J. Phys. Chem. 1996, 100, 6225-6230
6225
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 been studied. Self-consistent field and complete active space self-consistent field potential curves of these molecules have been calculated. Exact potential energy curves are constructed from experimental data using the RydbergKlein-Rees method. With appropriate definitions, the dynamical and nondynamical correlation energies are obtained and the variation of these with bond distance is calculated. Two definitions of nondynamical correlation are examined. Classifying the angular correlation as dynamical seems to be a better way to partition the correlation energy. The correlation functionals of density functional theory, VWN, LYP, and P86, are also evaluated and compared with the ab initio dynamical correlation energies. LYP appears to give the closest agreement with the dynamical correlation energy.
1. Introduction The electron correlation energy of a molecule, Ecorr, has been defined by Lo¨wdin1 as the difference between the exact nonrelativistic energy eigenvalue of the electronic Schro¨dinger equation, Eexact, and the basis limit energy of the single configuration (≡configuration state function, CSF) approximation, commonly called the Hartree-Fock energy, EHF. Thus,
Ecorr ) Eexact - EHF
(1)
While this definition is satisfactory near equilibrium, it becomes less satisfactory as molecular bonds are stretched. It is well known that for H2 at equilibrium Ecorr = 0.04Eh ) 1.1 eV ) 25 kcal/mol, and at infinite separation Ecorr = 0.25Eh ) 6.8 eV ) 156 kcal/mol. Of course, this problem is well-understood; at equilibrium 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 generally acknowledged 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-dynamical’ correlations, on the other hand, arise from degeneracies or near-degeneracies (first order CI).” Nondynamical correlation energy (NDCE) is associated with the lowering of the energy through interaction of the HF configuration with low-lying excited states. It is a neardegeneracy effect and may be specifically calculated by diagonalizing the appropriate secular matrix. An unambiguous definition is to include in the secular matrix all CSFs which arise from all possible occupancies of the valence orbitals, that is bonding, nonbonding, and antibonding orbitals. The number of such orbitals is the same as the number of basis functions in a minimum basis set (e.g., STO-3G) calculation on the molecule. To obtain a unique definition, the orbitals in such a calculation should then be optimized to self-consistency. Such a calculation was first carried through by Ruedenberg and Sundberg3 followed by Dombek and Ruedenberg4 and Ruedenberg et al.5 RuedenX
Abstract published in AdVance ACS Abstracts, March 1, 1996.
0022-3654/96/20100-6225$12.00/0
berg gave the name full optimized reaction space (FORS) to such calculations. In 1980 Roos6 also considered the same multiconfiguration self-consistent field (MCSCF) procedure and called 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
END ) ECASSCF - EHF
(2)
This near-degeneracy correlation is essential for the correct dissociation of a molecule into its constituent atoms, which is apparent from the argument that atomic orbitals will be a linear combination of the molecular orbitals. This correlation is therefore a long-range effect, sending electrons to individual atoms as the molecule dissociates, as easily appreciated from an understanding of the H2 molecule. Since the dynamical correlation energy (DCE), ED, must be such that
Ecorr ) ED + END
(3)
ED ) Ecorr - END
(4)
it follows that
It only seems possible to define DCE once NDCE has been defined, but it then follows that DCE is a short-range effect, and it is the reduction in the repulsion energy which arises from the reduction in the value of the wave function when two electrons approach one another. Specifically, we know that if the electrons have parallel spin, near r12 ) 0 the wave function obeys
ψ ∼ Ar122 (1 + 1/4r12)
(5)
and if they have opposite spin,
ψ ∼ B(1 + 1/2r12)
(6)
This DCE is more difficult to calculate because the above arguments make it clear that it will only be accurately obtained from wave functions which explicitly include the interelectronic distance, r12, linearly. In variational calculations this is almost impossible for systems with more than two electrons. One © 1996 American Chemical Society
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Mok et al.
associates the names of Hylleraas7 with the first such calculation on He and James and Coolidge8 for the first such calculation on H2. For He, the first shell is full, and there are no nearby electronic states. Thus, the correlation in He is only dynamical, so the Hylleraas calculations on He are for ED, because in this case He He EDHe ) Eexact - EHF He END )0
(7)
Of course, full configuration interaction (FCI) calculations on He will also obtain this correlation energy, although they will be slowly convergent. There are only a very small number of systems for which we can precisely define NDCE and DCE. For H2 at infinite separation, there is certainly no dynamical correlation in the separated H atoms, and thus
For the potential curves, the Rydberg-Klein-Rees (RKR) procedure11-14 is used to construct the curve if there is no published potential curve for the molecule under study. The RKR procedure is a well-established semiclassical method to construct the potential curve of a diatomic molecule from experimental vibrational energies, G(V), and rotational constants, B(V). The turning points of the potential curves at vibrational level V are defined by
[] [] p2 2µ
1/2
1 2µ 1 )2 2 r1(V) r2(V) p
1/2
r2(V) - r1(V) ) 2
dV′ [G(V) - G(V′)]1/2
∫VV
B(V′) dV′
min
min
[G(V) - G(V′)]1/2
(12)
The G(V) and B(V) are represented by polynomials of (V + 1/2) m
G(V) ) ∑ Yi0(V + 1/2)i
H2,∞ H2,∞ H2,∞ ) Eexact - EHF END
EDH2,∞ ) 0
∫VV
(13)
i)1 n
B(V) ) ∑ Yi1(V + 1/2)i
(8)
(14)
i)0
It must also hold that for the Ne atom there is zero NDCE, because there are no low-lying unoccupied orbitals. Thus Ne Ne - EHF EDNe ) Eexact Ne END
)0
Vmin ) - 1/2 - Y00/Y10 (9)
Such arguments do not extend so rigorously to heavier noble gas atoms because the unoccupied orbitals become closer in energy to the occupied orbitals. It may also be argued that the important model system, the uniform electron gas,9 has no degeneracy and therefore no NDCE. Thus, many consider that for this system U EG U EG - EHF EDU EG ) Eexact U EG END )0
(10)
The purpose of this paper is to examine DCE and NDCE for some diatomic molecules. To be able to perform such calculations, we must know the exact value for the correlation energy. In the next section, we shall describe how we have been able to acquire the required information from (a) Hartree-Fock calculations, (b) CASSCF calculations, (c) exact atomic energies, and (d) Rydberg-Klein-Rees potential energy curves. We shall therefore study DCE and NDCE for some diatomic molecules for a range of internuclear distances. 2. Methods The exact nonrelativistic energy, Eexact, of a molecule is almost impossible to calculate except very small systems such as H2. However, predictions for the ground state energies Eexact of atoms are much easier to obtained. In fact, Davidson et al.10 have predicted the ground state energies for atoms up to atomic number 10. With these exact ground state energies, the exact energies of the dissociation asymptotes of the diatomic molecules found between these atoms may be calculated. Once the dissociation asymptotes are obtained, the exact energies of the molecules in the equilibrium geometry are calculated from the experimental dissociation energies, Ediss(≡D0), and zero point energies, EZPE. That is,
Ere ) Er)∞ - Ediss - EZPE
Yij are the Dunham coefficients.15 The lower integration limit Vmin with the Kaiser correction16 is given by
(11)
Y00 )
(15)
Y20 + Y01 Y11Y10 (Y11Y10)2 + 4 12Y01 144Y 3
(16)
01
With the above equations, the potential well of a diatomic molecule can be constructed from G(V) and B(V). Values of Yij for all molecules under study are taken from Herzberg17 except for Be2.18 For all of the molecules studied, there are observed values of V for which r2(V) > 1.5re except for Be2, and N2. To determine the specific values of the energies given in the tables, we have interpolated the energy data of the provided vibrational energy levels. Another approach which is probably more accurate than RKR is the inverted perturbed analysis (IPA),19 whereby, with an initial guess, the potential energy curve is iteratively improved by comparing numerical solutions of the nuclear Schro¨dinger equation with observed data. IPA curves are available for Li2 20 and LiH.21 RKR and IPA methods define the potential wells with the well minimum as zero. Therefore, with eq 11 the exact potential curve is now defined. However, the RKR and IPA potential wells 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,
V(r) )
Cn n
r
+
Cm rm
(17)
where n, m ) 5, 6, or 8, depending on the case. We have determined Cn and Cm by least squares to the RKR or IPA potential curve and the dissociation asymptote. Of course, such values will only be approximate. The SCF and CASSCF energy curves calculations were performed with the software package MOLPRO.22-24 The basis set used is TZ2P except for Be, for which 6-311G* has been used. The basis set is expected to be sufficient for our purposes for both the CASSCF and SCF calculations. The TZ2P basis for first row atoms is Huzinaga’s25 primitive (10s6p) set
Dynamical and Nondynamical Correlation
J. Phys. Chem., Vol. 100, No. 15, 1996 6227
contracted by Dunning26 to [5s4p] with two polarization functions added. The exponents of the polarization functions are 0.4, 0.1 for Li, 1.2, 0.4 for C, 1.35, 0.45 for N, 1.35, 0.45 for 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 the choice of orbitals in the active space, we further examine the NDCE. Many scientists have discussed “in-out”, “angular”, and “left-right” correlation. In-out correlation refers to that correlation obtained from double excitations from occupied to excited orbitals of the same angular type, but with more nodes, such as s f s′, p f p′ or σg f σg′. In atoms this increases the radial separation of electrons. Angular correlation refers to that correlation obtained from double excitations from occupied to excited orbitals of different angular type, and it increases the angular separation of electrons. An example of angular correlation is that correlation which arises in Be by considering the interaction of the HF CSF 1s22s2 with the CSF 1s22p2. It may easily be shown that such a CSF explicitly introduces r122 into the wave function by consideration of the following model,
TABLE 1: Ground States and the Equilibrium Bond Lengths of the Studied Moleculesa no. of CSFs in the active space H2 LiH FH Li2 Be2 N2 F2 CO
∑+ g 1∑+ 1
1∑+
∑+ g ∑+ g 1 + ∑g 1 + ∑g 1 + ∑ 1 1
re
with angular
without angular
0.741 1.596 0.917 2.673 2.450 1.098 1.412 1.128
2 8 8 10 60 328 10 328
2 3 5 2 1 32 6 55
a The number of CSFs involved in the CASSCF calculations with and without angular correlation is also given.
4
Ψ ) ∑ ciΦi
(18)
Φi ) A(φi2)
(19)
i)1
φ1 ) e-2r, φ2 ) xe-2r, φ3 ) ye-2r, φ4 ) ze-2r (20) c1 ) 1, c2 ) c3 ) c4 ) c
(21)
Then it follows that
Ψ) Ψ)
(
( )
) [
Rβ - βR -2(r1+r2) e [1 + cr1‚r2] 21/2
Rβ - βR -2(r1+r2) c e 1 + (r12 + r22 - r122) 1/2 2 2
(22)
]
(23)
Thus, such correlation can also be considered dynamical because it introduces r122. Thus, it is possible to consider an alternative definition for NDCE in which it is assumed that promotion from primarily 2s-type orbitals to 2p-type orbitals are excluded. Under such a definition Be2 would have zero NDCE, which is a good definition, because a single CSF description of Be2 does dissociate to ground state Be atoms. The deletion of these CSFs from the CASSCF calculations leaves only interactions between CSFs constructed from orbitals (in a minimum basis description) involving 1s on H, 2s on Li, 2p on B-F, 3s on Na, 3p on AlCl, etc. Such correlation includes left-right correlation which is necessary to ensure the correct dissociation of molecules into constituent atoms. It explicitly includes the effects present in valence bond wave functions, which may be constructed from such configurations. The interelectronic distance does not arise from these CSFs. Under this definition atoms do not have any NDCE; e.g., C is described by 1s22s22p2, and 1s22p4 introduces DCE. In order to distinguish these two definitions of NDCE, and the subsequent definition of DCE, we shall refer to this new subset of NDCE as NDCE′, and the consequent DCE′. It is probable that NDCE′ and DCE′ are more satisfactory partitions of the correlation energy. Obviously, the selection of active orbitals depends on whether we are working with NDCE or NDCE′. To calculate NDCE and DCE, all molecular orbitals arising from the valence shells are included in the active space. For calculations of NDCE′ and DCE′, the selection is such that the angular correlation is not incorporated into the wave
Figure 1. RKR, SCF, and CAS potential energy curves of H2 and the variation of correlation energies, LYP, P86, and VWN, with internuclear distance.
function. The number of CSFs included in the CASSCF wave functions with and without the angular correlation are shown in Table 1. We admit that this distinction between DCE and DCE′ is more satisfactory for alkali and alkaline earth atoms than it is for those with more valence electrons, such as C, where hybridization mixing of the 2p and 2s orbitals occurs as the orbitals are optimized during the CASSCF calculation. To put our arguments another way, we are trying to separate molecular nondynamical correlation from atomic nondynamical correlation, using the more usual terminology. Many years ago Wahl and Das27 stressed the importance of distinguishing between intraatomic 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
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TABLE 2: Energies (-E, hartree) of the Molecules at re and 1.5re from RKR, SCF, CASSCF (Including Angular), and CASSCF (Excluding Angular)a with angular
TABLE 4: Correlation Energies (-E, hartree) for the Molecules at re and 1.5rea RKR-SCF CAS-SCF RKR-CAS
without angular
RKR
SCF
CAS
CAS+ LYP
CAS
CAS+ LYP
H2 LiH FH Li2 Be2 N2 F2 CO
1.1745 8.0705 100.4589 14.9952 29.3384 109.5427 199.5289 113.3272
1.1327 7.9830 100.0631 14.8702 29.1316 108.9826 198.7584 112.7812
1.0re 1.1514 8.0135 100.0874 14.8964 29.2206 109.1319 198.8376 112.9134
1.1896 8.1019 100.4500 15.0297 29.4182 109.6156 199.5129 113.3983
1.1514 7.9997 100.0870 14.8793 29.1316 109.1194 198.8353 112.9040
1.1897 8.0879 100.4495 15.0125 29.3276 109.6030 199.5106 113.3888
H2 LiH FH Li2 Be2 N2 F2 CO
1.5re 1.1299 1.0791 1.1113 1.1468 8.0374 7.9443 7.9799 8.0652 100.3682 99.9491 100.0022 100.3575 14.9743 14.8504 14.8781 15.0071 29.3351 29.1422 29.2297 29.4222 109.3052 108.5881 108.9002 109.3644 199.4768 198.5940 198.8151 199.4758 113.1102 112.4761 112.6962 113.1628
1.1113 7.9724 100.0017 14.8727 29.1422 108.8884 198.8147 112.6802
1.1468 8.0573 100.3570 15.0018 29.3332 109.3527 199.4753 113.1466
LYP
P86
VWN
H2 LiH FH Li2 Be2 N2 F2 CO
0.0418 0.0875 0.3958 0.1250 0.2068 0.5601 0.7705 0.5460
0.0187 0.0167 0.0239 0.0091 0.0000 0.1368 0.0769 0.1228
1.0re 0.0231 0.0708 0.3719 0.1158 0.2068 0.4233 0.6936 0.4232
0.0383 0.0882 0.3625 0.1332 0.1960 0.4837 0.6754 0.4847
0.0471 0.0918 0.3798 0.1334 0.1997 0.5059 0.6986 0.5033
0.0953 0.2171 0.7045 0.3303 0.4587 0.9467 1.3026 0.9501
H2 LiH FH Li2 Be2 N2 F2 CO
0.0508 0.0931 0.4191 0.1239 0.1930 0.7171 0.8828 0.6340
0.0322 0.0281 0.0526 0.0223 0.0000 0.3003 0.2207 0.2041
1.5re 0.0186 0.0650 0.3665 0.1016 0.1930 0.4168 0.6621 0.4300
0.0355 0.0849 0.3553 0.1290 0.1910 0.4643 0.6606 0.4664
0.0452 0.0877 0.3694 0.1290 0.1925 0.4795 0.6738 0.4804
0.0895 0.2112 0.6956 0.3255 0.4508 0.9277 1.2947 0.9316
a The six columns are E corr, NDCE′, DCE′, and the values of the correlation functionals LYP, P86, and VWN calculated with the CAS density.
a The LYP energy is also added to the CASSCF values to give an approximate total energy.
TABLE 3: Correlation Energies (-E, hartree) of the Molecules at re and 1.5rea RKR-SCF CAS-SCF RKR-CAS
LYP
P86
VWN
H2 LiH FH Li2 Be2 N2 F2 CO
0.0418 0.0875 0.3958 0.1250 0.2068 0.5601 0.7705 0.5460
0.0186 0.0305 0.0243 0.0262 0.0891 0.1493 0.0792 0.1322
1.0re 0.0232 0.0570 0.3715 0.0988 0.1178 0.4108 0.6913 0.4138
0.0383 0.0884 0.3625 0.1333 0.1976 0.4837 0.6753 0.4848
0.0471 0.0923 0.3798 0.1337 0.2004 0.5060 0.6962 0.5034
0.0953 0.2178 0.7045 0.3307 0.4567 0.9468 1.3026 0.9502
H2 LiH FH Li2 Be2 N2 F2 CO
0.0508 0.0931 0.4191 0.1239 0.1930 0.7171 0.8828 0.6340
0.0322 0.0355 0.0532 0.0277 0.0876 0.3121 0.2211 0.2200
1.5re 0.0186 0.0575 0.3659 0.0962 0.1054 0.4050 0.6617 0.4140
0.0355 0.0853 0.3553 0.1290 0.1925 0.4643 0.6607 0.4666
0.0452 0.0884 0.3696 0.1293 0.1928 0.4795 0.6738 0.4806
0.0895 0.2122 0.6956 0.3256 0.4528 0.9275 1.2947 0.9320
a The six columns are Ecorr, NDCE, DCE, and the values of the correlation functionals LYP, P86, and VWN calculated with the CAS density.
exchange-correlation energy functionals. In current terminology such local functionals are written
Exc[F] ) ∫Fxc(F,∇F) dr
(24)
and Fxc is written as a sum of separate parts
Fxc ) Fx + Fc
(25)
The usual representation of Fc 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 the inhomogeneous case P86,29 or to the correlation energy of the He-like systems, as with LYP,30 which was constructed from the correlated He wave function of Colle and Salvetti.31 Fx must include the effects of electron exchange and indeed the most popular Fx, due to Becke,32 has been derived from the UEG exchange energy expression, and an additional term which has been parametrized to the exchange energies of the noble
Figure 2. RKR, SCF, and CAS potential energy curves of Be2 and the variation of correlation energies, LYP, P86, and VWN, with internuclear distance.
gas atoms. This Fx indeed gives high-accuracy exchange energies for all atoms. Because such a Fx is a local expression, it must include some molecular left-right correlation effects, as we have argued elsewhere by considering the long-range effects.33 The above arguments suggest that the magnitude of the effect of Fx might reasonably be compared with NDCE′ plus the exchange energy and the magnitude of the effect of Fc will be
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J. Phys. Chem., Vol. 100, No. 15, 1996 6229
Figure 3. Variation of NDCE, DCE, NDCE′, DCE′, and LYP of LiH, Li2, FH, and CO with internuclear distance.
Ebasis error ) EFCI - ERKR
compared with DCE′. We shall therefore compute Ec,
Ec[F] ) ∫Fc(F,∇F) dr
(26)
using appropriate values for the density F. One logical choice is the density of the CASSCF wave function. As the CASSCF wave function already includes the NDCE (or NDCE′) and the Ec probably gives an accurate DCE (or DCE′), CASSCF and DFT together may be a useful method. In this work, the VWN, P86, and LYP correlation functionals have been evaluated with the optimized CASSCF spin averaged densities. 3. Results and Discussion Figure 1 shows the SCF, CASSCF, full CI (FCI), and RKR curves of H2. For this small system, the CASSCF wave function gives a very good description of the system. It has the same dissociation limit as the FCI wave function. The FCI calculations are included to show the basis set error,
(27)
Figure 1 also shows the variation with internuclear distance of NDCE, DCE, and the VWN, P86, and LYP correlation energies evaluated with the optimized CASSCF densities. The graph shows some basic characteristics of correlation energies, which are also shared by most of the molecules under study. The NDCE goes up as SCF fails to dissociate to the ground state atoms. The DCE is decreasing as the molecule dissociates since the hydrogen atom does not have any dynamical correlation energy. It is well-known that VWN overestimates the correlation energy, and this is clearly shown in the graph. None of the 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 list the correlation energies at the corresponding internuclear distances with and without angular correlation in the CASSCF wave functions, respectively. Comparing the values of the
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Mok et al.
TABLE 5: Error in the Total Correlation Energy, When Calculated as CAS+LYP, for the Cases When Angular Correlation Is/Is Not Included in the CASSCF Calculationsa with angular correlation H2 LiH FH Li2 Be2 N2 F2 CO a
without angular correlation
1re
1.2re
1.4re
1.5re
mean
1re
1.2re
1.4re
1.5re
mean
av Ecorr
0.0151 0.0314 -0.0090 0.0345 0.0798 0.0729 -0.0160 0.0711
0.0158 0.0304 -0.0113 0.0332 0.0841 0.0687 -0.0088 0.0647
0.0165 0.0276 -0.0115 0.0327 0.0865 0.0625 -0.0027 0.0567
0.0169 0.0278 -0.0107 0.0328 0.0871 0.0592 -0.0010 0.0526
0.0161 0.0293 -0.0106 0.0333 0.0844 0.0658 0.0071 0.0612
0.0152 0.0174 -0.0094 0.0174 -0.0108 0.0603 -0.0183 0.0616
0.0158 0.0190 -0.0119 0.0214 -0.0046 0.0546 -0.0102 0.0529
0.0165 0.0186 -0.0120 0.0255 -0.0024 0.0491 -0.0033 0.0420
0.0169 0.0199 -0.0112 0.0275 0.0019 0.0475 -0.0015 0.0364
0.0161 0.0187 0.0111 0.0229 0.0049 0.0529 0.0083 0.0482 0.0229
0.0452 0.0900 0.4084 0.1236 0.1979 0.6421 0.8307 0.5939 0.3665
The mean absolute errors are also given.
correlation functionals with RKR-CAS, which is the dynamical correlation energy, it is seen that the correlation functionals usually overestimate the value. Although the correlation functionals do not give a good approximation to the DCE of H2, they do give values much closer to DCE and DCE′ for heavier molecules (such as HF, CO, F2, etc.). Among the functionals used, LYP gives closest agreement and P86 gives values slightly greater than LYP, for all the molecules under study. Thus, in the following, the comparison will be focused on LYP. Figure 2 shows the SCF, CASSCF, and RKR curves of Be2. As has been previously mentioned, Be2 would have zero NDCE if the angular correlation is considered as dynamical. That is, the NDCE′ of Be2 is zero, and DCE′ equals the total correlation energy. Figure 2 shows the variations of NDCE (≡CAS-SCF), DCE (≡RKR-CAS) and DCE′ (≡Ecorr). The NDCE curve does not increase because a single CSF description of Be2 dissociates properly. The LYP gives much better approximation to DCE′ rather than to DCE. Indeed, the LYP curve nearly overlaps the DCE′ curve (which is the total correlation energy in this case). Figure 3 shows the variations of correlation energies of LiH, Li2, FH, and CO. For molecules containing heavier atoms, the difference of DCE and DCE′ is rather small. However, for LiH and Li2, the difference is pronounced, because the angular correlation is relatively more important. The definition of DCE′ and NDCE′ is most noticeable in molecules which contain Li and Be. In general, LYP agrees with DCE′ better than with DCE. Finally in Table 5 we examine whether the CASSCF energy plus the LYP energy (calculated with the CASSCF density) is a good approximation to the total energy. Table 5 gives the values for
ERKR - ECASSCF - ELYP for the molecules studied at re, 1.2re, 1.4re, and 1.5re bond lengths, for the CASSCF calculations with and without angular correlation. This table makes it clear that the latter is a better procedure, as measured by the fact that the mean absolute error is only 0.0229 hartree compared with 0.0385 hartree in the former case. Table 5 does indeed suggest that evaluating the LYP energy on top of such CASSCF energies may be a cheap and fairly reliable procedure for obtaining good total energies. We see that on average the error of the procedure is less than 7% of correlation energy. In conclusion, in this paper we have examined definitions for nondynamical and dynamical correlation energies. Using CASSCF calculations, and experimental (RKR) data, we have been able to compute values for these correlation energies along the entire potential energy curve. Because DFT correlation functionals are designed to calculate the dynamical correlation energy, we have compared values for these with the above determined 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 counted as dynamical. Acknowledgment. Dr. P. J. Knowles is acknowledged for the provision of the MOLPRO package and also for advice. Professor K. Ruedenberg is acknowledged for stimulating discussions. Prof. I. Shavitt is acknowledged for critical comments. D.K.W.M. wishes to thank the Croucher Foundation for financial support. R.N. acknowledges the support of a Human Capital and Mobility Fellowship of the EU. References and Notes (1) Lo¨wdin, P.-O. AdV. Chem. Phys. 1959, 2, 207. (2) Sinanogˇlu, O. AdV. Chem. Phys. 1964, 6, 358. (3) Ruedenberg, K.; Sundberg, K. R. In Quantum Science Calais, 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 State University, 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 of ab 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. In Modem 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. Acta 1975, 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.
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