational level (up to J = 8) from the ab initio calculated VRT states. As already mentioned above, the differences between the experimental q values derived from the fits and the values of the ab initio e/f energy differences do not exceed 7% for the three lightest NH 3 -Rg complexes but are as high as 23% in the case of NH 3 -Xe. This error in the determination of q is partly compensated by the distortion constant, which causes a notable error in this parameter: the comparison between experimental and ab initio values (Table 3)

shows differences ranging from 6 to 17% from Ne to Kr and higher than 100% for Xe, which confirms our assumption. Therefore, it is not surprising that the vdW stretching frequency of NH 3 -Xe is too large when the effective distortion constant is underestimated by a factor close to 2. In conclusion, the fit of of the NH 3 -Xe spectrum seems not to be predictive for the f perturbed state. An additional argument is provided by analyzing only the Q(J) lines of the ortho  e/f (j=1,k=0) f (j=0,k=0) transition to derive molecular constants and structural parameters for the unperturbed e symmetry component of the  e (j=1,k=0) state (Table 6). The rotational constants derived from the fit of the Q(J) lines are very close (~0.1%) to the effective values reported in Table 3 for the whole series of NH 3 -Rg complexes. The difference between the distortion constants and the effective values varies between 5 and 20% for Rg=Ne, Ar, Kr but reaches 90 % for Xe. Structural parameters obtained for the NH 3 -Rg series in the  e (j=1,k=0) state are consistent with those in the ground state, namely similar values of the stretching vdW frequency and an increase of  with the polarizability of the Rg atom. These results clearly show that the discrepancy originates from the perturbed  f state. Finally, the model of weak Coriolis coupling which determines the rotational energy in the excited vibrational states through Eqn (3) can be used to evaluate the doubling constant q as a function of experimental transition frequencies of the R and Q branches (by neglecting distortion) and of the J rotational quantum number : (4) While the   ab initio energy differences reported in Table S4 vary very little as J increases for all NH 3 -Rg complexes, Fig. 4 compares the evolution of the ab initio q value and that derived experimentally from Eqn (4) as a function of J. For NH 3 -Ne, NH 3 -Ar and NH 3 -Kr complexes, both   and the experimental q values are found to be nearly independent of J (relative variations not larger than 1-3% for NH 3 -Ne), showing that the quasi-constant behavior of q corresponding to the weak Coriolis coupling case is well reproduced. On the other hand, the experimental value of q for NH 3 -Xe increases by about 10 % for J=1-12 and differs from the ab initio one by 26 % in average.

All these additional analyses support the hypothesis that the Coriolis perturbations involving the  f state are so strong in the case of NH 3 -Xe that the weak Coriolis coupling model is unsuitable. Given that the rotational study of Wen et al. 17 clearly proved that there was no sign of a tunneling component and a Coriolis perturbation for the lowest ortho state  f (j=0,k=0) of NH 3 -Xe, the most probable explanation is the presence of Coriolis coupling between the v 2 =1,  f (j=1,k=0) state and another vibrational state. The energy difference between the  f (j=1,k=0) and  f (j=1,k=0) states of NH 3 -Xe, predicted to be around 7 cm -1 is very close to that of NH 3 -Ne (6.991 cm -1 ) for which the weak coupling model holds. Consequently, we have no proof of a Coriolis perturbation involving the  f (j=1,k=0) state.

Conclusion

The present work reports a high resolution IR spectroscopic study of NH 3 -Rg (Rg= Ne, Ar, Kr, Xe) heterodimers in the region of the umbrella  2 band of the NH 3 monomer. Recent instrumental improvements on our infrared tunable quantum cascade laser spectrometer coupled to a pulsed supersonic jet enabled us to largely increase the collection of spectroscopic data derived from a preliminary study of NH 3 -Ar, by extending it to the whole rare gas series. Thus we could investigate the intermolecular dynamics from both ortho  e/f (j=1,k=0) f (j=1,k=0) and  f (j=1,k=0) f (j=0,k=0) transitions with the aid of a pseudodiatomic model applied to a weak Coriolis coupling case.

Accurate vdW bond lengths , force constants , and intermolecular stretching frequencies  in the  2 state have been derived from rovibrational analyses for NH 3 -Ne, -Ar and -Kr. They compare well with ground state structural parameters and with ab initio calculations based on a 4D IPES that explicitly takes into account the umbrella motion of NH 3 . However, strong deviations are observed for  et in the  e/f (j=1,k=0) state in the case of NH 3 -Xe, which suggests that the applied model is not predictive in this case. The presence of another state coupling to the perturbed  f state has been suggested, but no evidence has been found that weak Coriolis coupling between  f and  f states could be responsible for these anomalous NH 3 -Xe results.

Acknowlegments

This work was supported by the MICHEM Labex at Sorbonne Université. 3: Spectroscopic constants (expressed in MHz, except for the band centers in cm -1 ) for the transitions v 2 =10  e/f (j=1,k=0)  f (j,k=0) and  f (j=1,k=0)  f (j,k=0) bands of ortho NH 3 -Rg with Rg =Ne, Ar, Kr and Xe. The molecular parameters of the  f (j,k=0) state are given by Nelson et al. 2 for NH 3 -Ar, by Van Wijngaarden et al 15,16 for NH 3 -Ne and NH 3 -Kr and by Wen et al. 17 for NH 3 -Xe. The parameters "calculated" in italics are obtained from fits to the VRT levels calculated for J values up to 10. The observed band center of the  f (j=1,k=0)  f (j,k=0) transition in NH Table 4: Comparison between experimental and ab initio band origins and q doubling constants of ortho  e/f (j=1,k=0) and  f (j=1,k=0) states of NH 3 -Rg complexes. The errors between parentheses of the ab initio q values correspond to the standard deviation for J = 1-8 (see Table 6). Table 5: Experimental and ab initio molecular parameters of the four NH 3-Rg complexes for the  e/f (j=1,k=0) state determined using the expressions of the pseudodiatomic model applied to either the experimental or the ab initio calculated transition frequencies. Using the same approach, molecular parameters of the  f (j=0,k=0) ground state obtained from microwave experiments are compared to our ab initio 4D PES results.

Figure 1 Figure 2 :

 12 Figure 1 EC-QCL jet-cooled spectra of NH 3 -Rg of the ortho  e/f (j=1,k=0) f (j=0,k=0) transition recorded in the 947-957 cm -1 range with 2 % NH 3 seeded in 4 bar Rg and 4 bar Helium for Rg = Ar, Kr, Xe, in 8 bar Ne for NH 3 -Ne (right). Expanded view of the Q branch fully resolved in rotation, which displays the asymmetry change between Ne and the heaviest Rg atoms (left).The spectra of NH 3 -Ne and NH 3 -Xe have been magnified by a factor of 7 and 18, respectively, to obtain a maximum absorption similar to that of NH 3 -Ar or NH 3 -Kr. An expanded view of the P(7) line of the NH 3 -Kr spectrum highlights the rich isotopic structure of krypton with 82 Kr, 83 Kr, 84 Kr and 86 Kr.Ten rotational lines marked with an asterisk belong to the  2 band of 14 NH 3 monomer. The rotational line marked with a cross corresponds to the transition v 2 =1(e)0(f), Q R(0,0) of 15 NH 3 at 947.79221 cm -1 line, according to the notation K J(J '' ,K '' ).[35]

Figure 3 :

 3 Figure 3: Rovibrational band contour simulations at T R equal to 6.5, 10, 12.5 K, respectively for Ne, Ar and Kr (top) compared to EC-QCL jet-cooled slit spectra of the ortho  f (j=1,k=0) f (j=0,k=0) transition of NH 3 -Rg heterodimers (middle), and ab initio frequencies derived from VRT states calculated on the IPES of Loreau et al. 14 (bottom). The P and R branches of ab initio spectra have been shifted by -0.323(20), -0.723(14) and -0.226(12) cm -1 , for Ne, Ar and Kr, respectively.

Figure 4 :

 4 Figure 4: Evolution of the ab initio energy difference 

Table 1 :

 1 Parameters used in the bound state calculations. The first three columns contain the radial grid parameters. The last two columns contain the parameters of the effective potential used to optimize the radial basis functions.

		D e (cm -1 )	D 0 (cm -1 )
	NH 3 -Ne	66.81	30.35
	NH 3 -Ar	147.6	94.22
	NH 3 -Kr	173.6	119.78
	NH 3 -Xe	197.7	144.36

Table 2 :

 2 Binding energy and dissociation energy of the ortho-NH 3 -Rg complexes.

  3 -Xe has been estimated by comparison of the observed and ab initio band centers of the three other NH 3 -Rg complexes (see text).

		NH 3 -Ne	NH 3 -Ar	NH 3 -Kr	NH 3 -Xe
		Exp	Ab initio	Exp	Ab initio	Exp	Ab initio	Exp	Ab initio
				 e/f (j=1,k=0) f (j=1,k=0)
	 0 (cm -1 ) 950.25492(9) 950.534 950.38809(8) 950.336 951.55658(6)	951.091 954.34849(5)	952.966
	q	282.08(16)	273.9(48) 90.845(67) 85.20(21)	66.177(43)	61.96(21) 84.104(89) 65.44(56)
				 f (j=1,k=0)  f (j=0,k=0)
	 0 (cm -1 )	957.2462(1)	957.591 960.73361(5) 961.470 960.13595(64)	960.370	-	959.011

20 Ne-NH 3 40 Ar-NH 3 84 Kr-NH 3 132 Xe-NH 3  f (j=0,k=0)

  from Van Wijngaarden et al. b from Nelson et al. c from Van Wijngaarden et al.

	R eff (Å)	Exp	3.7227 a	3.8358 b	3.9220 c	4.067 d
	Ab initio	3.71	3.86	3.95	4.09
	 s (cm -1 ) Exp.	23.2 a	34.6(1) b	35.0 c	36.5 d
	Ab initio	19.3	33.6	34.6	36.1
	k s (N/m)	0.291 a	0.840 b	1.02 c	1.18 d
				 e/f (j=1,k=0)
	R eff (Å) Exp	3.70487(4)	3.83839(2)	3.93434(6)	4.09369(4)
	Ab initio	3.77	3.87	3.97	4.16
	 s (cm -1 ) Exp	22.0(1)	33.9(1)	33.1(1)	26.6(2)
	Ab initio	20.4	32.0	32.3	32.8
	k s (N/m)		0.26	0.81	0.91	0.63

a d from Wen et al.

Table 6 :

 6 Spectroscopic constants and molecular parameters of the four NH 3 -Rg complexes for the  e (j=1,k=0) state derived from the band contour fitting of the Q(J) lines of the v 2 =10  e (j=1,k=0)  f (j,k=0) transition.

		20 Ne	Ar	84 Kr	132 Xe
			 e (j=1,k=0)
	B(MHz)	4003.134(50)	2872.313(339)	2307.083(22)	2000.316(17)
	D J (MHz)	0.49127(61)	0.0876(27)	0.04387(15)	0.02675(22)
	H J (10 5 MHz)	4.77(10)	1.59(63)	0.05(2)	-0.081(36)
	# lines	15	18	28	20
	RMS(MHz)	1.2	6.9	1.4	2.6
	R eff (Å)	3.705	3.838	3.934	4.094
	 s (cm -1 )	24.0	34.6	35.2	36.4