Multiconfiguration Pair-Density Functional Theory and Complete

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Multiconfiguration Pair-Density Functional Theory and Complete Active Space Second Order Perturbation Theory. Bond Dissociation Energies of FeC, NiC, FeS, NiS, FeSe, and NiSe Kamal Sharkas, Laura Gagliardi, and Donald G. Truhlar J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b09779 • Publication Date (Web): 14 Nov 2017 Downloaded from http://pubs.acs.org on November 19, 2017

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The Journal of Physical Chemistry for J. Phys. Chem. A, Nov. 13, 2017

Multiconfiguration Pair-Density Functional Theory and Complete Active Space Second Order Perturbation Theory. Bond Dissociation Energies of FeC, NiC, FeS, NiS, FeSe, and NiSe Kamal Sharkas, Laura Gagliardi,* and Donald G. Truhlar* Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, 207 Pleasant Street SE, Minneapolis, MN 55455-0431 Abstract. We investigate the performance of multi-configuration pair-density functional theory (MC-PDFT) and complete active space second-order perturbation theory for computing the bond dissociation energies of the diatomic molecules FeC, NiC, FeS, NiS, FeSe, and NiSe, for which accurate experimental data have become recently available [Matthew, D. J.; Tieu, E.; Morse, M. D. J. Chem. Phys. 2017, 146, 144310-144320]. We use three correlated participating orbitals (CPO) schemes (nominal, moderate, and extended) to define the active spaces (CAS), and we consider both the complete active space (CAS) and the separated-pair (SP) schemes to specific the configurations included for a given active space. We found that the moderate SP-PDFT scheme with the tPBE on-top density functional has the smallest mean unsigned error (MUE) of the methods considered. This level of theory provides a balanced treatment of the static and dynamic correlation energies for the studied systems. This is encouraging because the method is low in cost even for much more complicated systems.

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1. Introduction The bond dissociation energy (BDE) is a fundamental quantity used to describe a chemical bond.1 In transition metal chemistry, the accurate prediction of BDEs requires a balanced treatment of the static and dynamic correlation effects.2, 3, 4, 5, 6, 7 The static correlation effects in transition metal-containing compounds come from a combination of near-degeneracy correlation due to incomplete d subshells8 and weak overlap effects due to ligand-metal exchange repulsion.9 In the complete active space self-consistent field (CASSCF) method,10 which is widely used for accounting for static correlation, the wave function includes configuration interaction (CI) among configuration state functions with all possible occupancies of the active orbitals, which is called full configuration interaction (FCI) in the subspace. Computational savings are available by limiting the configuration interaction in the active space, and several procedures have been suggested; for example, the generalized valence bond perfect-pairing (GVB-PP) method11, 12,13 the restricted active space (RAS) method,14,15 the occupation-restricted-multiple-active-space (ORMAS),16,17 and the generalized active space (GAS)18,19,20 method. In the GAS method, the active space is partitioned into subspaces, and within each subspace, one uses all possible occupancies of a given number of electrons, and all or most interspace excitations are forbidden. In the present article, we consider a special case of the GAS method, namely the separated-pair21 (SP) wave function, which is built by partitioning an active space into subspaces that contain at most two orbitals while forbidding all interspace excitations; each subspace has either one or two electrons in one or two orbitals (typically two in two). This strategy can greatly reduce the number of configuration state functions in the CI expansion of the SP method as compared to the CAS method, and this results in a much less-expensive computation compared to the corresponding CASSCF calculation. In this work, we consider both CASSCF and SP, each of which is a special case of multi-configuration self-consistent-field (MCSCF) theory. In addition to deciding on which configurations to include in the CI for a given active space, one must also specify the active space. In the present work we do this with systematic procedures. In particular, we use three versions (nominal, moderate, and extended) of the correlated participating orbitals22,23,24 scheme. The correlation energy included in a CASSCF calculation corresponds to what


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Silverstone and Sinanoglu25 called internal correlation energy. It includes the static correlation energy and a part of the dynamic correlation energy. The SP approximation, by including a subset of the CAS configurations, is aimed at including the most important part of the internal correlation energy. The inclusion of internal correlation effects by using a multi-configuration wave function is not sufficient to provide quantitative energetics in reaction processes and/or spectroscopy. External correlation energy (the rest of the dynamic correlation energy, beyond that included in the CASSCF calculations) has to be included for quantitatively accurate results. In principle, the dynamic correlation energy can be converged by using active spaces of increasing sizes in the SCF step; however, the convergence of dynamic correlation energy is impractically slow with respect to the number of configurations included. Post-MCSCF wave function methods, such as the complete active space second-order perturbation theory26,27,28 (CASPT2) and multireference configuration interaction29,30,31 (MRCI), can be used to recover dynamic electron correlation more in a more practical manner. Unfortunately, the computational cost (both in memory and time) of these methods rapidly becomes unaffordable as one includes larger numbers of configuration state functions in the MCSCF step. In this respect, the recently developed multi-configuration pair-density functional theory32 (MC-PDFT) is a promising alternative as a post-MCSCF method because in this approach, the post-SCF step has negligible cost compared to the MCSCF step. The electronic energy is computed as the sum of the electronic kinetic energy and the classical Coulomb energy evaluated using a MCSCF wave function and an additional energy contribution computed using a functional (called an on-top functional) of the total one-electron density and the on-top pair density, which is the probability of finding two electrons at the same point in space. The density and on-top pair density used in the on-top functional are generated from the MCSCF wave function. Although MC-PDFT has now been tested quite widely, with generally encouraging results,21,32,33,34,35,36 the tests on transion metal systems are still sparser than one would like, in part because of the paucity of highly accurate test data. Therefore, the recent experiments of Morse and coworkers generating accurate data for a large number of transition-metal diatomic molecules is very welcome, and here we use the most recent of that data37 to test MC-PDFT



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for molecular dissociation energies, in particular we present such tests for six diatomics: FeC, NiC, FeS, NiS, FeSe, and NiSe. The electronegativities of Ni, Fe, S, Se, and C are respectively 1.91, 1.83, 2.458, 2.55, and 2.55. Thus the electronegativity differences of these bonds all fall in the narrow range of 0.64–0.75; and they may therefore be considered to be polar covalent bonds with a small but non-negligible amount of charge transfer. We present tests based on two kinds of MCSCF wave functions, namely CASSCF (leading to CAS-PDFT) and the SP approximation (leading to SP-PDFT). We apply each of these kinds of wave function with three schemes for selecting the active space, namely the nominal, moderate, and extended correlating participating orbital schemes. The on-top functionals we test have all been presented previously, namely three simply translated on-top functionals32 (tPBE, trevPBE, tBLYP) and three fully translated on-top functionals36 (ftPBE, ftrevPBE, ftBLYP); the precise protocols for translation and full translation are given in previous papers.32,36 For comparison with the MC-PDFT results, we also present calculations by CASPT2. 2. Computational Methods All computations were performed using a locally modified version of Molcas 8.1.38 We carried out gas-phase state-specific single-point multi-reference calculations by CASSCF, GASSCF (in particular SP), CASPT2 (with the standard empirical IPEA39 shift of 0.25 hartree), and MC-PDFT (including CAS-PDFT and SP-PDFT). The MC-PDFT calculations were done both with translated functionals32 (tPBE, trevPBE, tBLYP) and with fully translated functionals36 (ftPBE, ftrevPBE, ftBLYP). All of the calculations were performed in C1 symmetry using the ANO-RCC-VTZP basis40 set. This basis set (Table 1) is sufficient for reasonable convergence of the perturbative dynamic correlation of CASPT2. The scalar relativistic effects were treated by the Douglas–Kroll–Hess second-order Hamiltonian41,42,43 (DKH2). We used the ultrafine integration grid for MC-PDFT calculations. Since we compare the calculated BDEs against experimental values, the zero point vibrational energies (ZPEs) and the change of spin-orbit coupling energy ΔESO upon bond formation were included by calculating the dissociation energy as


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𝐷! = 𝐸 M + 𝐸 L − 𝐸 ML − ZPE + Δ𝐸!"

(1)

where E(M), E(L), and E(ML) are the Born-Oppenheimer potential energy for isolated neutral metal atom (M = Fe or Ni), isolated neutral atomic ligand (L = C, S, or Se), and diatomic ML at the equilibrium geometry without spin-orbit coupling. The sum of first two terms, E(M) + E(L), is approximated as the energy of a supermolecule with a M-X distance of 12 Å. The first three terms in equation (1) then represent the theoretical spin-orbit-free bond equilibrium dissociation energy (De) of ML. Table 2 gives the experimental D0 values, the equilibrium internuclear distances (re), the change Δ𝐸!" of spin-orbit coupling energies upon bond formation, and the ZPEs. For FeC, NiC, FeS, NiS we used experimental,44,45,46,47,48,49 equilibrium bond distances, and we calculated ZPE from experimental vibrational frequencies ω in the harmonic approximation (ZPE= 0.5ω). For FeSe and NiSe we optimized the geometries by M06-L50/def2-TZVP51 (using Gaussian 0952) and used the corresponding scaled ZPE53. For FeC, we used a previous54,24 value of ΔESO (-1.13 kcal/mol), while for other molecules the change of spin-orbit coupling energies was calculated as Δ𝐸!" = 𝐸!" M + 𝐸!" L − 𝐸!" ML

(2)

where ESO(X) is the (negative) amount by which spin-orbit coupling changes the energy for species X. The atomic limit terms, ESO(M) and ESO(L), were calculated using the atomic spectroscopic data.55 The spin-orbit coupling at equilibrium, ESO(ML), was approximated to zero for NiC, which is the only closed-shell singlet in this set, while for other diatomics (FeS, NiS, FeSe, and NiSe) this term was obtained at the RASSI56 level with the ANO-RCC-VTZP basis set.40 The RASSI computations are based on state-averaged SA-CASSCF calculations, in which the corresponding nominal CPO (nom-CPO, vide infra) active space is used with 10 roots for FeS and FeSe and 6 roots for NiS and NiSe. These roots were allowed to interact at the RASSI level in order to take into account the effect of the low-lying excited states. The ground-state configurations of these six molecules can be understood in terms of the atomic valence 3d and 4s orbitals of Fe and Ni interacting with the atomic valence orbitals of C (2s and 2p), S (3s and 3p), or Se (4s and 4p). The lowest molecular orbital in this reduced energy window is core-like and is dominated by the nonmetal ns orbital; it is labeled as the 1σ



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orbital. The bonding combinations of the metal 3dσ and 3dπ orbitals with the nonmetal npσ and npπ orbitals, respectively, form the next higher 2σ and 1π (with the πxz and πyz components, where the z-axis is along the bond) molecular orbitals. Next higher in energy are the 1δ (with the δxz and δyz components) and 3σ orbitals, which are formed from the metal 3dδ and 4sσ atomic orbitals and have mainly nonbonding character. The combinations of the metal 3dπ and 3dσ with the nonmetal npπ and npσ orbitals, respectively, yield the highest antibonding 2π* (with the πxz* and πyz* components) and 4σ* orbitals. A somewhat nonsystematic aspect of many MCSCF calculations is the process of choosing the active space. Here we use a sequence of systematic choices based on the correlated participating orbitals22,23,24 (CPO) scheme. For a given active space we then must choose which configurations to include in the CI, and we make two choices, namely CPO-CASSCF and CPO-SP. We adopt the same definitions as in ref 24 for three levels of CPO active space: Nominal CPO (nom-CPO). This active space includes all the bonding orbitals (2σ2 1π4, for all six diatomics), all singly occupied nonbonding orbitals (1δ1xz 3σ1 for FeC, and 1δ1xz 1δ1yz for FeS and FeSe), all occupied antibonding orbitals (2π*1xz 2π*1yz for FeS, NiS, FeSe, and NiSe) and correlates each orbital with its correlating orbital. All of the doubly occupied nonbonding orbitals are inactive. Moderate CPO (mod-CPO). In addition to the orbitals in the nom-CPO space, we add to the active space the d- and s-subshell doubly occupied nonbonding orbitals of the metal (1δ2yz for FeC, and 1δ4 3σ2 for NiC, NiS and NiSe, and 3σ2 for FeS and FeSe) and their correlating orbitals. Extended CPO (ext-CPO). In addition to the orbitals in the mod-CPO space, we add the core-like s-subshell doubly occupied orbitals of the nonmetal (1σ2, for all six diatomics) and their correlating orbitals. The correlating orbitals in the CPO schemes are defined as follows. For an occupied bonding orbital, if its antibonding orbital is empty, then the corresponding antibonding orbital is taken as the correlating orbital of the bonding orbital; if its antibonding orbital is occupied, then its correlating orbital is taken as an unoccupied orbital. For an occupied nonbonding orbital, atomic orbital, or antibonding orbital, its correlating orbital is taken as an empty


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orbital. For a nonbonding d orbital, we use the notation d′ to denote its correlating orbital. Based on the above definitions, the active spaces corresponding to nom-CPO, mod-CPO, and ext-CPO are listed in Table 3. The two orbitals in a given pair of parentheses are in one GAS subspace in the CPO-SP calculations. Intersubspace electronic excitations are not allowed in the SP calculations, except when there are occupied antibonding orbitals in the molecule. For FeS, NiS, FeSe and NiSe where the antibonding 2π* orbitals are occupied (2π*1xz and 2π*1yz), intersubspace excitations are allowed between the two GAS subspaces (πi, πi´) and (πi*, πi*´) where i = xz or yz. Without allowing intersubspace excitations for such cases, the bonding and antibonding orbitals cannot be well correlated, which would cause difficulty in obtaining the correct electronic state and introduce significant errors in the calculation of bond dissociation energy.24 Table 3 shows the number of configuration state functions for nom-, mod-, and ext-CPO CAS and SP calculations; it shows that the number of configuration state functions contained in an SP calculation is dramatically lower than the number in the CAS calculation with the same active space. 3. Results and discussion 3.1 Spin-orbit coupling The change of spin-orbit coupling energy upon bond formation, ΔESO, for each molecule is presented in Table 2. The effect of the spin-orbit coupling tends to systematically reduce the value of the bond dissociation energy compared to the spin-orbit-free one. 3.2 Bond dissociation energies We report signed errors as D0(theory) minus D0(experiment). We also compute the mean signed error (MSE) and the mean unsigned error (MUE). The MSEs and MUEs for the different methods with the nom-CPO, mod-CPO, and ext-CPO active spaces are presented in Tables 4, 5, and 6, respectively. 3.2.1 Bond dissociation energies of FeC and NiC The ground states of FeC (3Δ3) and NiC (1Σ+) have been experimentally44,47,57,58 determined with the configurations of 1σ2 2σ2 1π4 1δ3 3σ1 and 1σ2 2σ2 1π4 1δ4 3σ2,



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respectively. Several theoretical46,54,59,60,61,62,63,64,65,66,67 studies reported the bond dissociation energies of FeC and NiC using various wave function and density functional methods. The SP calculations for FeC give unusually small errors when compared to the corresponding CASSCF for each of the three CPO active schemes. At the equilibrium/dissociation limits, the total electronic energies of SP (Table S1 in the Supporting Information (SI)) are higher than those of CAS by 7.5/49.2, 12.0/36.5, and 19.8/36.3 kcal/mol for nom-, mod-, and ext-CPO, respectively. Although the CASSCF and SP calculations predict the correct ground separated-atom limit configurations (5D for Fe and 3P for C), this significant destabilization of SP energies at dissociation limit, may be due to the relative small number of configuration state functions compared to CASSCF for the case of FeC. The CAS-trevPBE, CAS-ftrevPBE, and CAS-ftBLYP methods perform slightly better than CASPT2 in reproducing the experimental bond dissociation energy across all three active spaces. With the mod-CPO active space, the absolute errors with SP-ftPBE, SP-trevPBE, and SP-ftrevPBE are less than 4 kcal/mol. For NiC, most SP-PDFT calculations have smaller errors when comparing them with the corresponding CAS-PDFT ones. Especially for the mod-CPO active space, SP-tPBE, SP-ftPBE, SP-trevPBE, and SP-ftrevPBE give much more reduced errors than the CAS counterparts. For ext-CPO, the SP-trevPBE, SP-ftrevPBE, and SP-ftBLYP approximations again outperform their CAS counterparts, yielding about 1 kcal/mol as an absolute error. 3.2.2 Bond dissociation energies of FeS and NiS The ground state of FeS has been experimentally68,69,70 determined to be 5Δ4 with the configuration of 1σ2 2σ2 1π4 1δ3 3σ1 2π*2, while some theoretical71,72,73 studies show that the ground state is 5Σ+ corresponding to the electronic configuration 1σ2 2σ2 1π4 1δ2 3σ2 2π*2. At the CASSCF and CASPT2 levels of theory using the mod-CPO active space, which is the largest affordable active space for this system (see Table 3), the 5Σ+ state is lower than 5Δ4 by 1.2 and 2.3 kcal/mol, respectively. With the nom-CPO active space, the absolute error for FeS with CAS-tPBE, CAS-ftrevPBE, CAS-tBLYP, and CAS-ftBLYP is quite similar to that of CASPT2 (about 6 kcal/mol) and it reduces to 1 kcal/mol with CAS-ftPBE and CAS-trevPBE. On increasing


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from nom-CPO to mod-CPO, the absolute error for CASPT2 and CAS-ftrevPBE decreases to less than 1 kcal/mol. With more than 2.5×108 configuration state functions (see Table 3), the CASSCF calculations for FeS with the ext-CPO active space are not affordable. By using the nom-CPO space, the SP-PDFT variants look similar to the corresponding CAS-PDFT ones, where their errors have the same signs, and the maximum absolute deviation (about 1.5 kcal/mol) occurs for ftBLYP. By increasing the active space to mod-CPO, the bond dissociation energy of FeS is underestimated by the SP-PDFT approximations, among which SP-tPBE gives the smallest deviation (-4.1 kcal/mol). Diatomic NiS is also experimentally49,74,75 and computationally71,76,77,78 known to have a 3Σ- ground state with electronic configuration 1σ2 2σ2 1π4 1δ4 3σ2 2π*2. We performed CASSCF calculations only with the nom-CPO active space, while the mod-CPO and ext-CPO ones were only considered at the SP level due to the high corresponding number of the configuration state functions (more than 5.3×107 and 7.3×108 CSFs, respectively, as shown in Table 3). With the nom-CPO active space, CASPT2, and all CAS-PDFT variants underestimate the BDE, and the CAS-tPBE and CASPT2 give comparable estimations with absolute errors of 2.7 and 2.1 kcal/mol, respectively. Among the SP-PDFT calculations, which also underestimate the BDE, the SP-tPBE variant has the best performance for nom-CPO, mod-CPO, and ext-CPO with errors of -4.4, -1.0, and -4.5 kcal/mol, respectively. 3.2.3 Bond dissociation energies of FeSe and NiSe The FeSe (5Σ+) and NiSe (3Σ-) diatomics have the same ground states71 as FeS and NiS, respectively. Similarly to the FeS and NiS cases, the CASSCF calculations for FeSe with ext-CPO and NiSe with mod-CPO and ext-CPO are not affordable, so we consider only the SP model in those cases. For FeSe with nom-CPO, the error with CAS-tPBE is -4.8 kcal/mol, which is smaller than that of CASPT2 (-13.6 kcal/mol). By increasing from nom-CPO to mod-CPO, the absolute errors of most translated and fully translated functionals decrease significantly, in particular for CAS-ftrevPBE, CAS-tBLYP, and CAS-ftBLYP (by 11.7, 12.3, and 9.2 kcal/mol, respectively). For SP-PDFT with nom-CPO and mod-CPO, the SP-tPBE variant has the best performance yielding the smallest errors for FeSe (-0.7 and -3.1 kcal/mol). Similarly to the CAS-PDFT case, the absolute errors of SP-ftrevPBE, SP-tBLYP, and



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SP-ftBLYP decrease significantly (by 10.7, 13.8, and 12.7 kcal/mol, respectively) when we increase the active space from mod-CPO to ext-CPO. For diatomic NiSe with the nom-CPO active space, the CAS-tPBE, CAS-ftPBE, CAS-trevPBE, CAS-ftrevPBE, and CAS-ftBLYP methods, with absolute errors of 1.2, 1.5, 2.4, 5.2 and 5.9 kcal/mol, outperform CASPT2, which gives an absolute error of 11.2 kcal/mol. The SP calculations underestimate BDEs across all active spaces. The significant underestimation with ext-CPO may be attributed to imbalanced treatment of static and dynamic correlation effects, similar to FeS case. Among the SP-PDFT variants, the absolute errors for SP-tPBE are the lowest with 6.0, 4.8, and 13.3 for nom-, mod- and ext-CPO, respectively. For FeSe and NiSe we used the calculated equilibrium geometries and zero-point energies because the experimental data are not available. Our conclusions do not differ qualitatively from the other cases where we used experimental geometries. 3.2.4 Average errors in bond dissociation energies: comparison of methods The MUE-6 and MSE-6 rows include all 6 molecules with the mean of six unsigned errors) and six signed errors. The MUE-4 and MSE-4 rows exclude NiS and NiSe because these molecules are unaffordable for CAS calculations with the mod-CPO (or larger) active space. The MSE-4 and MUE-4 mean errors provide an aid to the reader in judging the errors of methods that are not affordable for large systems by comparing them consistently to more affordable cases only for those cases where all methods are affordable to carry out. The need for this kind of limited comparison with more expensive methods underscores the motivation for developing more affordable methods like MC-PDFT. For all three levels of CPO spaces, the CASSCF and SP calculations underestimate the bond dissociation energy of all molecules, i.e. they yield only negative signed errors in all 30 cases. CASPT2 improves over CASSCF noticeably, as expected due to a significant increase in included dynamic correlation. Although CASSCF is more accurate than SP for a given active space, for the same active space the difference of MUE-6 between CAS-PDFT and SP-PDFT is smaller than or about equal to 1 kcal/mol, and these two approximations, for most translated and fully translated


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functionals, perform slightly better than CASPT2. This is a key result of the present paper, and it is very encouraging for MC-PDFT. CAS-trevPBE/nom-CPO gives the lowest MUE-6 of the CASSCF-based methods, in particular 4.8 kcal/mol. With the mod-CPO active space, the CASSCF calculation was affordable for only four of the six molecules, and CAS-trevPBE, CAS-tBLYP, and CAS-ftBLYP yield quite similar MUE-4 values compared to CASPT2, while the MUE-4 of CAS-ftrevPBE is 1.9 kcal/mol lower than that of CASPT2. The MUE-4 of SP-tPBE/mod-CPO is only 0.3 kcal/mol higher than the MUE-4 of CASPT2/mod-CPO. CAS-PDFT is about equally as accurate than SP-DFT for the nom-CPO active space, but it is more accurate (on average by 2 kcal/mol) than SP-PDFT with the mod-CPO and ext-CPO active spaces. But the CASSCF reference calculation rapidly becomes unaffordable. Across Table 3, there is a significant reduction of the number of configuration state functions when the separated-pair configuration selection is adopted compared to CASSCF. For cases where CASSCF and CASPT2 are unaffordable, the SP model provides a good enough reference wave function that the rest of the dynamic correlation can be reasonably well incorporated at the SP-PDFT level. This approach enabled us to calculate the bond dissociation energies for all 6 molecules at the mod-CPO and ext-CPO levels. In fact SP-PDFT can give comparable accuracy to CASPT2 when CASPT2 is affordable. 3.2.5. Choice of active space The previous subsection was primarily comparing SCF, PT2, and MC-PDFT errors to one another. A second objective of this study is to learn the active space requirements because the problem of active space selection has become an important topic of discussion in its own right. On increasing from nom-CPO to mod-CPO and then to ext-CPO, the signed and unsigned errors for CASSCF and SP both decrease systematically. But these SCF methods are not quantitatively accurate enough to be useful for thermochemistry. For CASPT2, the results improve in passing from nom-CPO to mod-CPO, but then degrade when the active space is increased to ext-CPPO. When we proceed from the nom-CPO active space to the mod-CPO active space to the ext-CPO active space, the MSE-6 for SP-tPBE (SP-ftPBE) changed from 2.8 (1.8) to -0.6



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(-2.5) to -1.3 (-2.6) kcal/mol, respectively. But the trend in MUE-6 is from 6.3 (6.4) to 3.7 (5.0) to 9.3 (11.4). Consequently, SP-tPBE with mod-CPO has the best performance for mean unsigned error with an MUE-6 of 3.7 kcal/mol. At this level of theory, the mod-CPO active space, which consists of all the valence electrons and valence orbitals with the corresponding correlating orbitals, and the translated on-top functional tPBE provides a balanced treatment of the static and dynamic correlation effects. We conclude, on the basis of both CASPT2 and SP-PDFT, that the nom-CPO active space is too small, and the mod-CPO active space provides the best accuracy. The size of the active space required for the nom-CPO scheme does not increase exponentially with the size of the molecule because the CPO scheme does not put all bonds or all valence orbitals in the active space.22,23,24 4.3 Dipole moments In order to judge the extent of charge transfer between the metal and the nonmetal, we also calculated dipole moments for all six molecules, and they are given in Table 7. The CASPT2 dipole is usually very close to the CASSCF one with a mean absolute deviation of only 0.07 Debye. This indicates that the CASSCF method, although it is not reliable for energies, gives good first moments of electron densities, which is important because the MC-PDFT calculations use the SCF densities. Simplifying the SCF calculation from CASSCF to SP does, however, sometimes have a large effect on the dipole moment; in five of the cases the absolute deviation of the two dipole moments is 0.53–0.86 Debye, whereas in two other cases it is 0.21–0.25 Debye, and in four other cases this deviation it 0.10 Debye or less. It is unknown which dipole moments are most accurate, especially since the calculated dipole moments by the CASPT2 method have a significant dependence on active space size. 4.4 Computation times Table 8 lists the running time of the SCF, PT2, and PDFT steps. For FeS and FeSe, the PT2 step is much more expensive (almost one day) than the PDFT step (less than one minute), and therefore the CAS-PDFT method has an appealing advantage over the CASPT2.


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For PT2 or PDFT calculations, the total computation time required is the sum of the SCF time and the listed post-SCF step. If we consider the most accurate method that is affordable for all six systems which is mod-CPO-SP-PDFT, and the most expensive system for which CASSCF is doable with this active space, which is FeSe, we find (by adding the two times in Table 8) that the total cost for a PDFT calculation, is 5.4 minutes, whereas the total time for a CASPT2 calculation is 3951 minutes; the time for the CASSCF step alone with this active space is 2700 minutes. One sees similar ratios of timings for FeS. We conclude that SP-PDFT gives accuracy better than or comparable to CASPT2 at a total cost much less than CASSCF. SP-PDFT, being affordable even for large systems with large active spaces, is a very promising theory. We also note that the memory required for the PDFT step is much smaller than that for the PT2 step,79 even with the same size active space and the same level of configuration interaction, i.e., CAS or SP. 4. Conclusions The bond dissociation energies of the diatomic molecules FeC, NiC, FeS, NiS, FeSe, and NiSe were calculated by employing multi-configuration pair-density functional theory (MC-PDFT and compared with recent experimental values [Matthew, D. J.; Tieu, E.; Morse, M. D. J. Chem. Phys. 2017, 146, 144310] and with MCSCF and CASPT2 results. In the CAS-PDFT and SP-PDFT calculations, the CASSCF and SP wave functions, respectively, are used to get reference wave functions with the correct spin and spatial symmetry. These calculations are carried out with active spaces defined by the correlated participating orbitals (CPO) scheme. For CASSCF-based calculations, CAS-PDFT with translated or fully translated functionals gives results with accuracies similar to CASPT2. For cases where CASSCF calculations are not affordable, SP and SP-PDFT are still practical, and they provide a computationally economical approach with equally good performance as for the smaller systems. We find the SP-tPBE calculation with the moderate CPO active space has the smallest mean unsigned error (3.7 kcal/mol). We also find that the mod-CPO active space performs well for CASPT2. The mod-CPO active space contains all the valence electrons and valence orbitals with the corresponding correlating orbitals, and this provides a practical way



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to treat the static correlation effects along with the dynamic correlation, which is conveniently treated by the tPBE on-top density functional. ! ASSOCIATED CONTENT Supporting Information. Total electronic energies. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/ ! AUTHOR INFORMATION Corresponding Authors *E-mails: [email protected] (LG) and [email protected] (DGT). ! ACKNOWLEDGMENTS The authors are grateful to Jianwei Lucas Bao for helpful assistance. This work was supported in part by the Air Force Office of Scientific Research under grant no. FA9550-16-1-0134 and NSF grant CHE-1464536.


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15

Table 1. The number of basis functions (N), the primitive functions, and the contracted functions of the ANO-RCC-VTZP basis set. Molecule

N

Primitive functions

Contracted functions

FeC

89

Fe: 21s15p 10d 6f 4g

C: 14s 9p 4d 3f

Fe: 6s 5p 3d 2f 1g

C: 4s 3p 2d 1f

NiC

89

Ni: 21s15p 10d 6f 4g

C: 14s 9p 4d 3f

Ni: 6s 5p 3d 2f 1g

C: 4s 3p 2d 1f

FeS

93

Fe: 21s15p 10d 6f 4g

S: 17s 12p 5d 4f

Fe: 6s 5p 3d 2f 1g

S: 5s 4p 2d 1f

NiS

93

Ni: 21s15p 10d 6f 4g

S: 17s 12p 5d 4f

Ni: 6s 5p 3d 2f 1g

S: 5s 4p 2d 1f

FeSe

102

Fe: 21s15p 10d 6f 4g

Se: 20s 17p 11d 4f

Fe: 6s 5p 3d 2f 1g

Se: 6s 5p 3d 1f

NiSe

102

Ni: 21s15p 10d 6f 4g

Se: 20s 17p 11d 4f

Ni: 6s 5p 3d 2f 1g

Se: 6s 5p 3d 1f

Table 2. Experimental Bond Dissociation Energy D0 (kcal/mol), Equilibrium Bond Length re (Å), Change of Spin-Orbit Coupling Energy ΔESO (kcal/mol), and Zero Point Energy Ezp (kcal/mol). Molecule

D0

re

ΔESO

ZPE

Ref.

FeC

91.34

1.596

-1.13

1.22

44,45,46

NiC

96.11

1.627

-2.92

1.25

47

FeS

74.72

2.025

-2.22

0.74

48

NiS

84.20

1.963

-3.13

0.73

49

FeSe

63.16

2.133

-3.99

0.55

–

NiSe

74.21

2.102 ACS Paragon Plus -5.05 Environment

0.52

–


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Table 3. The nom-, mod-, and ext-CPO Active Spaces. An active space consisting of n electrons in m orbitals is denoted as (n, m). The Number of Configuration State Functions [CAS/SP]. The two orbitals in a given pair of parentheses are in one GAS subspace in the CPO-SP calculations. nom-CPO FeC (8,10) [20790/ 504] (σ3dz , σ3dz *), 2(π, π*), (3d, 3d´), (4sFe, 4sFe´)

mod-CPO (10,12) [283140/ 2568] (σ3dz , σ3dz *), 2(π, π*), 2(3d, 3d´), (4sFe, 4sFe´)

ext-CPO (12,14) [3864861/ 13236] (σ3dz , σ3dz *), 2(π, π*), 2(3d, 3d´), (4sFe, 4sFe´), (2sC, 2sC´)

(6,6)[175/ 37] (σ3dz , σ3dz *), 2(π, π*)

(12,12) [226512/ 3012] (σ3dz , σ3dz *), 2(π, π*), 2(3d, 3d´), (4sNi, 4sNi´)

(14,14) [2760615/ 14445] (σ3dz , σ3dz *), 2(π, π*), 2(3d, 3d´), (4sNi, 4sNi´), (2sC, 2sC´)

(10,14)[975975/ 22912] (σ3dz , σ3dz *), 2(π, π´), 2(π*, π*´), 2(3d, 3d´)

(12,16)[17017000/ 134464] (σ3dz , σ3dz *), 2(π, π*´), 2(π, π*´), 2(3d, 3d´), (4sFe, 4sFe´)

(14,18)[282676680/ 774912] (σ3dz , σ3dz *), 2(π, π*´), 2(π, π*´), 2(3d, 3d´), (4sFe, 4sFe´), (3sS, 3sS´)

(8,10)[20790/ 2896] (σ3dz , σ3dz *), 2(π, π´), 2(π*, π*´)

(14,16)[53093040/ 392080] (σ3dz , σ3dz *), 2(π, π*´), 2(π, π*´), 2(3d, 3d´), (4sNi, 4sNi´)

(16,18)[734959368/ 2057488] (σ3dz , σ3dz *), 2(π, π*´), 2(π, π*´), 2(3d, 3d´), (4sNi, 4sNi´), (3sS, 3sS´)

(10,14)[975975/ 22912] (σ3dz , σ3dz *), 2(π, π´), 2(π*, π*´), 2(3d, 3d´)

(12,16)[17017000/ 134464] (σ3dz , σ3dz *), 2(π, π*´), 2(π, π*´), 2(3d, 3d´), (4sFe, 4sFe´)

(14,18)[282676680/ 774912] (σ3dz , σ3dz *), 2(π, π*´), 2(π, π*´), 2(3d, 3d´), (4sFe, 4sFe´), (3sSe, 3sSe´)

(8,10)[20790/ 2896] (σ3dz , σ3dz *), 2(π, π´), 2(π*, π*´)

(14,16)[53093040/ 392080] (σ3dz , σ3dz *), 2(π, π*´), 2(π, π*´), 2(3d, 3d´), (4sNi, 4sNi´)

(16,18)[734959368/ 2057488] (σ3dz , σ3dz *), 2(π, π*´), 2(π, π*´), 2(3d, 3d´), (4sNi, 4sNi´), (3sSe, 3sSe´)

2

NiC

2

FeS

2

NiS

2

FeSe

2

NiSe

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2


2

2

2

2

2

2

2

2

2

2

2

2

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Table 4. Signed Error (kcal/mol), Mean Signed Error (MSE), and Mean Unsigned Error (MUE) for the Bond Dissociation Energy, D0 (kcal/mol), Computed by Multireference Methods with nom-CPO Spaces.a CAS SCF

PT2

SP

tPBE

trevPBE

tBLYP

ftPBE

ftrevPBE

ftBLYP

SCF

tPBE

trevPBE

tBLYP

ftPBE

ftrevPBE

ftBLYP

FeC

-42.3

-4.3

5.3

0.2

-5.4

8.0

1.6

0.5

-0.6

10.9

5.6

0.1

10.4

3.6

2.3

NiC

-45.9

8.6

16.2

10.7

5.0

19.2

12.7

9.4

-64.4

10.4

4.8

-1.2

12.2

5.2

2.5

FeS

-46.1

-6.2

4.8

1.1

-6.8

1.2

-4.5

-6.3

-42.2

5.1

1.1

-7.6

2.0

-3.5

-7.8

NiS

-44.0

-2.1

-2.7

-6.5

-13.4

-3.9

-9.5

-11.0

-38.0

-4.4

-8.1

-15.3

-5.7

-11.3

-13.1

FeSe

-41.2

-13.6

-4.8

-8.1

-16.1

-7.1

-13.2

-13.0

-41.1

0.7

-3.2

-12.1

-1.6

-7.1

-11.3

NiSe

-59.4

-11.2

1.2

-2.4

-11.2

1.5

-5.2

-5.9

-37.6

-6.0

-9.5

-16.4

-6.4

-12.1

-12.9

MSE-6

-46.5

-4.8

3.3

-0.8

-8.0

3.2

-3.0

-4.4

-37.3

2.8

-1.6

-8.8

1.8

-4.2

-6.7

MUE-6

46.5

7.7

5.8

4.8

9.7

6.8

7.8

7.7

37.3

6.3

5.4

8.8

6.4

7.1

8.3

MSE-4a

-43.9

-3.9

5.4

1.0

-5.8

5.3

-0.9

-2.4

-37.1

6.8

2.1

-5.2

5.8

-0.5

-3.6

MUE-4a

43.9

8.2

7.8

5.0

8.3

8.9

8.0

7.3

37.1

6.8

3.7

5.3

6.6

4.9

6.0

a

To allow easy comparison with Table 5, the MSE-4 and MUE-4 rows are averaged over only the four molecules for which we have CAS results in Table 5.



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Table 5. Signed Error (kcal/mol), Mean Signed Error (MSE), and Mean Unsigned Error (MUE) for the Bond Dissociation Energy, D0 (kcal/mol), Computed by Multireference Methods with the mod-CPO Spaces. CAS

SP

SCF

PT2

tPBE

trevPBE

tBLYP

ftPBE

ftrevPBE

ftBLYP

SCF

tPBE

trevPBE

tBLYP

ftPBE

ftrevPBE

ftBLYP

FeC

-28.8

-6.0

4.0

-1.2

-6.6

6.1

-0.5

-1.1

-4.2

5.8

0.4

-4.7

3.7

-3.2

-4.2

NiC

-34.6

0.1

11.6

5.9

0.6

12.0

4.8

3.3

-47.8

3.4

-1.9

-5.6

3.6

-2.9

-3.1

FeS

-36.6

-0.3

8.9

4.8

-2.4

5.0

-0.8

-3.4

-25.5

-4.1

-8.3

-16.2

-7.1

-12.3

-17.5

NiS

NA

NA

NA

NA

NA

NA

NA

NA

-51.1

-1.0

-4.7

-11.0

-2.8

-8.7

-9.2

FeSe

-36.7

-8.7

7.5

3.5

-3.8

4.4

-1.5

-3.8

-34.3

-3.1

-6.9

-14.4

-5.8

-11.5

-14.0

NiSe

NA

NA

NA

NA

NA

NA

NA

NA

-30.4

-4.8

-8.4

-14.5

-6.7

-12.8

-12.0

MSE-6

-32.2

-0.6

-5.0

-11.1

-2.5

-8.6

-10.0

MUE-6

32.2

3.7

5.1

11.1

5.0

8.6

10.0

a

-34.2

-3.7

8.0

3.3

-3.1

6.9

0.5

-3.1

-28.0

0.5

-4.2

-10.2

-1.4

-7.5

-9.7

a

34.2

3.8

8.0

3.9

3.4

6.9

1.9

3.4

28.0

4.1

4.4

10.2

5.1

7.5

9.7

MSE-4

MUE-4 a

The MSE-4 and MUE-4 rows are averaged over only the four molecules for which we have CAS results with this active space.


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Table 6. Signed Errors (kcal/mol), Mean Signed Error (MSE), and Mean Unsigned Error (MUE) for the Bond Dissociation Energy, D0 (kcal/mol), Computed by Multireference Methods with ext-CPO Spaces. CAS

SP

SCF

PT2

tPBE

trevPBE

tBLYP

ftPBE

ftrevPBE

ftBLYP

SCF

tPBE

trevPBE

tBLYP

ftPBE

ftrevPBE

ftBLYP

FeC

-23.8

-3.3

4.9

-0.3

-5.9

7.5

1.0

0.2

-7.2

9.4

10.2

4.7

15.9

9.0

8.2

NiC

-28.2

-14.3

1.3

-4.1

-8.7

0.8

-6.6

-6.0

-52.7 4.6

-0.7

-4.4

5.3

-1.2

-1.1

FeS

NA

NA

NA

NA

NA

NA

NA

NA

-16.6 -13.9

-17.6

-24.5

-18.5 -24.1

-26.2

NiS

NA

NA

NA

NA

NA

NA

NA

NA

-36.7 -4.5

-8.3

-14.5

-7.1

-13.0

-13.6

FeSe

NA

NA

NA

NA

NA

NA

NA

NA

-15.3 9.9

6.1

-0.6

5.3

-0.8

-1.3

NiSe

NA

NA

NA

NA

NA

NA

NA

NA

-17.4 -13.3

-16.9

-22.8

-16.5 -22.5

-21.8

MSE-6

-24.3 -1.3

-4.5

-10.4

-2.6

-8.8

-9.3

MUE-6

24.3

9.3

10.0

11.9

11.4

11.8

12.0

MSE-4a

-23.0 2.5

-0.5

-6.2

2.0

-4.3

-5.1

MUE-4a

23.0

8.7

8.6

11.3

8.8

9.2

a

9.5

To allow easy comparison with Table 5, the MSE-4 and MUE-4 rows are averaged over only the four molecules for which we have CAS results in Table 5.



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Table 7. The Dipole Moments (Debye) of the nom-, mod-, and ext-CPO CASSCF, CASPT2, SP. Active spaces and methods nom

mod

ext

FeC

NiC

FeS

NiS

FeSe

NiSe

SCF

1.70

2.51

6.46

5.38

6.50

5.40

PT2

1.75

2.40

6.35

5.33

6.39

5.38

SP

SCF

1.17

2.54

6.56

5.59

6.64

5.40

CAS

SCF

1.71

1.67

5.33

NA

5.16

NA

PT2

1.75

1.68

5.48

NA

5.27

NA

SP

SCF

1.14

1.12

6.19

5.35

5.89

5.08

CAS

SCF

1.37

1.67

NA

NA

NA

NA

PT2

1.45

1.67

NA

NA

NA

NA

SCF

1.12

1.69

6.35

5.21

5.63

5.02

CAS

SP


Page 21 of 27


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Table 8. The running time (minutes) of the nom-, mod-, and ext-CPO CASSCF, CASPT2 CAS-PDFT, SP, and SP-PDFT for equilibrium/dissociation limits. The PT2 and PDFT times are just the post-SCF part; this must be added onto the SCF step for these calculations. Active

FeC

NiC

FeS

NiS

FeSe

NiSe

SCF

0.3/3.2

0.2/0.2

31/240

7.1/2.7

34/69

0.9/3.8

PT2

0.4/0.3

0.1/0.1

26/28

0.4/0.4

34/41

0.5/0.7

PDFT

0.1/0.1

0.03/0.05

0.2/0.2

0.1/0.1

0.1/0.1

0.1/0.1

SCF

0.2/0.3

0.1/1.1

1.4/4.3

4.2/0.4

0.9/0.3

0.7/1.2

PDFT

0.1/0.1

0.1/0.05

0.2/0.2

0.4/0.2

0.3/0.2

0.1/0.1

SCF

4.5/13.2

12/16

1620/4080

NA

2700/3360

NA

PT2

4.7/5.1

3.5/3.3

1302/1470

NA

1251/1311

NA

PDFT

0.2/0.2

0.2/0.2

0.5/0.4

NA

0.5/0.5

NA

SCF

1.9/3.7

3.5/8.9

4.1/14

22/20

4.9/7.8

20/87

PDFT

0.2/0.2

0.2/0.2

0.4/0.4

0.4/0.4

0.4/0.4

0.4/0.4

SCF

159/394

428/88

NA

NA

NA

NA

PT2

136/131

99/100

NA

NA

NA

NA

PDFT

0.2/0.2

0.3/0.3

NA

NA

NA

NA

SCF

6.0/38

23/22

33/89

343/817

36/133

313/476

PDFT

0.3/0.2

0.3/0.2

0.6/0.6

0.8/0.8

0.6/0.6

0.7/0.8

Method

space

CAS nom SP

CAS mod SP

CAS ext SP



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