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Second-Order Perturbation Theory for Generalized Active Space SelfConsistent-Field Wave Functions Dongxia Ma,*,† Giovanni Li Manni,*,† Jeppe Olsen,‡ and Laura Gagliardi*,§,∥ †

Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1, 70569 Stuttgart, Germany Department of Chemistry, Aarhus University, Langelandsgade 140, 8000 Aarhus, Denmark § Department of Chemistry, Supercomputing Institute, and ∥Chemical Theory Center, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States ‡

S Supporting Information *

ABSTRACT: A multireference second-order perturbation theory approach based on the generalized active space self-consistent-field (GASSCF) wave function is presented. Compared with the complete active space (CAS) and restricted active space (RAS) wave functions, GAS wave functions are more flexible and can employ larger active spaces and/or different truncations of the configuration interaction expansion. With GASSCF, one can explore chemical systems that are not affordable with either CASSCF or RASSCF. Perturbation theory to second order on top of GAS wave functions (GASPT2) has been implemented to recover the remaining electron correlation. The method has been benchmarked by computing the chromium dimer ground-state potential energy curve. These calculations show that GASPT2 gives results similar to CASPT2 even with a configuration interaction expansion much smaller than the corresponding CAS expansion.

1. INTRODUCTION An accurate quantum chemical method should be able to correctly describe electronic correlation effects of chemical systems. The term correlation was defined by Löwdin1 as the difference between the theoretical limit of the nonrelativistic energy within the Born−Oppenheimer approximation, namely, the full configuration interaction (FCI) energy, and the Hartree−Fock energy. Electron correlation is usually classified into dynamic and static correlations. Static correlation arises from (near) degeneracy effects, and it may play an important role in lanthanide, actinide, and transition-metal chemistry; in solids containing them; in bond dissociation processes; and in electronic excited states. Examples are spin crossover complexes,2,3 transition-metal systems with diradical character,4 ferroelectric materials,5 metal complexes featuring biological or biomimetic activity,6−11 single-molecule magnets,12,13 photocatalytic materials,14 most excited states of molecules, and some transition states. Degeneracy effects also occur in polynuclear aromatic hydrocarbons, which are of considerable interest as field-effect transistors15 and porphyrin-based functional materials.16 This form of correlation is usually described by multireference methods, which are based on multiconfigurational reference wave functions. The most commonly used multireference method is the complete active space selfconsistent field (CASSCF)17−20 because of its simple feature that the wave function is defined by choosing a set of active orbitals and electrons, namely, the active space. CASSCF treats the entire correlation effects confined in the active space. © 2016 American Chemical Society

The remaining dynamic correlation effects, which are essential for a quantitative treatment of chemical properties such as bond energies and electronic excitation energies, can be added by post-SCF methods, for example, multireference configuration interaction (MRCI) or second-order perturbation theory based on the CASSCF wave function (CASPT2).21,22 Recently, the multiconfiguration pair-density functional theory method (MC-PDFT) has been introduced as an alternative and cheap approach to recover dynamic correlation.23 The combined CASSCF/CASPT2 strategy is currently one of the most widely used ab initio methods for highly accurate calculations on multireference systems. However, there is a practical limitation associated with the size of the active space that can be employed in the CASSCF model. The CI expansion for CASSCF scales approximately exponentially with the number of active orbitals and electrons. Active spaces with 18 electrons in 18 orbitals in a singlet spin state already saturate the computational resources available to date. A complete active space bigger than that will result in CI expansions that are too large. Many strategies have been developed to reduce the configuration space of multiconfigurational self-consistent field (MCSCF) methods and to allow the use of larger active spaces. Examples are the restricted active space self-consistent field (RASSCF),24,25 the generalized active space self-consistent field (GASSCF),26 the SplitGAS approach,27,28 and the Received: April 15, 2016 Published: June 8, 2016 3208

DOI: 10.1021/acs.jctc.6b00382 J. Chem. Theory Comput. 2016, 12, 3208−3213

Article

Journal of Chemical Theory and Computation

containing all single and double replacements from |0⟩ not included in V0 or VK, and VTQ··· contains all of the other states coupled via higher level excitations. The first-order wave function is contained in VSD because this is the only space interacting with |0⟩. The zeroth-order Hamiltonian takes the form

occupation-restricted multiple active space (ORMAS) method.29 In the RASSCF formalism, the active space is divided into three distinct regions: RAS1, RAS2, and RAS3. The RAS2 region is equivalent to the active region in a CAS calculation; that is, all possible spin- and symmetry-adapted configuration state functions (CSFs) that can be constructed from the orbitals in RAS2 are included in the multiconfigurational wave function. The RAS1 and RAS3 spaces, on the other hand, permit the generation of additional CSFs subject to the restriction that a maximum number of excitations may occur from RAS1, which otherwise contains only doubly occupied orbitals, and a maximum number of excitations may occur into RAS3, which otherwise contains only external orbitals. The effects of dynamic electron correlation are included through the RASPT2 formalism. Methods based on the density matrix renormalization group (DMRG)30−34 and the FCI quantum Monte Carlo approaches35,36 are also able to access large active spaces. It is also important to mention other multireference perturbation theories such as the n-electron valence states for multireference perturbation theory (NEVPT2)37−43 and the multireference linearized coupled-cluster theory44 and the DMRG-PT2 method developed by Kurashige and Yanai and applied to the chromium dimer in 2011.45 Recently, a CI with reduced computational costs, based on nonorthogonal orbitals, has been proposed by Olsen and has been tested in the cases of the chromium dimer and trimer.46 The GAS concept, first proposed by Olsen and co-workers,24,47 is a generalization of the RAS approach. In recent years, this method has been efficiently implemented and has been made available in the Molcas quantum chemistry software package.26,48 In GAS, the user can, in principle, choose an arbitrary number of active spaces. Instead of a maximum number of holes in RAS1 and a maximum number of electrons in RAS3, accumulated minimum and maximum electron occupation numbers are used in GAS to define the wave function. Arbitrary restrictions over the interspace excitations are therefore possible. By the GAS strategy, most of the inefficient configurations can be removed while keeping the CASSCF accuracy. GASSCF is less computationally demanding than CASSCF, and the exponential scaling problem is partially circumvented. GASSCF has proven to be successful in many challenging cases, especially for the generation of reference wave functions to be used in combination with other approaches, such as SplitGAS or MC-PDFT.26,27,49,50 In this article, we describe the theory and implementation of GASSCF followed by second-order perturbation theory, the GASPT2 method. The remainder of this article is organized as follows: in section 2, the method is described; some algorithm details are discussed in section 3; in section 4, results on the Cr2 dimer are reported; and a discussion and conclusions are provided in sections 5 and 6.

̂ 0̂ + PK̂ FP ̂ K̂ + PSD ̂ SD ̂ TQ ̂ FP ̂ + PTQ ̂ ...FP ̂ ... Ĥ 0 = P0̂ FP

(1)

where operators P̂ are projector operators acting on the reference state, its complement, and the other higher order excitations, respectively, and F̂ is the one-particle generalized Fock operator F̂ =

∑ f pq ap†̂ aq̂ = ∑ f pq Epq̂ pq

(2)

pq

where f pq represents the spin-averaged expectation values of the operator Fpq̂ σ = ap̂ σ [Ĥ , aq̂†σ ] − ap†̂ σ [Ĥ , aq̂ σ ]

(3)

or more explicitly f pq =

∑ Dpt hqt + ∑ dprstgqrst t

(4)

rst

Here, the usual second quantization formalism has been employed. Because the CASSCF energy is invariant under unitary transformations of the orbitals within each subspace, one can block-diagonalize the inactive−inactive, active−active, and virtual−virtual blocks of matrix (f pq) to get a new set of orbitals, and the transformed operator F̂ will take the form F ̂ = FD̂ + FN̂

(5)

with diagonal part

FD̂ =

∑ ϵpEpp̂ (6)

p

and nondiagonal term FN̂ =

∑ (f ′it Eit̂ + f ′ti Etî ) + ∑ (f ′at Eat̂ + f ′ta Etâ ) it

at

(7)

Indices i, t, a, and p run over the inactive, active, virtual, and whole orbital spaces, respectively. The first-order wave function, C, is obtained by solving the equation (FD + FN − E0S)C = −V

(8)

where S is the overlap matrix and Vi = ⟨i|Ĥ |0⟩. FD and FN are the matrices corresponding to the operators defined in eqs 6 and 7, respectively. The second-order energy can be therefore obtained as † E2 = ⟨0|Ĥ |Ψ⟩ 1 = V C

2. THEORY 2.1. CASPT2. The GASPT2 is, in essence, an extension of the CASPT2 approach, and for many algorithmic aspects, it is similar to the RASPT2 method. A brief description of CASPT2 is provided in this section. For more details on the CASPT2 and RASPT2 formulations, we invite the reader to refer to the relevant papers.21,22,51−54 The configuration space spanned by the CASPT2 expansion is decomposed into four subspaces, V0, VK, VSD, and VTQ···, where V0 is the space expanded only by CAS reference state |0⟩, VK is the orthogonal complement space of |0⟩, VSD is the space

(9)

2.2. GAS. The GAS strategy allows the user to selectively remove numerous ineffective configurations from the CI expansion of a large CAS-type wave function while keeping the important ones in the CI space. The user defines a number of active spaces and sets constraints on the occupation numbers of the active spaces to reduce the CI space. The following input parameters need to be specified to define a GAS wave function: (1) number of GAS spaces (ngas), (2) number of orbitals in each GAS space in each irreducible representation, and (3) accumulated minimum and maximum 3209


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configuration space would not contribute to the density matrix evaluation.24 For the more general GAS spaces, the closure property does not hold anymore because in each GAS space arbitrary occupation constraints are specified. It follows that in computing the two- and three-body density matrix elements the first excitation could, in principle, bring about a configuration that violates the occupation number constraints specified by the user. Without considering that a second excitation might be able to restore an acceptable configuration, such excitation would be discarded, leading to erroneous density matrix elements. To cure the lack of closure in the GASPT2 method, the algorithm for the two- and three-body density matrices has been modified. In particular, an enlarged “intermediate space” has been constructed. Such a space contains configurations that exceed the GAS constraints by 2 units both in the minimum and in the maximum occupation numbers for each GAS space. By doing so, single and double excitations out of the GAS constraints are not discarded; instead, they are still within the intermediate space and still contribute to the density matrix elements.

number of electrons occupying the GAS spaces [minocc(igas) and maxocc(igas), igas runs from 1 through ngas]. One has thus to define the minimum and maximum electron occupation number for the first space, then the minimum and maximum electron occupation number for the first two spaces (GAS1 + GAS2), and so on. Note that the series {minocc(i)} and {maxocc(i)} are both rising and always fulfill the condition minocc(i) ≤ maxocc(i). Two GAS spaces can be either connected or disconnected depending on whether interspace excitations are allowed or not. CAS and RAS are special cases of GAS specifications, and by ad hoc input specifications, a GASSCF calculation can be reduced to a CASSCF or RASSCF calculation. It has been shown49 that one can get results of CASSCF quality with GASSCF in cases in which both methods are affordable and that within GASSCF one can use active spaces that are prohibitively expensive for CASSCF or RASSCF. It is worth mentioning that the GAS approach can also be used as a tool to constrain wave functions to specific states, which is otherwise hard or impossible to obtain by the RAS or CAS strategy. For example, with GAS, one can set constraints such that core holes are generated in the wave function, forcing the wave function to converge to a coreexcited state by avoiding its relaxation to the ground state. This special application of the GAS paradigm is currently under investigation. 2.3. GASPT2. Although GASSCF is successful in many cases, it fails when dealing with systems featuring strong dynamic correlation. In these cases, it is indispensable to perform a perturbation theory treatment on top of the GASSCF wave function. Analogously to CASPT2 and RASPT2, the zeroth-order Hamiltonian is chosen according to eq 1, and the first-order wave function arises only from the VSD space. In this implementation we used an approximation: the VSD space is derived not from the GAS specifications but from the corresponding CAS to simplify the construction of the perturbation parameters. This approximation implies that the VSD space is incomplete because the configurations that would be in the CAS but outside of the GAS are not included in the expansion. These configurations are called internally excited configurations, and their absence could result in the lack of accuracy in the perturbative treatment (vide infra). As in the CASPT2 and RASPT2 cases, the current implementation of GASPT2 is based on CSFs, and the graphic unitary group approach is used to generate and organize the CSFs.55−58 The GASPT2 energy is determined from a set of quasi-canonical orbitals by block-diagonalizing the f pq matrix for each orbital space, active as well as inactive and virtual spaces. In the limit of merging all of the GAS spaces, the GASPT2 energy would be equal to the corresponding CASPT2 energy. The method was implemented in the developer version of Molcas 848 (version 8.1).

4. APPLICATION TO THE Cr2 POTENTIAL ENERGY CURVE We calculated the ground-state potential energy curves (PECs) for the chromium dimer by using both CASPT2(12,12) and GASPT2(12,12). The ANO-RCC basis set was employed with the {21s15p10d6f4g2h} primitive basis contracted to {10s10p8d6f4g2h} functions. This basis set was chosen to allow direct comparison to other results already available in the literature.27,59 The calculations were performed by imposing the D2h point group symmetry constraints on the molecule, which was aligned along the z axis. Restricted Hartree−Fock canonical orbitals were used as starting orbitals for the MCSCF optimizations at geometries near equilibrium (1.65−1.75 Å), whereas optimized natural orbitals and wave functions of neighbor geometries were used as starting guesses for the other points along the curve. The active space consists of the full valence shell containing the molecular orbitals formed by linear combinations of the 4s and 3d orbitals of each chromium atom. For the GAS approach, each pair of bonding and antibonding molecular orbitals was confined in one GAS space, and only two active electrons could be excited within the pair. No interspace excitations were allowed. These noncoupled subspaces have been referred to as disconnected spaces in our original work on the GAS formulation.26 This approach has also been referred to as the separated-pairs approximation.50 The distribution of orbitals among the subspaces is reported in Table 1. Table 1. Distribution of Molecular Orbitals among Inactive, Active (GAS), and Secondary Spaces for Each Irreducible Representation

3. ALGORITHMIC DETAILS From an algorithmic standpoint, the GASPT2 is closely related to the RASPT2 code for most of the steps, except for the evaluation of the two- and three-body density matrices. In these steps, the RASPT2 relies upon the “closure” property of the wave function because of the special occupation constraints for RAS1 and RAS3; namely, for RAS1, the maximum occupation consists of all RAS1 orbitals doubly occupied, and for RAS3, the minimum occupation is zero. Given this property, by sorting the excitation operators, the excitation out of the

Inactive GAS 1 GAS 2 GAS 3 GAS 4 GAS 5 GAS 6 Secondary 3210

Ag

B3u

B2u

B1g

B1u

B2g

B3g

Au

5 1 1 1 0 0 0 58

2 0 0 0 1 0 0 41

2 0 0 0 0 1 0 41

0 0 0 0 0 0 1 25

5 1 1 1 0 0 0 58

2 0 0 0 1 0 0 41

2 0 0 0 0 1 0 41

0 0 0 0 0 0 1 25


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Journal of Chemical Theory and Computation The comparison of the CASPT2 results based on a complete valence active space and the GASPT2 results based on the truncated CI expansion of GAS type shows that second-order perturbation theory can provide quantitatively accurate answers even when numerous configurations are removed from the full valence complete active space reference wave function and ultimately can still dissociate the chromium dimer correctly. In the PT2 calculations, four choices for the IPEA parameter were made (0.0, 0.25, 0.45, and 1.0 au) to verify its effect on the GASPT2 results as compared to that on the CASPT2 results.59,60 The PECs obtained at CASPT2(12,12) and GASPT2-6(12,12) levels and with an IPEA shift value of 0.45 au are reported in Figure 1, and the computed dissociation

Figure 2. Active natural orbitals for GAS6-(12,12) for the chromium dimer at the equilibrium bond distance and their occupation numbers. In parenthesis the occupation number for the corresponding CAS(12, 12) natural orbitals are also reported. The CAS(12,12) natural orbitals show similar shapes.

GAS-6(12,12) (consisting only of 1516 CSFs), the GASSCF and GASPT2 PECs look quite similar to the corresponding CASSCF and CASPT2 ones. These results are encouraging in showing that PT2 theory is not oversensitive to changes in the reference wave function. Changes in the IPEA value cause similar shifts in the GASPT2 and CASPT2 curves. This is not surprising because in this case the same size active space was used in both the GAS and CAS frameworks. It is expected that, because in the GASPT2 framework one can use larger active spaces than in the CASPT2 framework, the need for IPEA shifts will be less of an issue, making the theory less empirical than CASPT2. An IPEA value of 0.45 au provides the best agreement with the experiment, as already established in ref 59. It is worth noticing that the dependence on the zeroth-order Hamiltonian can be reduced by enlarging the active space beyond the valence orbitals, as discussed already by Kurashige and Yanai45 and more recently by Vancoillie et al.30 It is also important to point out that the success of NEVPT2 in dissociating the chromium dimer is not related to the choice of the zeroth-order Hamiltonian. In fact, NEVPT2, which is not dependent on the parametrization of the zeroth-order Hamiltonian, is still dependent on the reference wave function, the CASSCF(12, 12), and as reported by Angeli et al.,61 large deviations from the second-order results are introduced by the NEVPT3 approach. These results clearly reveal that the inaccuracy of the method depends on the reference wave function rather than on the zeroth-order Hamiltonian. The dissociation energy values (Table 2) obtained at the GASPT2 level of theory are within 0.16 eV from the ones obtained at CASPT2 level. This is quite promising if one considers that the size of a CI expansion has been reduced by a factor of 20. The approximation due to truncation of the CI expansion is more enhanced at short internuclear distances. In fact, both GASSCF and GASPT2 curves are higher than the CAS counterpart at equilibrium, whereas at dissociation, the differences are negligible (on the order of 10−4 au or smaller). This result can be explained by considering that at dissociation, configurations due to interspace excitations are not contributing

Figure 1. PECs for the chromium dimer at CASPT2(12,12) and GASPT2(12,12) levels and an IPEA shift value of 0.45 au. Zero has been chosen as the CASPT2 energy at 10 Å.

Table 2. Equilibrium Distance (Å) and Dissociation Energy (eV) for Cr2 at CASPT2 and GASPT2 Levels of Theory and for Different Choices of the IPEA Shifta CASPT2(12,12) IPEA

Re

0.00 0.25 0.45 1.00

2.45 1.69 1.67 1.63

De (D0) 1.13 1.22 1.62 2.39

(1.12) (1.19) (1.59) (2.35)

GASPT2(12,12) Re 2.44 2.21 1.67 1.63

De (D0) 1.14 1.17 1.46 2.21

(1.13) (1.16) (1.42) (2.16)

a Dissociation energies including zero-point energy correction D0 (in parenthesis) are given. These values have been obtained from the interpolation of the data points with second-order splines using the VIBROT program as implemented in the Molcas package.

energies are reported in Table 2. PECs at CASSCF(12,12), GASSCF(12,12), CASPT2, and GASPT2 for the various choices of the IPEA shift analyzed in this study are reported in Figure S1 of the Supporting Information. Absolute values for all calculations are given in Table S1. In Figure 2, the natural orbitals and their occupation numbers for chromium dimer at the equilibrium geometry are shown for both the CASSCF and the GASSCF wave functions.

5. DISCUSSION As shown in Figure 1 (see also Figure S1), in going from CAS(12,12) (whose expansion consists of 28 784 CSFs) to 3211


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appreciably to the wave function. Therefore, the truncation of the CI expansion that follows the GAS constraints is correct in the sense that only truly negligible configurations have been removed. Near the equilibrium geometry, those configurations are not totally negligible, and their removal from the CI expansion sets the total GASSCF electronic energy at a higher value. Also, higher GASPT2 energies have been obtained. For comparison, we also report two experimental values of D0: a value of 1.44 eV reported by Hilpert and Ruthardt,62 also used by Casey and Leopold63 to evaluate the binding energy of the anion, and a value of 1.53 eV by Simard et al.64 The GASPT2 value is closer to the first experimental value, whereas the CASPT2 value is closer to the second experimental one.

6. CONCLUSIONS A quasi-degenerate second-order multireference perturbation theory method based on the GASSCF wave function was derived. In the current implementation of GASPT2, the internally excited configurations are missing from VSD. An intermediate space was introduced to cope with the lack of closure of the GAS type of wave functions. The method was employed to compute the ground-state chromium dimer PEC. A reduction of a factor of 20 in the CI expansion was achieved. The CASPT2 and GASPT2 curves look quite similar. The dissociation energy differs by at most 0.16 eV between the two methods. The dependence on the IPEA shift is similar in GASPT2 and CASPT2. The method has some interesting potential. It can be employed to truncate the CI expansion corresponding to active spaces that cannot be reached by CAS or RAS constraint. It can also be used as a tool to optimize wave functions for core excitations by imposing core holes in selected active subspaces and as a benchmark for other methods starting from a GASSCF-type wave function, for example, the MC-PDFT.23,50

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jctc.6b00382. PECs (Figure S1) and absolute energy values (Table S1) at CASSCF(12,12), GASSCF(12,12), CASPT2, and GASPT2 for various choices of the IPEA shift (PDF).

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (D.M.). *E-mail: [email protected] (G.L.M.). *E-mail: [email protected] (L.G.). 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, under SciDAC grant no. DE-SC0008666. Support by the Max Planck Society is gratefully acknowledged.

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REFERENCES

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