Multiconfiguration Pair-Density Functional Theory: A New Way To

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Multiconfiguration Pair-Density Functional Theory: A New Way To Treat Strongly Correlated Systems Laura Gagliardi,* Donald G. Truhlar,* Giovanni Li Manni,† Rebecca K. Carlson, Chad E. Hoyer, and Junwei Lucas Bao Department of Chemistry, Chemical Theory Center, and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States CONSPECTUS: The electronic energy of a system provides the Born−Oppenheimer potential energy for internuclear motion and thus determines molecular structure and spectra, bond energies, conformational energies, reaction barrier heights, and vibrational frequencies. The development of more efficient and more accurate ways to calculate the electronic energy of systems with inherently multiconfigurational electronic structure is essential for many applications, including transition metal and actinide chemistry, systems with partially broken bonds, many transition states, and most electronically excited states. Inherently multiconfigurational systems are called strongly correlated systems or multireference systems, where the latter name refers to the need for using more than one (“multiple”) configuration state function to provide a good zero-order reference wave function. This Account describes multiconfiguration pair-density functional theory (MC-PDFT), which was developed as a way to combine the advantages of wave function theory (WFT) and density functional theory (DFT) to provide a better treatment of strongly correlated systems. First we review background material: the widely used Kohn−Sham DFT (which uses only a single Slater determinant as reference wave function), multiconfiguration WFT methods that treat inherently multiconfigurational systems based on an active space, and previous attempts to combine multiconfiguration WFT with DFT. Then we review the formulation of MC-PDFT. It is a generalization of Kohn−Sham DFT in that the electron kinetic energy and classical electrostatic energy are calculated from a reference wave function, while the rest of the energy is obtained from a density functional. However, there are two main differences with respent to Kohn-Sham DFT: (i) The reference wave function is multiconfigurational rather than being a single Slater determinant. (ii) The density functional is a function of the total density and the on-top pair density rather than being a function of the spin-up and spin-down densities. In work carried out so far, the multiconfigurational wave function is a multiconfiguration self-consistent-field wave function. The new formulation has the advantage that the reference wave function has the correct spatial and spin symmetry and can describe bond dissociation (of both single and multiple bonds) and electronic excitations in a formally and physically correct way. We then review the formulation of density functionals in terms of the on-top pair density. Finally we review successful applications of the theory to bond energies and bond dissociation potential energy curves of main-group and transition metal bonds, to barrier heights (including pericyclic reactions), to proton affinities, to the hydrogen bond energy of water dimer, to ground- and excited-state charge transfer, to valence and Rydberg excitations of molecules, and to singlet−triplet splittings of radicals. We find that that MC-PDFT can give accurate results not only with complete-active-space multiconfiguration wave functions but also with generalized-active-space multiconfiguration wave functions, which are practical for larger numbers of active electrons and active orbitals than are complete-active-space wave functions. The separated-pair approximation, which is a special case of generalized active space self-consistent-field theory, is especially promising. MC-PDFT, because it requires much less computer time and storage than pure WFT methods, has the potential to open larger and more complex strongly correlated systems to accurate simulation.

I. INTRODUCTION

A qualitatively correct description of the electronic structure of an MR molecule requires two or more configuration state functions (CSFs), where a CSF is an antisymmetric manyelectron function composed of the minimum number of Slater determinants needed to obtain the correct spatial and spin

Molecular systems with degenerate or nearly degenerate states are usually called multireference (MR) systems or strongly correlated systems. Examples of MR systems include ground states of many molecules and solids that contain transition metals,1−3 systems with partially broken bonds, most excited states,4 and some transition states. © 2016 American Chemical Society

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to active orbitals are included variationally in the wave function. When using the CASSCF method, one can take advantage of certain simplifications due to the fact that the many-electron wave function is independent of unitary transformations within the spaces of inactive, active, or external orbitals. Modern extensions of these methods forego these simplifications by not including all possible CSFs that can be obtained by distributing active electrons into active orbitals. This allows larger active spaces. Examples of methods employing incomplete configuration lists are restricted active space SCF (RASSCF),13 generalized active space SCF (GASSCF),17−20 and occupationrestricted-multiple-active-space SCF (ORMAS).21,22 For example, in the GASSCF model,17−20 the active orbitals are placed in distinct subspaces, and accumulated minimum and maximum electron occupation numbers are applied to each subspace. Within a subspace, one includes all possible spin- and symmetry-adapted CSFs that can be constructed with this occupation number constraint, and optionally one includes a restricted set of intersubspace excitations. An interesting special case of the GAS method is the separated pair (SP) approximation,23 in which no more than two orbitals are included in any GAS subspace and in which intersubspace excitations are excluded. Further restricting the number of configurations such that each subspace is coupled as a singlet leads to the generalized valence bond perfect-pairing24 (GVBPP) method. The SP approximation typically leads to many less configurations than CASSCF, but more than GVB-PP; a key advantage of CASSCF and SP over GVB-PP is that they allow size-consistent dissociation to high-spin fragments. The treatment of larger active spaces is possible also through other methods including the density matrix renormalization group (DMRG).25 An MCSCF wave function will yield both nondynamic correlation energy and some dynamic correlation energy, but it is impractical to include in this step enough dynamic correlation energy for chemical accuracy because of the cost. For example, the cost of a CASSCF calculation increases steeply (exponentially26) as the size of the active space increases. One can add additional dynamic correlation by multireference perturbation theory (MRPT2), such as complete-active-space second-order perturbation theory (CASPT2),27 or by multireference configuration interaction (MRCI)28 methods, in either case using the MCSCF wave function as a reference. The cost and memory requirements of these methods are even greater than for CASSCF. Although the incomplete MCSCF methods like GASSCF reduce the computational demand of both the SCF and post-SCF steps, MRPT2 and MRCI are still restricted to small systems. For large systems, we need a more affordable approach.

symmetry (it can be either single-determinantal or multideterminantal), and each CSF corresponds to a particular way of distributing the electrons in orbitals.5 Correctly predicting the energy of MR systems is a frontier area for quantum chemistry.3,6 This Account is about a new way to calculate electronic energies for MR systems. One can distinguish between two types of electron correlation, although the border between the two is not strict. One kind of electron correlation is called “dynamic” correlation; this arises from electrons avoiding one another at short-range to minimize repulsion or correlating their motion at large distances to produce dispersion interactions.7 The other kind of electron correlation is caused by near degeneracies of the electronic state of interest with other electronic states of the same symmetry; it is sometimes described as “static”, “left− right”, or, most generally, “nondynamic” correlation.8−11 It is usually negligible in closed-shell ground states with large energy gaps to the first excited state, but otherwise it is present to a greater or lesser extent, depending on the system, the state, and the geometry. Simply stated, MR systems are those with significant nondynamic correlation. There are two kinds of electronic structure theory. Methods based on electron density are called density functional theory (DFT), while methods based on wave functions are called wave function theory (WFT). Both kinds of theory have their advantages and disadvantages. This Account is about a new way to combine them for treating MR systems.

II. BACKGROUND II.A. Background: WFT Methods for MR Systems

WFT methods for calculating correlation energy usually have two steps; first, one obtains a reference wave function, usually with orbitals that are variationally optimized by a self-consistent field (SCF) calculation. The reference wave function should include the important nondynamic correlation; therefore, for MR systems, it should be multiconfigurational. In the second step, one treats dynamic correlation (e.g., by configuration interaction, coupled clusters, or perturbation theory) to treat dynamic correlation. The usual way to generate reference wave functions for MR systems is multiconfiguration self-consistentfield (MCSCF) theory. An MCSCF wave function is a multiconfigurational wave function in which one variationally optimizes both the orbitals used in the CSFs and the coefficients of the CSFs in a configuration interaction expansion of the wave function. An MCSCF wave function includes only a portion of the dynamic correlation energy, and a more complete treatment is necessary to quantitatively describe bond energies, electronic excitation energies of valence or core electrons, and transition state barrier heights. The choice of which configurations are included in the MCSCF step is often based on the concept of an active space.12−15 In a complete-active-space SCF (CASSCF) calculation,16 the molecular orbitals are divided into three strata: inactive, active, and external. Inactive orbitals are doubly occupied in all CSFs that are used to build the multiconfigurational wave function, and the external orbitals are unoccupied in all such CSFs. The remaining electrons (beyond those in inactive orbitals) are the active electrons, and they are distributed in a number of active orbitals, which may be doubly occupied, singly occupied, or unoccupied. In a CASSCF calculation, all CSFs with spatial and spin symmetry of the state of interest that can be formed by assigning the active electrons

II.B. Background: Kohn−Sham DFT

Kohn−Sham DFT (KS-DFT) involves what may be called a reference wave function, namely, a Slater determinant. KS-DFT calculates the kinetic energy and classical Coulomb interactions from the Slater determinant and the density that it generates and obtains the rest of the energy from a functional of the density, called the exchange−correlation (XC) functional. If one knew the exact XC functional, one could obtain the exact results for any system, even one that is not well described by a single Slater determinant in wave function theory. However, the exact XC functional of KS-DFT is essentially unknowable, and current approximations to it work better when the systems are well described by a Slater determinant. If the Slater determinant 67

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We call it multiconfiguration pair-density functional theory (MC-PDFT), and it is a generalization of KS-DFT. KS-DFT requires full variational optimization of the Slater determinant, including allowing different orbitals for different spins (DODS) even in nominally doubly occupied orbitals; DODS is called spin polarization. Spin-polarized Kohn−Sham theory,67,68 defined in terms of ρ and m, yields brokensymmetry Slater determinants for MR systems. Although the resulting energies are sometimes reasonable, in other cases they are not, and the broken-symmetry nature of the solution can even make it ambiguous as to which state is being approximated. If one represents the density by CSFs with the correct spin and spatial symmetries, conventional density functionals do not yield accurate energies for MR systems. Thus, the net spin densities are not appropriate variables for density functionals if the correct spin symmetry is imposed. The problem of the generalization of ζ to proper-spin wave functions has been overcome by considering a functional in terms of the on-top two-particle density Π rather than ζ,67−70 and that is the method we shall use. The on-top density at point r is the probability that two electrons are found at this point; because wave functions are antisymmetric under electron interchange, the two electrons must have opposite spin. Using ρ and Π, we have proposed an approach for treating strongly correlated systems at much lower computational costs than existing MR methods, by combining MCSCF with DFT in a way that differs fundamentally from the approaches described in section II.C. The new approach is called multiconfiguration pair-density functional theory (MC-PDFT).71 It may be considered to be a multiconfigurational analog of KS-DFT, as indicated schematically in Figure 1. The energy is computed as

representing the density does not provide a realistic description of the system, an approximate XC functional will not be able to correct it. One hopes that representing the density in a more physical way will allow simpler approximations to the density functional to perform well. A widely used kind of XC functional is known as a generalized gradient approximation (GGA). A GGA depends on the spin densities, ρα and ρβ, and their gradients. Alternatively one can use the total density ρ given by ρ(r) = ρα (r) + ρβ (r)

(1)

and the spin magnetization density, m (which may also be called the net spin density), given by m(r) = ρα (r) − ρβ (r)

(2)

A third choice would be ρ and the spin polarization, ζ, defined as ζ(r) = (ρα (r) − ρβ (r))/(ρα (r) + ρβ (r))

(3)

and a fourth would be the singlet and triplet charge densities.29 GGAs are known to be poor for certain kinds of problems such as charge transfer and Rydberg states. These problems are ameliorated by including a fixed percentage of nonlocal Hartree−Fock exchange in the density functionals, which are then called global-hybrid functionals. II.C. Background: Combining MCSCF Theory with DFT

Several methods have also been proposed for combining MCSCF with DFT. One can add some amount of density functional correlation energy to the energy computed in a multiconfigurational wave function calculation.15,30−59 There are two main problems with this combination.31 The first problem is that an MCSCF calculation inevitably contains some dynamic correlation, but existing correlation functionals are designed to treat the entire dynamic correlation (and not to treat nondynamic correlation60). Thus, the density functional correlation must be reduced so that a portion of dynamic correlation energy is not counted twice; fully successful methods to treat this have not been developed. The second problem with the approach is that the fundamental variables used to define functionals in KS-DFT, namely, the spin densities ρα and ρβ, need to be replaced when a multiconfiguration reference wave function with the correct symmetry is employed. In KS-DFT, one can formally separate the Coulomb interaction into a short-range part and a long-range part. This range separation can also be applied to multiconfigurational methods as a way to combine them with DFT.61,62 Several workers have proposed using multiconfiguration WFT for the long-range part and KS-DFT for the short -range part.38,61−65 Sharkas et al.66 designed a combination of MCSCF and DFT based on a simple linear decomposition of the Coulombic electron−electron interaction. The idea of decomposing the Coulomb interaction potential is limited by the assumption that the short-range correlation potential will always result in only the short-range correlation effects.38,46−48 That said, range separation is a promising approach.

Figure 1. Comparison of KS-DFT to MC-PDFT.

the sum of the kinetic energy and classical electrostatic energy of an MCSCF reference wave function and a density functional. The density functional is called the on-top density functional; it includes electron exchange, the electron−electron repulsion contribution to electron correlation, and the difference between the exact kinetic energy and that computed from the reference wave function. Notice that the MCSCF energy is not used; thus the correlation energy is not computed by WFT, and so the question of separating residual correlation energy from that included in the MCSCF calculation does not arise, although one does have to be careful that the required correction of the MCSCF kinetic energy does depend on the amount of correlation energy included in the MCSCF calculation. Next we discuss the choice of on-top density functional. If one approximates an exchange−correlation functional using only local densities and their gradients, an exchange− correlation functional for a spin-polarized system can be written as a functional of ρ and m, and the magnitudes, ρ′ 

III. MULTICONFIGURATION PAIR-DENSITY FUNCTIONAL THEORY We have recently proposed a simple way to generalize density functional theory to multiconfigurational reference functions. 68

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Accounts of Chemical Research |∇ρ| and m′  |∇m|, of their gradients. For a singledeterminant wave function, m can be related to Π and ρ by67 m(r) = ρ(r)[1 − R(r)]1/2

for various transition metal dimers where we compared the performance of tPBE, ftPBE, CASSCF, CASPT2, and PBE for dissociation energies and equilibrium geometries.73 Both tPBE and ftPBE correct the overbinding of PBE and have comparable performance to CASPT2. Using the same reference wave function as CASPT2, both types of MC-PDFT functionals recover correlation missing in CASSCF, improving the equilibrium geometry and generating quantitatively accurate energetics (Figure 2).

(4)

where R(r) =

4Π(r) [ρ(r)]2

(5)

with R ≤ 1 at all points in space. MC-PDFT uses ρ and Π to define the on-top density functional.71 Ultimately we hope to develop new on-top density functionals specifically for use with MC-PDFT. For multiconfiguration wave functions, eq 4 is not true, and in fact R can be greater than 1. Nevertheless, as a first step, we simply translated previously developed exchange−correlation functionals of spin densities by using eq 4 as a guide. In particular, given Exc(ρ, m, ρ′, m′), we proposed the following translation prescription: ⎛ ⎧ ρ(r)(1 − R )1/2 if R ≤ 1⎫ ⎬, Eot[ρ(r), Π(r)] = Exc⎜⎜ρ(r), ⎨ ⎩0 if R > 1⎭ ⎝ ⎪

⎪

⎪

⎪

⎧ ρ′(r )(1 − R )1/2 if R ≤ 1⎫⎞ ⎬⎟⎟ ρ′(r), ⎨ ⎩0 if R > 1⎭⎠ ⎪

⎪

⎪

⎪

(6)

Figure 2. Potential energy curves for Cu2 with dissociation energies in parentheses (eV).

Our most extensively tested on-top density functional is called tPBE, which is a translation by eq 6 of the PBE72 GGA functional. We also proposed a second kind of translation73 (called “fully translated”, e.g., ftPBE) that is continuous with continuous first and second derivatives and that includes the gradient of the on-top density. Garza et al.74 explored a similar idea by combining a pair coupled cluster doubles (pCCD) wave function with DFT, resulting in the pCCD+DFT approach, which greatly improves upon pCCD in the description of some typical problems where static and dynamic correlations are both important.

We also used MC-PDFT to study ground- and excited-state charge-transfer processes.76 We computed the dissociation energy of seven ground-state charge transfer complexes and the electronic excitation energy of three intermolecular chargetransfer complexes, namely, H2N−H···H−NO2, C2H4···C2F4, and NH3···F2. Results were compared with KS-DFT and with linear-response time-dependent KS-DFT77 calculations employing the PBE gradient approximation,72 the global-hybrid PBE0 functional,78 and two long-range-corrected hybrid functionals (CAM-B3LYP 79 and ωB97X 80 ), and with CASPT2. For ground-state charge transfer, tPBE performs better than either the PBE exchange−correlation functional or CASPT2. For intermolecular charge-transfer excitations, tPBE is superior to PBE, PBE0, and CAM-B3LYP and is comparable to, if not better than, ωB97X. This is especially notable when we consider that the energy functional is calculated without any component of nonlocal exchange, and the current on-top density functional used in MC-PDFT is a first-generation gradient approximation. We also tested the method for valence and Rydberg excitations of organic molecules and atoms,81,82 for which it is equally promising, as shown in Figure 3. We found that MCPDFT is insensitive to the addition of extra diffuse basis functions, which is not often the case for KS-DFT.82 We also tested MC-PDFT on 23 molecular excitations to see if MCPDFT would be a useful tool for spectroscopy.81 To gauge general performance, we included valence, Rydberg, intermolecular charge transfer, and intramolecular charge transfer excitations. We found that MC-PDFT is as accurate as CASPT2, which is extremely promising since MC-PDFT is much less expensive than CASPT2. The above-mentioned tests used CASSCF reference functions. We also tested the dependence of the performance of MC-PDFT on the active space choice; in particular we examined whether we can obtain accurate results with the much

IV. PERFORMANCE OF MC-PDFT We have implemented MC-PDFT in a locally modified version of Molcas 8.1.26 The MC-PDFT energy calculation has four steps: (i) a standard MCSCF calculation is performed to generate the one- and two-body density matrices, (ii) the energy contribution from the on-top density functional is evaluated from these density matrices, (iii) the MC-PDFT energy contributions that come from the MCSCF wave function, namely, the kinetic energy, the electron−nuclear potential energy, and the electron−electron classical electrostatic potential energy, are computed in the standard MCSCF way, and (iv) the energy contributions of steps ii and iii are added. We have tested MC-PDFT on several systems. We applied it to the calculation of bond energies of main-group molecules and transition metal complexes, barrier heights and reaction energies for diverse chemical reactions, proton affinities, and the water dimerization energy.75 Averaged over 56 data points, the performance of MC-PDFT was found to be superior to that of KS-DFT theory with a comparable density functional, and the accuracy was even found to be comparable to CASPT2. Bond dissociations of diatomic molecules are good test cases for MC-PDFT because the wave function changes character as the bond breaks and different types of correlation effects will dominate along the curve. We computed the potential curves 69

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diradicals, the results show that the MC-PDFT is a systematic improvement over the corresponding weighted-average brokensymmetry (WABS) Kohn−Sham density functional calculations, and the performance of the current implementation of CAS-tPBE can be better than some widely used Kohn−Sham schemes. We have tested MC-PDFT using several translated and fully translated functionals, namely, the translated functionals tPBE, tLSDA, tBLYP, and trevPBE and their fully translated analogs. The results are moderately dependent on the functional choice, but the promise of the method can be well understood by considering only tPBE and ftPBE.

V. CONCLUDING REMARKS AND FUTURE DIRECTIONS One of the most compelling challenges for modern quantum chemists is to develop quantum chemical methods able to accurately treat, at an affordable computational cost, large systems with inherently multiconfigurational electronic structures. One possibility is to combine multireference wave functions with density functional theory. We reviewed the state of the art of methods that explore this avenue, and we summarized the features of a recently developed method, multiconfiguration pair-density functional theory, MC-PDFT, which may be considered to be a multiconfigurational analog of Kohn−Sham density functional theory. This method is promising for handling multireference systems, and it is applicable to both ground and excited states, including Rydberg and charge transfer states. The results to date are very encouraging and in many cases competitive with the much more resource-demanding CASPT2 method. They are also of better quality than those obtained with Kohn−Sham DFT for multireference systems. Future work will include testing the performance of simplified wave functions, like the separatedpair approximation, for transition metal complexes and developing better on-top functionals of the MCSCF density and on-top pair density.

Figure 3. First electronic excitation of He atom and average mean unsigned errors for a variety of molecular excitations.

more affordable SP approximation.23 The accuracy of the SPPDFT method was tested for predicting the structural properties and bond dissociation energies of 12 diatomic molecules and two triatomic molecules. SP-PDFT (MC-PDFT based on the SP wave function) reproduces the accuracy of CAS-PDFT (MC-PDFT based on the CASSCF wave function) for predicting C−H bond dissociation energies, the reaction barriers of pericyclic reactions and the properties of open-shell singlet systems, all at only a small fraction of the computational cost of the full CASSCF wave function. It is promising that the SP approximation agrees quite well with CASSCF, at greatly reduced cost, and can thus be extended to bigger systems. It is especially encouraging that the quality of results obtained from MC-PDFT calculations remains largely unchanged even with drastic reductions in the number of included CSFs. Recently we investigated the performance of MC-PDFT for computing the adiabatic singlet−triplet splitting for small maingroup divalent radicals for which accurate experimental data are available.83 In order to define theoretical model chemistries that can be assessed consistently, we proposed three correlated participating orbital schemes,84 in particular, nominal, moderate, and extended active spaces, to define the constitution of complete active spaces, and we tested them systematically. Figure 4 shows methylene as an example. Considering 13

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Laura Gagliardi: 0000-0001-5227-1396 Donald G. Truhlar: 0000-0002-7742-7294 Chad E. Hoyer: 0000-0003-2597-9502 Present Address †

G.L.M.: Max-Planck-Institut fü r Festkö r perforschung, Heisenbergstraße 1, 70569 Stuttgart, Germany.

Notes

The authors declare no competing financial interest. Biographies Laura Gagliardi was born in Bologna and received her Ph.D. from the University of Bologna. Since 2009, she has been on the Chemistry faculty of University of Minnesota where she is now Distinguished McKnight Professor. She develops electronic structure theories and applies them to study sustainable energy challenges. Don Truhlar was born in Chicago and received a Ph.D. from Caltech. Since 1969, he has been on the Chemistry faculty of the University of

Figure 4. Singlet−triplet gap and key orbitals for methylene. 70

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Accounts of Chemical Research Minnesota where he is now Regents Professor. His research fields are chemical dynamics and computational quantum mechanics.

(12) Roos, B. O.: Multiconfigurational quantum chemistry. In Theory and Applications of Computational Chemistry: The First Forty Years; Dykstra, C. E., Frenking, G., Kim, K. S., Scuseria, G. E., Eds.; Elsevier: Amsterdam, 2005; pp 725−764. (13) Malmqvist, P. Å.; Pierloot, K.; Shahi, A. R. M.; Cramer, C. J.; Gagliardi, L. The restricted active space followed by second-order perturbation theory method: Theory and application to the study of CuO2 and Cu2O2 systems. J. Chem. Phys. 2008, 128, 204109. (14) Schmidt, M. W.; Gordon, M. S. The construction and interpretation of MCSCF wavefunctions. Annu. Rev. Phys. Chem. 1998, 49, 233−266. (15) Grafenstein, J.; Cremer, D. The combination of density functional theory with multi-configuration methods - CAS-DFT. Chem. Phys. Lett. 2000, 316, 569−577. (16) Roos, B. O.; Taylor, P. R.; Siegbahn, P. E. M. A Complete Active Space Scf Method (CASSCF) Using a Density-Matrix Formulated Super-Ci Approach. Chem. Phys. 1980, 48, 157−173. (17) Ma, D.; Li Manni, G.; Gagliardi, L. The Generalized Active Space concept in Multiconfigurational Self-Consistent-Field methods. J. Chem. Phys. 2011, 135, 044128. (18) Olsen, J.; Roos, B. O.; Jorgensen, P.; Jensen, H. J. A. Determinant based Configuration-Interaction algorithms for complete and restricted Configuration-Interaction spaces. J. Chem. Phys. 1988, 89, 2185−2192. (19) Fleig, T.; Olsen, J.; Marian, C. M. The generalized active space concept for the relativistic treatment of electron correlation. I. Kramers-restricted two-component configuration interaction. J. Chem. Phys. 2001, 114, 4775−4790. (20) Ma, D. X.; Li Manni, G.; Olsen, J.; Gagliardi, L. Second-Order Perturbation Theory for Generalized Active Space Self ConsistentField Wave Functions. J. Chem. Theory Comput. 2016, 12, 3208−3213. (21) Ivanic, J. Direct configuration interaction and multiconfigurational self-consistent-field method for multiple active spaces with variable occupations. I. Method. J. Chem. Phys. 2003, 119, 9364−9376. (22) Ivanic, J.; Ruedenberg, K. Identification of deadwood in configuration spaces through general direct configuration interaction. Theor. Chem. Acc. 2001, 106, 339−351. (23) Odoh, S. O.; Li Manni, G.; Carlson, R. K.; Truhlar, D. G.; Gagliardi, L. Separated-pair approximation and separated-pair pairdensity functional theory. Chem. Sci. 2016, 7, 2399−2413. (24) Hunt, W. J.; Hay, P. J.; Goddard, W. A., III Self-Consistent Procedures for Generalized Valence Bond Wavefunctions. Applications H3, BH, H2O, C2H6, and O2. J. Chem. Phys. 1972, 57, 738. (25) Chan, G. K. L.; Sharma, S.: The Density Matrix Renormalization Group in Quantum Chemistry. In Annual Review of Physical Chemistry; Leone, S. R., Cremer, P. S., Groves, J. T., Johnson, M. A., Eds.; Annual Reviews: Palo Alto, CA, 2011; Vol. 62; pp 465−481. (26) Aquilante, F.; Autschbach, J.; Carlson, R. K.; Chibotaru, L. F.; Delcey, M. G.; De Vico, L.; Galvan, I. F.; Ferre, N.; Frutos, L. M.; Gagliardi, L.; Garavelli, M.; Giussani, A.; Hoyer, C. E.; Li Manni, G.; Lischka, H.; Ma, D. X.; Malmqvist, P. A.; Muller, T.; Nenov, A.; Olivucci, M.; Pedersen, T. B.; Peng, D. L.; Plasser, F.; Pritchard, B.; Reiher, M.; Rivalta, I.; Schapiro, I.; Segarra-Marti, J.; Stenrup, M.; Truhlar, D. G.; Ungur, L.; Valentini, A.; Vancoillie, S.; Veryazov, V.; Vysotskiy, V. P.; Weingart, O.; Zapata, F.; Lindh, R. Molcas 8: New capabilities for multiconfigurational quantum chemical calculations across the periodic table. J. Comput. Chem. 2016, 37, 506−541. (27) Andersson, K.; Malmqvist, P.-Å.; Roos, B. O. 2nd-order perturbation-theory with a complete active space self-consistent field reference function. J. Chem. Phys. 1992, 96, 1218−1226. (28) Siegbahn, P. E. M.; Almlof, J.; Heiberg, A.; Roos, B. O. The complete active space scf (CASSCF) method in a Newton-Raphson formulation with application to the HNO molecule. J. Chem. Phys. 1981, 74, 2384−2396. (29) Staroverov, V. N.; Davidson, E. R. Charge densities for singlet and triplet electron pairs. Int. J. Quantum Chem. 2000, 77, 651−660. (30) Savin, A.: Correlation contributions from density functionals. In Density functional methods in Chemistry; Labanowski, J. K., Andzelm, J. W., Eds.; Springer: Berlin, 1991; pp 213−230.

Giovanni Li Manni was born in Palermo, Italy, and received his Ph.D. from the University of Geneva in 2013. Since 2015, he is scientist at the MPI where he continues to develop advanced electronic structure methods and applies them to bioinspired catalysts. Rebecca Carlson was born in Edina, MN, and received a B.S. in chemistry from Bethel University. She is a recipient of the National Science Foundation Graduate Research Fellowship and is currently a Ph.D. student with Professor Gagliardi. Her research interests include MC-PDFT and studying bimetallic complexes. Chad Hoyer was born in Lebanon, PA, and received a B.S. in chemistry and physics from Shippensburg University. In 2012, he joined the groups of Professors Gagliardi and Truhlar as a Ph.D. student. His current research work includes diabatization and MCPDFT. Junwei Lucas Bao was born in Nanjing, China, and received a B.S. in chemistry from Nanjing Normal University. In 2013, he joined Professor Truhlar’s group as a Ph.D. student. His current research interests include variational transition state theory, KS-DFT, and MCPDFT.

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ACKNOWLEDGMENTS We acknowledge the contributions of Soumen Ghosh, Roland Lindh, Andy Luo, Dongxia Ma, Samuel Odoh, Jeppe Olsen, Andrew Sand, and Kamal Sharkas to MC-PDFT development. The development of MC-PDFT is supported in part by NSF Grant CHE-1464536; work on charge transfer is supported by AFOSR Grant FA9550-16-1-0134. R.K.C. acknowledges an NSF Fellowship.

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