A Hierarchy of Static Correlation ModelsDeborah L. Crittenden*

Department of Chemistry, University of Canterbury, Christchurch, New Zealand

*S Supporting Information

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

■ INTRODUCTIONGiven its restricted Hartree−Fock (RHF) and full configurationinteraction (FCI) energies in a complete basis set (CBS),1 asystem’s correlation energy is defined2 as

= −E E EcFCI/CBS RHF/CBS

(1)

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

Ec values have been deduced for various small atoms4 andmolecules5 by judiciously blending computational and exper-imental results, but the single greatest challenge in modernquantum chemistry is to find a generally applicable method thatefficiently generates accurate correlation energies formedium-sizemolecules both at and away from their equilibrium structures.Although many exact and approximate schemes have beenproposed and explored, each has shortcomings, and none has yet

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

= +E E Ec stat dyn (2)

where the static term Estat arises from near-degeneracies betweenoccupied and unoccupied orbitals and the dynamic term Edynoriginates from short-range electron−electron interactions nearthe cusps of the wave function. It is well-known that wavefunctions (e.g., MCSCF6−8) that capture static correlation arevery different from those (R12-based schemes, e.g., MP2-R12and CCSD-R129) that efficiently account for dynamiccorrelation. Computationally, however, it has proven difficultto directly exploit this partition, and this can be traced to thevariety of definitions that have been proposed6−8,10−27 for Estat.Most mainstream quantum chemical calculations lie some-

where in the shaded area of the Pople chart, where the correlationand basis treatments are more or less balanced. In contrast, thisLetter focuses on EFCI/MBS, wherein a full treatment of correla-tion is combined with an atomic minimal basis set (MBS). Thisrather unpopular corner of the Pople chart is exploited to yielda series of robust and cost-effective static correlation modelsby invoking systematic approximations to the treatment ofelectron correlation. For completeness, different ways of definingor constructing MBSs are also considered, for example, using a

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

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

Article

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minimal set of molecular orbitals (MOs) rather than a minimalset of atomic orbitals (AOs). The performance of all staticcorrelationmodels is tested for a range of molecular systems bothat and away from their equilibrium structures.

■ 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 valenceelectrons in a minimal basis, that is

= −E E EstatFCI(val)/MBS RHF/MBS

(3)

Valence Orbital Flexibility. The essential differencebetween various “flavors” of the static correlation modelgenerally lies in how the MBS is defined, but the advantages ofFCI(val)/MBS are universal:• All determinants required to describe all possible bond

dissociations are included but not those that would model thewave function cusps. In CH4, for example, all configurationsof the eight valence electrons in the eight valence orbitals areaccounted for.• All determinants needed to account for near-degeneracy

effects are included. In the Be-like ions, for example, both 2s2 and2p2 configurations are present.• The requirements of a theoretical model chemistry

are satisfied; FCI(val)/MBS is well-defined, unbiased, size-consistent, and variational and varies continuously with mole-cular geometry.33

Perhaps the most obvious MBS to use is a preoptimized fixed-exponent atomic minimal basis such as STO-3G or MINI.Although a Hartree−Fock (HF) calculation with an atomicminimal basis will yield valence MOs of the correct shape andsymmetry, it is clear that the MOs constructed from an atomicminimal basis will not have the flexibility required to properlydescribe 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

minimalMO basis be constructed from a larger secondary atomicbasis and optimized during a full valence CASSCF calculation.6−8

In principle, this minimal MO basis can be rotated back to thespace of AOs, giving a minimal set of AOs that are optimallyadapted to their molecular environment (“molecule-polar-ized”).30 Unfortunately, obtaining MOs using this approachincurs exponential scaling of computational cost as the orbitaloptimization and configuration interaction procedures must becarried out concurrently.Recently, a raft of intermediate alternatives has been pro-

posed.29−31,34−39 These are all based upon the observation thatHF yields “good” occupied valence MOs that are similar to theirCASSCF counterparts but “poor” virtual valence MOs, withincorrect shape and symmetry properties.40,41 Therefore, it isdesirable to obtain improved virtual valence orbitals without thecomputational cost of full valence CASSCF calculations. Suchmethods fall into one of two broad categories, based on whetherthe improved virtual orbitals are constructed from AOs orobtained directly as MOs. AO-based methods, such as the EPAOapproach of Lee and Head-Gordon29,35 or the QUAMBOalgorithm of Lu et al.,30 involve extracting molecule-polarizedAOs from a HF wave function and using them to reconstructappropriate symmetry-projected virtual orbitals. MO-basedmethods,31,34,36−39 on the other hand, rely on extractingimproved virtual orbitals from post-HF correlated calculationse.g. for example,MP2 natural orbitals. Thesemethods are naturally

more expensive than AO-based methods but less expensive thanfull valence CASSCF calculations.In the present work, we quantify the effect of increasing

flexibility in the underlying atomic basis on the static correlationenergy by considering only the two extremes

= −

= −

= −

E E E

E E E

E E

(4a)

(4b)

(4c)

statFCI FCI(val)/MBS(atomic) RHF/MBS(atomic)

statCAS CASSCF(val)/CBS RHF/CBS

FCI(val)/MBS(molecular) RHF/MBS(molecular)

as the lack of a single optimal and widely implemented methodfor extracting polarized AOs or improved virtual MOs, whichmeans that the intermediate “semiflexible” atomic basis methodsare not uniquely defined. Here, MBS refers to a fixed-exponentatomic 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 ofEstat rapidly becomes impractical with increasing molecule size asthe computational cost grows exponentially with the number ofvalence orbitals and electrons.

Electron Correlation Models. A practical static correlationestimator must approximate eq 3 accurately and economically.42

Coupled cluster methods43−45 offer a promising way forward asthey include the determinants needed to describe most chemicalphenomena, and yet, their computational costs scale only poly-nomially with system size. We therefore propose the followinghierarchy of static correlation approximations:

= −E E EstatD CCSD(val)/MBS HF/MBS

(5a)

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

(5b)

= −E E EstatT CCSDT(val)/MBS HF/MBS

(5c)

= −E E EstatQ CCSDTQ(val)/MBS HF/MBS

(5d)

and so forth, where MBS represents a preoptimized fixed-exponent 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 thevirtual MOs, that is, by using Bruckner orbitals.46 This yields, inprinciple, the simplest wave function capable of capturing staticcorrelation effects.

= −E E EstatBD BCCD(val)/MBS RHF/MBS

(6)

However, in practice, EstatBD = Estat

D and the extra computationalcost associated with constructing Bruckner orbitals areunwarranted in this case.

Valence Orbital Flexibility and Electron CorrelationModels. Finally, it remains to consider the case in whichapproximate correlation models are employed in conjunctionwith a fully flexible atomic basis or, equivalently, a minimal set ofoptimized molecular valence orbitals:

= −

= −

E E E

E E

(7a)

(7b)

statVOD VOD/CBS RHF/CBS

VOD/MBS(molecular) RHF/MBS(molecular)

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

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Combining all correlation treatment approximations andMBSdefinitions gives a static correlation analogue (Figure 2) of the

Pople chart (Figure 1). It is important to note that this is notdirectly equivalent to the Pople chart. The basis set axis definesthe minimal basis, ranging from a fixed-exponent minimal atomicbasis through to a completely optimizedminimal molecular basis.The correlation method axis starts with the simplest method thatincludes doubly excited determinants, as required to account fororbital near-degeneracies, and continues to the full configurationinteraction limit.Each method discussed above lies in a different corner of this

chart. It is clear that using a sophisticated treatment of electroncorrelation with a sufficiently flexible atomic basis will closelyreproduce the benchmark CASSCF(val)/CBS static correlationenergies (bottom right corner), but we posit that evensignificantly 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 atomsand molecules in Pople’s G1 data set,49 the ozone molecule,the DBH24 reaction database,50 and a pair of notoriouslymultireference atomic insertion reactions (Al into O2 and Beinto H2).

51−53

■ COMPUTATIONAL DETAILSFCI and CASSCF calculations were carried out using theGAMESS suite of quantum chemical software,54 using arestricted (open-shell where necessary) HF reference wavefunction in all cases. VOD, CCSD, and CCSD(T) energies wereobtained using Q-Chem,55 and CCSDT and CCSDTQ energieswere calculated using NWChem.56

In the interest of computational expedience, the STO-3Gbasis57−59 was used as the minimal atomic basis for all FCIcalculations reported here. Additional testing has shown that theresults are not very sensitive to this choice. For example, usingSTO-6G yields a change in Estat energies of less than 2% across theG1 data set, which is equivalent to a maximum deviation of 2.9 mEhin systems with large static correlation energies and a mean absolutedeviation 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 hasconfirmed that Estat values obtained using the pc-2 and pc-3 basissets 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 asSupporting Information).

■ DISCUSSION

Static Correlation Energies. The usefulness of eq 3 can begauged from theEstat values that it yields, and these are listed in the firstcolumn of Tables 1 (atoms) and 2 (molecules and reactive systems).

Most of the atoms have Estat = 0, but those in Groups 2, 13, and14 have |Estat| > 0 because of near-degeneracy effects discussed byLinderberg and Shull.61

The σ-bonded molecules (Table 2, top block) followpredictable patterns, in which each covalent bond and eachnear-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 anomalouslylarge Estat values, as is well-known. The π-bonded molecules(the lower block) have low-lying valence excited states andconsequently exhibit much larger Estat values, culminating in thestrong multireference O3 molecule. We note also that systemswith second-row atoms usually have smaller Estat than their first-row 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-degeneraciesarising from bond stretching. The exception to this rule is thetransition structure for HCN ↔ HNC isomerization, whose staticcorrelation energy lies between that of the reactants and products.

Valence Orbital Flexibility. The effect of valence orbitalflexibility on Estat is most clearly illustrated by the bond energyand bond correlation energy curves for H2, N2, O2, and F2presented in Figure 3. The bond energy is calculated as

= −E E E(bond) (molecule) (atoms) (8)

Experimental dissociation energy curves62 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 evenqualitative agreement with the exact curves, which are derivedfrom experimental data. Substantial basis set superposition error(BSSE) is responsible for overstabilization of H2 and F2 near theirequilibrium geometries, bringing the FCI/STO-3G bond energycurve 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 correlationenergy profiles depicted on the right-hand side of Figure 3.

= −E E E(bond) (molecule) (atoms)statFCI

statFCI

statFCI

(9a)

= −E E E(bond) (molecule) (atoms)statCAS

statCAS

statCAS

(9b)

For reference, the bond dynamic correlation energy is alsodetermined.

= −E E Edynexact

statCAS

(10a)

= −E E E(bond) (molecule) (atoms)dyn dyn dyn (10b)

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

Table 1. Near-Exact (EstatCAS, CBS = pc-2) Static Correlation

Energies and Basis Set Errors (mEh) for Atoms

atom −Estat δFCI δVOD

Be 42.5 −9.0 0.2B 33.8 −6.2 −3.0C 18.8 −1.4 −4.4Mg 30.8 −5.4 0.0Al 23.8 −1.8 −1.9Si 13.2 −0.2 −3.1

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

increases with bond length as the HOMO and LUMO becomedegenerate.It is also clear that static correlation energy is sensitive to the

flexibility of the underlying atomic basis set, particularly as eachmolecule is stretched. This can be attributed to the fact that the

Table 2. Near-Exact (EstatCAS, CBS = pc-2) Static Correlation Energies (mEh), δ

FCI and δVOD Errors for the G1Molecules, Along withAll Reactants, Transition States, and Products from the DBH24 Data Set, Plus Atomic Insertion Reactions

|Estat| δFCI δVOD |Estat| δFCI δVOD

H2 18.5 −2.1 0.0 Li2a 9.4 1.1 0.0

LiHa 16.1 1.6 0.0 Na2a 10.6 −6.9 0.0

LiFa 5.5 0.9 0.0 NaCla 8.4 5.7 0.0BeH 26.4 4.1 0.0 CH 42.2 −0.1 −3.5NH 25.9 −1.2 −5.0 OH 24.7 −0.9 −4.5FH 24.2 −1.6 0.0 ClH 17.3 −2.4 0.0CH2 (

3B1) 38.3 −4.4 −3.9 SiH2 (3B1) 32.7 −8.8 −3.5

CH2 (1A1) 60.8 0.6 0.2 SiH2 (

1A1) 48.4 −3.4 0.2CH3 58.8 −3.1 −2.9 SiH3 42.3 −11.3 −2.2CH4 82.3 3.9 0.5 SiH4 54.8 −13.1 0.2NH2 48.7 0.5 −3.9 PH2 34.6 −8.2 −4.0NH3 74.3 9.7 0.6 PH3 48.1 −11.1 0.2OH2 53.3 3.7 0.2 SH2 34.2 −5.4 0.1CH3OH

b 117.8 −0.1 CH3SH 103.9 −2.3 0.8CH3Cl 86.4 0.3 0.7 FCl 37.8 −9.5 0.0C2H6 149.9 4.3 1.0 Si2H6 91.8 −25.0 0.5O2H2 106.9 0.5 0.8 N2H4 134.1 14.3 1.0O2 104.7 −7.1 −15.4 S2 46.8 −15.8 −10.1F2 79.1 −3.1 0.1 Cl2 23.5 −5.0 0.0ClO 42.9 0.9 −6.5 HOCl 66.3 −2.8 0.4SO 62.9 −22.4 −19.9CN 151.4 −24.0 −3.0 HCN 150.2 −15.0 2.6N2 148.1 −8.5 3.1 P2 93.1 −31.7 2.7CO 131.1 −7.4 2.9 CS 104.7 −16.1 4.8C2H4 121.2 −39.6 0.8 C2H2 130.9 −40.6 2.0HCO 125.4 −12.7 −10.4 H2CO 145.2 2.4 1.5NO 120.4 −12.7 −6.6 CO2 176.2 −34.7 9.7SiO 121.3 −33.5 2.1 Si2 82.9 −28.8 −1.1O3 237.2 −24.4 17.3 SO2 131.7 −90.9 11.0

H• + •OH 24.7 −0.9 −4.5 H• + H2S 34.2 −5.4 0.1[HOH]•• 40.4 0.3 −6.3 [SH3]

• 41.0 −8.9 −2.2H2 + O•• 18.5 −2.1 −6.2 H2 +

•SH 37.1 −4.8 −4.8CH4 + OH• 107.1 3.0 −4.0 H• + HCl 17.3 −2.4 0.0[HOCH4]

• 127.0 5.2 −7.5 [HClH]• 37.9 −13.8 −2.3CH3OH

b + H• 117.8 −0.1 − HCl + H• 17.3 −2.4 0.0N2O + H• 214.7 −7.2 11.9 FCl + •CH3 96.5 −12.6 −2.8[HN2O]

• 223.8 −35.0 3.3 [CH3FCl]• 120.6 −7.7 1.1

N2 +•OH 172.8 −9.3 −1.4 CH3F + •Cl 93.0 −1.2 −4.4

HN2• 129.6 −9.3 −5.6 Cl− + CH3Cl 86.4 0.3 0.7

[HN2]• 157.8 −13.7 −1.1 [ClCH3Cl]

− 79.5 −15.6 1.9N2 + H• 148.1 −8.5 3.1 CH3Cl + Cl− 86.4 0.3 0.7C2H4 + H• 121.2 −39.6 0.8 F− + CH3Cl 86.4 0.3 0.7[HC2H4]

• 147.4 −17.5 −3.5 [FCH3Cl]− 146.9 6.7 1.9

CH3CH2• 127.9 −0.7 −2.7 CH3F + Cl− 93.0 −1.2 0.9

HCN 150.2 −15.0 2.6 OH− + CH3F 116.5 −1.5 1.0[HCN] 141.2 −15.8 4.1 [HOCH3F]

− 159.1 −0.1 2.2HNC 139.6 −5.4 2.5 CH3OH

b + F− 117.8 −0.1Be + H2 61.0 −11.0 0.1 Al + O2 128.4 −7.2 −17.3[BeH2] 84.8 −44.8 47.5 [AlO2] 227.9 10.3 −1.3HBeH 32.7 −2.6 0.1 OAlO 213.3 −50.3 50.0

aEstatCAS, Estat

FCI, and EstatVOD were calcuted using the (2,2) active space, with unoccupied alkali metal AOs and doubly occupied halide AOs. bEstat

CAS and EstatFCI

were calculated using the (10,10) active space, constraining oxygen AOs to remain doubly occupied.

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STO-3G basis set is constructed for optimal performance nearequilibrium but lacks the flexibility to adapt to its molecularenvironment. Therefore, in FCI/STO-3G calculations formolecules far from their equilibrium structures, additionaldeterminants can play two roles, compensating for the fixed-exponent nature of the minimal atomic basis and capturingstatic correlation effects associated with bond dissociation andother near-degeneracy situations. Thus, FCI/STO-3G predicts aspuriously large value for Estat as it is not solely capturing staticcorrelation effects.It is possible to further subdivide the bond static correla-

tion energy into a component that can be recovered by allowingspin symmetry breaking of the orbitals, that is, using a UHFwave function and a remainder term. As discussed by Hollett

and Gill,63 UHF wave functions can capture static correlationeffects in cases of absolute near-degeneracy, such as occursduring bond breaking where the energy difference betweenthe HOMO and LUMO tends to zero. However, for cases ofrelative near-degeneracy, for example, during bond stretching orfor “anomalous” atoms like Be, where the HOMO−LUMO gapis nonvanishing, UHF struggles and often fails entirely to recoverthe static correlation energy. As a result, UHF-partitionedbond static correlation energy profiles display complicatedand 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

δ| | = | | −E EstatFCI

statCAS FCI

(11)

Figure 3. Bond energy (left-side panels) and bond correlation energy (right-side panels) curves for H2, N2, O2, and F2. Bond energies are calculatedaccording to eq 8, while bond correlation energies are defined by eqs 9 and 10. (Left-side panels) Black curves are “exact” (derived from experimentaldata62), RHF curves are shown as dotted lines, while the dashed curves also incorporate a static correlation correction. (Right-side panels) Bonddynamic correlation energy (dashed line) and different bond static correlation energy estimates; the gray curve represents Estat

FCI(bond), and the solidblack curve represents Estat

CAS(bond).

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as presented in Tables 1 and 2. The predominantly negativevalues of δFCI indicate that FCI/STO-3G tends to overestimateEstat, as discussed above.The δFCI error for SO2 is anomalously large due primarily to its

unexpectedly low EstatCAS value. In general, molecules containing

a single second-row substitution (e.g., CH3SH compared to

CH3OH) have lower EstatCAS values by 10−20 mEh compared to

their first-row counterparts. However, the EstatCAS value of SO2 is

smaller than that of O3 by around 100 mEh. This unexpectedand somewhat perplexing result is well-attested by a variety ofdifferent quantum chemical software packages and orbitaloptimization algorithms. In this case, it appears that accounting

Table 3. EstatFCI static correlation energies (mEh), δ

D, δ(T), δT and δQ errors for the G1 molecules, and all reactants, transition statesand products from the DBH24 data set, plus atomic insertion reactions

EstatFCI δD δ(T) δT δQ Estat

FCI δD δ(T) δT δQ

H2 20.6 0 0 0 0 Li2a 8.2 0 0 0 0

LiHa 14.5 0 0 0 0 Na2a 17.5 0 0 0 0

LiFa 4.6 0 0 0 0 NaCla 2.8 0 0 0 0BeH 22.2 0.5 0.2 0 0 CH 42.3 0.1 0.1 0.1 0.0NH 27.1 0 0 0 0 OH 25.6 0 0 0 0FH 25.8 0 0 0 0 ClH 19.7 0 0 0 0CH2 (

3B1) 42.7 0.2 0.1 0.0 0.0 SiH2 (3B1) 41.4 0.4 0.1 0.0 0.0

CH2 (1A1) 60.2 0.5 0.3 0.1 0.0 SiH2 (

1A1) 51.8 0.3 0.1 0.1 0.0CH3 61.8 0.2 0.1 0.0 0.0 SiH3 53.6 0.5 0.2 0.0 0.0CH4 78.5 0.2 0.1 0.0 0.0 SiH4 67.9 0.5 0.2 0.0 0.0NH2 48.1 0.1 0.0 0.0 0.0 PH2 42.8 0.1 0.0 0.0 0.0NH3 64.6 0.2 0.1 0.0 0.0 PH3 59.1 0.2 0.0 0.0 0.0OH2 49.6 0.1 0.0 0.0 0.0 SH2 39.7 0.0 0.0 0.0 0.0CH3OH 117.9 1.0 0.3 0.1 0.0 CH3SH 106.1 0.6 0.2 0.1 0.0CH3Cl 86.1 0.6 0.2 0.1 0.0 FCl 47.3 0 0 0 0C2H6 145.6 0.7 0.3 0.1 0.0 Si2H6 116.8 1.1 0.3 0.0 0.0O2H2 106.4 0.8 0.3 0.2 0.0 N2H4 119.8 0.9 0.3 0.1 0.0O2 111.8 2.1 1.4 1.2 0.0 S2 62.6 1.3 0.8 0.7 0.0F2 82.1 0 0 0 0 Cl2 28.5 0 0 0 0ClO 42.0 1.1 0.5 0.0 0.0 HOCl 69.1 0.4 0.2 0.1 0.0SO 85.4 1.9 1.1 0.9 0.0CN 175.5 5.8 3.5 2.3 0.3 HCN 165.1 3.3 2.1 2.0 0.0N2 156.6 3.9 2.2 2.0 0.0 P2 124.8 4.3 2.7 2.7 0.1CO 138.5 8.0 0.9 −0.2 0.1 CS 120.7 9.1 1.0 −0.2 0.1C2H4 160.7 1.2 0.5 0.4 0.0 C2H2 171.5 2.5 1.9 1.8 0.0HCO 138.1 5.6 1.3 0.2 0.1 H2CO 142.9 2.6 0.7 0.3 0.0NO 133.1 5.0 1.7 0.8 0.1 CO2 210.9 20.2 −0.3 −0.2 −0.1SiO 154.9 14.5 0.6 −0.7 0.4 Si2 111.7 6.5 3.3 2.4 0.2O3 261.5 23.7 3.9 2.6 0.3 SO2 222.5 21.0 1.1 1.0 0.2

H• + •OH 25.6 0.0 0.0 0 0 H• + H2S 39.7 0.0 0.0 0.0 0.0[HOH]•• 40.1 0.1 0.0 0.0 0.0 [SH3]

• 49.9 0.2 0.0 0.0 0.0H2 + O•• 20.6 0 0 0 0 H2 +

•SH 41.9 0.0 0.0 0.0 0.0CH4 + OH• 104.0 0.2 0.1 0.0 0.0 H• + HCl 19.7 0.0 0.0 0.0 0.0[HOCH4]

• 121.8 0.7 0.2 0.1 0.0 [HClH]• 51.7 0.0 0.0 0.0 0.0CH3OH + H• 117.9 1.0 0.3 0.1 0.0 HCl + H• 19.7 0.0 0.0 0.0 0.0N2O + H• 221.9 14.3 5.7 2.9 0.1 FCl + •CH3 109.2 0.2 0.1 0.0 0.0[HN2O]

• 258.8 18.3 2.2 2.6 0.2 [CH3FCl]• 128.3 0.5 0.1 0.1 0.0

N2 +•OH 182.2 3.9 2.2 2.0 0.0 CH3F + •Cl 94.1 1.1 0.3 0.0 0.0

HN2• 138.9 3.4 1.3 0.7 0.1 Cl− + CH3Cl 86.1 0.6 0.2 0.1 0.0

[HN2]• 171.6 5.0 2.5 2.0 0.1 [ClCH3Cl]

− 95.1 1.9 0.2 0.0 0.0N2 + H• 156.6 3.9 2.2 2.0 0.0 CH3Cl + Cl− 86.1 0.6 0.2 0.1 0.0C2H4 + H• 160.7 1.2 0.5 0.4 0.0 F− + CH3Cl 86.1 0.6 0.2 0.1 0.0[HC2H4]

• 164.9 1.4 0.6 0.4 0.0 [FCH3Cl]− 140.2 4.1 −0.9 −0.1 0.0

CH3CH2• 128.6 0.8 0.2 0.1 0.0 CH3F + Cl− 94.1 1.1 0.3 0.0 0.0

HCN 165.1 3.3 2.1 2.0 0.0 OH− + CH3F 118.0 1.1 0.3 0.0 0.0[HCN] 156.9 5.9 1.8 1.4 0.1 [HOCH3F]

− 159.2 5.1 0.6 −0.1 0.0HNC 145.0 6.6 1.0 −0.1 0.1 CH3OH + F− 117.9 1.0 0.3 0.1 0.0Be + H2 72.0 0 0 0 0 Al + O2 135.6 2.1 1.4 1.2 0.0[BeH2] 129.6 4.4 −4.4 −2.2 0.0 [AlO2] 217.6 24.8 −4.2 4.5 0.7HBeH 35.3 0.4 0.2 0.0 0.0 OAlO 263.6 26.3 5.5 3.5 0.9

aEstatFCI, Estat

D , EstatT , and Estat

Q are calculated using (2,2) valence space.

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for valence orbital flexibility strongly stabilizes the HOMOrelative to the LUMO, significantly decreasing the multireferencecharacter of the CASSCF/CBS wave function relative to itsFCI/MBS counterpart. This is supported by inspection of theFCI/MBS and CASSCF/CBS CI coefficients provided in theSupporting Information. Unfortunately, it appears impossibleto predict a priori which molecules will exhibit this behavior,although experience suggests that this should occur only rarely.Electron CorrelationModels.The usefulness of the correla-

tion treatment approximations in eq 5, measured by δX,

δ| | = | | −E EstatX

statFCI X

(12)

are shown in Table 3. To exclude orbital flexibility effects, aminimal atomic basis has been used throughout. The errors arealmost always positive, indicating that the approximations usuallyunderestimate Estat

FCI.The Estat

D model reproduces the exact EstatFCI energies of the

molecules in the G1 data set well overall (MAD= 2.7 mEh), but itis 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-bondedmolecules such as CO2, SiO, SO2, and O3. These results areconsistent with the fact that CCSD includes only thedeterminants needed to describe single-bond dissociations.The Estat

D model performs equally well for the DBH24 data setand atomic insertion reactions (MAD = 3.5 mEh), although themajority of the error comes from N2O (which is isoelectronicwith CO2), a nearby [HN2O]

• transition state, linear aluminumoxide (OAlO), and a nearby AlO2 transition state. Transitionstates per se do not appear to pose particular problems for theEstatD approximation.Formally, in order to obtain comparable accuracy for σ- and

π-bonded systems, it is necessary to include connected quadrupleexcitations.64 However, the Estat

(T) energies (MAD = 0.6 and1.0 mEh for G1 and expanded DBH24, respectively) reveal thatCCSD(T) is an excellent approximation to FCI in a minimalbasis. In particular, the perturbative triples correction recoversalmost all of the missing static correlation energy in CO2 andSO2 and significantly improves the treatment of O3 and otherπ-bonded systems.The Estat

T errors in Table 3 indicate that CCSDT yieldscomparable results (MAD = 0.4 and 0.7 mEh for both G1 andDBH24) to those obtained using CCSD(T). However, becauseof the much greater computational cost of the former, weconclude that the Estat

(T) model lies at a price-performance “sweetspot”.The Estat

Q errors in Table 3 show that CCSDTQ is almostidentical (MAD = 0.04 and 0.1 mEh for G1 and DBH24,respectively) to FCI for all of the systems considered. When itscomputational cost is manageable, CCSDTQ is a very accuratecorrelation method.Valence-Orbital-Optimized Coupled Cluster Doubles.

The effect of approximating the treatment of electron correlationwhile allowing valence orbital optimization is measured by

δ| | = | | −E EstatVOD

statCAS VOD

(13)

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 correla-tion in obtaining accurate static correlation energies, reducing themean absolute deviation across all data from 10.6 to 4.1 mEh.

Counterintuitively, VOD/pc-2 often slightly overestimates themagnitude of Estat, with a series of small negative values of δ

VOD

appearing in Tables 1 and 2. Closer inspection reveals that all ofthese cases involve radicals or diradicals, and the overstabiliza-tion in these cases is due to spin symmetry breaking of theapproximate VOD wave function, as established previously.65

Otherwise, VOD/pc-2 slightly underestimates the magnitudeof Estat in σ-bonded systems and more significantly under-estimates Estat in strong multireference systems such as O3, SO2,CO2, N2O, BeH2, and AlO2, as expected.

Side Note: The EstatD Model as a CASSCF Diagnostic.

Pragmatically, the EstatD model provides surprisingly good

estimates of EstatCAS due to partial cancellation of AO inflexibility

and correlation treatment errors. However, because orbitalflexibility tends to play a larger role in determining Estat thaninclusion of higher excitations in the coupled cluster wavefunction, the Estat

D model generally overestimates Estat. Therefore,the Estat

D model provides a cost-effective but nonrigorous upperestimate of Estat

CAS, which may be used to verify that full-valenceCASSCF calculations recover only Estat.If |Estat

CAS| > |EstatD |, it is likely that the full valence CASSCF model

is capturing some dynamic correlation effects, that is, that theadditional determinants preferentially stabilize the system byaccounting for interelectronic wave function cusps rather thannear-degeneracies between bonding and antibonding orbitals.One unexpected example from the present work is methanol,CH3OH, in which the dynamic correlation energy of each oxygenlone pair exceeds the static correlation energy of a pair ofσ-bonded electrons. Therefore, it is necessary to constrain thelone pair oxygen orbitals to remain doubly occupied throughoutto ensure that only static correlation effects are captured.On the other hand, as the Estat

D model provides surprisinglygood estimates of Estat

CAS, particularly for σ-bonded moleculescomprised of atoms from the first and second rows of theperiodic table (H−Ne), it may be used to confirm that theCASSCF(val)/CBS wave function has converged to the optimalsolution. If Estat

CAS is not within 15 mEh of EstatD for these systems,

this is a strong indication that the CASSCF orbitals are incorrect.Even for more problematic molecules (involving third-rowelements or multiple bonds), Estat

CAS usually lies within 40 mEhof Estat

D .These results reinforce the well-known conclusion that CASSCF

is not a black-box method for calculating static correlation energiesas it depends strongly on both appropriate choice of active spaceand initial guess of MOs. Even with a well-defined choice of activespace (full valence), the potential pitfalls of inadvertently capturingdynamic correlation effects and converging to nonoptimal orbitalsremain. In this work, we have encountered both of these issues andtaken advantage of Estat

D in deciding to further refine the CASSCFwave function in these cases.

■ SUMMARY AND CONCLUSIONS

The real utility of approximate models is as stand-alone staticcorrelation estimators because the exponential scaling ofcomputational cost with number of electrons makes routineCASSCF calculations impractical for most molecules. Bycontrast, the computational costs of the Estat

VOD, EstatD , Estat

(T), EstatT ,

and EstatQ models scale asO(n6),O(n6),O(n7),O(n8), andO(n10),

respectively, where n is the number of electrons in the molecules’valence space. We note, however, the Estat

VOD and EstatD prefactors

are substantially different, that is, EstatVOD calculations are much

more computationally taxing than EstatD calculations.

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EstatVOD, Estat

T , and EstatQ estimates of the correlation energy may be

readily obtained for systems with n ≤ 14, while the EstatD and Estat

(T)

models can be extended to systems with n ≈ 60. Although thisis already a significant improvement over the performance ofconventional full-valence CASSCF, further developments inlinear-scaling correlated methods, for example, the localized AOplus pair natural orbital based coupled cluster method recentlypublished by Riplinger and Neese,66 are required to obtainuniversally applicable static correlation models.Nonetheless, the results presented here provide clear proof

of principle that it is often possible to obtain reasonable staticcorrelation energies without exponential scaling of computa-tional cost. This provides a foundation for uncoupled, or onlyweakly coupled, modeling of the dynamic and static contribu-tions to the total electronic energy. For example, one mayenvisage applying R129 or F12 theory67 in conjunction with anapproximate static correlation model like Estat

(T). Alternatively, itmay be easier to parametrize density functional68 or intraculefunctional69 methods to recover only the dynamic component ofthe correlation energy rather than the total correlation energy.Further accuracy may be obtained at little extra computationalcost by allowing intermediate flexibility in the AO basis, forexample, by extracting molecule-polarized AOs29 or “improved”virtual valence orbitals30 from a large basis HF calculation.

■ ASSOCIATED CONTENT*S Supporting InformationEnergy values for various basis sets and FCI/MBS and CASSCF/CBS CI coefficients. This material is available free of charge viathe Internet at http://pubs.acs.org.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: deborah.crittenden@canterbury.ac.nz.NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThe author would like to thank Professor Peter Gill and Dr.David Brittain for helpful discussions and the referees for theirhelpful comments and feedback.

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