Dynamical and Nondynamical Correlation

Dynamical and Nondynamical Correlation

Daniel K. W. Mok, Ralf Neumann, and Nicholas C. Handy*Department of Chemistry, UniVersity of Cambridge, Cambridge CB2 1EW, United Kingdom

ReceiVed: September 20, 1995; In Final Form: December 5, 1995X

The variation of correlation energies with bond distances of various first row diatomic molecules has beenstudied. Self-consistent field and complete active space self-consistent field potential curves of these moleculeshave been calculated. Exact potential energy curves are constructed from experimental data using the Rydberg-Klein-Rees method. With appropriate definitions, the dynamical and nondynamical correlation energies areobtained and the variation of these with bond distance is calculated. Two definitions of nondynamicalcorrelation are examined. Classifying the angular correlation as dynamical seems to be a better way to partitionthe correlation energy. The correlation functionals of density functional theory, VWN, LYP, and P86, arealso evaluated and compared with theab initio dynamical correlation energies. LYP appears to give theclosest agreement with the dynamical correlation energy.

1. Introduction

The electron correlation energy of a molecule,Ecorr, has beendefined by Lowdin1 as the difference between the exactnonrelativistic energy eigenvalue of the electronic Schro¨dingerequation,Eexact, and the basis limit energy of the singleconfiguration (≡configuration state function, CSF) approxima-tion, commonly called the Hartree-Fock energy,EHF. Thus,

While this definition is satisfactory near equilibrium, it becomesless satisfactory as molecular bonds are stretched. It is wellknown that for H2 at equilibriumEcorr = 0.04Eh ) 1.1 eV) 25kcal/mol, and at infinite separationEcorr = 0.25Eh ) 6.8 eV)156 kcal/mol. Of course, this problem is well-understood; atequilibrium one CSF (σg2) is sufficient, but, at infinite separation,two CSFs are necessary (σg2, σu2).It is usual to recognize that the correlation energy so defined

may be split into two parts, which Sinanogˇlu2 is generallyacknowledged to be the first to recognize:“Correlation effects may be divided into ‘dynamical’ and

‘nondynamical’ ones. Dynamical correlation occurs with a‘tight pair’ of electrons as in He or in the (2pz)2 in Ne, etc.There is no one configuration in the Configuration Interaction,CI, wavefunction which mixes strongly with the Hartree Fock,HF, configuration and CI is slowly convergent. ‘Non-dynami-cal’ correlations, on the other hand, arise from degeneracies ornear-degeneracies (first order CI).”Nondynamical correlation energy (NDCE) is associated with

the lowering of the energy through interaction of the HFconfiguration with low-lying excited states. It is a near-degeneracy effect and may be specifically calculated by diago-nalizing the appropriate secular matrix. An unambiguousdefinition is to include in the secular matrix all CSFs whicharise from all possible occupancies of the valence orbitals, thatis bonding, nonbonding, and antibonding orbitals. The numberof such orbitals is the same as the number of basis functions ina minimum basis set (e.g., STO-3G) calculation on the molecule.To obtain a unique definition, the orbitals in such a calculationshould then be optimized to self-consistency. Such a calculationwas first carried through by Ruedenberg and Sundberg3 followedby Dombek and Ruedenberg4 and Ruedenberget al.5 Rueden-

berg gave the name full optimized reaction space (FORS) tosuch calculations. In 1980 Roos6 also considered the samemulticonfiguration self-consistent field (MCSCF) procedure andcalled it the complete active space self-consistent field(CASSCF) method, and it is this latter name which has held.Thus, we define the NDCE,END, as

This near-degeneracy correlation is essential for the correctdissociation of a molecule into its constituent atoms, which isapparent from the argument that atomic orbitals will be a linearcombination of the molecular orbitals. This correlation istherefore a long-range effect, sending electrons to individualatoms as the molecule dissociates, as easily appreciated froman understanding of the H2 molecule.Since the dynamical correlation energy (DCE),ED, must be

such that

it follows that

It only seems possible to define DCE once NDCE has beendefined, but it then follows that DCE is a short-range effect,and it is the reduction in the repulsion energy which arises fromthe reduction in the value of the wave function when twoelectrons approach one another. Specifically, we know that ifthe electrons have parallel spin, nearr12 ) 0 the wave functionobeys

and if they have opposite spin,

This DCE is more difficult to calculate because the abovearguments make it clear that it will only be accurately obtainedfrom wave functions which explicitly include the interelectronicdistance,r12, linearly. In variational calculations this is almostimpossible for systems with more than two electrons. OneX Abstract published inAdVance ACS Abstracts,March 1, 1996.

Ecorr ) Eexact- EHF (1)

END ) ECASSCF- EHF (2)

Ecorr ) ED + END (3)

ED ) Ecorr - END (4)

ψ ∼ Ar122 (1+ 1/4r12) (5)

ψ ∼ B(1+ 1/2r12) (6)

6225J. Phys. Chem.1996,100,6225-6230

0022-3654/96/20100-6225$12.00/0 © 1996 American Chemical Society

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associates the names of Hylleraas7 with the first such calculationon He and James and Coolidge8 for the first such calculationon H2.For He, the first shell is full, and there are no nearby electronic

states. Thus, the correlation in He is only dynamical, so theHylleraas calculations on He are forED, because in this case

Of course, full configuration interaction (FCI) calculations onHe will also obtain this correlation energy, although they willbe slowly convergent.There are only a very small number of systems for which

we can precisely define NDCE and DCE. For H2 at infiniteseparation, there is certainly no dynamical correlation in theseparated H atoms, and thus

It must also hold that for the Ne atom there is zero NDCE,because there are no low-lying unoccupied orbitals. Thus

Such arguments do not extend so rigorously to heavier noblegas atoms because the unoccupied orbitals become closer inenergy to the occupied orbitals.It may also be argued that the important model system, the

uniform electron gas,9 has no degeneracy and therefore noNDCE. Thus, many consider that for this system

The purpose of this paper is to examine DCE and NDCE forsome diatomic molecules. To be able to perform such calcula-tions, we must know the exact value for the correlation energy.In the next section, we shall describe how we have been ableto acquire the required information from (a) Hartree-Fockcalculations, (b) CASSCF calculations, (c) exact atomic energies,and (d) Rydberg-Klein-Rees potential energy curves. Weshall therefore study DCE and NDCE for some diatomicmolecules for a range of internuclear distances.

2. Methods

The exact nonrelativistic energy,Eexact, of a molecule is almostimpossible to calculate except very small systems such as H2.However, predictions for the ground state energiesEexact ofatoms are much easier to obtained. In fact, Davidsonet al.10

have predicted the ground state energies for atoms up to atomicnumber 10. With these exact ground state energies, the exactenergies of the dissociation asymptotes of the diatomic mol-ecules found between these atoms may be calculated. Oncethe dissociation asymptotes are obtained, the exact energies ofthe molecules in the equilibrium geometry are calculated fromthe experimental dissociation energies,Ediss(≡D0), and zero pointenergies,EZPE. That is,

For the potential curves, the Rydberg-Klein-Rees (RKR)procedure11-14 is used to construct the curve if there is nopublished potential curve for the molecule under study. TheRKR procedure is a well-established semiclassical method toconstruct the potential curve of a diatomic molecule fromexperimental vibrational energies,G(V), and rotational constants,B(V). The turning points of the potential curves at vibrationallevel V are defined by

TheG(V) andB(V) are represented by polynomials of (V + 1/2)

Yij are the Dunham coefficients.15 The lower integration limitVmin with the Kaiser correction16 is given by

With the above equations, the potential well of a diatomicmolecule can be constructed fromG(V) andB(V). Values ofYijfor all molecules under study are taken from Herzberg17 exceptfor Be2.18 For all of the molecules studied, there are observedvalues ofV for which r2(V) > 1.5re except for Be2, and N2. Todetermine the specific values of the energies given in the tables,we have interpolated the energy data of the provided vibrationalenergy levels.Another approach which is probably more accurate than RKR

is the inverted perturbed analysis (IPA),19 whereby, with aninitial guess, the potential energy curve is iteratively improvedby comparing numerical solutions of the nuclear Schro¨dingerequation with observed data. IPA curves are available for Li2

20

and LiH.21

RKR and IPA methods define the potential wells with thewell minimum as zero. Therefore, with eq 11 the exact potentialcurve is now defined. However, the RKR and IPA potentialwells are usually only available in the region near the minimum(perhaps 0.5re < r < 1.5re). At large internuclear distances,the potential curve is fitted to the following expression,

wheren, m ) 5, 6, or 8, depending on the case. We havedeterminedCn andCm by least squares to the RKR or IPApotential curve and the dissociation asymptote. Of course, suchvalues will only be approximate.The SCF and CASSCF energy curves calculations were

performed with the software package MOLPRO.22-24 The basisset used is TZ2P except for Be, for which 6-311G* has beenused. The basis set is expected to be sufficient for our purposesfor both the CASSCF and SCF calculations. The TZ2P basisfor first row atoms is Huzinaga’s25 primitive (10s6p) set

EDHe ) Eexact

He - EHFHe

ENDHe ) 0 (7)

ENDH2,∞ ) Eexact

H2,∞ - EHFH2,∞

EDH2,∞ ) 0 (8)

EDNe ) Eexact

Ne - EHFNe

ENDNe ) 0 (9)

EDU EG ) Eexact

U EG - EHFU EG

ENDU EG ) 0 (10)

Ere ) Er)∞ - Ediss- EZPE (11)

r2(V) - r1(V) ) 2[p2

2µ]1/2∫VminV dV′[G(V) - G(V′)]1/2

1r1(V)

- 1r2(V)

) 2[2µp2]1/2∫VminV B(V′) dV′

[G(V) - G(V′)]1/2(12)

G(V) ) ∑i)1

m

Yi0(V + 1/2)i (13)

B(V) ) ∑i)0

n

Yi1(V + 1/2)i (14)

Vmin ) - 1/2 - Y00/Y10 (15)

Y00 )Y20 + Y01

4-Y11Y1012Y01

+(Y11Y10)

2

144Y013

(16)

V(r) )Cn

rn+Cm

rm(17)

6226 J. Phys. Chem., Vol. 100, No. 15, 1996 Mok et al.

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contracted by Dunning26 to [5s4p] with two polarizationfunctions added. The exponents of the polarization functionsare 0.4, 0.1 for Li, 1.2, 0.4 for C, 1.35, 0.45 for N, 1.35, 0.45for O, and 2.0, 0.6667 for F. For hydrogen, the TZ2P is (5s)/[3s] with polarization function exponents 1.5 and 0.5. For thechoice of orbitals in the active space, we further examine theNDCE.Many scientists have discussed “in-out”, “angular”, and

“left-right” correlation. In-out correlation refers to thatcorrelation obtained from double excitations from occupied toexcited orbitals of the same angular type, but with more nodes,such as sf s′, pf p′ or σg f σg′. In atoms this increases theradial separation of electrons. Angular correlation refers to thatcorrelation obtained from double excitations from occupied toexcited orbitals of different angular type, and it increases theangular separation of electrons. An example of angular cor-relation is that correlation which arises in Be by consideringthe interaction of the HF CSF 1s22s2 with the CSF 1s22p2. Itmay easily be shown that such a CSF explicitly introducesr122

into the wave function by consideration of the following model,

Then it follows that

Thus, such correlation can also be considered dynamical becauseit introducesr122. Thus, it is possible to consider an alternativedefinition for NDCE in which it is assumed that promotion fromprimarily 2s-type orbitals to 2p-type orbitals are excluded.Under such a definition Be2 would have zero NDCE, which isa good definition, because a single CSF description of Be2 doesdissociate to ground state Be atoms. The deletion of these CSFsfrom the CASSCF calculations leaves only interactions betweenCSFs constructed from orbitals (in a minimum basis description)involving 1s on H, 2s on Li, 2p on B-F, 3s on Na, 3p on Al-Cl, etc. Such correlation includes left-right correlation whichis necessary to ensure the correct dissociation of molecules intoconstituent atoms. It explicitly includes the effects present invalence bond wave functions, which may be constructed fromsuch configurations. The interelectronic distance does not arisefrom these CSFs. Under this definition atoms do not have anyNDCE; e.g., C is described by 1s22s22p2, and 1s22p4 introducesDCE.In order to distinguish these two definitions of NDCE, and

the subsequent definition of DCE, we shall refer to this newsubset of NDCE as NDCE′, and the consequent DCE′. It isprobable that NDCE′ and DCE′ are more satisfactory partitionsof the correlation energy. Obviously, the selection of activeorbitals depends on whether we are working with NDCE orNDCE′. To calculate NDCE and DCE, all molecular orbitalsarising from the valence shells are included in the active space.For calculations of NDCE′ and DCE′, the selection is such thatthe angular correlation is not incorporated into the wave

function. The number of CSFs included in the CASSCF wavefunctions with and without the angular correlation are shownin Table 1. We admit that this distinction between DCE andDCE′ is more satisfactory for alkali and alkaline earth atomsthan it is for those with more valence electrons, such as C, wherehybridization mixing of the 2p and 2s orbitals occurs as theorbitals are optimized during the CASSCF calculation. To putour arguments another way, we are trying to separate molecularnondynamical correlation from atomic nondynamical correlation,using the more usual terminology. Many years ago Wahl andDas27 stressed the importance of distinguishing between intra-atomic and interatomic correlation.One of the principal purposes behind this research is to

attempt a further understanding of density functional theory(DFT). In particular, we are interested in the nature of

Ψ ) ∑i)1

4

ciΦi (18)

Φi ) A(æi2) (19)

æ1 ) e-2r, æ2 ) xe-2r, æ3 ) ye-2r, æ4 ) ze-2r (20)

c1 ) 1, c2 ) c3 ) c4 ) c (21)

Ψ ) (Râ - âR21/2 )e-2(r1+r2)[1 + cr1‚r2] (22)

Ψ ) (Râ - âR21/2 )e-2(r1+r2)[1+ c

2(r1

2 + r22 - r12

2)] (23)

TABLE 1: Ground States and the Equilibrium BondLengths of the Studied Moleculesa

no. of CSFs in the active space

re with angular without angular

H21∑g

+ 0.741 2 2LiH 1∑+ 1.596 8 3FH 1∑+ 0.917 8 5Li2 1∑g

+ 2.673 10 2Be2 1∑g

+ 2.450 60 1N2

1∑g+ 1.098 328 32

F2 1∑g+ 1.412 10 6

CO 1∑+ 1.128 328 55

a The number of CSFs involved in the CASSCF calculations withand without angular correlation is also given.

Figure 1. RKR, SCF, and CAS potential energy curves of H2 and thevariation of correlation energies, LYP, P86, and VWN, with internucleardistance.

Dynamical and Nondynamical Correlation J. Phys. Chem., Vol. 100, No. 15, 19966227

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exchange-correlation energy functionals. In current terminol-ogy such local functionals are written

andFxc is written as a sum of separate parts

The usual representation ofFc is derived from fits to DCE.Specifically fits are made to the uniform electron gas, UEG,correlation energy, as with VWN,28 and extended to theinhomogeneous case P86,29 or to the correlation energy of theHe-like systems, as with LYP,30 which was constructed fromthe correlated He wave function of Colle and Salvetti.31

Fx must include the effects of electron exchange and indeedthe most popularFx, due to Becke,32 has been derived from theUEG exchange energy expression, and an additional term whichhas been parametrized to the exchange energies of the noble

gas atoms. ThisFx indeed gives high-accuracy exchangeenergies for all atoms. Because such aFx is a local expression,it must include some molecular left-right correlation effects,as we have argued elsewhere by considering the long-rangeeffects.33

The above arguments suggest that the magnitude of the effectof Fx might reasonably be compared with NDCE′ plus theexchange energy and the magnitude of the effect ofFc will be

TABLE 2: Energies (-E, hartree) of the Molecules atreand 1.5re from RKR, SCF, CASSCF (Including Angular),and CASSCF (Excluding Angular)a

with angular without angular

RKR SCF CASCAS+LYP CAS

CAS+LYP

1.0reH2 1.1745 1.1327 1.1514 1.1896 1.1514 1.1897LiH 8.0705 7.9830 8.0135 8.1019 7.9997 8.0879FH 100.4589 100.0631 100.0874 100.4500 100.0870 100.4495Li2 14.9952 14.8702 14.8964 15.0297 14.8793 15.0125Be2 29.3384 29.1316 29.2206 29.4182 29.1316 29.3276N2 109.5427 108.9826 109.1319 109.6156 109.1194 109.6030F2 199.5289 198.7584 198.8376 199.5129 198.8353 199.5106CO 113.3272 112.7812 112.9134 113.3983 112.9040 113.3888


a The LYP energy is also added to the CASSCF values to give anapproximate total energy.

TABLE 3: Correlation Energies (-E, hartree) of theMolecules at re and 1.5rea

RKR-SCF CAS-SCF RKR-CAS LYP P86 VWN



a The six columns areEcorr, NDCE, DCE, and the values of thecorrelation functionals LYP, P86, and VWN calculated with the CASdensity.

Exc[F] )∫Fxc(F,∇F) dr (24)

Fxc ) Fx + Fc (25)

TABLE 4: Correlation Energies (-E, hartree) for theMolecules at re and 1.5rea

RKR-SCF CAS-SCF RKR-CAS LYP P86 VWN



a The six columns areEcorr, NDCE′, DCE′, and the values of thecorrelation functionals LYP, P86, and VWN calculated with the CASdensity.

Figure 2. RKR, SCF, and CAS potential energy curves of Be2 andthe variation of correlation energies, LYP, P86, and VWN, withinternuclear distance.


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compared with DCE′. We shall therefore computeEc,

using appropriate values for the densityF. One logical choiceis the density of the CASSCF wave function. As the CASSCFwave function already includes the NDCE (or NDCE′) and theEc probably gives an accurate DCE (or DCE′), CASSCF andDFT together may be a useful method. In this work, the VWN,P86, and LYP correlation functionals have been evaluated withthe optimized CASSCF spin averaged densities.

3. Results and Discussion

Figure 1 shows the SCF, CASSCF, full CI (FCI), and RKRcurves of H2. For this small system, the CASSCF wave functiongives a very good description of the system. It has the samedissociation limit as the FCI wave function. The FCI calcula-tions are included to show the basis set error,

Figure 1 also shows the variation with internuclear distance ofNDCE, DCE, and the VWN, P86, and LYP correlation energiesevaluated with the optimized CASSCF densities. The graphshows some basic characteristics of correlation energies, whichare also shared by most of the molecules under study. TheNDCE goes up as SCF fails to dissociate to the ground stateatoms. The DCE is decreasing as the molecule dissociates sincethe hydrogen atom does not have any dynamical correlationenergy. It is well-known that VWN overestimates the correla-tion energy, and this is clearly shown in the graph. None ofthe three functionals used gives a very good prediction for DCE.Table 2 tabulates the RKR, SCF, and CASSCF energies at 1

and 1.5 times the equilibrium bond length. Tables 3 and 4 listthe correlation energies at the corresponding internucleardistances with and without angular correlation in the CASSCFwave functions, respectively. Comparing the values of the

Figure 3. Variation of NDCE, DCE, NDCE′, DCE′, and LYP of LiH, Li2, FH, and CO with internuclear distance.

Ec[F] )∫Fc(F,∇F) dr (26)

Ebasis error) EFCI - ERKR (27)

Dynamical and Nondynamical Correlation J. Phys. Chem., Vol. 100, No. 15, 19966229

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correlation functionals with RKR-CAS, which is the dynamicalcorrelation energy, it is seen that the correlation functionalsusually overestimate the value. Although the correlationfunctionals do not give a good approximation to the DCE ofH2, they do give values much closer to DCE and DCE′ forheavier molecules (such as HF, CO, F2, etc.). Among thefunctionals used, LYP gives closest agreement and P86 givesvalues slightly greater than LYP, for all the molecules understudy. Thus, in the following, the comparison will be focusedon LYP.Figure 2 shows the SCF, CASSCF, and RKR curves of Be2.

As has been previously mentioned, Be2 would have zero NDCEif the angular correlation is considered as dynamical. That is,the NDCE′ of Be2 is zero, and DCE′ equals the total correlationenergy. Figure 2 shows the variations of NDCE (≡CAS-SCF),DCE (≡RKR-CAS) and DCE′ (≡Ecorr). The NDCE curve doesnot increase because a single CSF description of Be2 dissociatesproperly. The LYP gives much better approximation to DCE′rather than to DCE. Indeed, the LYP curve nearly overlapsthe DCE′ curve (which is the total correlation energy in thiscase). Figure 3 shows the variations of correlation energies ofLiH, Li 2, FH, and CO. For molecules containing heavier atoms,the difference of DCE and DCE′ is rather small. However, forLiH and Li2, the difference is pronounced, because the angularcorrelation is relatively more important. The definition of DCE′and NDCE′ is most noticeable in molecules which contain Liand Be. In general, LYP agrees with DCE′ better than withDCE.Finally in Table 5 we examine whether the CASSCF energy

plus the LYP energy (calculated with the CASSCF density) isa good approximation to the total energy. Table 5 gives thevalues for

for the molecules studied atre, 1.2re, 1.4re, and 1.5re bondlengths, for the CASSCF calculations with and without angularcorrelation. This table makes it clear that the latter is a betterprocedure, as measured by the fact that the mean absolute erroris only 0.0229 hartree compared with 0.0385 hartree in theformer case. Table 5 does indeed suggest that evaluating theLYP energy on top of such CASSCF energies may be a cheapand fairly reliable procedure for obtaining good total energies.We see that on average the error of the procedure is less than7% of correlation energy.In conclusion, in this paper we have examined definitions

for nondynamical and dynamical correlation energies. UsingCASSCF calculations, and experimental (RKR) data, we havebeen able to compute values for these correlation energies alongthe entire potential energy curve. Because DFT correlationfunctionals are designed to calculate the dynamical correlationenergy, we have compared values for these with the abovedetermined values. We find that the gradient-corrected LYP

functional gives reasonable agreement for the heavier diatomics,but that agreement lessens for the lighter atoms (expecially H2).Better agreement is obtained if angular correlation is countedas dynamical.

Acknowledgment. Dr. P. J. Knowles is acknowledged forthe provision of the MOLPRO package and also for advice.Professor K. Ruedenberg is acknowledged for stimulatingdiscussions. Prof. I. Shavitt is acknowledged for criticalcomments. D.K.W.M. wishes to thank the Croucher Foundationfor financial support. R.N. acknowledges the support of aHuman Capital and Mobility Fellowship of the EU.

References and Notes

(1) Lowdin, P.-O.AdV. Chem. Phys. 1959, 2, 207.(2) Sinanogˇlu, O. AdV. Chem. Phys. 1964, 6, 358.(3) Ruedenberg, K.; Sundberg, K. R. InQuantum ScienceCalais, J.

L., Goscinski, O., Linderberg, J., Ohrn, Y., Eds.; Plenum Press: New York,1976; p 505.

(4) Dombek, M. G. Ph.D. thesis, Chemistry Department, Iowa StateUniversity, 1977.

(5) Ruedenberg, K.; Cheung, L. M.; Elbert, S. T.Int. J.Quantum Chem.1979, 16, 1069.

(6) Roos, B. O.Int. J. Quantum Chem. Symp. 1980, 14, 175.(7) Hylleraas, E. A.Z. Phys. 1929, 54, 347.(8) James, H. M.; Coolidge, A. S.J. Chem. Phys. 1933, 18, 1561.(9) Parr, R. G.; Yang, W.Density-Functional Theory of Atoms and

Molecules; Clarendon Press: Oxford, U.K., 1986.(10) Davidson, E. R.; Hagstrom, S. A.; Chakravorty, S. J.; Umar, V.

M.; Fisher, C. F.Phys. ReV. A 1991, 44, 7071.(11) Rydberg, R.Z. Phys. 1931, 73, 376.(12) Klein, O.Z. Phys. 1932, 76, 226.(13) Tellinghuisen, J.J. Mol. Spectrosc. 1972, 44, 194.(14) Tellinghuisen, J.Comput. Phys. Commun. 1974, 6, 221.(15) Dunham, J. L.Phys. ReV. 1932, 41, 721.(16) Kaiser, E. W.J. Chem. Phys. 1970, 53, 1686.(17) Huber, K. P.; Herzberg, G.Constants of Diatomic Molecules; Von

Nostrand Reinhold: New York, 1979.(18) Bondybey, V. E.Chem. Phys. Lett. 1984, 109, 436.(19) Kosman, W. M.; Hinze, J.J. Mol. Spectrosc. 1975, 56, 93.(20) Hessel, M. M.; Vidal, V. R.J. Chem. Phys. 1979, 70, 4439.(21) Vidal, C. R.; Stwalley, W. C.J. Chem. Phys. 1982, 77, 883.(22) MOLPRO is a package ofab initio programs written by H. J.

Werner and P. J. Knowles, with constribution from J. Almlo¨f, R. D. Amos,M. J. O. Deegan, S. T. Elbert, C. Hampel, W. Meyer, K. Peterson, R. Pitzer,A. J. Stone, and P. R. Taylor.

(23) Werner, H.-J.; Knowles, P. J.J. Chem. Phys. 1985, 82, 5053.(24) Knowles, P. J.; Werner, H.-J.Chem. Phys. Lett. 1985, 115, 259.(25) Huzinaga, S.J. Chem. Phys. 1965, 42, 1293.(26) Dunning, T. H.J. Chem. Phys. 1971, 55, 716.(27) Wahl, A. C.; Das, G. InModem Theoretical Chemistry; Schaefer,

H. F., III, Ed.; Plenum Press: New York, 1977; Vol. 3, p 51.(28) Vosko, S. J.; Wilk, L.; Nusair, M.Can. J. Phys. 1980, 58, 1200.(29) Perdew, J. P.; Wang, Y.Phys. ReV. B 1986, 33, 8822.(30) Lee, C.; Yang, W.; Parr, R. G.Phys. ReV. B 1988, 37, 785.(31) Colle, R.; Salvetti, D.Theor. Chim. Acta1975, 37, 329.(32) Becke, A. D.J. Chem. Phys. 1988, 88, 2547.(33) Neumann, R.; Nobes, R. H.; Handy, N. C.Mol. Phys., in press.

JP9528020

TABLE 5: Error in the Total Correlation Energy, When Calculated as CAS+LYP, for the Cases When Angular CorrelationIs/Is Not Included in the CASSCF Calculationsa

with angular correlation without angular correlation

1re 1.2re 1.4re 1.5re mean 1re 1.2re 1.4re 1.5re mean avEcorr

H2 0.0151 0.0158 0.0165 0.0169 0.0161 0.0152 0.0158 0.0165 0.0169 0.0161 0.0452LiH 0.0314 0.0304 0.0276 0.0278 0.0293 0.0174 0.0190 0.0186 0.0199 0.0187 0.0900FH -0.0090 -0.0113 -0.0115 -0.0107 -0.0106 -0.0094 -0.0119 -0.0120 -0.0112 0.0111 0.4084Li2 0.0345 0.0332 0.0327 0.0328 0.0333 0.0174 0.0214 0.0255 0.0275 0.0229 0.1236Be2 0.0798 0.0841 0.0865 0.0871 0.0844 -0.0108 -0.0046 -0.0024 0.0019 0.0049 0.1979N2 0.0729 0.0687 0.0625 0.0592 0.0658 0.0603 0.0546 0.0491 0.0475 0.0529 0.6421F2 -0.0160 -0.0088 -0.0027 -0.0010 0.0071 -0.0183 -0.0102 -0.0033 -0.0015 0.0083 0.8307CO 0.0711 0.0647 0.0567 0.0526 0.0612 0.0616 0.0529 0.0420 0.0364 0.0482 0.5939

0.0229 0.3665

a The mean absolute errors are also given.

ERKR - ECASSCF- ELYP


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Dynamical and Nondynamical Correlation