Nadia Ben Amor

Adrien Soupart

Marie-Catherine Heitz
email: heitz@irsamc.ups-tlse.fr

Valence excited states of metalloporphyrins studied with the CASPT2 method: the case of an Iron complex

Keywords: Iron-porphyrins, CASSCF, MS-CASPT2, IPEA shift Iron-porphyrins, CASSCF, MS-CASPT2, IPEA shift

The singlet valence excited states of an iron-porphyrin-pyrazine-carbonyl complex are investigated up to the Soret band (about 3.eV) using Multi-State Complete Active Space with Perturbation at the Second Order (MS-CASPT2). This complex is a model for the active site of carboxy-hemoglobin/myoglobin. The spectrum of the excited states is rather dense, comprising states of different nature: d→pi* transitions, d→d states, pi→pi* excitations of the porphyrin and doubly excited states involving simultaneous intra-porphyrin pi→pi* and d→d transitions. Specific features of the MS-CASPT2 method are investigated. The effect of varying the number of roots in the state average calculation is quantified as well as the consequence of targeted modifications of the active space. The effect of inclusion of standard Ionization Potential-Electron Affinity (IPEA) shift in the perturbation treatment is also investigated.

Introduction

Heme proteins are responsible for many biological functions like transport, storage and detection of ligands and catalytic activities. For example, hemoglobin and myoglobin are well known for assuming the transport and the storage of the O2 molecule. But their active centers can also interact with other diatomics, in particular with the CO poisoning molecule which can strongly bind to iron porphyrin sites. Advantage of the IR-spectroscopic response of CO can be taken to perform vibrational ladder climbing [START_REF] Ventalon | Coherent vibrational climbing in carboxyhemoglobin[END_REF] and to probe the ligand dissociation and transfer inside the protein [START_REF] Treuffet | Direct observation of ligand transfer and bond formation in cytochrome c oxidase by using mid-infrared chirped-pulse upconversion[END_REF][START_REF] Polack | CO Vibration as a Probe of Ligand Dissociation and Transfer in Myoglobin[END_REF][START_REF] Nuernberger | Multiply Excited Vibration of Carbon Monoxide in the Primary Docking Site of Hemoglobin Following Photolysis from the Heme[END_REF]. Photodissociation of the iron-CO bond following excitation with visible or near UV light has been reported experimentally [START_REF] Petrich | Photophysics and reactivity of heme proteins: a femtosecond absorption study of hemoglobin, myoglobin, and protoheme[END_REF][START_REF] Franzen | Heme Photolysis Occurs by Ultrafast Excited State Metal-to-Ring Charge Transfer[END_REF] and theoretically investigated with TD-DFT [START_REF] Dreuw | Characterization of the Relevant Excited States in the Photodissociation of CO-Ligated Hemoglobin and Myoglobin[END_REF][START_REF] Dunietz | Initial Steps of the Photodissociation of the CO Ligated Heme Group[END_REF][START_REF] Ohta | Excited State Property of Hardly Photodissociable Heme-CO Adduct Studied by Time-Dependent Density Functional Theory[END_REF]. However, further investigations are needed to get a deeper insight of the nature of the excited states involved in the dissociation process and, as a perspective, in the accompanying spin relaxation process from S=0 to S=2 (unligated active site). In this paper, we present Multi-State Complete Active Space with Perturbation at the Second Order [START_REF] Finley | The multi-state CASPT2 method[END_REF] (MS-CASPT2) calculations of the valence singlet excited states (< 3.5eV) of a model of the active site of carboxyhemoglobin or carboxymyoglobin. The model complex consists of an iron center in a porphyrin (P), with CO and pyrazine (pz) as axial ligands (FeP(pz)CO).

In the 1960s, Gouterman proposed a four orbitals model to rationalize the absorption spectra of porphyrins in terms of combinations of one-electron transitions between frontier π orbitals of the macrocycle [START_REF] Gouterman | Study of the Effects of Substitution on the Absorption Spectra of Porphin[END_REF][START_REF] Gouterman | Spectra of porphyrins[END_REF]. Since this early work, interest for a theoretical description of porphyrin-based systems has not decreased, taken advantage of the development of ab initio methods for the treatment of electronic excited states [START_REF] Manni | Combining the Complete Active Space Self-Consistent Field Method and the Full Configuration Interaction Quantum Monte Carlo within a Super-CI Framework, with Application to Challenging Metal-Porphyrins[END_REF][START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF][START_REF] Perun | Singlet and Triplet Excited States and Intersystem Crossing in Free-Base Porphyrin: TDDFT and DFT/MRCI Study[END_REF][START_REF] Aspuru-Guzik | Quantum Monte Carlo for electronic excitations of free-base porphyrin[END_REF][START_REF] Sundholm | Interpretation of the electronic absorption spectrum of free-base porphin using time-dependent density-functional theory[END_REF][START_REF] Sauri | Multiconfigurational Second-Order Perturbation Theory Restricted Active Space (RASPT2) Method for Electronic Excited States: A Benchmark Study[END_REF][START_REF] Merchán | Theoretical determination of the electronic spectrum of free base porphin[END_REF][START_REF] Rubio | Theoretical study of the electronic spectrum of magnesium-porphyrin[END_REF][START_REF] Serrano-Andrés | Interpretation of the electronic absorption spectrum of free base porphin by using multiconfigurational second-order perturbation theory[END_REF][START_REF] Fransson | Four-Component Damped Density Functional Response Theory Study of UV/Vis Absorption Spectra and Phosphorescence Parameters of Group 12 Metal-Substituted Porphyrins[END_REF].

Iron-porphyrin complexes belong to the class of so-called irregular porphyrins where the partially filled d-shell contributes significantly to the excited state spectrum by introducing excitations from/to metallic orbitals. Particular attention must thus be given to the description of the interaction of the metal and the ligand(s), porphyrin ligand and axial ligand(s) if present.

A recent overview on multiconfiguration ab initio methods applied on heme-related systems, summarizing ground and low-lying states calculations, can be found in Reference [START_REF] Chen | Multireference and Multiconfiguration Ab Initio Methods in Heme-Related Systems: What Have We Learned So Far?[END_REF].

CASPT2 determination of spin-state energetics [START_REF] Vancoillie | Performance of CASPT2 and DFT for Relative Spin-State Energetics of Heme Models[END_REF][START_REF] Radoń | Electronic Structure of Selected {FeNO} 7 Complexes in Heme and Non-Heme Architectures: A Density Functional and Multireference ab Initio Study[END_REF] and of binding energies of small molecules on heme [START_REF] Radoń | Binding of CO, NO, and O2 to Heme by Density Functional and Multireference ab Initio Calculations[END_REF] have been reported and in some cases used to assess the validity of various density functionals. Particular interest is shown for the addition reaction of O2 to heme [START_REF] Ribas-Ariño | The mechanism for the reversible oxygen addition to heme. A theoretical CASPT2 study[END_REF][START_REF] Kitagawa | A DFT and Multi-Configurational Perturbation Theory Study on O2 Binding to a Model Heme Compound Via the Spin-Change Barrier[END_REF] and for the characterization of the iron-O2 bond [START_REF] Jensen | O-binding to heme: electronic structure and spectrum of oxyheme, studied by multiconfigurational methods[END_REF][START_REF] Chen | Nature of the Fe-O2 Bonding in Oxy-Myoglobin: Effect of the Protein[END_REF]. Relative energies of spin-states of manganese complexes involving either a porphyrin ligand [START_REF] Kepenekian | Toward Reliable DFT Investigations of Mn-Porphyrins through CASPT2/DFT Comparison[END_REF] or related corrole or corrolazine ligands [START_REF] Alcover-Fortuny | Role of the Imide Axial Ligand in the Spin and Oxidation State of Manganese Corrole and Corrolazine Complexes[END_REF] are also reported in the literature. In all these CASPT2 calculations, the active space includes pure metallic and metal-ligand bonding/antibonding orbitals and possibly axial ligand orbitals. Massive inclusion of π and π* orbitals of the porphyrin ring is not possible in a Complete Active Space Self Consistent Field [START_REF] Roos | A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach[END_REF] (CASSCF) active space. A possible approach consists in a set-up of the active space in order to take into account only the most important orbitals, at least those for which the occupation numbers vary significantly.

Another solution consists in performing a Restricted Active Space Self Consistent Field (RASSCF) [START_REF] Roos | A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach[END_REF] scheme, which allows for considering more active orbitals while the restriction is put on the degree of excitation of the determinants. In the recent years, several studies at the RASPT2 [START_REF] Malmqvist | The restricted active space followed by second-order perturbation theory method: Theory and application to the study of CuO2 and Cu2O2 systems[END_REF] level have investigated the relative energies of low-lying (mainly spin) excited states of iron, manganese, cobalt/porphyrin or corrole complexes [START_REF] Vancoillie | Multiconfigurational Second-Order Perturbation Theory Restricted Active Space (RASPT2) Studies on Mononuclear First-Row Transition-Metal Systems[END_REF][START_REF] Jannuzzi | Spin State Energetics and Oxyl Character of Mn-Oxo Porphyrins by Multiconfigurational ab Initio Calculations: Implications on Reactivity[END_REF][START_REF] Radoń | DFT and Ab Initio Study of Iron-Oxo Porphyrins: May They Have a Low-Lying Iron(V)-Oxo Electromer?[END_REF][START_REF] Zhao | Low-Energy States of Manganese-Oxo Corrole and Corrolazine: Multiconfiguration Reference ab Initio Calculations[END_REF].

In the present work, we intend to gain information on the valence excited states of the metallic center of carboxy-hemoglobin/myoglobin by means of the CASSCF/MS-CASPT2 method. The heme site is bound to the protein through the imidazole (im) group of a histidine residue. Hence, a complex restricted to the ligands directly coordinated to the iron comes down to FeP(im)CO. This standard model, used in Symmetry Adapted Cluster -CI (SAC-CI) [START_REF] Nakatsuji | Ground and excited states of carboxyheme: a SAC/SAC-CI study[END_REF][START_REF] Tokita | Ground and Excited States of Hemoglobin CO and Horseradish Peroxidase CO: SAC/SAC-CI Study[END_REF] and TD-DFT [START_REF] Dreuw | Characterization of the Relevant Excited States in the Photodissociation of CO-Ligated Hemoglobin and Myoglobin[END_REF][START_REF] Dunietz | Initial Steps of the Photodissociation of the CO Ligated Heme Group[END_REF] investigations of the absorption spectra of carboxyhemoglobin, has Cs symmetry. Due to the high density of excited states, we choose to replace the imidazole (C3H4N2) group by a pyrazine (C4H4N2) group, leading to a FeP(pz)CO complex of C2v symmetry. We believe that the conclusions drawn from our calculations are not strongly affected by this simplification while the increase of symmetry makes the calculation more tractable. In this paper, we address the problem of the description of the singlet valence excited states of FeP(pz)CO, up to the Soret band (about 3eV). Dependence of the results on the number of roots and on modifications of the active space, as well as the behavior of the basis set with respect to the correlation of the 3s electrons of the iron, are examined. Concerning the perturbation step, the default value (0.25 a.u.) of the Ionization Potential-Electron Affinity (IPEA) shift for the zeroth order Hamiltonian [START_REF] Ghigo | A modified definition of the zeroth-order Hamiltonian in multiconfigurational perturbation theory (CASPT2)[END_REF] is controversially discussed in the literature [START_REF] Kerridge | A RASSCF study of free base, magnesium and zinc porphyrins: accuracy versus efficiency[END_REF][START_REF] Vela | On the zeroth-order hamiltonian for CASPT2 calculations of spin crossover compounds: Spin State Energetics of FeN6 -Based SCO Compounds[END_REF][START_REF] Kepenekian | What zeroth-order Hamiltonian for CASPT2 adiabatic energetics of Fe(II)N6 architectures?[END_REF][START_REF] Radoń | How can [Mo IV (CN)6] 2-, an apparently octahedral (d) 2 complex, be diamagnetic? Insights from quantum chemical calculations and magnetic susceptibility measurements[END_REF][START_REF] Ruipérez | Complete vs Restricted Active Space Perturbation Theory Calculation of the Cr2 Potential Energy Surface[END_REF], both for iron compounds [START_REF] Vela | On the zeroth-order hamiltonian for CASPT2 calculations of spin crossover compounds: Spin State Energetics of FeN6 -Based SCO Compounds[END_REF][START_REF] Kepenekian | What zeroth-order Hamiltonian for CASPT2 adiabatic energetics of Fe(II)N6 architectures?[END_REF] and for porphyrin ones [START_REF] Kerridge | A RASSCF study of free base, magnesium and zinc porphyrins: accuracy versus efficiency[END_REF]. Thus, the excited states of the iron-porphyrin complex studied here have been calculated with and without IPEA shift, and the differences analyzed.

I. Computational details

The calculations on the iron complex have been performed at a nuclear configuration optimized at the DFT level, using the PBE0 functional [START_REF] Adamo | Toward reliable density functional methods without adjustable parameters: The PBE0 model[END_REF] with the Def2QZVPP basis set [START_REF] Weigend | Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy[END_REF][START_REF] Weigend | Accurate Coulomb-fitting basis sets for H to Rn[END_REF] on the iron atom and the Def2TZVP basis sets [START_REF] Weigend | Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy[END_REF][START_REF] Weigend | Accurate Coulomb-fitting basis sets for H to Rn[END_REF] on the remaining atoms, employing the Gaussian 09 package [START_REF] Frisch | Gaussian 09[END_REF]. The final structure, displayed in Figure 1 (plotted with MacMolPlt[51]), is a stable minimum of C2v symmetry, as was confirmed by a harmonic frequencies calculation. In the chosen axis system, the Fe-C-O line is the z axis and the pyrazine ligand lies in the xz plane, bisecting the two planes defined as containing opposite iron-N(porphyrin) bonds. Table 1 presents some of the interatomic distances, around the metallic center. Complete coordinates of the optimized structure can be found as Supplementary Information. A diagram of the valence orbitals of the complex with their occupations as obtained by the DFT calculation is also displayed in Figure 1 (right).

Fe -C (CO) and for H a (8s4p) set is contracted to [2s1p]. The ANO-RCC basis set for the iron atom has been developed without correlating the 3s electrons, i.e., these electrons are described with minimal basis quality [START_REF] Roos | New Relativistic ANO Basis Sets for Transition Metal Atoms[END_REF]. It is mentioned in the original publication that these electrons should not be included in any correlation treatment, because that could cause large basis set superposition errors [START_REF] Roos | New Relativistic ANO Basis Sets for Transition Metal Atoms[END_REF]. However, in most of the iron studies using ANO-RCC basis sets, the 3s electrons are correlated. Thus, an additional test calculation has been performed to check the validity of correlating the 3s electrons, using another relativistic all-electron correlation consistent basis set for the iron atom, the aug-cc-pwCVTZ-DK [9s8p6d3f2g] [START_REF] Balabanov | Systematically convergent basis sets for transition metals. I. All-electron correlation consistent basis sets for the 3d elements Sc-Zn[END_REF],

which takes into account the semi-core correlation of the [3s,3p] electrons, while the ligands are still described by the ANO-RCC basis set.

Concerning the active space, a straightforward inclusion of all valence π and π* orbitals of the porphyrin and of all orbitals with significant 3d metallic contribution would result in a 36 electrons in 36 orbitals active space which exceeds by far the current computational possibilities of the CASSCF method. Compromises have thus to be done. Our smallest active space includes the 3d orbitals of Fe, the 3d' double shell orbitals correlating the three (ground state) occupied 3d orbitals of Fe and the two metal-ligand σ-bonding orbitals counterpart of the empty 3d iron orbitals, supplemented by four orbitals of the porphyrin ring. These latter are the so-called Gouterman's orbitals which consist in the four frontier orbitals of the porphyrin ring (two occupied and two unoccupied). This leads to 14 electrons in 14 orbitals, noted CAS [START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF][START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF]. Figure 2 presents a sketch of the active orbitals (plotted with Molekel [60]). Note that the * orbitals of the porphyrin ring have a metallic contribution and that the 3d'xz and 3d'yz orbitals are mixed with the * orbitals of CO. Such an active space is intended to include all the orbitals whose occupation varies significantly between the ground and the excited states, for states below the Soret band (the second set of intraporphyrin →* transitions) and fulfill the basic recommendations for first row transition metal compounds. The situation is more complicated for the Soret band. Even if excitations among the Gouterman's set are still dominant, the results of SAC-CI studies [START_REF] Nakatsuji | Ground and excited states of carboxyheme: a SAC/SAC-CI study[END_REF][START_REF] Tokita | Ground and Excited States of Hemoglobin CO and Horseradish Peroxidase CO: SAC/SAC-CI Study[END_REF] point out excitations from the two next occupied  orbitals of the macrocycle. On this basis, attempts have been made to include the second highest occupied porphyrin  orbitals (one of A1, the other of A2 symmetry) in the active space leading to a cumbersome but still tractable 18 electrons in 16 orbitals calculation, but it was never possible to stabilize the targeted a1 orbital in the active space.

Three other active spaces resulting from the inclusion of two unoccupied orbitals have been tested. A-CAS [START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF][START_REF] Aspuru-Guzik | Quantum Monte Carlo for electronic excitations of free-base porphyrin[END_REF] includes the two remaining 3d' orbitals which correlate the 3d orbitals at most singly occupied in all the states considered. B-CAS [START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF][START_REF] Aspuru-Guzik | Quantum Monte Carlo for electronic excitations of free-base porphyrin[END_REF] includes the two * orbitals of CO while C-CAS [START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF][START_REF] Aspuru-Guzik | Quantum Monte Carlo for electronic excitations of free-base porphyrin[END_REF] includes two unoccupied * orbitals of the porphyrin ring, of A1 and A2 symmetry respectively (see Supplementary Information for the representation of these latter orbitals). We intend to investigate the excited states of the FeP(pz)CO complex up to the Soret band. Ten roots at the CASSCF level are thus calculated for each symmetry. State specific orbitals are sometimes impossible to obtain and average orbitals are generally a good compromise to get the whole set of states involved in the electronic spectrum. In order to estimate the effect of using state-average orbitals, we perform also, in addition to the ten roots MS-CASPT2, three or four roots state-average calculations for selected cases. Vertical excitations energies were always calculated considering the ground and excited states at the same level of calculation (same number of roots, same active space).

In all cases, Cholesky decomposition technique [START_REF] Aquilante | Low-cost evaluation of the exchange Fock matrix from Cholesky and density fitting representations of the electron repulsion integrals[END_REF][START_REF] Aquilante | Cholesky Decomposition-Based Multiconfiguration Second-Order Perturbation Theory (CD-CASPT2): Application to the Spin-State Energetics of Co III (diiminato)(NPh)[END_REF] which allows a drastic reduction of the number of bielectronic integrals, is used, with a 10 -6 a.u. threshold. To avoid intruder states, a level shift of 0.4 a.u. has been used in all the MS-CASPT2 calculations, which is the lowest common value that gives similar reference weights for all states.

II. Results and Discussion

Correlation of the 3s electrons

In CASPT2 calculations, core electrons, like the 1s, 2s, 2p electrons of Fe and the 1s electrons of C,N and O, are usually frozen. As mentioned in the computational details, the ANO-RCC basis set of the iron atom has been set up without including the correlation of the 3s electrons [START_REF] Roos | New Relativistic ANO Basis Sets for Transition Metal Atoms[END_REF]. As these electrons are often correlated, a test calculation has been performed on the first A1 excited states of the compound. In Table 2, we present excitation energies for two basis sets, in the case of either frozen or correlated 3s electrons. For the aug-cc-pwCVTZ-DK basis sets, which takes into account the semi-core correlation of the [3s,3p] electrons and not only the [3p], the two sets of excitation energies differ from at most 0.12 eV. This is small but not negligible, therefore the 3s electrons should be correlated, as expected. As the transition energies between the two basis sets differ from at most 0.01 eV, for frozen and for correlated 3s electrons, we can conclude that, when the ANO-RCC basis is used for iron, the correlation of the 3s electrons does not cause basis set superposition errors in this molecular system. In all the rest of the study, these electrons are thus included in the correlation calculations.

State average orbitals

The results discussed in this subsection are all obtained without inclusion of the IPEA shift.

As state specific evaluation of all the states of interest is often problematic, state average (SA) orbitals -which provide most of the time a well-balanced description of the different states-are used. The number of states for which density matrices are averaged can have a quite important effect on the total energies. The goal is here to analyze the effect on the transition energies. We intend to calculate the states up to the Soret band (about 3. eV), which corresponds to the second set of π→π* excitations. At the CASSCF [START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF][START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF] level, these states correspond to the 8th roots of B1 and B2 symmetries while in the different CAS [START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF][START_REF] Aspuru-Guzik | Quantum Monte Carlo for electronic excitations of free-base porphyrin[END_REF] active spaces these states correspond either to the 8 th , 9 th or 10 th root. Hence, we calculate the excited states on the basis of 10-roots SA-CASSCF/MS-CASPT2. In the A1 and A2

symmetries, where a fewer number of mono-excitations are involved, it is possible to obtain the lowest valence states by averaging over a smaller number of states. Table 3 configurations have balanced weights. Also included in the same table are the results of a three roots calculation, in principle slightly biased in favor of d 5 states, which are similar to those of the four roots calculation. A comparison of 10-roots and 3-roots transition energies for the MLCT states shows difference of as much as 0.3-0.35 eV. However, the effect of varying the number of roots is different for the A2 symmetry (Table 4). There, a 3-roots calculation (again two MLCT and one MC states) is compared to a 10-roots calculation. The three states are close in energy in both calculations but the nature of the states changes. In the three roots calculation, MCLT and MC characters are mixed for two of the states. Hence, the assignment of the states is changed but the excitation energies vary at most of 0.

Inclusion of the IPEA shift

In the CASPT2 method, the zeroth-order Hamiltonian is based on Fock-type one-electron operators. It means that the two-electron terms are not taken into account in the Hamiltonian.

A systematic error is then present in the original formulation of the CASPT2 method and open-shell configurations are favored compared to closed-shell one. Several corrections have been developed such as the G1, G2 and G3 [START_REF] Andersson | Different forms of the zeroth-order Hamiltonian in second-order perturbation theory with a complete active space self-consistent field reference function[END_REF] or the IPEA shift [START_REF] Ghigo | A modified definition of the zeroth-order Hamiltonian in multiconfigurational perturbation theory (CASPT2)[END_REF]. The IPEA shift modifies the zeroth order Hamiltonian [START_REF] Ghigo | A modified definition of the zeroth-order Hamiltonian in multiconfigurational perturbation theory (CASPT2)[END_REF] to correct this systematic error, using a single parameter whose value has been selected from benchmark calculations. But, the default value (0.25 a.u.) of this shift is regularly disputed [START_REF] Kerridge | A RASSCF study of free base, magnesium and zinc porphyrins: accuracy versus efficiency[END_REF][START_REF] Vela | On the zeroth-order hamiltonian for CASPT2 calculations of spin crossover compounds: Spin State Energetics of FeN6 -Based SCO Compounds[END_REF][START_REF] Kepenekian | What zeroth-order Hamiltonian for CASPT2 adiabatic energetics of Fe(II)N6 architectures?[END_REF][START_REF] Radoń | How can [Mo IV (CN)6] 2-, an apparently octahedral (d) 2 complex, be diamagnetic? Insights from quantum chemical calculations and magnetic susceptibility measurements[END_REF][START_REF] Ruipérez | Complete vs Restricted Active Space Perturbation Theory Calculation of the Cr2 Potential Energy Surface[END_REF]. Tables 5 and6 Excitation energies of MLCT states increase of about 0.5-0.9 eV while states dominated by →* excitations between porphyrin frontier orbitals are shifted up from about 1 eV. The effect on doubly excited states is even larger, between 0.9 and 1.4 eV. As a consequence, the energetic order of the states is markedly altered. For instance, in A1 symmetry, the d→d transition is now lower than the doubly excited states and in B1 the relative order of the d→d and →* transitions is reversed. This last remark holds also for the states of B2 symmetry which are very similar to those of B1. In the A2 symmetry, the lowest excited state is now a d→d transition instead of a MLCT state. In Figure 3, we present a graph of the excitation energies of the states of all symmetries with IPEA shifts of 0., 0.1, 0. In the original publication introducing IPEA [START_REF] Ghigo | A modified definition of the zeroth-order Hamiltonian in multiconfigurational perturbation theory (CASPT2)[END_REF], excitation energies of N2 and benzene molecules are reported for IPEA shifts varying from 0.0 to 0.5 a.u. In these examples, a value of IPEA of 0.25 a.u. increases the excitation energies from about 0.2 to 0.4 eV [START_REF] Ghigo | A modified definition of the zeroth-order Hamiltonian in multiconfigurational perturbation theory (CASPT2)[END_REF]. This is in the order of magnitude of the shifts observed for the d→d transitions of the iron porphyrin complex, while the MLCT and intra-porphyrin states are more destabilized. As mentioned above, the effect is even larger for some doubly excited states. No experimental results are available for FeP(pz)CO in the gas phase, but the experimental absorption spectrum of carboxyhemoglobin, for which the minimal description of the active site consists in a FeP(im)CO complex, is available in the literature [START_REF] Makinen | Polarized single crystal absorption spectra of carboxyand oxyhemoglobin[END_REF]. The lowest absorption bands (the Q band at 2.18 and 2.30 eV and the strong Soret band at 2.96 eV [START_REF] Makinen | Polarized single crystal absorption spectra of carboxyand oxyhemoglobin[END_REF]) are usually assigned to intra porphyrin →* transitions mainly due to excitation amongst the four Gouterman frontiers orbitals [START_REF] Dunietz | Initial Steps of the Photodissociation of the CO Ligated Heme Group[END_REF][START_REF] Nakatsuji | Ground and excited states of carboxyheme: a SAC/SAC-CI study[END_REF]. The →* states are of B1 and B2 symmetry. As can be seen in experimentally is thus probably due to the environmental effect of the protein which namely tilts slightly the Fe-CO bond with respect to the porphyrin plane normal. Due to the very large effect observed for most of the transitions, between the inclusion of standard IPEA (0.25 a.u) and its omission, it seems reasonable to choose, for this particular system, the omission. CASPT2 is not intended to turn into a parametrical method. However, we mention that setting the IPEA value at 0.1 a.u. would result in the first π→π* transitions to be at 2.19 and 2.22 eV and the second ones to be at 3.24 and 3.28 eV, in reasonable agreement with the closest available experimental data, and with TD-DFT on FeP(im)CO [START_REF] Dunietz | Initial Steps of the Photodissociation of the CO Ligated Heme Group[END_REF].

Recent RASPT2 study compares excitation energies with no IPEA shift and with IPEA = 0.25 a.u. for FBP, MgP and ZnP [START_REF] Kerridge | A RASSCF study of free base, magnesium and zinc porphyrins: accuracy versus efficiency[END_REF] and concludes that the calculations without IPEA shift are in better agreement with experiments and reports that the discrepancy is more pronounced when the active space is truncated. The lowest doubly excited states are found energetically below some mono-excitations at the MS-CASPT2 level, whatever the value of IPEA shift. The existence of such rather low-lying doubly excited states among the valence excited states comes from the large conjugated π system of the porphyrin ligand which results in a low energy HOMO-LUMO intra-ligand gap and could be specific to partially filled d-shell metalloporphyrins. It is useful to know the energetics of such states with respect to the mono-excited states in order to assess their role in the excited states relaxation processes following excitation to the Soret band for instance. In the B1 and B2 symmetries, the lowest doubly excited states involve a π→π* excitation and the dx2-y2→dxy excitation which correspond to the lowest MC state. Without IPEA, two such states are found at 2.40 and 2.38 eV, well below the second π→π* states. Inclusion of standard IPEA increases the excitation energies of slightly less than 1 eV and these states are still found below the second π→π* set. In A1 and A2 symmetries, the doubly excited states are not based on the lowest MC state and their occurrence at low energy cannot be justified qualitatively. Note that, in these cases, the effect of IPEA shift is larger, up to 1.4 eV.

In the present calculations, the effect of IPEA depends strongly on the nature of the leading configurations of the different states. The effect can be very large for states involving excitation from or/and to the π system of the porphyrin, and thus an effect of the remaining  orbitals of the porphyrin cannot be excluded.

Active spaces

The CAS [START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF][START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF] active space includes all orbitals for which the occupation number varies significantly. However, some other orbitals might play a non-negligible role. Several possibilities have been considered by adding different orbitals to the CAS(14,14): i) the two missing 3d' orbitals of the iron atom, noted A-CAS [START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF][START_REF] Aspuru-Guzik | Quantum Monte Carlo for electronic excitations of free-base porphyrin[END_REF] ; ii) the anti-bonding π* orbitals of the carbonyl, noted B-CAS [START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF][START_REF] Aspuru-Guzik | Quantum Monte Carlo for electronic excitations of free-base porphyrin[END_REF]; iii) two additional π* orbitals of the porphyrin (see supplementary material for a graphical representation) which are low lying and might affect the d→π* states, noted C-CAS [START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF][START_REF] Aspuru-Guzik | Quantum Monte Carlo for electronic excitations of free-base porphyrin[END_REF]. The results are presented in Table 7. Note that, unlike for the other tables, the states in the different columns are classified according to the dominant configurations and not according to increasing energies. 7). The addition of two π* orbitals of the porphyrin (C-CAS [START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF][START_REF] Aspuru-Guzik | Quantum Monte Carlo for electronic excitations of free-base porphyrin[END_REF])) has limited effects on the mono-excited states (at most 0.13 eV on the second * transition) but has slightly larger effects on the doubly excited states with a maximum of 0.18 eV. To summarize, on average, CAS [START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF][START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF] results are not significantly affected by inclusion of the different couples of orbitals tested.

A1 CAS (14,14) 10-roots A-CAS(14,16) 3d' 10-roots B

Conclusion

This work presents MS-CASPT2 calculations of the singlet valence excited states of the FeP(pz)CO complex up to the Soret band (about 3.eV). The small HOMO-LUMO gap on the porphyrin ligand itself and the metal-ligand interaction induces a large diversity of excitations: d* transitions, dd states, * excitation of the porphyrin and some doubly excited states involving simultaneous intra-porphyrin * and d→d transitions. As a consequence, the valence excited states spectrum is rather dense. Several tests on the number of roots and targeted enlargement of the active space have shown moderate effects.

Omitting or including the standard IPEA shift is seen to have drastic effects on relative positions of the excited states. The importance of the effect of the IPEA is strongly correlated to the nature of the electronic excitation: d→d states are only moderately affected (<0.5 eV), while iron-to-porphyrin charge transfer states (0.5-1 eV), intra-porphyrin * (about 1. eV) and doubly excited states ( ≥ 0.9 eV) are more significantly affected. As a consequence, the relative positions of d→d and * states are changed when applying the standard IPEA shift. In the literature, excited states of regular porphyrins are reported to be best described without IPEA [START_REF] Kerridge | A RASSCF study of free base, magnesium and zinc porphyrins: accuracy versus efficiency[END_REF] while spin-state energy differences in iron compounds are reported to require higher than standard IPEA [START_REF] Vela | On the zeroth-order hamiltonian for CASPT2 calculations of spin crossover compounds: Spin State Energetics of FeN6 -Based SCO Compounds[END_REF][START_REF] Kepenekian | What zeroth-order Hamiltonian for CASPT2 adiabatic energetics of Fe(II)N6 architectures?[END_REF]. Concerning our calculations on FeP(pz)CO, inclusion of the standard IPEA has a too strong effect on most of the singly excited states and it seems reasonable to either omit the shift or to reduce its value. There is a need for further investigations on the effect of IPEA on excited states of transition metal complexes. A connection between the effect of IPEA and the limitation of the active space in CASSCF/MS-CASPT2 is not to be excluded. In the present study, the active space used is rather complete for the iron but is restricted to the minimal (the Gouterman's set) for the porphyrin. Further

Investigations at the RASSCF/RASPT2 level will help to clarify this point by including other orbitals of the porphyrin ring. Table S3: IPEA shift. Excitation energies calculated at the MS-CASPT2 level based on a 10roots SA CASSCF [START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF][START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF] without and with the inclusion of the IPEA shift of 0.25 a.u. (standard value), for the A2 symmetry.

Figure S4: Additional active orbitals for the active space C-CAS [START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF][START_REF] Aspuru-Guzik | Quantum Monte Carlo for electronic excitations of free-base porphyrin[END_REF]

Figure 1 :

 1 Figure 1: FeP(pz)CO optimized structure (see text) and occupation of the valence orbitals at the DFT level.

Figure 2 :

 2 Figure 2: Active orbitals from a 10-roots state average CASSCF (CAS(14,14)) of A1 symmetry.

  displays excitation energies of the electronic states dominated by mono-excitations of A1 symmetry, at the MS-CASPT2 level, based on a SA-CASSCF involving either 10, 4 or 3 roots. The fourteen active orbitals issued from the 3-and 4-roots calculations have very similar shapes as compared to those displayed in Figure2(issued from a 10-roots average), except for the occupied  orbital of A1 symmetry which has a significant contribution of iron-carbonyl bonding type. The 4-roots calculation results in three excited states of clearly identified characters, two Metal to Ligand Charge Transfer (MLCT) states and a ligand field transition to the dz 2 orbital (Metal Centered -MCstates). The two groups of states are rather well separated in energy and when additional roots are included in the CASSCF procedure, mainly doubly excited states supplement the set. These doubly excited states can be described as two mono-excitations localized on two subsystems, namely a d→d transition and an intraporphyrin →* transition. After the MS-CASPT2 step, two such states are found below the ligand field excitation. This type of states will be discussed in the next subsection. Including more roots in the calculation has a greater effect on the MLCT excitation energies than on the d→d transition. This can be explained by considering the varying oxidation numbers of the iron in the different states. In the MLCT states, the iron has mainly a d 5 configuration while all other states are d6 . So the 10-roots averaged CASSCF orbitals might be less adapted to the description of the MLCT states than those of the 4-roots calculation where d 5 and d6

  present a comparison of the transition energies calculated at the MS-CASPT2 level based on a 10-roots SA CASSCF with and without the inclusion of the IPEA shift of 0.25 a.u. (standard value), for the A1 and B1 symmetries respectively. The corresponding tables for the B2 and A2 symmetries are enclosed as supplementary material. The states of transition energies below 3.3 eV (value without IPEA) are reported in the tables. The spectrum of the excited states is rather dense, comprising states of different nature. There are six d→* states, six d→d states, four →* states and fifteen doubly excited states, four of these latter having a porphyrin to metal charge transfer additional contribution. It is found that the effect on the excitation energies of inclusion of IPEA can be very large and is correlated to the type of excitation. Standard IPEA increases excitation energies of d→d states of an amount ranging from 0.11 to 0.45 eV.

  2 and 0.25 a.u, with a color code corresponding to the nature of the electronic excitation. The effect is clearly differentiated according the type of excitation. The blue curves corresponding to d→d transitions cross the other curves which are more or less parallel. The most striking effect when introducing IPEA is the change of relative position of d→d and π→π* states.

Figure 3 :

 3 Figure 3: Effect of IPEA. Excitations energies of the states of A1,B1,B2 and A2 symmetries for different values of IPEA shift (0., 0.1, 0.2 and 0.25 a.u.) classified according to the nature of dominant electronic configurations: d→d (blue), π→π* (red), d→π* (green) and doubly excited states (black).

Table 1 :

 1 Selected interatomic distances of the optimized structure, in Ångström.

		1.784
	Fe -N (pyrazine)	2.080
	Fe -N (porphyrin)	2.006
	C -O	1.135

Table 2 :

 2 Effect of the correlation of the 3s electrons of the iron on the excitation energies of the first A1 excited states of the FeP(pz)CO molecule (eV), for different basis sets on the iron and different frozen orbital schemes.

		ANO-RCC [7s6p5d2f1g]	aug-cc-pwCVTZ [9s8p6d3f2g]
		3s correlated	3s frozen	3s correlated	3s frozen
		ΔE (eV)	ΔE (eV)	ΔE (eV)	ΔE (eV)
	d  *	1.64	1.68	1.64	1.68
	d  *	1.93	1.96	1.93	1.96
	d  d	2.90	3.02	2.89	3.01
	MS-CASPT2 on 4-roots CAS(14,14) (see text) and IPEA=0.0 a.u.

  16 eV (0.1 eV for the pure MLCT state).

	A1	10-roots	4-roots		3-roots
	Ground state
	(dx2-y2) 2 (dxz) 2 (dyz) 2 (a2) 2 (a1) 2	Energy (a.u.) Weight Energy (a.u.) Weight Energy (a.u.) Weight
		-2635.374664	0.81 -2635.369656	0.81	-2635.368812	0.80
	Excited states	ΔE (eV)	Weight	ΔE (eV)	Weight	ΔE (eV)	Weight
	dyz    b2	1.39	0.40	1.64	0.38	1.76	0.39
	dxz    b1		0.30		0.41		0.44
	dyz    b2	1.72	0.25	1.93	0.40	2.01	0.42
	dxz    b1		0.36		0.36		0.37
	dx2-y2  dz2	2.79	0.68	2.90	0.74

Table 3 : State average orbitals.

 3 Excitation energies of A1 symmetry states from MS-CASPT2 based on SA-CASSCF[START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF][START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF] performed with different number of roots. The value of the IPEA shift is 0. a.u.. Configurations with a weight larger than 10% are listed.

	A2	10-roots		3-roots
	State	ΔE (eV) Weight	ΔE (eV)	Weight
	dyz   * b1	1.86	0.14	1.89	0.25
	dxz   * b2		0.54		< 0.01
	dx2-y2  dxy		0.05		0.58
	dyz   * b1	1.89	0.56	1.96	0.12
	dxz   * b2		0.13		0.69
	dx2-y2  dxy		< 0.01		0.02
	dyz   * b1	2.05	< 0.01	1.98	0.47
	dxz   * b2		0.04		0.14
	dx2-y2  dxy		0.76		0.22

Table 4 :

 4 State average orbitals. Excitation energies of A2 symmetry states from MS-CASPT2 based on state average CAS(14,14) performed with different number of roots. The value of the IPEA shift is 0. a.u..

Such a study is not possible for the B1 symmetry where there is only one d→π* configuration and where two doubly excited states are energetically in between the five mono-excitations at the CASSCF level. In B1, calculations with 7, 8 and 10 roots give very close results (between 0.01 and 0.04 eV of deviation) at the MS-CASPT2 level.

Table 6 ,

 6 the value of the second →* transition (2.75 eV) without IPEA shift is in much better agreement with the experimental Soret band maximum than the value with standard IPEA (3.86 eV). But for the first →* transition, the experimental value lies in between the results with and without IPEA (2.74 and 1.78 eV respectively). Note that excitation energies of B2 symmetry states are very close to those of B1. The splitting of the Q band reported

Table 5 :

 5 IPEA shift. Excitation energies calculated at the MS-CASPT2 level based on a 10roots SA CASSCF[START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF][START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF] without and with the inclusion of the IPEA shift of 0.25 a.u. (standard value), for the A1 symmetry.

	B1	No IPEA	Standard IPEA (0.25 a.u.)
	State	ΔE (eV) Weight		ΔE (eV) Weight
	dx2-y2  *b1	1.50 0.73	dx2-y2  *b1	2.00	0.73
	a1  *b1	1.78 0.54	dxz  dz2	2.67	0.65
	a2  *b2	0.35	dyz  dxy		0.21
	dx2-y2 a1  *b2 dxy	2.40 0.61	a1  *b1	2.74	0.51
	a1 dx2-y2  *b2 dxy	0.20	a2  *b2		0.38
	dxz  dz2	2.45 0.66	dx2-y2 a1  *b2 dxy	3.35	0.63
	dyz  dxy	0.19	a1 dx2-y2  *b2 dxy		0.21
	a1  *b1	2.75 0.27	dxz  dz2	3.57	0.19
	a2  *b2	0.43	dyz  dxy		0.60
	dx2-y2 a2  *b1 dxy	2.77 0.53	dx2-y2 a2  *b1 dxy	3.64	0.63
	a2 dx2-y2  *b1 dxy	0.18	a2 dx2-y2  *b1 dxy		0.21
	dx2-y2 a1  dz2 *b1	3.04 0.58	a1  *b1	3.86	0.35
	a1 dx2-y2  dz2 *b1	0.19	a2  *b2		0.47
	dxz  dz2	3.27 0.17	dx2-y2 a1  dz2 *b1	4.00	0.61
	dyz  dxy	0.62	a1 dx2-y2  dz2 *b1		0.20

Table 6 :

 6 IPEA shift. Excitation energies calculated at the MS-CASPT2 level based on a 10roots SA CASSCF[START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF][START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF] without and with the inclusion of the IPEA shift of 0.25 a.u. (standard value), for the B1 symmetry.

Table 7 :

 7 Active space. Excitation energies calculated at the MS-CASPT2 level based on 10 roots for A1 and B1 states for various active spaces defined in the text. IPEA shift value of 0. a.u. . respectively, being the largest deviations of the whole table. Although the *CO orbitals are not directly involved in these states, they are in the same symmetries as the π* orbitals of the porphyrin of the active space and influence the whole set of orbitals through their d metallic component. Note that in B1 symmetry, a MLCT d  *CO state is found at high energy (4.07 eV, not reported in Table

	Including the missing 3d' orbitals, which correlate the 3dz2 and 3dxy orbitals, in the active
	space (A-CAS(14,16)), increases the excitation energies of the d d states of at most 0.11
	eV, which is not more than the deviation observed for the other mono-excited states. The
	effect on the doubly excited states is larger in average, with a maximum deviation from
	CAS(14,14) of 0.22 eV. The effect of the π*CO (B-CAS(14,16)) is less important on average
	(0.093 eV vs 0.109 eV ) with the notable exceptions of a MLCT state of A1 symmetry and the
	second →* state of B1 symmetry with differences from the CAS(14,14) results of 0.23 and
	0.30 eV

Table S1 :

 S1 Coordinates of the Optimized structure (in Angströms)

		H		2.059971	0.000000 -2.264572
		H		-2.059971	0.000000 -2.264572
		H		2.058639	0.000000 -4.760403
		H		-2.058639	0.000000 -4.760403
		H		2.612054	4.520112 -0.139158
		H		-2.612054	4.520112 -0.139158
		H		2.612054 -4.520112 -0.139158
		H		-2.612054 -4.520112 -0.139158
		H		4.519599	2.620206	0.050942
		Fe H		0.000000 -4.519599	0.000000 -0.044953 2.620206 0.050942
		C H		0.000000 4.519599 -2.620206 0.000000	1.738656 0.050942
		C H		2.767584 -4.519599 -2.620206 1.229204	0.007914 0.050942
		C		-2.767584	1.229204	0.007914
		C		2.767584 -1.229204	0.007914
		C		-2.767584 -1.229204	0.007914
		C		1.229179	2.761585 -0.134930
		C		-1.229179	2.761585 -0.134930
	B2	C C		1.229179 -2.761585 -0.134930 -1.229179 -2.761585 -0.134930 No IPEA Standard IPEA (0.25 a.u.)
	State	C	2.491938 ΔE (eV) Weight	3.447272 -0.101547	ΔE (eV) Weight
	dx2-y2 -> *b2	C	1.52	-2.491938 0.73	3.447272 -0.101547 dx2-y2 -> *b2	2.01	0.73
	a2 -> *b1	C	1.80	2.491938 -3.447272 -0.101547 0.37 dyz -> dz2	2.67	0.65
	a1 -> *b2	C		-2.491938 -3.447272 -0.101547 0.52 dxz -> dxy		0.21
	dx2-y2 a1 -> *b1 dxy	C	2.38	3.448090 0.61	2.495181 -0.004897 a2 -> *b1	2.76	0.40
	a1 dx2-y2 -> *b1 dxy	C		-3.448090 0.20	2.495181 -0.004897 a1 -> *b2		0.48
	dyz -> dz2	C	2.45	3.448090 -2.495181 -0.004897 0.67 dx2-y2 a1 -> *b1 dxy	3.34	0.63
	dxz -> dxy	C		-3.448090 -2.495181 -0.004897 0.19 a1 dx2-y2 -> *b1 dxy		0.21
	a2 -> *b1	C	2.79	0.000000 0.35	3.394071 -0.180465 dyz -> dz2	3.58	0.18
	a1 -> *b2	C		0.000000 -3.394071 -0.180465 0.25 dxz -> dxy		0.60
	dx2-y2 a2 -> *b2 dxy	C		3.401747 0.15	0.000000	0.064482
	dx2-y2 a2 -> *b2 dxy	C	2.80	-3.401747 0.46	0.000000 dx2-y2 a2 -> *b2 dxy 0.064482	3.68	0.63
	a2 dx2-y2 -> *b2 dxy	C		1.133575 0.15	0.000000 -2.823100 a2 dx2-y2 -> *b2 dxy		0.21
	a1 -> *b2	C		-1.133575 0.12	0.000000 -2.823100 a1 -> *b2		< 0.01
	dx2-y2 a1 -> dz2 *b2	C	3.07	1.123867 0.57	0.000000 -4.208894 a2 -> *b1	3.90	0.44
	a1 dx2-y2 -> dz2 *b2	C		-1.123867 0.18	0.000000 -4.208894 a1 -> *b2		0.36
		O		0.000000	0.000000 dx2-y2 a2 -> *b2 dxy 2.873911		<0.01
	dyz -> dz2	N	3.29	0.000000 0.17	0.000000 -2.124772 dx2-y2 a1 -> dz2 *b2	4.02	0.60
	dxz -> dxy	N		0.000000 0.62	0.000000 -4.912649 a1 dx2-y2 -> dz2 *b2		0.19
		N		1.421423	1.415583 -0.077949
		N		-1.421423	1.415583 -0.077949
		N		1.421423 -1.415583 -0.077949
		N		-1.421423 -1.415583 -0.077949
		H		4.483096	0.000000	0.132927
		H		-4.483096	0.000000	0.132927
		H		0.000000	4.476824 -0.220913
		H		0.000000 -4.476824 -0.220913

Table S2 :

 S2 IPEA shift. Excitation energies calculated at the MS-CASPT2 level based on a 10roots SA CASSCF[START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF][START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF] without and with the inclusion of the IPEA shift of 0.25 a.u. (standard value), for the B2 symmetry.

	A2	No IPEA		Standard IPEA (0.25 a.u.)
	State	ΔE (eV) Weight		ΔE (eV) Weight
	dyz -> *b1	1.86	0.14	dx2-y2 -> dxy	2.16	0.83
	dxz -> *b2		0.54
	dyz -> *b1	1.89	0.56	dyz -> *b1	2.38	0.71
	dxz -> *b2		0.13	dxz -> *b2		0.02
	dx2-y2 -> dxy	2.05	0.76	dyz -> *b1	2.40	0.02
				dxz -> *b2		0.71
	a1 -> dxy	2.23	0.25	a2 -> dz2	3.47	0.25
	dyz a1 -> *b2 dxy		0.18	a2 dyz -> dz2 *b2		0.30
	dxz a1 -> *b1 dxy		0.16	a2 dxz -> dz2 *b1		0.30
	a2 -> dz2	2.55	0.24	a1 -> dxy	3.52	0.29
	a2 dyz -> dz2 *b2		0.29	dyz a1 -> *b2 dxy		0.20
	a2 dxz -> dz2 *b1		0.27	dxz a1 -> *b1 dxy		0.19
	dyz a1 -> *b1 dz2	2.97	0.26	dyz a1 -> *b1 dz2	4.40	0.16
	dyz a1 -> dz2 *b1		0.09	dyz a1 -> dz2 *b1		0.06
	a1 dxz -> dz2 *b2		0.17	a1 dxz -> dz2 *b2		0.27
	dyz a1 -> *b1 dz2	3.02	0.22	dyz a1 -> *b1 dz2	4.44	0.35
	dyz a1 -> dz2 *b1		0.09	dyz a1 -> dz2 *b1		0.14
	a1 dxz -> dz2 *b2		0.30	a1 dxz -> dz2 *b2		0.24
	dxz a1 -> dz2 *b2		0.12	dxz a1 -> dz2 *b2		0.10
	dyz a2 -> *b1 dxy	3.27	0.13	dyz a2 -> *b1 dxy	4.55	0.14
	a2 dyz -> dz2 *b2		0.24	a2 dyz -> dz2 *b2		0.31
	a2 dxz -> dz2 b1*		0.27	a2 dxz -> dz2 b1*		0.31

Acknowledgments

The authors thank Thomas Applencourt for technical help and acknowledge the computational resources provided by CALMIP (Toulouse).

Supporting Information

Coordinates of the optimized geometry (Table S1).

Excitation energies calculated at the MS-CASPT2 level based on a 10-roots SA CASSCF [START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF][START_REF] Kumar | A density matrix renormalization group method study of optical properties of porphines and metalloporphines[END_REF] without and with the inclusion of the IPEA shift of 0.25 a.u., for the B2 symmetry (Table S2) and the A2 symmetry (Table S3).

Additional active orbitals for the active space C-CAS(14,16) (Figure S4).

Supplementary Information about

"Valence excited states of metalloporphyrins studied with the CASPT2 method: the case of an Iron complex"


Kremena Makasheva

Xiaoning Jiang

Reuven Gordon

James Spicer

2021 IEEE 16th Nanotechnology Materials and Devices Conference (NMDC) Welcome from the General Chairs

  the IEEE Nanotechnology Council (IEEE NTC) and the Conference Organizing Committee, it is our pleasure to welcome all authors, reviewers, session chairs, and participants to the 16 th IEEE Nanotechnology Materials and Devices Conference (IEEE NMDC) in Vancouver, BC, Canada, in December 12 -15, 2021. The IEEE NMDC was founded in 2005 and since that time it has developed as one of the major IEEE conferences in the rapidly growing field of Nanotechnology fundamental research and applications. The IEEE NMDC is held annually on a rotational principle in different locations in IEEE Regions 1-7 & 9 (North & South America), Region 8 (Europe & Africa) and Region 10 (Asia & Pacific), with the latest editions taking place in Anchorage, AK, USA (2015), Toulouse, France (2016), Singapore (2017), Portland, OR, USA (2018), Stockholm, Sweden (2019) and Nanjing, China (2020, virtually due to the COVID-19 pandemic). IEEE NMDC aims to develop critical assessment of existing work and future directions in nanotechnology research including nanomaterials and fabrications, nanoelectronics, nanophotonics, nanobio medical applications, devices, and integration. Its main purpose is to foster communication between physicists, chemists, microbiologists and engineers from academia and industry, interested in nanodevices and nanostructured materials, advanced preparation techniques, new material properties, standards and safety issues of nanotechnology, computer simulations and theoretical developments. The interdisciplinary exchange between academic scientists and contributions from industrial researchers, during the conference, stimulates gathering knowledge and helps inspiring new perspectives in applications on this exciting area. IEEE NMDC 2021 is primarily sponsored by the IEEE Nanotechnology Council. Valued patrons of the conference are NC State University (NC, USA), University of Victoria (BC, Canada), CMC Microsystems (Canada), Biosensors and Micromachines by MDPI, as well as the conference media partners: IEEE TNANO, IEEE OJ-NANO and IEEE Nanotechnology Magazine. The 16 th IEEE NMDC 2021 has a particular flavor since for the first time one of the main IEEE NTC conferences is organized in co-location with one of the oldest IEEE conferences, namely the 96 th IEEE Conference on Electrical Insulation and Dielectric phenomena (IEEE CEIDP 2021) of the IEEE Dielectrics and Electrical Insulation Society (IEEE DEIS). Given the challenges imposed by the pandemic of our time, COVID-19, the NMDC & CEIDP organizing committees succeeded to work in concert, in the effort to offer the conference attendees many parallel sessions, covering an enlarged panel of scientific topics and their industrial applications, organized in an excellent environment offered by the Pinnacle Harbourfront Hotel, located in the heart of Vancouver. Conference attendees that experience travel

restrictions in this difficult time will be given the possibility to remotely present their contributions.

Apart from the possibility of the registered participants to freely access the contributions and sessions of both conferences, they are highly encouraged to attend the NMDC & CEIDP two joint sessions and the joint workshop dedicated to topics of common interest.

The IEEE NMDC 2021 would not be successful without the commitment and contribution from many volunteers, authors, reviewers, conference participants, and partners. All are highly acknowledged for their help and support.

We wish you a productive and interesting conference and an agreeable time in Vancouver.


Elisa Miragliotta
email: elisa.miragliotta@unipv.it

Giulia Lisarelli
email: giulia.lisarelli@dm.unipi.it

Did you know you can draw a huge number of infinite heights? The students' realization tree of the heights of a triangle

Keywords: Realization tree, commognition, heights, triangle

In this paper we report on the analyses of the mathematical discourse of 7th-grade students about the solution of a task on the recognition of the heights of a triangle. Our aim is to describe through the commognitive lens the realizations that appear in the mathematical discourse of the classroom and to observe possible interactions between them. For this purpose, we construct and compare the expected and the actual students' realization trees, showing the richness of the realizations addressed by the participants in the mathematical discourse.

Introduction

A wide literature in mathematics education has documented students' difficulties in drawing and recognizing the heights of a triangle and has highlighted that the most common difficulties are related to obtuse triangles and right triangles [START_REF] Hershkowitz | Visualization in geometry: Two sides of the coin[END_REF][START_REF] Mariotti | Images and concepts in geometrical reasoning[END_REF]. Among the several reasons for explaining these difficulties, there is the influence of specific representations of the height, that are presented at school, and which become so popular and widespread that they are considered the concept itself by many students [START_REF] Hershkowitz | Visualization in geometry: Two sides of the coin[END_REF]. In particular, at least in the Italian context, at the end of primary school students' idea of "height of a polygon" is strongly related to stereotyped representations of it and this is often the result of a univocal definition of height [START_REF] Sbaragli | Convinzioni di allievi e docenti sul concetto di altezza di poligoni. [Students and teachers' beliefs about the height of polygons[END_REF].

Based on this scenario, and on the assumption that the notion of height is involved in many important mathematical results that students will face during their learning experiences, this study investigates the effects of a didactical approach for introducing students to the notion of heights of a triangle based on the presentation of multiple characterizations and representations. The effects are described in terms of the features of the height of a triangle, with respect to one side, that have become part of the classroom discourse.

Theoretical framework and research questions

We adopt Sfard's commognitive lens (2008) according to which discourse and its development are analyzed on their own, as main objects of research rather than as means to explore other constructs. We share the underlying assumption of this theory that thinking is not ontologically different from communicating and learning is a way of changing one's discourse. Commognition describes mathematical discourse as involving continuous transitions between signifiers that are "words or symbols that function as nouns in utterances" (Sfard, 2008, p. 154) and their realizations that are "perceptually accessible object[s] that may be operated upon" for producing narratives about the signifiers (Sfard, 2008, p. 154). These terms are used to emphasize that nothing is "there" to be represented, since mathematical objects are discursive objects. Moreover, some signifiers are more commonly used than others, but there is no one signifier that represents the object. This hierarchical role of realizations can be visually represented through realization trees [START_REF] Sfard | Thinking as communicating: Human development, the growth of discourses, and mathematizing[END_REF]. In this study, we are interested in the classroom discourse on the heights of a triangle. The construction of the students' realization tree allows us to gain insights into its growth and richness as resulted from a didactical sequence, thus, giving information on the formation of the mathematical object "height of a triangle" in students' discourse. Usually when students deal with a new signifier, they use different realizations separately; the conscious reference to the same signifier by using many different realizations is a learning achievement which is the result of a saming process [START_REF] Sfard | Thinking as communicating: Human development, the growth of discourses, and mathematizing[END_REF].

One of the main elements characterizing the mathematical discourse are narratives, i.e. statements (written or spoken) formulated as descriptions of objects, relationships between objects or processes, which are subject to possible approval or rejection. The shared narratives of a classroom are those that are considered true; they may be different from those accepted by the scientific community. Moreover, mathematical discourse involves a consistent use of routines, i.e. specific repetitive patterns [START_REF] Lavie | Routines we live by: From ritual to exploration[END_REF]) and of visual mediators, i.e. objects of symbolic, iconic, gestural nature to which we refer [START_REF] Nachlieli | Growing mathematical objects in the classroom -The case of function[END_REF]. For example, in geometry, the drawing of a figure can be a visual mediator in both paper-and-pencil and dynamic geometry environments.

In light of this theoretical lens, the research questions guiding our investigation are: (a) What realizations of the signifier height of a triangle can be identified in students' discourse? (b) Are there interactions between different realizations? If so, when do they occur?

Methodology

The data we present are collected in a 7th-grade class, during a video recorded lesson on the recognition of the heights of a triangle. The students were introduced to the mathematical object height through different possible realizations. In particular, two main realizations were proposed by the teacher, who was participating in a larger project aimed at designing and experimenting inclusive mathematical activities at middle school. We describe these two realizations as follows: T1) Segment perpendicular to a side, or to its extension, passing through the vertex not contained in that side (opposite). It was discussed with students that this realization can be realized in two ways: as a segment drawn from the vertex (T1-a) or from the side (T1-b). The teacher also promoted the construction of heights using both physical (ruler and set square) and digital (GeoGebra) artifacts. T2) Height of the strip in which the triangle is inscribed. This realization addresses the height of a triangle in terms of the height of the strip into which the polygon can be inscribed. A strip is determined by two parallel lines: one line contains a side of the triangle and the other the vertex that does not belong to the chosen side.

These two realizations are chosen because T1 is very familiar among students at the end of primary school and it is largely proposed in school textbooks at different levels; on the contrary, T2 was intentionally introduced for the first time by the teacher because it lays the foundation for looking at the length of the height as the class of congruent segments and for overcoming the common difficulty consisting in placing the height outside an obtuse triangle.

In this study we focus on a 1-hour lesson, conducted by the second author of the paper with the regular mathematics teacher. Students are given a task (Figure 1), involving an obtuse triangle.

Observe the figure. Are all the four dotted segments heights of the triangle? Explain why. The task is designed for triggering the students' discussion on the four dotted segments. In order to answer the question, students have to argue why each of them can or cannot be a height of the given triangle. The explanations will inform us on the realizations that the class has developed until then.

The a priori realization tree

Building on Weingarden and colleagues' (2019) elaborations, the theoretical realization tree (Figure 2) is a priori constructed according to all the possible realizations of the height that students, who were exposed to specific geometrical activities on the height of a triangle, might refer to. The tree has the root in the signifier "height of the triangle with respect to a side". From this root, two main branches with T1-realizations (on the left) and T2-realizations (on the right) develop; each branch contains both verbal and visual realizations. At the first level under the root, there are the main realizations that are explicitly used during the geometrical activities.

Focusing on the left side of the tree, the node at the first level splits into two branches (a and b). At this level the realizations (a.1, a.2, b.1, b.2) maintain some traces of the construction rituals. T1-a realizations characterize the height by starting from all the straight lines passing through a vertex and choosing the line that perpendicularly intersects the straight line upon which the side of the triangle lies; the second extreme of the height is reached in this way. Within the tree, we reported these realizations by referring to the bundle of straight lines centered in a vertex of the triangle. In these realizations the starting point is the chosen vertex. T1-b realizations characterize the height by starting from all the straight lines perpendicular to the side of the triangle and choosing the line that passes through the non-consecutive vertex. Within the tree, we reported these realizations by referring to the bundle of straight lines perpendicular to the chosen side of the triangle. In these realizations the starting point is the chosen side.

Focusing on the right side of the tree, starting from a realization with many heights (1.1, 2.1), following the branch we can reach a realization where only one height passing through a vertex is considered (1.3, 2.3). The coexistence of these different realizations in a student's tree presumes that a saming process was accomplished, as for the expert mathematicians.

The leaves at the lower level make explicit the realizations of the heights with respect to the given task. Although they are visualized only as visual realizations, since the given drawing provides the student with a common visual mediator, we use them also for verbal realizations such as "CE" or "BG" which make sense only coupled with the given drawing.

How the analyses were conducted

Data were cyclically analyzed, passing through several rounds of analyses. The preliminary step is the transcription of the classroom discussion by reporting all the components of students' discourse, i.e. verbal utterances, drawings, gestures. Working on this transcription, the first round is aimed at identifying the instances of realizations -labeled using a progressive number in square bracketswithin the students' discourse. For example, an utterance as "if you put the ruler on BA, you have also the height CE" is coded as a verbal realization of the height with respect to the side AB.

The second round focuses on these instances of realization. The analysis aims at identifying the type of realization that students refer to. In particular, the coding is conducted according to the closeness of students' discourse with one of the descriptions of the height as a T1 or T2 realization. For example, a statement such as "it is a height because it goes to the opposite vertex" is coded as a verbal T1realization of the segment traced from the side to the vertex, and the reference to the movement towards the opposite vertex allows us to label it as a T1-b realization. The closeness is also established according to the inferences of the researcher who knows the endorsed narratives developed by the class during the previous activities. For example, the expression "Paolo's method" is coded as a T2realization because it was developed by the students for shortly referring to the ritual of drawing the heights inside the strip. This round allows us to qualitatively describe the characteristics of each realization and therefore provides useful information for answering the first research question.

These two rounds of analyses are fundamental for gathering the essential information on the realizations to be exploited for the construction of the actual realization tree. For the subsequent round, we start from an empty structure of the a priori realization tree; we read again the labeled transcription for filling the leaves with the corresponding realizations. For example, since the first instance of realization emerges in Alessio's discourse and it is coded as T1-b, we write "[1]Alessio" in the leaf named b (Figure 3). The unexpected realizations are included as new leaves with the label NEW. After this round of analyses we obtain a new tool for observing at a glance how the realizations were spread during the discussion. Moreover, this round allows us to observe which realizations of the height have become part of the classroom discourse and which not yet. The progressive numbers coupled with the name of the students allow us to keep track of the temporal development of the realizations during the lesson and of the students who participated in the mathematical discourse. The actual realization tree provides us with a tool for observing the possible interactions among different realizations, and therefore for answering the second research question.

Preliminary findings: the realization tree as a tool for analyses

The actual realization tree (Figure 3) highlights that there are some empty leaves for the expected realizations that are not addressed by students and new leaves for unexpected realizations that emerged within the classroom discourse.

Comparing the a priori and the actual realization trees

As regards the expected realizations that are not addressed by students, we notice the bundle of straight lines centered in a vertex of the triangle. Although students refer to realizations of height that are drawn from the vertex towards the side of the triangle ([20], [22]), in their discourse we do not find elements that suggest the choice of the perpendicular line among the many passing through the vertex. This aspect can be justified by frequent references to the construction ritual through ruler and set square. Among the T1-realizations, this ritual shifts much of the students' focus toward realizations that implicitly make use of the bundle of parallel lines ([2], [19]); in this way, the perpendicularity is embedded into the use of the set square. Among the T2-realizations, the empty leaf in Figure 3 shows that there are no explicit references to the strip in which the triangle is inscribed. Rather, the students' discourse seems to be linked to the construction ritual, that is Paolo's method, which involves a line parallel to the side in focus.

As regards the unexpected realizations, there are leaves (see NEW in Figure 3) which report on the discourse on segments different from CE as realizations of the height for the side AB, and realizations which refer to triangles different from ABC. In the former case, students' discourse is focused on the segment BD, after that Lavinia triggers the attention on it ([6]) as a possible T2-realization of the height. Then she adds further details ([15]): Lavinia:

If you put the ruler on the segment AB, okay, are you with me? And then you take the set square. You start drawing some heights and BD is one of those.

Roberto accepts, to some extent, Lavinia's idea of considering BD as a height, but he addresses a different triangle ([16]): Roberto:

I think you're looking at a hidden triangle ABD and you think BD is a height.

Claims like this lead us to include another leaf in the actual tree (see New2 Figure 3). This is a T1realization since Roberto's discourse does not contain any reference to the strip.

The last new T2-realization is introduced by Alice who proposes a refined version of Lavinia's realization of the height. Alice starts describing how to draw a height for AB which passes through point B ([31]); the researcher supports her description by adding new visual elements on the visual mediator, which is shared with the whole classroom. Comparing the new realization with the segment BD, Alice explains why BD is not a suitable realization of height for AB ([33]):

Alice: Therefore, BD cannot be a height, because it has to reach at least... the intersection.

Building on Anita and Lavinia's discourse, students consider the possibility of having more than one height for AB and continue arguing on these T2-realizations.

Another unexpected realization is the one proposed by Fiorella. She considers another triangle, ABG, and shares her drawing with the classmates (see New3 Figure 3). In particular, Fiorella uses this new visual mediator to justify the possibility for BD to be accepted as a realization of height, and in line with Roberto's discourse she stresses the need to change the reference triangle. However, she disagrees with Roberto concerning the choice of such a triangle.

Moving from T1 to T2 realizations

By looking at the progressive numbering of the labels in the realization tree (Figure 3) it is possible to observe the distribution of T1 and T2 realizations within the classroom discourse. Generally speaking, each student seems to be tied to a certain type of realization. Indeed, during the entire lesson some students, as Alessio, Roberto, and Giulio, focus on T1-realizations while other students, as Lavinia and Ciro, focus on T2-realizations. Although most of the students remain anchored in a certain type of realization and they do not spontaneously move back and forth between different types of realizations, we can notice that when Roberto moves from the T1 branch to the T2 ([23], [24]) some other students start reporting on T2-realizations in their discourse. Therefore, the global classroom discourse goes through a shift from T1 to T2 realizations. This shift in discourse develops around the segment BD. Lavinia claims that BD is a height, addressing T2-realizations ([15]). She refers to the use of the set square, describing the construction ritual for drawing "several heights" (T2-realization). She does not speak about moving the set square in order to trace the only segment that intercepts the vertex of the triangle. On the other hand, other students argue that BD is not a height, bringing arguments based on T1-realizations. In this respect, the exchange between Lavinia and Roberto is very interesting:

Lavinia: So BA is the segment. Did you know you can draw a huge number of infinite heights? Roberto:

But if they don't go to the opposite vertex... Lavinia:

But that doesn't matter! They don't have to go!

The turning point occurs when Roberto changes perspective and switches from the left to the right side of the tree, that is, when a T2-realization of height enters his discourse ([24]). This shift is shown in the tree: between [23] and [24] Roberto changes side. The corresponding discourse is:

Roberto: Do you know how to make a height? You have to put a ruler and a set square, from the opposite vertex you have to draw a line to the opposite side or to its extension, that's what the EC side is for. Roberto:

If you really want to make a height here [he points at a position on AB], at least you have to use Paolo's method.

The reference to "Paolo's method" seems to be key in shifting the attention of the class towards T2realizations. Indeed, after Roberto's intervention there are many references to T2-realizations: At this point Roberto tries to convince Lavinia that in order for the segment BD to be a height it lacks an important feature: the point D does not intercept the strip.

Roberto:

Oh thanks [sarcastic tone], but then DB is not a height if it doesn't reach there.

The first realizations of height in his discourse involve a segment with endpoints at one vertex and at the opposite side of the triangle, so his interventions are all placed in the left branch of the realization tree. Then Roberto seems to include in his tree, through a saming process, also segments not passing through any vertex of the triangle, and this happens when he starts participating in Lavinia's discourse.

The transition from T1 to T2 realizations seems to play a fundamental role in the classroom discourse. By remaining anchored to only T1-realizations of height, students fail to convince Lavinia that BD is not acceptable. When the focus of Roberto's discourse shifts to T2-realizations also the other students become participants in Lavinia's discourse and begin to use the same words and visual mediators. Finally, the classroom discourse finds a synthesis in the realization constructed by the researcher under the students' guidance (see New4 Figure 3).

Conclusion

The analyses of the classroom discourse allow us to conclude that students spontaneously refer to both T1 and T2 realizations of height. By looking at the realization tree (Figure 3) it is possible to grasp the richness of the classroom discourse, involving different realizations of height: students refer to most of the expected realizations and they also introduce new ones. Each student remains essentially tied to one type of realization; therefore, we do not observe in their discourse interactions between T1 and T2 realizations. However, we can notice how the interaction is promoted by the attempt to match two discourses on two different realizations. This is evident in the discussion between Roberto and Lavinia, who refer to a T1 and a T2-realization of the height, respectively, while they are searching for a shared narrative about BD. The interaction becomes necessary for them to participate in the same discourse and Roberto plays a key role in moving from a T1 towards a T2realization. In this way, he succeeds in finding a common ground with Lavinia and also in involving the whole classroom in the discourse. In particular, the "use of the set square" and "Paolo's method" are shared narratives referring to rituals that have been culturally constructed by the classroom and they are linked to both T1 and T2-realizations.

Comparing the two realization trees, we can claim that students include in their discourse most of the expected realizations. From the educational point of view, this is relevant since the richness of the realization tree seems to be a marker of the effectiveness of an unconventional -but in line with the solid findings of the research in mathematics education -introduction to the height of a polygon which explicitly works on different realizations. From the theoretical point of view, the study presents a first attempt of using realization trees as tools for a priori and a posteriori analysis of the classroom discourse in the domain of geometry, where the visual mediator (i.e. the drawing) has a specific and prominent role [START_REF] Mariotti | Images and concepts in geometrical reasoning[END_REF]. In this perspective, the classroom realization tree needs further elaboration to become a more effective tool in terms of reporting on the students' discourse behind the construction of the tree and the role of the teacher's intervention in supporting such a discourse.

Although this is an exploratory study and the findings are quite local, we hope that considerations emerging from this paper can provide educators and researchers with food for thought, in particular concerning the didactical value of promoting a widespread use of many different realizations for the same signifier as a means for triggering the students' participation into mathematical discourse.

Figure 1 :

 1 Figure 1: The heights recognition task

Figure 2 :

 2 Figure 2: The a priori realization tree of the signifier height of the triangle with respect to a side

Figure 3 :

 3 Figure 3: The classroom realization tree of the signifier height of the triangle with respect to a side

Acknowledgment

We are deeply endowed to Anna Baccaglini-Frank for having pointed us in this direction of research. Moreover, we wish to warmly thank the teacher, Federica Poli, for carrying out and documenting the activities, and all the students involved in this study.