Quantum mechanical study of transition metal hydrides: Comparison of determined molecular properties with experimental data

This study compares results of four relativistic pseudopotential basis sets, which differ mainly by their size: double‐zeta introduced by Hay and Wadt from Los Alamos National Laboratory (LANL2DZ), triple‐zeta based on Stuttgart energy‐consistent scalar‐relativistic pseudopotential (SDD3), its extension with 2fg polarization functions, and combination of Stuttgart pseudopotentials with quintuple‐zeta cc‐pV5Z base (SDD5). Hydrides of transition metals from Cr to Zn group are chosen as reference molecules. The coupled cluster method (CCSD(T)) is used for evaluation of selected molecular characteristics. Interatomic distances, dissociation energies, vibration modes, and anharmonicity constants are determined and compared with available experimental data. As expected, the accuracy of basis depends mainly on its size. However, only moderate modification of SDD3 basis set significantly improves its accuracy, which becomes comparable to the largest basis set. Nevertheless, the time consumption is significantly lower.

method was suggested in Reference 22, which was applied also for TM (Ni, Co) interactions with CO 2 and N 2 O molecules. 23Experimental study on hydrides and deuterated hydrides of coinage metals was published by Le Roy group 24,25 where a careful isotopomer analysis of metals is also presented.Based on vibrational-rotational parameters dissociation energies and equilibrium distances are presented: D 0 = 65.8 (Cu), 55.2 (Ag), and 77.8 (Au) kcal/mol; r O = 1.463 (Cu), 1.618 (Ag), and 1.524 (Au) Å.
Recently Li et al. 26 published magneto-infrared spectra of NiH measured in several noble gas matrices supported by ab initio or DFT calculations demonstrating a very good match with Herzberg Table 2. Another interested study dealt with tungsten hydrides and deuterated hydrates WH/WD À WH 6 /WD 6 27 measured in rare gas matrices (Ne, Ag).
We are aware of the fact that a large number of theoretical calculations were performed on the topics of transition metal hydrides, however, just a few most relevant computational studies are mentioned here.One of the very early papers on this topics comes from Baerends laboratory 29 where authors explored relativistic effects on hydrides of coinage and zinc group metals.The predicted distances, dissociation energies, and vibrational frequencies were in good accord to experimental values from Handbook of Molecular Constants. 30A year later, Lee and McLean 31 published similar paper on Ag and Au hydrides by Dirac-HF method.One of the first applications of pseudopotential on gold hydride was performed by Schwerdtfeger et al. 32 using CISD and CEPA methods.Much more recent results were published in methodological paper from Head-Gordon laboratory 33 where dissociation energies of the first-row transition metal hydrides were explored.Their CCSDTQ results differ from our approach using the S2FG basis set for D eA values by RMSD = 2.3 kcal/mol.(In this case, it was not possible correctly compare SDD5 approach due to a small number of available values because of missing basis sets of several elements, cf.below.)In similar methodological study employing large set of DFT functional, 34 authors claimed that larger amount of HF exchange is necessary (up to 50%) for hydrides of TM and therefore, these hydrides represent specifically challenging systems for DFT.Nevertheless, CAM-B3LYP performed as one of the best functionals together with some variants of B97 functional.group of three quazi-relativistic small-core pseudopotentials is chosen with corresponding basis sets taken from the Basis Set Exchange Library [40][41][42] ; namely Hay-Wadt LANL2DZ, 43,44 Stuttgart relativistic small core (RSC 1997) pseudopotentials and corresponding basis sets (labeled as SDD3 here after), [45][46][47] and another basis set, which can be found at the Basis Set Exchange Library under acronym cc-pV5Z-PP, 48,49 18 From those fitted potentials, all examined molecular parameters (optimal distances, bonding energies, frequencies, and anharmonicity constants) are also evaluated.The received parameters are compared with available experimental data.

| COMPUTATIONAL DETAILS
The computationally fastest pseudopotentials and basis sets used in this work are Hay-Wadt LANL2DZ, 43,44  Stuttgart relativistic small core pseudopotentials with atomic basis set is of extended triple-zeta quality (SDD3), [45][46][47] which describes valence shell plus sp-electrons from the previous lower shell.
Also, projector operators of the pseudopotentials contain one additional set of polarization projectors (i.e., f-projectors for third row of TM, up to the h-projector in the case of third row of TM).
The third approach is based on modified pseudopotentials (Stuttgart RSC 1997) in combination with Dunning's cc-pV5Z-PP basis set, 48,49 with substantially extended description of outer electrons including up to one set of i-polarization functions.[50][51] Polarization functions consist of uncontracted 4f3g2h1i orbitals.Hydrogen atom is considered consistently to individual pseudoorbitals; that is, Dunning Hay (31s)/[2s] DZ base 64,65 is used in LANL2DZ calculations, Pople's 6-311G(d,p) base, 60 in the case of SDD3, extended 6-311G(2df,2pd) 66 for the S2FG group, and with cc-pV5Z base 67 in calculation where metals were treated by SDD5 basis sets.
These four different basis sets are used for optimization at the CCSD level of theory and for scanning of the CCSD(T) energies in dependence on interatomic distances.All the calculations are performed with Gaussian 09 program package.In the scan evaluation, energies of ca.70 points with united step of 0.05 Å are determined.Then, the scanned potential energy points are fitted by the Morse potential (M) 15 : From the a parameter and/or k parameter of the harmonic potential (Equation 14), vibrational frequencies of the given hydrides are evaluated using equations: and/or where M is reduced mass of the given hydride.Since the Morse curve is not symmetrical the anharmonicity constant is defined by formula: and then the energy of eigenstates of Morse oscillator can be evaluated as: where the ZPVE values can be obtained by assuming n = 0. Finally, for the comparison with experimental data, the anharmonicity constants were also evaluated: In order to increase the accuracy of explored molecular parameters two modified Morse potential functions are considered.First, the so-called Morse polynomial to the fourth degree (MP) in the form: Second, the Hulburt-Hirschfelder 5-parameter potential function (HH) 18 : Here, the first term in the square bracket represents the original Morse potential.Please, note that D eX denotes dissociation energy obtained based on corresponding method without Zero-Point Vibration Energies (ZPVE).If ZPVE is included label D 0X is used instead.
An arbitrary potential energy function can be expanded about r e in powers of ξ ¼ x À r e ð Þ=r e , This expression, introduced by Dunham, 16 defines coefficients a n .
They are related to the parameters c n of the Morse polynomial (Equation 6) as follows (Coolidge et al. 17 ): Concerning the H-H function (Equation 7), they are found to be determined by the relations: Frequencies and anharmonic corrections can be evaluated using Dunham's coefficients as follows 18 : where B e ¼ h= 8π 2 μr 2 e c À Á is a rotational constant.ZPVE corrected DE for an arbitrary potential can be evaluated as: as approximated by Dunham 16 and Hirschfelder. 18

| Determination of equilibrium interatomic distances and dissociation energies
The equilibrium distances are determined at the harmonic approximation using the optimization algorithm (r O ) and also by fitting procedure of the scanned energies (applying the least square methods) to the above-mentioned potentials.
For the determination of DEs, two different approaches are utilized.The first one is based on definition of dissociation energy, that is, energy difference between the given hydride (SP calculation using CCSD(T) method with CCSD optimized bond length) and both atoms (in their ground states) D eA and with ZPVE correction À D 0A .The second approach uses fits of: (a) the Morse potential (Equation 1), (b) the Morse polynomial function (Equation 6), and (c) the H-H modification of the Morse potential (Equation 7).

| Vibration analysis and first anharmonicities
The above-mentioned fitted potentials are employed also for determination of vibrational frequencies and anharmonicity constants.However, to get more accurate fitted data, different strategy is adopted.Here, only relevant part of the points in the close vicinity of the potential minimum (usually about 20-30 points) are employed within the fitting procedure.
In previous fit for determination of DEs, the whole energy scan is used so that fitted potential curves are somehow worse reproduced.The bond lengths are always slightly underestimated and vibration parameters are also not so accurate.Therefore, in this second (or partial) fit, bond lengths and vibrational parameters are evaluated with substantially higher accuracy (overestimating bonding energy).
Moreover, from the closest points to the potential energy minimum (usually eight lowest points) also harmonic frequencies are determined, fitted by parabolic curve: at the same CCSD(T) computational level for all four basis sets.In this way, other optimal bonding distances (r H , r M , r MP , and r HH ) are obtained and will be used in further discussion.
3 | RESULTS AND DISCUSSION

| Search for the ground state multiplicity
It is very important to optimize correct ground state geometry.Therefore, special accent is laid on the correct multiplicity and corresponding electronic structure.In the first step, the optimal geometries for several possible multiplicities were searched at the CCSD/SDD3 computational level and the global minimum was found (the electronic structure corresponds to single sigma bond, usually slightly enhanced by some π-backdonation to p-AO of hydrogen).If the energy difference between the lowest and the second lowest multiplicities was greater than 0.25 eV (ca.7 kcal/mol) the same spin state for the other basis sets was expected to be the ground state as in the SDD3 basis.The energy differences for searched multiplicities are displayed at the CCSD/SDD3 computational level in Table 2.When the energy differences among the calculated multiplicities were smaller than 0.25 eV the optimization procedure was repeated for the remaining basis sets independently.This situation occurred only for two hydrides: TcH and ReH.In the TcH case, calculations in all basis sets confirmed quintuplet as the ground state, with the only exception of SDD5 basis set where the preferred ground state is septuplet.In the case of ReH, all basis sets prefer quintuplet despite small differences between both states.

| Interatomic distances
In T A B L E 2 Energy difference between ground state and other considered multiplicities ΔE X (X = 1 for singlet, 2 for doublet, etc.) determined at the CCSD/SDD3 computational level for each hydride (in eV).or Li et al., 26,27 mentioned in Section 1.

| Dissociation energies
The DEs are estimated based on two approaches.First, as already mentioned, according to the definitionas a difference between than for the larger S2FG and SDD5 ones.Nevertheless, the RMSD values of all the basis sets are relatively similar (with exception of LAN) to make any essential conclusions.The complete collection of determined anharmonicity constants for all the basis sets and both hydrides and deuterated hydrides are collected in Table S4.
The H-H potential gives the smallest anharmonicities for almost all hydrides, nevertheless, there are some exceptions when MP anharmonicities are negligibly smaller (by ca. 1 cm À1 ).In contrast, the anharmonicities obtained from the fit of the Morse potential are visibly higher leading to better agreement with experimental data.This is also confirmed by the RMSD values for the SDD3, S2FG, and SDD5 computational settings where the anharmonicities obtained by Morse potential exhibits the lowest deviations.
Calculated frequencies are also in reasonable match with the above-mentioned experimental data from Li et al. 26 7).

| CONCLUSIONS
In this study, performance of four basis sets is compared for transi- Comparing the quazi-relativistic basis sets, it can be concluded the best results (in comparison with measured data) are received using T A B L E 7 RMSD to experimental data for interatomic distances r e (in Å), dissociation energies a D 0 (in kcal/mol), vibrational frequencies ω e (in cm À1 ), and for anharmonicity constants ω e x e (in cm À1 ).
who found the ω e (WH) frequencies about 1853 and WD 1328 cm À1 and Wang and Andrews, 27,28 evaluating the same WH frequencies: ω e = 1911 (WH) and 1354 (WD) cm À1 and frequencies for manganese and rhenium hydrides: ω e = 1478 (MnH), 1066 (MnD), 1985 (ReH), and 1423 (ReD) cm À1 (Table tion metal hydrides.Besides one of the most frequently used basis set for TM (LANL2DZ), effectivity of Stuttgart energy-consistent relativistic (small core) pseudopotentials with triple-zeta quality base (SDD3) is examined and compared with moderately modified/ extended basis by 2fg polarization functions (S2FG), and with substantially larger quintuple-zeta cc-pV5Z basis set (SDD5) containing extended set of SDD3 pseudopotentials by additional two sets of s, p, and d projection operators.Application of relativistic pseudopotentials is really crucial for employing methods based on the nonrelativistic Schrödinger equation.The highly correlated CCSD(T) method is applied on determination of examined physico-chemical properties-interatomic distances, DEs, and vibrational analysis considering not only stretching vibration mode but also the anharmonicity constants.In preliminary part of this study, it was shown that application of extended well-tempered all-electron basis sets (WTBS) leads to completely incorrect molecular characterization even for the first row of the lightest TM arriving to too short interatomic distance and other shortcomings known from literature.
they do not consider relativistic effect and in this way, they failed to predict experimental data with sufficient accuracy.In order to demonstrate this fact, the well-tempered basis set for TM (WTBS References 37,38) and ANO-RCC for H (Reference 39) were used for the first row of transition metal hydrides.Even such a very large basis sets were not able to predict bond distances of the studied hydrides even qualitatively correctly; large difference of equilibrium distance for MnH: r exp = 1.731 and r opt = 1.450Å led to too high bonding energy (D 0M = 68 kcal/mol vs. experimental estimation D e < 55 kcal/mol) at the CCSD level of calculations.Similarly, also in the CuH case, the measured equilibrium distance (r exp = 1.463) is much longer in comparison with AE-calculations: r opt = 1.289Å with analogous overestimation of the calculated DE: D 0M = 115 kcal/mol versus 60 kcal/mol determined experimentally.Therefore, only a 36Despite we could not find experimental data systematically for all transition metal hydrides, the calculations are performed for the TM from the sixth (Cr) to the twelfth (Zn) group.Great attention is also paid to determination of the correct ground state multiplicity of the explored hydrates.Selection of compared basis sets follows from general availability.The all-electron (AE) basis sets were excluded in preliminary part of this study since this study, several different sets of interatomic distances are deter- a If basis set available.
68,69ly longer by several hundredth of Å, the distances from SDD5 and S2FG bases give distinctly more accurate estimations.This trend can be quantitatively expressed in RMSD values displayed in the first section of Table7.The interatomic distances of the complete set of all the explored hydrides in all the basis sets are summarized in TableS1.In that table, different sets of bond lengths are collected: from (a) the CCSD optimization r O , (b) fit of the harmonic potential r H , (c) fit of the Morse potential r M , (d) fit of the Morse polynomial r MP , and (e) fit with H-H potential r HH .The interatomic distances obtained from the fit of the harmonic potential are very close to distancesbased on the fitted Morse potential (due to manner of fitting procedure).In the case of TcH and ReH, the predicted multiplicity is quintuplet.However, the septuplet state is only about 0.15 eV higher.As one could expect evaluated distances are relatively different (higher multiplicity has about ca.0.15Å longer bond length) so that it would be possible to make decision based also purely on the experimental data (at least in the Re case).Unfortunately, according to our knowledge no experimental evidence exists in literature.Another fact can be noticed in TableS1where for mercury hydride distances obtained by LAN and SDD3 basis sets are longer than analogous distances of cadmium hydride (r O (CdH) = 1.833 and r O (HgH) = 1.883 in LAN and r O (CdH) = 1.804, and r O (HgH) = 1.820 in SDD3).In all other hydrides and all the considered basis sets, the "second periodicity" as recognized by Pyykkö68,69is fulfilled (i.e., the longer distances for hydrides from second row of TM in comparison with third row).Regarding various types of PES potentials, it can be concluded that the Morse poly- a For all basis sets except SDD5.b For all basis sets.usually corresponding hydride and both atoms in their ground states.Similarly to hydrides, energies of metal atoms in several multiplicities were calculated and compared with experimental values.The second possibility follows from the fitted potentials (Morse function and polynomial, and H-H function).In Table4, there are listed DEs for experimentally known hydrides and compared with performance of all four basis sets.All the computed energies are corrected with ZPVE to be comparable with experimental data.RMSD values in Table7show again comparable accuracy of the S2FG and SDD5 levels of calculations.Nevertheless, S2FG basis set saves large amount of computational time.Regarding various types of PES potentials, it can be concluded that all of them give very similar results that usually do not differ by more than 1 kcal/mol.However, DEs obtained based on the H-H function are systematically slightly smaller than values from the Morse polynomial and, in the majority cases, also from the original Morse potential.RMSDs of the individual potentials in a particular basis set are very close, especially in the S2FG and SDD5 basis sets, where they do not differ more than 0.2 kcal/mol, cf.Table7.Therefore, the RMSDs to experimental data depend mainly on the used basis set rather than on the potential curve.Complete set of DEs are for all hydrides collected in TableS2.
Comparison of dissociation energies D 0 (in kcal/mol) obtained from the fitted H-H potential and values obtained from the definition for some experimentally known hydrides.e (MnH) = 1478 cm À1 , ω e (MnD) = 1066 cm À1 Reference 28.
Only for determination of DEs complete set of scanned points was used, cf.discussion in the part "Vibrational analysis."TA B L E 6 Comparison of experimentally 1,2 found anharmonic constants ω e x e -with calculated results obtained from the fitted H-H potentials (in cm À1 ).