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Predicting Bond Dissociation Energies of Transition-Metal Compounds by Multiconfiguration Pair-Density Functional Theory and Second-Order Perturbation Theory Based on Correlated Participating Orbitals and Separated Pairs Junwei Lucas Bao, Samuel O. Odoh,† Laura Gagliardi,* and Donald G. Truhlar* Department of Chemistry, Chemical Theory Center, and Supercomputing Institute, University of Minnesota, 207 Pleasant Street SE, Minneapolis, Minnesota 55455-0431, United States S Supporting Information *

ABSTRACT: We study the performance of multiconfiguration pair-density functional theory (MC-PDFT) and multireference perturbation theory for the computation of the bond dissociation energies in 12 transition-metal-containing diatomic molecules and three small transition-metal-containing polyatomic molecules and in two transition-metal dimers. The first step is a multiconfiguration selfconsistent-field calculation, for which two choices must be made: (i) the active space and (ii) its partition into subspaces, if the generalized active space formulation is used. In the present work, the active space is chosen systematically by using three correlated-participating-orbitals (CPO) schemes, and the partition is chosen by using the separated-pair (SP) approximation. Our calculations show that MC-PDFT generally has similar accuracy to CASPT2, and the active-space dependence of MCPDFT is not very great for transition-metal−ligand bond dissociation energies. We also find that the SP approximation works very well, and in particular SP with the fully translated BLYP functional SP-ftBLYP is more accurate than CASPT2. SP greatly reduces the number of configuration state functions relative to CASSCF. For the cases of FeO and NiO with extendedCPO active space, for which complete active space calculations are unaffordable, SP calculations are not only affordable but also of satisfactory accuracy. All of the MC-PDFT results are significantly better than the corresponding results with broken-symmetry spin-unrestricted Kohn−Sham density functional theory. Finally we test a perturbation theory method based on the SP reference and find that it performs slightly worse than CASPT2 calculations, and for most cases of the nominal-CPO active space, the approximate SP perturbation theory calculations are less accurate than the much less expensive SP-PDFT calculations.

1. INTRODUCTION The accurate prediction of the bond dissociation energy of transition-metal compounds is a challenge for modern quantum-chemical methods.1−6 This is generally due to the partially filled character of the d shell of the metal atoms. To obtain a quantitative description of these compounds, it is necessary to account for static and dynamic correlation effects in a balanced way. The complete active space self-consistent field (CASSCF) method7 is one approach widely used for accounting for static correlation. Dynamic correlation energy can also, in principle, be progressively captured in the CASSCF method by using active spaces of increasing sizes; however, the convergence of dynamic correlation energy in CASSCF is very slow with respect to active space size, and this would necessitate the use of impractically large active spaces to obtain quantitative accuracy. Post-CASSCF wave function methods, such as the complete active space second-order perturbation theory8−10 (CASPT2) and multireference configuration interaction11−13 (MRCI), can be used to recover dynamic electron correlation more practically than using larger active spaces; nevertheless, such © XXXX American Chemical Society

methods are also often too expensive due to the two bottlenecks: the computationally unaffordable huge number of configuration state functions in the CASSCF step and the unaffordable (both in memory and in computational time) requirements of the post-CASSCF step. We will show that these limitations can be circumvented. The first limitation can be lifted by smartly choosing a selected active space that is small enough for practical computations yet large enough to capture the requisite static correlation; in particular, we propose systematic ways for choosing an active space based on the principle of correlated-participating-orbitals14,15 (CPO). This provides a way to specify an active space consistently across a set of applications. The second barricade (i.e., the postCASSCF step for approximating the full dynamic correlation) can be torn down by using a post-SCF density functional method instead of a post-SCF wave function methodin particular, by applying the recently developed multiconfiguration pair-density functional theory16 (MC-PDFT). Received: November 15, 2016 Published: December 21, 2016 A

DOI: 10.1021/acs.jctc.6b01102 J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

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Journal of Chemical Theory and Computation

partitioning or specif ically partitioning an active space, we employ systematic procedures for selecting the active space; in particular, we use three versions (nominal, moderate, and extended) of the correlated participating orbitals14,15 scheme. Recently, a generalized active space self-consistent-field second-order perturbation theory30 (GASPT2) has been developed; it involves an approximation in that the orthogonal complement of the GAS subspace in the CAS space is omitted from the perturbed wave function. We also test the performance of this method by using the SP wave function as the specific case of GAS; we call this SPPT2.

In MC-PDFT, the Born−Oppenheimer electronic energy (which includes nuclear repulsion) is computed as the sum of (1) the electronic kinetic energy and the classical Coulomb energy evaluated using a multiconfiguration self-consistent field (MCSCF) wave function and (2) an “on-top energy”, which is computed post-SCF using an “on-top density functional” of the one-electron density and the on-top pair density of the MCSCF wave function. This may be compared to Kohn−Sham density functional theory in which the electronic energy is computed as the sum of (1) the electronic kinetic energy and the classical Coulomb energy evaluated using a Slater determinant and (2) an exchange−correlation energy that is included self-consistently in the optimization of the determinant. MC-PDFT was originally applied to both main-group compounds15−19 and transition-metal compounds20 by using CASSCF calculations as the MCSCF first step. To extend the applicability of the MC-PDFT framework to larger systems, we considered obtaining the total and on-top pair electronic densities from multiconfiguration wave functions that are more compact than the analogous CASSCF wave functions, in particular, that use a limited set of configuration state functions with variable occupancy of the active orbitals, as opposed to the characteristic feature of complete-active-space methods, which is that they include configuration state functions with all possible occupancies of the active orbitals. The generalized valence bond perfect-pairing (GVB-PP) wave function is a familiar example of a wave function with a limited configuration interaction in the active space,21−23 but it suffers from an inability to dissociate a molecule into open-shell fragments with more than one unpaired electron. More general procedures for limiting the configuration interaction in the active space are the generalized active space (GAS) method,24 the restricted active space (RAS) method,25,26 and the occupation-restrictedmultiple-active-space (ORMAS)27,28 method. Here we chose to use a particular case of the GAS method, which we have named the separated-pair29 (SP) wave function; the SP wave function is intermediate in completeness between the corresponding GVB-PP and CASSCF wave functions with the same active orbitals. In the previous work we applied SP to main-group bonds,29 and in the present work we apply it to transition-metal bonds. An SP wave function is built by partitioning an active space into subspaces that contain at most two orbitals while forbidding all or most interspace excitations. Within each subspace, one uses all possible occupancies of a given number of electrons, which is called full configuration interaction (FCI) in the subspace. Each subspace has either one or two electrons in one or two orbitals (typically two in two). The advantage of using an SP reference wave function is that it can greatly reduce the number of configuration state functions in the CI expansion, which results in a much less-expensive computation compared to the corresponding CASSCF calculation, yet in several tested cases SP-PDFT has a similar accuracy to CASPDFT. When the total density and the on-top pair density of an SP wave function are used in the MC-PDFT framework, the resulting method is called separated-pair pair-density functional theory29 (SP-PDFT). In the present work, our goal is to test the accuracy of the CAS-PDFT method and the SP-PDFT method for the computation of bond dissociation energies of transition-metal complexes. CASSCF involves FCI in the whole active space, whereas SP involves a partitioned active space as discussed above. In addition to these systematic procedures for not

2. COMPUTATIONAL DETAILS 2.1. Correlated Participating Orbitals Scheme for Selecting the Active Space Orbitals. Here, we extend our previous14,15 definitions of the correlated participating orbitals (CPO) scheme to describe the bond dissociation of transitionmetal containing molecules. We define three levels of active space: nominal, moderate, and extended. Nominal CPO (nom-CPO). For the chemical bond that is being broken or formed, include in the active space all the bonding orbitals, all singly occupied nonbonding orbitals, and all occupied antibonding orbitals (if any) and correlate 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, for metal, add to the active space the dsubshell doubly occupied nonbonding orbitals of the metal and their correlating orbitals and add the p-subshell doubly occupied nonbonding orbitals of the nonmetal (if any) involved in the bond and their correlating orbitals. Extended CPO (ext-CPO). In addition to the orbitals in the mod-CPO space, for metal, add the s-subshell doubly occupied nonbonding orbitals (if any) on the metal and their correlating orbitals in the active space and add the s-subshell doubly occupied nonbonding orbitals of the nonmetal (if any) and their correlating orbitals. The correlating orbitals 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 orbital. For a nonbonding d orbital, we use the notation d′ to denote its correlating orbital, although d′ is not necessarily of d-orbital character (i.e., d′ could be of s-, p-, d-, or s/p/d-hybridized character). 2.2. CPO-based GAS Subspaces in the Separated-Pair SCF Calculations. The separated-pair SCF calculations (SP) that use GAS subspaces of an active space defined by a CPO scheme are called CPO-SP calculations. In the current work, each GAS subspace contains two orbitals. We distribute n occupied active orbitals (in particular the bonding, antibonding, and singly and doubly occupied nonbonding active orbitals chosen in each level of the CPO scheme) in n GAS subspaces, and then we pair each of these orbitals with a corresponding correlating orbital. Within each GAS subspace, this is equivalent to doing an (m, 2) CASSCF calculation (i.e., m electrons in two orbitals), where m = 1 or 2. Note that when m = 2, we include both the singlet and triplet coupling of the two orbitals, and when we combine subspace wave functions into an overall configuration state function, we include all spin couplings B


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Journal of Chemical Theory and Computation consistent with the given total electron spin (neglecting spin− orbit coupling). Intersubspace electronic excitations are not allowed in the SP calculations, except when there are occupied antibonding orbitals in the molecule. If there are occupied σ* orbitals, intersubspace excitations (including all the possible excitations) are allowed between the two GAS subspaces (σ, σ′) and (σ*, σ*′); and if there are occupied πi* orbitals, intersubspace excitations are allowed between the two GAS subspaces (πi, πi′) and (πi*, πi*′), where the subscript i represents the type of the π orbitals (i.e., the π orbitals in the xz plane or in the yz plane, where the z-axis is along the bond). 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. Note that, as explained elsewhere,24 the GASSCF calculation is set up by specifying cumulative minimal and maximal occupation numbers, i.e., the minimal and maximal number of electrons in GAS1, GAS1+GAS2, GAS1+GAS2+GAS3, etc., where 1, 2, and 3 number the user-defined GAS spaces. 2.3. Separated-Pair Self-Consistent-Field SecondOrder Perturbation theory. As mentioned in the Introduction, an implementation for approximately carrying out secondorder perturbation theory based on a GASSCF wave function30 has become available. We test this here, and we label the theory that uses an SP zeroth-order wave function in a GASPT2 calculation as separated-pair second-order perturbation theory (SPPT2). 2.4. Database of Bond Energies. In this work, we examine a database of M−L bond energies of 17 transitionmetal complexes whose experimental dissociation energies are known. For metal−ligand complexes, M is the metal (neutral for diatomics and a +1 cation for polyatomics) and L is the ligand (a neutral atom or radical); for metal dimers, M and L are both metal atoms. Table 1 gives the experimental equilibrium bond length (re), the bond dissociation energy (De), the change of spin−orbit coupling energy upon bond formation, and the experimental term symbol for the molecules we considered in this work. Note that ΔESO = ESO(M) + ESO(L) − ESO(ML)

Table 1. Experimental Equilibrium Bond Length, Bond Dissociation Energy, The Change of Spin-Orbit Coupling Energy, And the Electronic Ground State for the Molecules Considered in This Work re (Å)

De (kcal/mol)

ΔESO (kcal/mol)

CrH MnH FeH CoH FeC ScN VN CrN TiO FeO NiO V2 Cr2 CrF CoOH+ MnCH3+ NiCH2+

1.656 1.731 1.610 1.531 1.592 1.687 1.563 1.563 1.620 1.616 1.627 1.77 1.679 1.784 a a a

46.8 31.1 36.9 45.6 88.4 113 116.7 87.6 161 102.6 90.4 64.2 33.9 106.3 73.8 51.9 76.3

0 0 −0.12 −0.37 −1.13 −0.29 −0.44 0 −0.58 −0.09 −3.02 −1.82 0 −0.39 −2.19 0 −1.72

ground state

ref

Σ 7 Σ 4 Δ 3 Φ 3 Δ 1 Σ 3 Δ 4 Σ 3 Δ 5 Δ 3 Σ 3 Σ 1 Σ 6 Σ 4 A 6 A 2 A

31 31 32 32 33 31 34 35 31 32 31 36 37 31 32 32 32

6

a

These geometries are optimized by using M06-L/aug-cc-pVTZ-DK with DKH2 and are given in the Supporting Iinformation.

We will compute signed errors as De(theory) minus De(experiment). We will also compute the mean signed error (MSE) and the mean unsigned error (MUE). 2.5. Details of CPO Active Spaces. The definition of the CPO active spaces is shown in Table 2. The two orbitals in a given pair of parentheses are in one GAS subspace in the CPOSP calculations. Table 3 shows the number of configuration state functions for nom-, mod-, and ext-CPO CAS and SP calculations; we see that the number of configuration state functions contained in the SP calculation is significantly lower than the number in the corresponding CAS calculation. 2.6. Multireference Calculations. To assess the multireference character of the molecules studied, we computed the M diagnostic,14 whose deviation from 0 indicates the multiconfigurational extent of a wave function based on natural occupation numbers of the most correlated nominally unoccupied orbital, most correlated nominally doubly occupied orbital, and the singly occupied orbitals (it would be zero for a single-configuration wave function). The value of this diagnostic is shown in Table 3 for all the molecules, along with a second diagnostic, namely, the percentage weight (the square of the CI coefficient) of the dominant configuration; these diagnostics were computed from calculations with nomCPO CAS spaces. As we can see from Table 3, the majority of the molecules studied in this work have M values larger than 0.05, and the percentage weight lower than 95%, both of which indicate that they have a significant amount of multireference character. We carried out multireference calculations by the following methods: • CASSCF, • GASSCF (in particular SP), • CASPT2 (with the standard empirical IPEA38 shift of 0.25 hartree),

(1)

where ESO(X) represents the spin−orbit coupling energy for species X. The references31−37 for all these data are also given in Table 1. For testing theory, the database is used as follows. First, single-point energies are computed without spin−orbit coupling using the geometries and spin multiplicities in Table 1. The theoretical spin−orbit-free equilibrium bond dissociation energy is then computed as the difference of the electronic energy at the dissociation limit and at the equilibrium geometry of ML. Then the theoretical De (for comparison to the experimental value in Table 1) is computed as De = E(M) + E(L) − E(ML) + ΔESO

ML

(2)

where E(M), E(L), and E(ML) are the electronic energy for isolated M, isolated L, and ML at equilibrium geometry, and ΔESO is from Table 1. For wave function and PDFT calculations, E(M) + E(L) is approximated as the energy of a supermolecule with a M−L distance of 12 Å. C


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Journal of Chemical Theory and Computation Table 2. Definition of the CPO Active Spacesa CrH MnH FeH CoH FeC ScN VN CrN TiO FeO NiO V2 Cr2 CrF CoOH+ MnCH3+ NiCH2+ a

nom-CPO

mod-CPO

ext-CPO

(7, 12): (σ, σ*), 5(3d, 3d′) (8, 14): (σ, σ*), 5(3d, 3d′), (4sMn, 4sMn′) (5, 8): (σ, σ*), 3(3d, 3d′) (4, 6): (σ, σ*), 2(3d, 3d′) (8, 10): (σ, σ*), 2(π, π*), (3d, 3d′), (4sFe, 4sFe′) (6, 6): (σ, σ*), 2(π, π*) (8, 10): (σ, σ*), 2(π, π*), 2(3d, 3d′) (9, 12): (σ, σ*), 2(π, π*), 2(3d, 3d′), (4sCr, 4sCr′) (8, 10): (σ, σ*), 2(π, π*), (3d, 3d′), (4sTi, 4sTi′) (12, 16): (σ3dz2, σ3dz2*), (σ4s, σ4s*), 2(π, π′), 2(π*, π*′), 2(3d, 3d′) (10, 12): (σ3dz2, σ3dz2*), (σ4s, σ4s*), 2(π, π′), 2(π*, π*′) (10, 12): (σ3dz2, σ3dz2*), (σ4s, σ4s*), 2(π, π*), 2(3d, 3d′) (12, 12): (σ3dz2, σ3dz2*), (σ4s, σ4s*), 2(π, π*), 2(3δ, 3δ′) (7, 12): (σ, σ*), 4(3d, 3d′), (4sCr, 4sCr′)

(7, 12): (σ, σ*), 5(3d, 3d′) (8, 14): (σ, σ*), 5(3d, 3d′), (4sMn, 4sMn′) (7, 10): (σ, σ*), 4(3d, 3d′) (8, 10): (σ, σ*), 4(3d, 3d′) (10, 12): (σ, σ*), 2(π, π*), 2(3d, 3d′), (4sFe, 4sFe′) (6, 6): (σ, σ*), 2(π, π*) (8, 10): (σ, σ*), 2(π, π*), 2(3d, 3d′) (9, 12): (σ, σ*), 2(π, π*), 2(3d, 3d′), (4sCr, 4sCr′) (8, 10): (σ, σ*), 2(π, π*), (3d, 3d′), (4sTi, 4sTi′)

(7, 12): (σ, σ*), 5(3d, 3d′) (8, 14): (σ, σ*), 5(3d, 3d′), (4sMn, 4sMn′) (9, 12): (σ, σ*), 4(3d, 3d′), (4sFe, 4sFe′) (10, 12): (σ, σ*), 4(3d, 3d′), (4sCo, 4sCo′) (12, 14): (σ, σ*), 2(π, π*), 2(3d, 3d′), (4sFe, 4sFe′), (2sC, 2sC′) (8, 8): (σ, σ*), 2(π, π*), (2sN, 2sN′) (10, 12): (σ, σ*), 2(π, π*), 2(3d, 3d′), (2sN, 2sN′) (11, 14): (σ, σ*), 2(π, π*), 2(3d, 3d′), (4sCr, 4sCr′), (2sN, 2sN′) (10, 12): (σ, σ*), 2(π, π*), (3d, 3d′), (4sTi, 4sTi′), (2sO, 2sO′) (14, 18): (σ3dz2, σ3dz2*), (σ4s, σ4s*), 2(π, π′), 2(π*, π*′), 2(3d, 3d′), (2sO, 2sO′) (16,18): (σ3dz2, σ3dz2*), (σ4s, σ4s*), 2(π, π′), 2(π*, π*′), 2(3d, 3d′), (2sO, 2sO′) (10, 12): (σ3dz2, σ3dz2*), (σ4s, σ4s*), 2(π, π*), 2(3d, 3d′) (12, 12): (σ3dz2, σ3dz2*), (σ4s, σ4s*), 2(π, π*), 2(3δ, 3δ′) (13, 18): (σ, σ*), 4(3d, 3d′), (4sCr, 4sCr′), 2(pF, pF′), (2sF, 2sF′) (13, 16): (σ, σ′), (σ*, σ*′), (π, π′), (π*, π*′), 3(3d, 3d′), (2pO, 2pO′) (7, 12): (σ, σ*), 5(3d, 3d′) (11, 12): (σ, σ*), (π, π*), 3(3d, 3d′), (4sNi, 4sNi′)

(7, 10): (σ, σ′), (σ*, σ*′), (π, π′), (π*, π*′), (3d, 3d′) (7, 12): (σ, σ*), 5(3d, 3d′) (5, 6): (σ, σ*), (π, π*), (3d, 3d′)

(12, 16): (σ3dz2, σ3dz2*), (σ4s, σ4s*), 2(π, π′), 2(π*, π*′), 2(3d, 3d′) (14, 16): (σ3dz2, σ3dz2*), (σ4s, σ4s*), 2(π, π′), 2(π*, π*′), 2(3d, 3d′) (10, 12): (σ3dz2, σ3dz2*), (σ4s, σ4s*), 2(π, π*), 2(3d, 3d′) (12, 12): (σ3dz2, σ3dz2*), (σ4s, σ4s*), 2(π, π*), 2(3δ, 3δ′) (11, 16): (σ, σ*), 4(3d, 3d′), (4sCr, 4sCr′), 2(pF, pF′) (13, 16): (σ, σ′), (σ*, σ*′), (π, π′), (π*, π*′), 3(3d, 3d′), (2pO, 2pO′) (7, 12): (σ, σ*), 5(3d, 3d′) (9, 10): (σ, σ*), (π, π*), 3(3d, 3d′)

An active space consisting of n electrons in m orbitals is denoted as (n, m).

Table 3. M Diagnostic Values and Percentage Weight of the Dominant Configuration (Which Are Computed Using nom-CPO CASSCF), and the Number of Configuration State Functions for nom-, mod-, and ext-CPO CASSCF and SP nom-CPO CrH MnH FeH CoH FeC ScN VN CrN TiO FeO NiO V2 Cr2 CrF CoOH+ MnCH3+ NiCH2+

mod-CPO

ext-CPO

M

weight

CAS

SP

CAS

SP

CAS

SP

0.08 0.04 0.57 0.48 0.22 0.09 0.10 0.16 0.06 0.15 0.16 0.25 0.40 0.03 0.04 0.11 0.11

94% 97% 58% 67% 70% 90% 85% 77% 92% 84% 85% 71% 47% 97% 96% 92% 92%

10296 45045 504 105 20790 175 20790 151008 20790 17017000 283140 283140 226512 10296 9240 10296 210

256 576 48 20 504 37 504 1504 504 134464 14704 2568 3012 256 1504 256 34

10296 45045 9240 20790 283140 175 20790 151008 20790 17017000 53093040 283140 226512 5834400 35395360 10296 27720

256 576 272 504 2568 37 504 1504 504 134464 392080 2568 3012 11264 251520 256 708

10296 45045 151008 283140 3864861 1764 283140 2342340 283140 282676680 734959368 283140 226512 113070672 35395360 10296 339768

256 576 1504 2568 13236 150 2568 8232 2568 774912 2057488 2568 3012 69280 251520 256 3408

• CASPT2-0 (which denotes CASPT2 without an IPEA shift), • GASPT2 (in particular SPPT2 and SPPT2-0), and • MC-PDFT (including CAS-PDFT and SP-PDFT) To carry out the MC-PDFT calculations, we need on-top density functionals, which are functionals of the total electron density and the on-top pair density, where the latter is the probability of finding two electrons at the same point in space. The required on-top functionals were obtained by protocols that we label “translation” and “full translation”. These protocols yield translated on-top functionals16 (tPBE, trevPBE,

tBLYP) and fully translated on-top functionals20 (ftPBE, ftrevPBE, ftBLYP); the precise protocols for translation and full translation are given in previous papers,16,20 and here we give a brief summary. The new functionals were obtained by translating the PBE, revPBE, and BLYP generalized gradient approximation (GGA) exchange−correlation functionals of Kohn−Sham theory. A GGA exchange−correlation functional may be written as Exc[ρ(r), m(r), ∇ρ(r), ∇m(r)], where ρ(r) is the total electron density, m(r) is the spin magnetization density, ∇ρ(r) is the gradient of the total density, and ∇m(r) is the gradient of the D


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Journal of Chemical Theory and Computation spin magnetization density. Motivated by the way the up-spin and down-spin densities are related to the total density and the on-top pair density in the case of single Slater determinant, we have defined a translated (t) functional as16

3. RESULTS AND DISCUSSION 3.1. Kohn−Sham Density Functional Theory. Table 4 shows the signed errors (in kcal/mol), mean signed errors

⎡ ⎧ ρ(r)(1 − R )1/2 if R ≤ 1⎫ ⎬, Π(r)] = Exc⎢ρ(r), ⎨ ⎢⎣ ⎩0 if R > 1⎭ ⎧|∇ρ(r)|(1 − R )1/2 if R ≤ 1⎫⎤ ⎬⎥ |∇ρ(r)| , ⎨ ⎩0 if R > 1⎭⎥⎦ (3)

Table 4. Signed Error (kcal/mol), Mean Signed Error (MSE), and Mean Unsigned Error (MUE) for the Bond Dissociation Energy, De (kcal/mol), Computed by BrokenSymmetry Spin-Unrestricted Kohn−Sham Density Functional

Eott[ρ(r),

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

CrH MnH FeH CoH FeC ScN VN CrN TiO FeO NiO V2 Cr2 CrF CoOH+ MnCH3+ NiCH2+ MSE-17a MUE-17a MSE-15b MUE-15b

where Π(r) is the on-top pair density and the ratio R is defined as R(r) = Π(r)/[ρ(r)/2]2

(4)

The translated functionals do not involve the gradient of the on-top pair density and have a discontinuous derivative with respect to the density and on-top pair density when R = 1. The fully translated (ft) functionals are defined to include the gradient of the on-top pair density and to have continuous first and second derivatives; in particular,20 Eotft[ρ(r), Π(r)] = Exc[ρ(r), ρ(r)χ (R ), |∇ρ(r)| , χ (R )|∇ρ(r)| + ρ(r)|∇χ (R )|] (5)

where χ(R) is defined as ⎧(1 − R )1/2 R ∈ [0, R ) 0 ⎪ ⎪ χ (R ) = ⎨ χft (R ) R ∈ [R 0 , R1] ⎪ ⎪0 R ∈ (R1 , ∞) ⎩

a

(6)

where R0 is 0.9 and R1 is 1.15, and χft(R) is a fifth-order polynomial function of R. We note that these are firstgeneration functionals, and they could almost surely be improved by adding more ingredients, by parametrization to accurate data, by nonempirical theoretical considerations, or by some combination of these strategies. All of the MC-PDFT calculations in the present article are single-point calculations carried out in C1 symmetry using a locally modified version of the Molcas 8.139 program. We chose C1 symmetry on the basis of the considerations that (a) symmetry is often not present in important practical applications, and (b) the full point group symmetry of a diatomic molecule is unavailable in the computer code. The calculations are performed using the ANO-RCC-VTZP basis40 and with the Douglas−Kroll−Hess second-order scalar relativistic Hamiltonian41 (DKH2). We use the ultrafine integration grid for MC-PDFT calculations. 2.7. Kohn−Sham Density Functional Calculations. Spin-unrestricted Kohn−Sham density functional42,43 calculations are performed with PBE,44 revPBE,45 and BLYP46 exchange−correlation functionals by using Gaussian 0947 program with “stable = (opt, xqc)” keyword that allows one to reach a stable solution,48,49 even if that involves breaking symmetry. The Douglas−Kroll−Hess second-order scalar relativistic Hamiltonian (DKH2) was used with the aug-ccpVTZ-DK basis50−52 and an ultrafine integration grid. Calcualtions on diatomics are carried out at experimental geometries, and calculations on polyatomics are carried out at geometries optimized by M06-L53/aug-cc-pVTZ-DK.

PBE

revPBE

BLYP

6.7 12.8 19.1 17.0 30.2 5.7 29.9 21.4 25.5 25.3 21.9 38.2 4.5 13.7 13.5 9.5 21.0 18.6 18.6 17.9 17.9

6.0 13.0 17.3 14.7 22.4 −0.9 21.6 13.9 17.4 17.4 13.8 26.4 −5.7 8.2 7.1 5.0 14.4 12.5 13.2 12.1 12.9

11.6 7.2 21.0 18.4 22.2 −1.0 22.1 24.3 16.6 19.2 16.7 25.6 9.1 15.1 9.2 8.2 20.4 15.6 15.8 15.3 15.5

Including all 17 molecules. bExcluding FeO and NiO.

(MSEs), and mean unsigned errors (MUEs) for the computed bond dissociation energy for the transition-metal complexes by using the PBE, revPBE, and BLYP exchange−correlation functionals. These three functionals were chosen for comparison because they are the starting points for the translated and fully translated functionals used in the PDFT calculations. These three density functionals tend to overestimate the bond dissociation energies, and the MUEs for PBE, revPBE, and BLYP are respectively 18.6, 13.2, and 15.8 kcal/mol for the 17 molecules. Although Kohn−Sham density functional method is a single-reference method, it is not always true that it works better for single-configuration-dominated systems than for systems with significant multireference characters. For CrF, which is the molecule with the smallest M diagnostic value (M = 0.03) in our testing set, the absolute errors for PBE, revPBE, and BLYP are 13.7, 8.2, and 15.1 kcal/mol, whereas for Cr2, whose M value is very large (0.40, which indicates very large multireference character), the absolute errors for PBE, revPBE, and BLYP are only 4.5, 5.7, 9.1 kcal/mol, all of which are smaller than the ones for CrF. 3.2. Multireference Calculations. Tables 5, 6, and 7 show the MSEs and MUEs for multireference methods with the nomCPO, mod-CPO, and ext-CPO active spaces, respectively. The MUE-17 and MSE-17 rows include all 17 molecules, and the MUE-15 and MSE-15 rows exclude FeO and NiO, because these two molecules are unaffordable for CAS calculations in the ext-CPO space. For all three levels of CPO spaces, the MUEs of the MCPDFT calculations are significantly smaller than the correE


a

F

4.1 6.5 −5.6 −6.8 3.5 5.3 −2.2 −8.6 −2.3 8.0 −4.5 −5.9 −8.3 5.0 3.0 −1.9 5.8 −0.3 5.1 −0.6 5.0

−10.1 6.9 −19.9 −28.6 −33.2 −29.6 −41.8 −25.7 −24.5 −30.6 −1.7 −46.1 −65.5 −25.1 −8.2 −25.8 2.5 −23.9 25.0 −25.0 26.2

4.0 6.3 −4.2 −5.5 1.6 −1.9 −6.2 −14.2 −5.1 0.0 −3.9 −16.0 −15.9 3.3 3.3 −3.1 6.1 −3.0 5.9 −3.2 6.4

PT2-0 3.4 10.5 −5.7 9.5 10.4 3.5 9.0 −2.0 9.6 −1.4 14.3 −13.4 −20.6 6.4 1.9 −2.1 14.1 2.8 8.1 2.3 8.1

tPBE



PT2

SCF 3.4 9.6 −6.3 7.7 5.1 0.7 4.6 −5.6 4.3 −7.3 9.2 −18.0 −24.2 3.0 −3.7 −4.9 8.1 −0.9 7.4 −1.1 7.3

trevPBE

CAS 5.4 8.2 −7.1 4.3 −0.6 −4.9 −0.5 −7.9 0.4 −10.2 7.1 −22.1 −24.3 4.3 −1.8 −7.8 10.4 −2.8 7.5 −2.9 7.3

tBLYP 7.0 9.1 −5.3 9.5 11.6 8.7 10.2 4.0 7.7 −6.7 11.9 0.1 −7.0 9.2 −2.9 0.5 8.1 4.4 7.0 4.7 6.7

ftPBE 6.5 9.0 −5.6 7.7 5.2 3.0 3.5 −2.8 0.9 −14.5 4.8 −9.2 −15.9 4.6 −8.6 −3.4 1.3 −0.8 6.3 −0.3 5.8

ftrevPBE 11.7 5.8 −5.5 5.2 3.0 5.2 3.3 12.1 0.4 −13.3 7.8 2.4 2.8 10.5 −6.7 −2.0 4.7 2.8 6.0 3.5 5.4

ftBLYP −18.0 5.5 −20.4 −28.7 −40.2 −30.3 −32.2 −30.8 −26.7 −6.7 −12.3 −55.2 −69.6 −27.9 −8.3 −30.4 2.1 −25.3 26.2 −27.4 28.4

SCF 1.8 8.8 −7.0 9.4 7.8 2.9 9.1 −2.9 13.9 16.0 11.2 −13.8 −21.4 1.5 1.9 −4.0 14.1 2.9 8.7 1.5 8.0

tPBE 1.8 8.0 −7.7 7.6 2.6 0.2 4.5 −6.4 8.3 10.6 6.0 −18.1 −25.1 −1.8 −3.7 −6.8 8.2 −0.7 7.5 −1.9 7.4

trevPBE 4.1 6.7 −8.5 4.2 −3.5 −5.4 −2.0 −8.5 4.1 9.5 4.2 −21.0 −25.1 −0.2 −1.8 −9.5 10.3 −2.5 7.6 −3.8 7.7

tBLYP

SP 5.7 7.8 −6.6 9.4 9.4 8.3 10.8 3.1 11.9 12.2 9.8 0.4 −8.1 4.5 −3.1 −1.0 8.5 4.9 7.1 4.1 6.6

ftPBE 5.3 8.0 −6.8 7.6 3.2 2.5 4.6 −3.5 4.8 4.4 2.7 −8.4 −16.9 0.0 −8.8 −4.8 1.7 −0.3 5.5 −0.8 5.8

ftrevPBE

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

10.1 4.1 −6.8 5.2 0.3 4.7 1.7 5.6 4.3 9.5 6.0 4.4 1.5 6.4 −6.9 −3.2 5.0 3.0 5.0 2.4 4.7

ftBLYP

Journal of Chemical Theory and Computation Article


a

G

4.1 6.5 −2.8 −0.6 3.1 5.3 −2.2 −8.6 −2.3 8.0 2.8 −5.9 −8.3 2.8 3.1 −1.9 5.6 0.5 4.3 −0.1 4.2

−10.1 6.9 −16.0 −5.4 −24.8 −29.6 −41.8 −25.7 −24.5 −30.6 −6.7 −46.1 −65.5 3.4 −15.9 −25.8 10.7 −20.4 22.9 −20.7 23.5

4.0 6.3 −1.8 −0.9 1.3 −1.9 −6.2 −14.2 −5.1 0.0 2.1 −16.0 −15.9 1.1 3.0 −3.1 5.2 −2.5 5.2 −2.9 5.7

PT2-0 3.4 10.5 6.1 −0.1 12.6 3.5 9.0 −2.0 9.6 −1.4 13.7 −13.4 −20.6 −0.5 7.0 −2.1 8.2 2.6 7.3 2.1 7.2

tPBE



PT2

SCF 3.4 9.6 4.8 −1.0 7.2 0.7 4.6 −5.6 4.3 −7.3 8.6 −18.0 −24.2 −4.1 1.5 −4.9 2.4 −1.1 6.6 −1.3 6.4

trevPBE

CAS 5.4 8.2 3.1 −3.1 1.4 −4.9 −0.5 −7.9 0.4 −10.2 6.4 −22.1 −24.3 −2.5 2.9 −7.8 4.7 −3.0 6.8 −3.1 6.6

tBLYP 7.0 9.1 5.9 3.6 14.5 8.7 10.2 4.0 7.7 −6.7 11.0 0.1 −7.0 1.8 3.5 0.5 3.3 4.5 6.2 4.9 5.8

ftPBE 6.5 9.0 4.2 2.5 8.1 3.0 3.5 −2.8 0.9 −14.5 4.0 −9.2 −15.9 −2.9 −2.0 −3.4 −3.4 −0.7 5.6 −0.1 5.2

ftrevPBE 11.7 5.8 4.3 0.9 5.8 5.2 3.3 12.1 0.4 −13.3 6.9 2.4 2.8 3.3 −0.4 −2.0 0.3 2.9 4.8 3.7 4.0

ftBLYP −18.0 5.5 −11.8 −8.6 −35.7 −30.3 −32.2 −30.8 −26.7 −6.7 −12.9 −55.2 −69.6 −4.2 −19.0 −30.4 9.5 −22.2 23.9 −23.8 25.8

SCF 1.8 8.8 3.4 −6.0 9.9 2.9 9.1 −2.9 13.9 16.0 11.5 −13.8 −21.4 −6.7 2.1 −4.0 8.2 1.9 8.4 0.4 7.7

tPBE 1.8 8.0 2.4 −6.6 4.6 0.2 4.5 −6.4 8.3 10.6 6.2 −18.1 −25.1 −10.1 −3.6 −6.8 2.4 −1.6 7.4 −3.0 7.3

trevPBE 4.1 6.7 1.9 −8.9 −1.4 −5.4 −2.0 −8.5 4.1 9.5 4.5 −21.0 −25.1 −8.4 −2.6 −9.5 4.6 −3.4 7.5 −4.8 7.6

tBLYP

SP 5.7 7.8 3.6 −4.1 12.0 8.3 10.8 3.1 11.9 12.2 10.2 0.4 −8.1 −4.0 −2.7 −1.0 3.5 4.1 6.4 3.1 5.8

ftPBE 5.3 8.0 2.7 −5.1 5.6 2.5 4.6 −3.5 4.8 4.4 3.1 −8.4 −16.9 −8.6 −8.8 −4.8 −3.3 −1.1 5.9 −1.7 6.2

ftrevPBE

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

10.1 4.1 3.4 −6.9 3.1 4.7 1.7 5.6 4.3 9.5 6.4 4.4 1.5 −2.1 −6.5 −3.2 0.5 2.4 4.6 1.6 4.1

ftBLYP



a

H

−17.3 19.5

0.7 4.5

4.1 6.5 −3.3 3.1 1.3 5.0 6.5 −3.0 −6.0 NA NA −5.9 −8.3 3.7 3.1 −1.9 5.2

−10.1 6.9 −7.2 −1.0 −19.5 −13.7 −35.3 −13.8 −16.6 NAa NA −46.1 −65.5 10.1 −15.9 −25.8 −5.7

−1.8 5.6

4.0 6.3 −3.0 2.6 −1.6 −2.0 5.2 −7.0 −7.1 NA NA −16.0 −15.9 2.3 3.0 −3.1 4.9

PT2-0

1.2 7.2

3.4 10.5 2.5 3.0 8.8 −0.5 7.3 −7.5 10.4 NA NA −13.4 −20.6 −1.4 7.0 −2.1 10.3

tPBE

CAS

−2.2 6.7

3.4 9.6 1.4 2.1 3.5 −3.6 2.9 −11.1 4.7 NA NA −18.0 −24.2 −5.1 1.5 −4.9 4.8

trevPBE

−3.9 7.2

5.4 8.2 0.3 0.7 −2.0 −8.2 −2.4 −13.2 1.0 NA NA −22.1 −24.3 −3.5 2.9 −7.8 6.1

tBLYP

3.9 5.1

7.0 9.1 2.9 6.3 11.5 5.2 7.1 −2.0 7.7 NA NA 0.1 −7.0 0.9 3.5 0.5 6.1

ftPBE

−1.1 4.8

6.5 9.0 1.6 4.5 5.0 −0.3 0.4 −8.9 0.7 NA NA −9.2 −15.9 −3.7 −2.0 −3.4 −0.4

ftrevPBE

NA denotes CASSCF calculation is not affordable. bIncluding all 17 molecules. cExcluding FeO and NiO.

CrH MnH FeH CoH FeC ScN VN CrN TiO FeO NiO V2 Cr2 CrF CoOH+ MnCH3+ NiCH2+ MSE-17b MUE-17b MSE-15c MUE-15c

PT2

SCF

2.7 3.1

11.7 5.8 3.4 6.1 4.2 1.1 0.0 0.5 0.2 NA NA 2.4 2.8 2.2 −0.4 −2.0 3.0

ftBLYP −18.0 5.5 −14.0 0.1 −38.3 −18.1 −69.2 −27.9 −24.8 −3.6 −6.9 −55.2 −69.6 1.1 −19.0 −30.4 −8.5 −23.3 24.1 −25.8 26.6

SCF 1.8 8.8 −3.4 8.2 2.6 −1.8 −1.2 −3.4 11.2 12.1 7.5 −13.8 −21.4 −8.3 2.1 −4.0 9.4 0.4 7.1 −0.9 6.8

tPBE 1.8 8.0 −4.1 6.9 −2.4 −4.9 −6.3 −6.9 5.6 6.8 2.3 −18.1 −25.1 −11.7 −3.6 −6.8 3.9 −3.2 7.4 −4.2 7.7

trevPBE 4.1 6.7 −4.1 2.6 −7.6 −9.4 −12.0 −8.4 1.9 5.8 0.6 −21.0 −25.1 −9.8 −2.6 −9.5 5.4 −4.8 8.0 −5.9 8.7

tBLYP

SP 5.7 7.8 −2.6 9.5 5.0 3.8 −0.4 2.7 9.3 8.6 6.0 0.4 −8.1 −5.9 −2.7 −1.0 5.7 2.6 5.0 1.9 4.7

ftPBE 5.3 8.0 −3.4 7.3 −1.6 −1.8 −7.9 −3.9 2.1 1.0 −1.0 −8.4 −16.9 −10.5 −8.8 −4.8 −0.9 −2.7 5.5 −3.1 6.1

ftrevPBE

Table 7. Signed Errors (kcal/mol), Mean Signed Error (MSE), and Mean Unsigned Error (MUE) for the Bond Dissociation Energy, De (kcal/mol), Computed by Multireference Methods with ext-CPO Spaces

10.1 4.1 −1.0 4.9 −1.8 −0.1 −9.8 6.2 1.9 5.8 2.0 4.4 1.5 −4.1 −6.5 −3.2 2.8 1.0 4.1 0.6 4.2

ftBLYP



Article


and because an MCSCF calculation includes not only static correlation but also some dynamic correlation, this raises the possibility of double counting of dynamic correlation energy, a problem that would become more serious as the active space is increased in size. But MC-PDFT does not follow this procedure. Instead, the only energy terms retained from the MCSCF energy are the kinetic energy, the electron−nuclear attraction, and the electron−electron electrostatic Coulomb energy computed from the one-electron density. Thus, there is no double counting of the two-electron correlation energy. However, the kinetic energy does depend on the active space size, so the question still arises of whether MC-PDFT calculations depend sensitively on active space size. Our past work has indicated that this is not a discouraging problem in practical calculations, but additional testing can be instructive. Here we provide such additional testing by examining the dependence of the MC-PDFT accuracy on active-space choice. Each of the three CPO active space schemes represents a systematic way to specify an active space, and an important objective of our current work is to test such systematic schemes, which qualify in the language of Pople54 as verifiable theoretical model chemistries. For eight of the 17 molecules studied, the size of the active space increases as one proceeds from the nom-CPO active space to the mod-CPO active space to the ext-CPO active space. However, for CrH, MnH, V2, Cr2, and MnCH3+, the size is the same for all three schemes; for ScN, CrN, TiO, and FeO, the size is the same for nom-CPO and mod-CPO; and for CoOH+, the size is the same for modCPO and ext-CPO. On increasing from nom-CPO to ext-CPO, the MUE-15 for CASSCF decreases from 26 to 20 kcal/mol, and the MUE-17 for SP decreases from 26 to 24 kcal/mol; for CASPT2 and CASPT2-0, the MUE-15 decreases by 0.5 and 0.8 kcal/mol, respectively. For CAS-PDFT and SP-PDFT, from nom-CPO to ext-CPO, for most translated or fully translated functionals, the change of MUE-15 is smaller or about 1 kcal/mol; to be more specific, the changes of MUE-15 for CAS-tPBE, CAS-trevPBE, CAS-tBLYP are 0.9, 0.6, and 0.1 kcal/mol. For SP-ftBLYP, which has excellent performance across all active spaces, the MUE-17 decreases by 0.9 kcal/mol from the nom-CPO to the ext-CPO space. These observations are very encouraging. They indicate that, for the systems studied here and for the property that we explored, namely, dissociation energy, the accuracy of MC-PDFT is not sensitive to the size of the active space, and in fact, this insensitivity is close to that of CASPT2-0. This is a promising feature because (a) small active spaces are preferable for practical work, (b) in the future, the development of density functionals that are specially optimized for MC-PDFT theory can benefit from this feature, and (c) this will allow us to apply the methods to much larger systems. (The systems studied here are small so that we can test the methods on systems where accurate values are available, but applications for practical purposes generally involve much larger systems.) It is perhaps not surprising in these cases that the small active spaces give reasonable results because we are computing dissociation energies and the orbitals involved in the dissociation processes are present by design in the small active spaces. If one instead were to compute electronic excitation energies involving several excited states, the picture would be different. So even though one has some success making the active space choice more of a black-box procedure, one has to keep in mind that the choice of active space may depend on the property calculated because, for example, a reasonable choice for ground state dissociation

sponding broken-symmetry spin-unrestricted Kohn−Sham density functional mean errors. For instance, although BLYP has an MUE-17 of 15.8 kcal/mol for 17 molecules and an MUE-15 of 15.5 kcal/mol for 15 molecule excluding FeO and NiO, the MUE-17 for CAS-ftBLYP with nom-CPO and modCPO are 6.0 and 4.8 kcal/mol, and the MUE-15s for CASftBLYP with nom-, mod-, and ext-CPO are 5.4, 4.0, and 3.1 kcal/mol, respectively. This clearly shows MC-PDFT is a significant improvement over Kohn−Sham density functional theory for this kind of problem when one uses density functionals of comparable complexity. With the nom-CPO active space, the MUEs for all 17 molecules for CASPT2 and CASPT2-0 are 5.1 and 5.9 kcal/ mol; the corresponding MUEs for CAS-ftrevPBE, CAS-ftBLYP, SP-ftrevPBE, and SP-ftBLYP are 6.3, 6.0, 5.5, and 5.0 kcal/mol, all of which are close to the accuracy of CASPT2 or CASPT2-0. Generally, for all three kinds of CPO active space, the fully translated functionals have smaller MUEs than the translated functionals, and their accuracies are close to or even better than CASPT2 or CASPT2-0. The PT2 step is significantly more expensive than the PDFT step (both in computational time and in memory requirements), and therefore the MC-PDFT method is very promising in terms of both the accuracy and the computational cost. The separated-pair partitioning significantly reduces the number of configuration state functions in the SCF step; and generally, as we can see from Table 3, SP has about 2 orders of magnitude less configuration state functions than the corresponding CASSCF. One of the advantages of SP and SP-PDFT is that, for cases where CASSCF is unaffordable (with CASPT2 then being even less affordable), one can still use SP to obtain a balanced description of the multireference wave function and then carry out the PDFT step to capture dynamic correlation because the SP and SP-PDFT calculations are very efficient and timesaving. In the present work, FeO and NiO are examples that show the power of the SP approximation. For FeO and NiO, the extCPO space has more than 2 × 108 configuration state functions, which is not affordable for CASSCF calculations. However, SP calculations are affordable, and SP-PDFT can be very accurate with fully translated functionals. The absolute errors for FeO and NiO with SP-ftrevPBE are all 1.0 kcal/mol, which are significantly smaller than the errors of Kohn−Sham revPBE calculations for FeO (17.4 kcal/mol) and NiO (13.8 kcal/mol). If we do not consider the empirical IPEA shift of the CASPT2 method, i.e., if we are comparing CASPT2-0 with SPPDFT, which is a fair comparison because there is no adjustable or empirical parameter in MC-PDFT, we find that SP-PDFT can perform better than CASPT2-0; such is the case for SPftrevPBE and SP-ftBLYP with the nom-CPO active space. In all three CPO active spaces, the SP-ftBLYP calculations perform better than CASPT2 or CASPT2-0, and this is the most successful MC-PDFT method tested in this work. SP-PDFT is generally able to maintain the accuracy of the corresponding CAS-PDFT, but with significant reductions of the number of configuration state functions. 3.3. Active Space Dependence of MC-PDFT. Most previous attempts to combine multiconfiguration wave function theory with density functional theory involve adding densityfunctional-based correlation energy to the MCSCF energy. Because density-functional correlation functionals are usually designed to approximate the entire dynamic correlation energy, I


Article


SP-PDFT is able to give accuracy similar to that of CASPT2 and that SP-PDFT provides systematic improvement over the spin-unrestricted Kohn−Sham density functional calculations. In particular, SP-ftBLYP performs the best on average, and it is even better than CASPT2 without an empirical IPEA shift. For cases where CASSCF calculations are not affordable, SP and SP-PDFT are still practical, and they provide a computationally economical approach with good performance/cost balance. For the systems studied here, the accuracy of MC-PDFT results is not sensitive to the variation of the size of the active space. The currently available approximate version of separatedpair second-order perturbation theory does not perform as well as CASPT2 with the same active space, and in most cases, it is not better than SP-PDFT. We therefore conclude that MCPDFT theory and especially SP-PDFT offers a powerful tool for efficiently calculating the electronic structure of transition-metal compounds; this can be particularly useful for understanding inorganometallic and organometallic catalysis.

energies might give poor results for bond lengths or electronic excitation energies. This is not a feature only of MC-PDFT; such consideration applies to all multireference calculations. 3.4. SPPT2. It is also of interest to test the performance of PT2 based on SP wave functions, so we also investigated the performance of SPPT2 in the current work, but only for nomCPO spaces. This is because the current implementation30 of the GASPT2 method has a similar cost when compared to the corresponding CASPT2 calculation, and the accuracy of SPPT2 is, on average, worse than that of CASPT2. Table 8 lists the Table 8. Signed Errors (kcal/mol), Mean Signed Error (MSE), and Mean Unsigned Error (MUE) for the Bond Dissociation Energy, De (kcal/mol), Computed by CASPT2 and SPPT2 Using nom-CPO Spaces CrH MnH FeH CoH FeC ScN VN CrN TiO FeO NiO V2 Cr2 CrF CoOH+ MnCH3+ NiCH2+ MSE-17a MUE-17a MSE-15b MUE-15b a

CASPT2

CASPT2-0

SPPT2

SPPT2-0

4.1 6.5 −5.6 −6.8 3.5 5.3 −2.2 −8.6 −2.3 8.0 −4.5 −5.9 −8.3 5.0 3.0 −1.9 5.8 −0.3 5.1 −0.6 5.0

4.0 6.3 −4.2 −5.5 1.6 −1.9 −6.2 −14.2 −5.1 0.0 −3.9 −16.0 −15.9 3.3 3.3 −3.1 6.1 −3.0 5.9 −3.2 6.4

5.7 3.5 −6.7 −7.0 −2.3 4.8 −4.6 −11.7 −1.6 15.4 −14.5 −7.1 −10.0 0.7 2.8 −5.2 5.2 −1.9 6.4 −2.2 5.3

4.8 2.8 −5.0 −5.5 −3.5 −2.4 −7.4 −17.3 −4.0 14.2 −13.9 −14.4 −17.5 −1.4 3.1 −6.4 5.6 −4.0 7.6 −4.6 6.7

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jctc.6b01102. Geometries of the three polyatomic molecules and their fragments and absolute energies in hartrees for the tPBE and ftrevPBE calculations (PDF)

■

AUTHOR INFORMATION

Corresponding Authors

*L.G. E-mail: [email protected]. *D.G.T. E-mail: [email protected]. ORCID

Samuel O. Odoh: 0000-0002-6633-8229 Laura Gagliardi: 0000-0001-5227-1396 Donald G. Truhlar: 0000-0002-7742-7294 Present Address


†

Department of Chemistry, 1664 N. Virginia Street, University of Nevada Reno, Reno, NV 89557.

signed errors, the MSEs, and the MUEs for SPPT2 and SPPT20, along with the corresponding CASPT2 and CASPT2-0 with the nom-CPO space. Notice that, although for SPPT2 the optimal IPEA shift value is not necessarily the same as the one used in CASPT2 (i.e., 0.25 hartree), we still use the same value to have a consistent comparison. The MUE-17s for CASPT2, CASPT2-0, SPPT2, and SPPT2-0 are respectively 5.1, 5.9, 6.4, and 7.6 kcal/mol; the SPPT2 performs worse than the corresponding CASPT2. The accuracy of SPPT2-0 (7.6 kcal/ mol) is not better than SP-PDFT results; all SP-PDFT results with the nom-CPO space, except for tPBE, are better than SPPT2-0.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported in part by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, under award DE-SC0012702 (Inorganometallic Catalyst Design Center).

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REFERENCES

(1) Cramer, C. J.; Truhlar, D. G. Phys. Chem. Chem. Phys. 2009, 11, 10757. (2) Kepp, K. P. Coord. Chem. Rev. 2013, 257, 196. (3) Tsipis, A. C. Coord. Chem. Rev. 2014, 272, 1. (4) Lupp, D.; Christensen, N. J.; Fristrup, P. Dalton Trans. 2014, 43, 11093. (5) Peverati; Truhlar, D. G. Philos. Trans. R. Soc., A 2014, 372, 20120476. (6) (a) Cheng, G.-J.; Zhang, X.; Chung, L. W.; Xu, L.; Wu, Y.-D. J. Am. Chem. Soc. 2015, 137, 1706. (b) Bao, J. L.; Yu, H. S.; Duanmu, K.; Makeev, M. A.; Xu, X.; Truhlar, D. G. ACS Catal. 2015, 5, 2070. (7) Roos, B. O.; Taylor, P. R.; Siegbahn, P. E. M. Chem. Phys. 1980, 48, 157.

4. CONCLUSIONS In the current work, the performance of multiconfiguration pair-density functional theory for calculating the bond dissociation energy of transition-metal complexes is tested over 17 small molecules against experimental data. We propose systematic ways for choosing the active space, namely, three correlated-participating-orbitals schemes, and a systematic way of partitioning the active space, namely, the separated pair approximation. Because these methods are systematic, we could test them as theoretical model chemistries. We have found that J


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