A Hierarchy of Static Correlation Models - The Journal of Physical

Article pubs.acs.org/JPCA

A Hierarchy of Static Correlation Models Deborah L. Crittenden* Department of Chemistry, University of Canterbury, Christchurch, New Zealand S Supporting Information *

ABSTRACT: It is commonly accepted in the scientific literature that the static correlation energy, Estat, of a system can be defined as the exact correlation energy of its valence electrons in a minimal basis. Unfortunately, the computational cost of calculating the exact correlation energy within a fully optimized minimal basis grows exponentially with system size, making such calculations intractable for all but the smallest systems. However, analogous to single-reference methods, it is possible to systematically approximate both the treatment of electron correlation and flexibility of the minimal basis to reduce computational cost. This yields a hierarchy of methods for calculating Estat, ranging from coupled cluster methods in a minimal atomic basis up to full valence complete active space methods with a minimal molecular orbital basis constructed from a near-complete atomic orbital basis. By examining a variety of dissociating diatomics, along with equilibrium and transition structures for polyatomic systems, we show that standard coupled cluster models with minimal atomic basis sets (e.g., STO-3G) offer a convenient and cost-effective hierarchy of black box estimates for Estat in small- to medium-sized systems near their equilibrium geometries. To properly describe homolytic bond dissociation, it is better to use a more flexible basis set expansion so that each atomic orbital can effectively adapt to its molecular environment.

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INTRODUCTION Given its restricted Hartree−Fock (RHF) and full configuration interaction (FCI) energies in a complete basis set (CBS),1 a system’s correlation energy is defined2 as Ec = EFCI/CBS − ERHF/CBS

emerged as a universal solution. Generally, one is obliged to choose between sophisticated wave-function-based approaches that are accurate but expensive and density-based models that are cheap but less reliable. Conceptually, it is helpful to consider the partition

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Ec = Estat + Edyn

and this corresponds to the difference between the top-right and bottom-right corners of the familiar Pople chart3 (Figure 1).

where the static term Estat arises from near-degeneracies between occupied and unoccupied orbitals and the dynamic term Edyn originates from short-range electron−electron interactions near the cusps of the wave function. It is well-known that wave functions (e.g., MCSCF6−8) that capture static correlation are very different from those (R12-based schemes, e.g., MP2-R12 and CCSD-R12 9 ) that efficiently account for dynamic correlation. Computationally, however, it has proven difficult to directly exploit this partition, and this can be traced to the variety of definitions that have been proposed6−8,10−27 for Estat. Most mainstream quantum chemical calculations lie somewhere in the shaded area of the Pople chart, where the correlation and basis treatments are more or less balanced. In contrast, this Letter focuses on EFCI/MBS, wherein a full treatment of correlation is combined with an atomic minimal basis set (MBS). This rather unpopular corner of the Pople chart is exploited to yield a series of robust and cost-effective static correlation models by invoking systematic approximations to the treatment of electron correlation. For completeness, different ways of defining or constructing MBSs are also considered, for example, using a

Figure 1. Pople chart showing the correlation and basis set dimensions in quantum chemistry3.

Ec values have been deduced for various small atoms4 and molecules5 by judiciously blending computational and experimental results, but the single greatest challenge in modern quantum chemistry is to find a generally applicable method that efficiently generates accurate correlation energies for medium-size molecules both at and away from their equilibrium structures. Although many exact and approximate schemes have been proposed and explored, each has shortcomings, and none has yet © XXXX American Chemical Society

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Received: January 20, 2013 Revised: April 2, 2013

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dx.doi.org/10.1021/jp400669p | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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more expensive than AO-based methods but less expensive than full valence CASSCF calculations. In the present work, we quantify the effect of increasing flexibility in the underlying atomic basis on the static correlation energy by considering only the two extremes

minimal set of molecular orbitals (MOs) rather than a minimal set of atomic orbitals (AOs). The performance of all static correlation models is tested for a range of molecular systems both at and away from their equilibrium structures.

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STATIC CORRELATION MODELS Following a large body of work in the scientific literature,6−8,28−32 we define Estat as the exact correlation energy of the valence electrons in a minimal basis, that is Estat = E

FCI(val)/MBS

−E

RHF/MBS

FCI Estat = EFCI(val)/MBS(atomic) − ERHF/MBS(atomic)

(4a)

CAS Estat = E CASSCF(val)/CBS − ERHF/CBS

(4b)

=E

FCI(val)/MBS(molecular)

−E

RHF/MBS(molecular)

(4c)

(3)

as the lack of a single optimal and widely implemented method for extracting polarized AOs or improved virtual MOs, which means that the intermediate “semiflexible” atomic basis methods are not uniquely defined. Here, MBS refers to a fixed-exponent atomic minimal basis, and CBS refers to a complete atomic basis, from which a minimal MO basis is constructed. Although employing a MBS makes CASSCF(val) and FCI(val) calculations possible for small to medium systems (up to 14 valence electrons in 14 MOs), the exact calculation of Estat rapidly becomes impractical with increasing molecule size as the computational cost grows exponentially with the number of valence orbitals and electrons. Electron Correlation Models. A practical static correlation estimator must approximate eq 3 accurately and economically.42 Coupled cluster methods43−45 offer a promising way forward as they include the determinants needed to describe most chemical phenomena, and yet, their computational costs scale only polynomially with system size. We therefore propose the following hierarchy of static correlation approximations:

Valence Orbital Flexibility. The essential difference between various “flavors” of the static correlation model generally lies in how the MBS is defined, but the advantages of FCI(val)/MBS are universal: • All determinants required to describe all possible bond dissociations are included but not those that would model the wave function cusps. In CH4, for example, all configurations of the eight valence electrons in the eight valence orbitals are accounted for. • All determinants needed to account for near-degeneracy effects are included. In the Be-like ions, for example, both 2s2 and 2p2 configurations are present. • The requirements of a theoretical model chemistry are satisfied; FCI(val)/MBS is well-defined, unbiased, sizeconsistent, and variational and varies continuously with molecular geometry.33 Perhaps the most obvious MBS to use is a preoptimized fixedexponent atomic minimal basis such as STO-3G or MINI. Although a Hartree−Fock (HF) calculation with an atomic minimal basis will yield valence MOs of the correct shape and symmetry, it is clear that the MOs constructed from an atomic minimal basis will not have the flexibility required to properly describe situations where the size of the orbitals varies significantly, for example, during bond dissociation or isomerization. Accordingly, Ruedenberg and co-workers have proposed that a minimal MO basis be constructed from a larger secondary atomic basis and optimized during a full valence CASSCF calculation.6−8 In principle, this minimal MO basis can be rotated back to the space of AOs, giving a minimal set of AOs that are optimally adapted to their molecular environment (“molecule-polarized”).30 Unfortunately, obtaining MOs using this approach incurs exponential scaling of computational cost as the orbital optimization and configuration interaction procedures must be carried out concurrently. Recently, a raft of intermediate alternatives has been proposed.29−31,34−39 These are all based upon the observation that HF yields “good” occupied valence MOs that are similar to their CASSCF counterparts but “poor” virtual valence MOs, with incorrect shape and symmetry properties.40,41 Therefore, it is desirable to obtain improved virtual valence orbitals without the computational cost of full valence CASSCF calculations. Such methods fall into one of two broad categories, based on whether the improved virtual orbitals are constructed from AOs or obtained directly as MOs. AO-based methods, such as the EPAO approach of Lee and Head-Gordon29,35 or the QUAMBO algorithm of Lu et al.,30 involve extracting molecule-polarized AOs from a HF wave function and using them to reconstruct appropriate symmetry-projected virtual orbitals. MO-based methods,31,34,36−39 on the other hand, rely on extracting improved virtual orbitals from post-HF correlated calculations e.g. for example, MP2 natural orbitals. These methods are naturally

D Estat = E CCSD(val)/MBS − EHF/MBS

(5a)

(T) Estat = E CCSD(T)(val)/MBS − EHF/MBS

(5b)

T Estat = E CCSDT(val)/MBS − EHF/MBS

(5c)

Q Estat = E CCSDTQ(val)/MBS − EHF/MBS

(5d)

and so forth, where MBS represents a preoptimized fixedexponent minimal AO basis as defined above. It is well-known that the effects of single excitations in coupled cluster wave functions may be absorbed into the definition of the virtual MOs, that is, by using Brückner orbitals.46 This yields, in principle, the simplest wave function capable of capturing static correlation effects. BD Estat = E BCCD(val)/MBS − ERHF/MBS

EBD stat

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EDstat

However, in practice, = and the extra computational cost associated with constructing Brü ckner orbitals are unwarranted in this case. Valence Orbital Flexibility and Electron Correlation Models. Finally, it remains to consider the case in which approximate correlation models are employed in conjunction with a fully flexible atomic basis or, equivalently, a minimal set of optimized molecular valence orbitals: VOD Estat = EVOD/CBS − ERHF/CBS

= EVOD/MBS(molecular) − ERHF/MBS(molecular)

(7a) (7b)

where VOD is the valence orbital optimized coupled cluster doubles method of Sherrill et al.47,48 B



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Combining all correlation treatment approximations and MBS definitions gives a static correlation analogue (Figure 2) of the

Article

DISCUSSION

Static Correlation Energies. The usefulness of eq 3 can be gauged from the Estat values that it yields, and these are listed in the first column of Tables 1 (atoms) and 2 (molecules and reactive systems). Table 1. Near-Exact (ECAS stat , CBS = pc-2) Static Correlation Energies and Basis Set Errors (mEh) for Atoms atom

−Estat

δFCI

δVOD

Be B C Mg Al Si

42.5 33.8 18.8 30.8 23.8 13.2

−9.0 −6.2 −1.4 −5.4 −1.8 −0.2

0.2 −3.0 −4.4 0.0 −1.9 −3.1

Most of the atoms have Estat = 0, but those in Groups 2, 13, and 14 have |Estat| > 0 because of near-degeneracy effects discussed by Linderberg and Shull.61 The σ-bonded molecules (Table 2, top block) follow predictable patterns, in which each covalent bond and each near-degenerate lone pair (such as that in singlet CH2) contributes 20−30 mEh but ionic bonds contribute much less. Bonds between electron-rich atoms (e.g., F2) have anomalously large Estat values, as is well-known. The π-bonded molecules (the lower block) have low-lying valence excited states and consequently exhibit much larger Estat values, culminating in the strong multireference O3 molecule. We note also that systems with second-row atoms usually have smaller Estat than their firstrow analogues (e.g., SiH4 versus CH4). The Estat values of the transition-state structures are generally larger than either reactants or products due to near-degeneracies arising from bond stretching. The exception to this rule is the transition structure for HCN ↔ HNC isomerization, whose static correlation energy lies between that of the reactants and products. Valence Orbital Flexibility. The effect of valence orbital flexibility on Estat is most clearly illustrated by the bond energy and bond correlation energy curves for H2, N2, O2, and F2 presented in Figure 3. The bond energy is calculated as

Figure 2. Static correlation analogue of the well-known Pople chart (Figure 1).

Pople chart (Figure 1). It is important to note that this is not directly equivalent to the Pople chart. The basis set axis defines the minimal basis, ranging from a fixed-exponent minimal atomic basis through to a completely optimized minimal molecular basis. The correlation method axis starts with the simplest method that includes doubly excited determinants, as required to account for orbital near-degeneracies, and continues to the full configuration interaction limit. Each method discussed above lies in a different corner of this chart. It is clear that using a sophisticated treatment of electron correlation with a sufficiently flexible atomic basis will closely reproduce the benchmark CASSCF(val)/CBS static correlation energies (bottom right corner), but we posit that even significantly more approximate methods (upper left corner) will often give useful estimates of Estat. In the following section, we test this conjecture for dissociating diatomics, the atoms and molecules in Pople’s G1 data set,49 the ozone molecule, the DBH24 reaction database,50 and a pair of notoriously multireference atomic insertion reactions (Al into O2 and Be into H2).51−53

E(bond) = E(molecule) − E(atoms)

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Experimental dissociation energy curves are taken to be “exact”. From the bond energy curves on the left-hand side of Figure 3, it is clear that accounting for static correlation is required for even qualitative agreement with the exact curves, which are derived from experimental data. Substantial basis set superposition error (BSSE) is responsible for overstabilization of H2 and F2 near their equilibrium geometries, bringing the FCI/STO-3G bond energy curve closer to experiment than anticipated. However, BSSE effects will cancel in the calculation of Estat. Therefore, to quantify the importance of valence orbital flexibility, excluding BSSE, it is necessary to examine the bond static correlation energy profiles depicted on the right-hand side of Figure 3. 62

COMPUTATIONAL DETAILS FCI and CASSCF calculations were carried out using the GAMESS suite of quantum chemical software,54 using a restricted (open-shell where necessary) HF reference wave function in all cases. VOD, CCSD, and CCSD(T) energies were obtained using Q-Chem,55 and CCSDT and CCSDTQ energies were calculated using NWChem.56 In the interest of computational expedience, the STO-3G basis57−59 was used as the minimal atomic basis for all FCI calculations reported here. Additional testing has shown that the results are not very sensitive to this choice. For example, using STO-6G yields a change in Estat energies of less than 2% across the G1 data set, which is equivalent to a maximum deviation of 2.9 mEh in systems with large static correlation energies and a mean absolute deviation of 0.9 mEh (data available as Supporting Information). Likewise, the pc-2 basis set60 was used as the CBS for all VOD and CASSCF calculations reported here. Additional testing has confirmed that Estat values obtained using the pc-2 and pc-3 basis sets agree to within 4.3 mEh for all molecules in the G1 data set, with a mean absolute deviation of 0.8 mEh (data available as Supporting Information).

FCI FCI FCI Estat (bond) = Estat (molecule) − Estat (atoms)

(9a)

CAS CAS CAS Estat (bond) = Estat (molecule) − Estat (atoms)

(9b)

For reference, the bond dynamic correlation energy is also determined.

C

CAS Edyn = E exact − Estat

(10a)

Edyn(bond) = Edyn(molecule) − Edyn(atoms)

(10b)



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FCI Table 2. Near-Exact (ECAS and δVOD Errors for the G1 Molecules, Along with stat , CBS = pc-2) Static Correlation Energies (mEh), δ All Reactants, Transition States, and Products from the DBH24 Data Set, Plus Atomic Insertion Reactions

|Estat|

δFCI

δVOD

H2 LiHa LiFa BeH NH FH CH2 (3B1) CH2 (1A1) CH3 CH4 NH2 NH3 OH2 CH3OHb CH3Cl C2H6 O2H2 O2 F2 ClO SO CN N2 CO C2H4 HCO NO SiO O3

18.5 16.1 5.5 26.4 25.9 24.2 38.3 60.8 58.8 82.3 48.7 74.3 53.3 117.8 86.4 149.9 106.9 104.7 79.1 42.9 62.9 151.4 148.1 131.1 121.2 125.4 120.4 121.3 237.2

−2.1 1.6 0.9 4.1 −1.2 −1.6 −4.4 0.6 −3.1 3.9 0.5 9.7 3.7 −0.1 0.3 4.3 0.5 −7.1 −3.1 0.9 −22.4 −24.0 −8.5 −7.4 −39.6 −12.7 −12.7 −33.5 −24.4

0.7 1.0 0.8 −15.4 0.1 −6.5 −19.9 −3.0 3.1 2.9 0.8 −10.4 −6.6 2.1 17.3

H• + •OH [HOH]•• H2 + O•• CH4 + OH• [HOCH4]• CH3OHb + H• N2O + H• [HN2O]• N2 + •OH HN2• [HN2]• N2 + H• C2H4 + H• [HC2H4]• CH3CH2• HCN [HCN] HNC Be + H2 [BeH2] HBeH

24.7 40.4 18.5 107.1 127.0 117.8 214.7 223.8 172.8 129.6 157.8 148.1 121.2 147.4 127.9 150.2 141.2 139.6 61.0 84.8 32.7

−0.9 0.3 −2.1 3.0 5.2 −0.1 −7.2 −35.0 −9.3 −9.3 −13.7 −8.5 −39.6 −17.5 −0.7 −15.0 −15.8 −5.4 −11.0 −44.8 −2.6

−4.5 −6.3 −6.2 −4.0 −7.5 − 11.9 3.3 −1.4 −5.6 −1.1 3.1 0.8 −3.5 −2.7 2.6 4.1 2.5 0.1 47.5 0.1

0.0 0.0 0.0 0.0 −5.0 0.0 −3.9 0.2 −2.9 0.5 −3.9 0.6 0.2

|Estat|

δFCI

δVOD

a

Li2 Na2a NaCla CH OH ClH SiH2 (3B1) SiH2 (1A1) SiH3 SiH4 PH2 PH3 SH2 CH3SH FCl Si2H6 N2H4 S2 Cl2 HOCl

9.4 10.6 8.4 42.2 24.7 17.3 32.7 48.4 42.3 54.8 34.6 48.1 34.2 103.9 37.8 91.8 134.1 46.8 23.5 66.3

1.1 −6.9 5.7 −0.1 −0.9 −2.4 −8.8 −3.4 −11.3 −13.1 −8.2 −11.1 −5.4 −2.3 −9.5 −25.0 14.3 −15.8 −5.0 −2.8

0.0 0.0 0.0 −3.5 −4.5 0.0 −3.5 0.2 −2.2 0.2 −4.0 0.2 0.1 0.8 0.0 0.5 1.0 −10.1 0.0 0.4

HCN P2 CS C2H2 H2CO CO2 Si2 SO2

150.2 93.1 104.7 130.9 145.2 176.2 82.9 131.7

−15.0 −31.7 −16.1 −40.6 2.4 −34.7 −28.8 −90.9

2.6 2.7 4.8 2.0 1.5 9.7 −1.1 11.0

H• + H2S [SH3]• H2 + •SH H• + HCl [HClH]• HCl + H• FCl + •CH3 [CH3FCl]• CH3F + •Cl Cl− + CH3Cl [ClCH3Cl]− CH3Cl + Cl− F− + CH3Cl [FCH3Cl]− CH3F + Cl− OH− + CH3F [HOCH3F]− CH3OHb + F− Al + O2 [AlO2] OAlO

34.2 41.0 37.1 17.3 37.9 17.3 96.5 120.6 93.0 86.4 79.5 86.4 86.4 146.9 93.0 116.5 159.1 117.8 128.4 227.9 213.3

−5.4 −8.9 −4.8 −2.4 −13.8 −2.4 −12.6 −7.7 −1.2 0.3 −15.6 0.3 0.3 6.7 −1.2 −1.5 −0.1 −0.1 −7.2 10.3 −50.3

0.1 −2.2 −4.8 0.0 −2.3 0.0 −2.8 1.1 −4.4 0.7 1.9 0.7 0.7 1.9 0.9 1.0 2.2 −17.3 −1.3 50.0

a CAS Estat ,

b CAS VOD FCI EFCI stat , and Estat were calcuted using the (2,2) active space, with unoccupied alkali metal AOs and doubly occupied halide AOs. Estat and Estat were calculated using the (10,10) active space, constraining oxygen AOs to remain doubly occupied.

The first thing that is immediately obvious from Figure 3 is that a significant fraction of each molecule’s total bond correlation energy is derived from its static correlation component, even at equilibrium bond lengths. This is particularly noticeable for the N2 molecule. In all cases, the static correlation energy

increases with bond length as the HOMO and LUMO become degenerate. It is also clear that static correlation energy is sensitive to the flexibility of the underlying atomic basis set, particularly as each molecule is stretched. This can be attributed to the fact that the D



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Figure 3. Bond energy (left-side panels) and bond correlation energy (right-side panels) curves for H2, N2, O2, and F2. Bond energies are calculated according to eq 8, while bond correlation energies are defined by eqs 9 and 10. (Left-side panels) Black curves are “exact” (derived from experimental data62), RHF curves are shown as dotted lines, while the dashed curves also incorporate a static correlation correction. (Right-side panels) Bond dynamic correlation energy (dashed line) and different bond static correlation energy estimates; the gray curve represents EFCI stat (bond), and the solid black curve represents ECAS stat (bond).

and Gill,63 UHF wave functions can capture static correlation effects in cases of absolute near-degeneracy, such as occurs during bond breaking where the energy difference between the HOMO and LUMO tends to zero. However, for cases of relative near-degeneracy, for example, during bond stretching or for “anomalous” atoms like Be, where the HOMO−LUMO gap is nonvanishing, UHF struggles and often fails entirely to recover the static correlation energy. As a result, UHF-partitioned bond static correlation energy profiles display complicated and nonintuitive behavior, which will not be discussed further here. The importance of valence orbital flexibility may also be quantified for atoms and molecules by δFCI

STO-3G basis set is constructed for optimal performance near equilibrium but lacks the flexibility to adapt to its molecular environment. Therefore, in FCI/STO-3G calculations for molecules far from their equilibrium structures, additional determinants can play two roles, compensating for the fixedexponent nature of the minimal atomic basis and capturing static correlation effects associated with bond dissociation and other near-degeneracy situations. Thus, FCI/STO-3G predicts a spuriously large value for Estat as it is not solely capturing static correlation effects. It is possible to further subdivide the bond static correlation energy into a component that can be recovered by allowing spin symmetry breaking of the orbitals, that is, using a UHF wave function and a remainder term. As discussed by Hollett

FCI CAS |Estat | = |Estat | − δ FCI

E

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D (T) T Q Table 3. EFCI stat static correlation energies (mEh), δ , δ , δ and δ errors for the G1 molecules, and all reactants, transition states and products from the DBH24 data set, plus atomic insertion reactions

EFCI stat

δD

δ(T)

δT

δQ

EFCI stat Li2a Na2a

H2 LiHa LiFa BeH NH FH CH2 (3B1) CH2 (1A1) CH3 CH4 NH2 NH3 OH2 CH3OH CH3Cl C2H6 O2H2 O2 F2 ClO SO CN N2 CO C2H4 HCO NO SiO O3

20.6 14.5 4.6 22.2 27.1 25.8 42.7 60.2 61.8 78.5 48.1 64.6 49.6 117.9 86.1 145.6 106.4 111.8 82.1 42.0 85.4 175.5 156.6 138.5 160.7 138.1 133.1 154.9 261.5

0 0 0 0.5 0 0 0.2 0.5 0.2 0.2 0.1 0.2 0.1 1.0 0.6 0.7 0.8 2.1 0 1.1 1.9 5.8 3.9 8.0 1.2 5.6 5.0 14.5 23.7

0 0 0 0.2 0 0 0.1 0.3 0.1 0.1 0.0 0.1 0.0 0.3 0.2 0.3 0.3 1.4 0 0.5 1.1 3.5 2.2 0.9 0.5 1.3 1.7 0.6 3.9

0 0 0 0 0 0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.1 0.2 1.2 0 0.0 0.9 2.3 2.0 −0.2 0.4 0.2 0.8 −0.7 2.6

0 0 0 0 0 0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0 0.0 0.0 0.3 0.0 0.1 0.0 0.1 0.1 0.4 0.3

H• + •OH [HOH]•• H2 + O•• CH4 + OH• [HOCH4]• CH3OH + H• N2O + H• [HN2O]• N2 + •OH HN2• [HN2]• N2 + H• C2H4 + H• [HC2H4]• CH3CH2• HCN [HCN] HNC Be + H2 [BeH2] HBeH

25.6 40.1 20.6 104.0 121.8 117.9 221.9 258.8 182.2 138.9 171.6 156.6 160.7 164.9 128.6 165.1 156.9 145.0 72.0 129.6 35.3

0.0 0.1 0 0.2 0.7 1.0 14.3 18.3 3.9 3.4 5.0 3.9 1.2 1.4 0.8 3.3 5.9 6.6 0 4.4 0.4

0.0 0.0 0 0.1 0.2 0.3 5.7 2.2 2.2 1.3 2.5 2.2 0.5 0.6 0.2 2.1 1.8 1.0 0 −4.4 0.2

0 0.0 0 0.0 0.1 0.1 2.9 2.6 2.0 0.7 2.0 2.0 0.4 0.4 0.1 2.0 1.4 −0.1 0 −2.2 0.0

0 0.0 0 0.0 0.0 0.0 0.1 0.2 0.0 0.1 0.1 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0 0.0 0.0

a FCI Estat ,

δD

δ(T)

δT

δQ

NaCla CH OH ClH SiH2 (3B1) SiH2 (1A1) SiH3 SiH4 PH2 PH3 SH2 CH3SH FCl Si2H6 N2H4 S2 Cl2 HOCl

8.2 17.5 2.8 42.3 25.6 19.7 41.4 51.8 53.6 67.9 42.8 59.1 39.7 106.1 47.3 116.8 119.8 62.6 28.5 69.1

0 0 0 0.1 0 0 0.4 0.3 0.5 0.5 0.1 0.2 0.0 0.6 0 1.1 0.9 1.3 0 0.4

0 0 0 0.1 0 0 0.1 0.1 0.2 0.2 0.0 0.0 0.0 0.2 0 0.3 0.3 0.8 0 0.2

0 0 0 0.1 0 0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.1 0 0.0 0.1 0.7 0 0.1

0 0 0 0.0 0 0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0 0.0 0.0 0.0 0 0.0

HCN P2 CS C2H2 H2CO CO2 Si2 SO2

165.1 124.8 120.7 171.5 142.9 210.9 111.7 222.5

3.3 4.3 9.1 2.5 2.6 20.2 6.5 21.0

2.1 2.7 1.0 1.9 0.7 −0.3 3.3 1.1

2.0 2.7 −0.2 1.8 0.3 −0.2 2.4 1.0

0.0 0.1 0.1 0.0 0.0 −0.1 0.2 0.2

H• + H2S [SH3]• H2 + •SH H• + HCl [HClH]• HCl + H• FCl + •CH3 [CH3FCl]• CH3F + •Cl Cl− + CH3Cl [ClCH3Cl]− CH3Cl + Cl− F− + CH3Cl [FCH3Cl]− CH3F + Cl− OH− + CH3F [HOCH3F]− CH3OH + F− Al + O2 [AlO2] OAlO

39.7 49.9 41.9 19.7 51.7 19.7 109.2 128.3 94.1 86.1 95.1 86.1 86.1 140.2 94.1 118.0 159.2 117.9 135.6 217.6 263.6

0.0 0.2 0.0 0.0 0.0 0.0 0.2 0.5 1.1 0.6 1.9 0.6 0.6 4.1 1.1 1.1 5.1 1.0 2.1 24.8 26.3

0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.3 0.2 0.2 0.2 0.2 −0.9 0.3 0.3 0.6 0.3 1.4 −4.2 5.5

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.1 0.0 0.1 0.1 −0.1 0.0 0.0 −0.1 0.1 1.2 4.5 3.5

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.7 0.9

EDstat, ETstat, and EQstat are calculated using (2,2) valence space.

CH3OH) have lower ECAS stat values by 10−20 mEh compared to their first-row counterparts. However, the ECAS stat value of SO2 is smaller than that of O3 by around 100 mEh. This unexpected and somewhat perplexing result is well-attested by a variety of different quantum chemical software packages and orbital optimization algorithms. In this case, it appears that accounting

as presented in Tables 1 and 2. The predominantly negative values of δFCI indicate that FCI/STO-3G tends to overestimate Estat, as discussed above. The δFCI error for SO2 is anomalously large due primarily to its unexpectedly low ECAS stat value. In general, molecules containing a single second-row substitution (e.g., CH3SH compared to F



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for valence orbital flexibility strongly stabilizes the HOMO relative to the LUMO, significantly decreasing the multireference character of the CASSCF/CBS wave function relative to its FCI/MBS counterpart. This is supported by inspection of the FCI/MBS and CASSCF/CBS CI coefficients provided in the Supporting Information. Unfortunately, it appears impossible to predict a priori which molecules will exhibit this behavior, although experience suggests that this should occur only rarely. Electron Correlation Models. The usefulness of the correlation treatment approximations in eq 5, measured by δX, X FCI |Estat | = |Estat | − δX

Counterintuitively, VOD/pc-2 often slightly overestimates the magnitude of Estat, with a series of small negative values of δVOD appearing in Tables 1 and 2. Closer inspection reveals that all of these cases involve radicals or diradicals, and the overstabilization in these cases is due to spin symmetry breaking of the approximate VOD wave function, as established previously.65 Otherwise, VOD/pc-2 slightly underestimates the magnitude of Estat in σ-bonded systems and more significantly underestimates Estat in strong multireference systems such as O3, SO2, CO2, N2O, BeH2, and AlO2, as expected. Side Note: The EDstat Model as a CASSCF Diagnostic. D model provides surprisingly good Pragmatically, the Estat CAS estimates of Estat due to partial cancellation of AO inflexibility and correlation treatment errors. However, because orbital flexibility tends to play a larger role in determining Estat than inclusion of higher excitations in the coupled cluster wave function, the EDstat model generally overestimates Estat. Therefore, the EDstat model provides a cost-effective but nonrigorous upper estimate of ECAS stat , which may be used to verify that full-valence CASSCF calculations recover only Estat. D If |ECAS stat | > |Estat|, it is likely that the full valence CASSCF model is capturing some dynamic correlation effects, that is, that the additional determinants preferentially stabilize the system by accounting for interelectronic wave function cusps rather than near-degeneracies between bonding and antibonding orbitals. One unexpected example from the present work is methanol, CH3OH, in which the dynamic correlation energy of each oxygen lone pair exceeds the static correlation energy of a pair of σ-bonded electrons. Therefore, it is necessary to constrain the lone pair oxygen orbitals to remain doubly occupied throughout to ensure that only static correlation effects are captured. On the other hand, as the EDstat model provides surprisingly good estimates of ECAS stat , particularly for σ-bonded molecules comprised of atoms from the first and second rows of the periodic table (H−Ne), it may be used to confirm that the CASSCF(val)/CBS wave function has converged to the optimal D solution. If ECAS stat is not within 15 mEh of Estat for these systems, this is a strong indication that the CASSCF orbitals are incorrect. Even for more problematic molecules (involving third-row elements or multiple bonds), ECAS stat usually lies within 40 mEh of EDstat. These results reinforce the well-known conclusion that CASSCF is not a black-box method for calculating static correlation energies as it depends strongly on both appropriate choice of active space and initial guess of MOs. Even with a well-defined choice of active space (full valence), the potential pitfalls of inadvertently capturing dynamic correlation effects and converging to nonoptimal orbitals remain. In this work, we have encountered both of these issues and taken advantage of EDstat in deciding to further refine the CASSCF wave function in these cases.

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are shown in Table 3. To exclude orbital flexibility effects, a minimal atomic basis has been used throughout. The errors are almost always positive, indicating that the approximations usually underestimate EFCI stat . The EDstat model reproduces the exact EFCI stat energies of the molecules in the G1 data set well overall (MAD = 2.7 mEh), but it is clear that the errors are not uniformly distributed. Specifically, the errors are zero for all systems with two valence electrons, almost zero in other σ-bonded molecules, but large in π-bonded molecules such as CO2, SiO, SO2, and O3. These results are consistent with the fact that CCSD includes only the determinants needed to describe single-bond dissociations. The EDstat model performs equally well for the DBH24 data set and atomic insertion reactions (MAD = 3.5 mEh), although the majority of the error comes from N2O (which is isoelectronic with CO2), a nearby [HN2O]• transition state, linear aluminum oxide (OAlO), and a nearby AlO2 transition state. Transition states per se do not appear to pose particular problems for the EDstat approximation. Formally, in order to obtain comparable accuracy for σ- and π-bonded systems, it is necessary to include connected quadruple excitations.64 However, the E(T) stat energies (MAD = 0.6 and 1.0 mEh for G1 and expanded DBH24, respectively) reveal that CCSD(T) is an excellent approximation to FCI in a minimal basis. In particular, the perturbative triples correction recovers almost all of the missing static correlation energy in CO2 and SO2 and significantly improves the treatment of O3 and other π-bonded systems. The ETstat errors in Table 3 indicate that CCSDT yields comparable results (MAD = 0.4 and 0.7 mEh for both G1 and DBH24) to those obtained using CCSD(T). However, because of the much greater computational cost of the former, we conclude that the E(T) stat model lies at a price-performance “sweet spot”. The EQstat errors in Table 3 show that CCSDTQ is almost identical (MAD = 0.04 and 0.1 mEh for G1 and DBH24, respectively) to FCI for all of the systems considered. When its computational cost is manageable, CCSDTQ is a very accurate correlation method. Valence-Orbital-Optimized Coupled Cluster Doubles. The effect of approximating the treatment of electron correlation while allowing valence orbital optimization is measured by VOD CAS |Estat | = |Estat | − δ VOD

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SUMMARY AND CONCLUSIONS The real utility of approximate models is as stand-alone static correlation estimators because the exponential scaling of computational cost with number of electrons makes routine CASSCF calculations impractical for most molecules. By D (T) T contrast, the computational costs of the EVOD stat , Estat, Estat , Estat, 6 6 7 8 10 Q and Estat models scale as O(n ), O(n ), O(n ), O(n ), and O(n ), respectively, where n is the number of electrons in the molecules’ D valence space. We note, however, the EVOD stat and Estat prefactors VOD are substantially different, that is, Estat calculations are much more computationally taxing than EDstat calculations.

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and these values are given in Tables 1 and 2. Comparing δFCI and δVOD errors, it is clear that valence orbital flexibility is more important than treatment of electron correlation in obtaining accurate static correlation energies, reducing the mean absolute deviation across all data from 10.6 to 4.1 mEh. G



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T Q EVOD stat , Estat, and Estat estimates of the correlation energy may be readily obtained for systems with n ≤ 14, while the EDstat and E(T) stat models can be extended to systems with n ≈ 60. Although this is already a significant improvement over the performance of conventional full-valence CASSCF, further developments in linear-scaling correlated methods, for example, the localized AO plus pair natural orbital based coupled cluster method recently published by Riplinger and Neese,66 are required to obtain universally applicable static correlation models. Nonetheless, the results presented here provide clear proof of principle that it is often possible to obtain reasonable static correlation energies without exponential scaling of computational cost. This provides a foundation for uncoupled, or only weakly coupled, modeling of the dynamic and static contributions to the total electronic energy. For example, one may envisage applying R129 or F12 theory67 in conjunction with an approximate static correlation model like E(T) stat . Alternatively, it may be easier to parametrize density functional68 or intracule functional69 methods to recover only the dynamic component of the correlation energy rather than the total correlation energy. Further accuracy may be obtained at little extra computational cost by allowing intermediate flexibility in the AO basis, for example, by extracting molecule-polarized AOs29 or “improved” virtual valence orbitals30 from a large basis HF calculation.

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ASSOCIATED CONTENT

S Supporting Information *

Energy values for various basis sets and FCI/MBS and CASSCF/ CBS CI coefficients. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The author would like to thank Professor Peter Gill and Dr. David Brittain for helpful discussions and the referees for their helpful comments and feedback.

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REFERENCES

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A Hierarchy of Static Correlation Models - The Journal of Physical

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