Calculations on model systems using quasi-degenerate variational perturbation theory with an average pair correction

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL. 42,273-285 (1992)

Calculations on Model Systems Using Quasi-Degenerate Variational Perturbation Theory

with an Average Pair Correction

CHRISTOPHER MURRAY, STEPHEN C. RACINE, AND ERNEST R. DAVIDSON

Department of Chemistry, Indiana University, Blwmington, Indiana 47405

Abstract

A correction to quasi-degenerate variational perturbation theory is proposed that is similar to the average coupled pair functional method. The correction is shown to lack solid theoretical justification when applied to importance-selected multireference calculations and to systems with irreducible interpair contributions to the correlation energy. The method is shown to give improved results when applied to small model systems.

I. Introduction

The search for an accurate, size-consistent multireference method has recently been the subject of intense research [l-61. A size-consistent method is one in which the energy scales linearly with the number of electrons [7], and this condi- tion is becoming increasingly important as ab initio quantum chemistry is applied to larger systems. Multireference methods are appropriate for systems where the self-consistent field (SCF) method gives a poor zeroth-order description of the wave function. This is often the case when looking at excited states or when working on transition states of reactions. It is also desirable that the methods be of a similar complexity to the multireference singles and doubles configuration interaction (MRSDCI).

In this paper we shall discuss techniques that fit these requirements and shall assess which methods are most accurate. A correction to quasi-degenerate variational perturbation theory (QDVARPT) [l] is introduced that is similar to the multireference average coupled pair functional (MRACPF) method of Gdanitz and Ahlrichs [3], and the theoretical justification of this approach will be discussed. The method will be tried out on some small molecular systems and its performance compared with full CI calculations. It will also be tested on a novel model system designed to approximate the metallic environment. Our calculations of the correlation energy for this system will be compared with those obtained from density functional theory.

0 1992 John Wiley & Sons, Inc. CCC 0020-7608/92/020273-13$04.00

274 MURRAY, RACINE, A N D DAVIDSON

11. Theory

In several MRSD methods, the following matrix equation is solved:

The wave function, C, is partitioned into two parts: C,, which spans the reference space and is assumed to give a good zeroth-order description of the wave function, and C,, which spans the space of single and double excitations from the reference space. The wave function is intermediate normalized, so CPTC, = 1. H is the Hamiltonian matrix over the configurations and is partitioned in a similar manner. The diagonal elements of H in this formulation have the SCF energy subtracted. E is a unit of matrix of appropriate dimension multiplied by the correlation energy, and Ex is a unit matrix multiplied by E x , which is a parameter that changes from method to method.

If Ex is E, the correlation energy, then the equations reduce to MRSDCI. Other choices of Ex yield approximately size-consistent results. If Ex = Eo, which is the reference space energy defined by CPTH,C,, then the method is the QDVARPT [l] or the multireference linearized coupled cluster (MRLCC) [2], the difference being that in MRLCC the C, vector is kept fixed and is obtained from diagonalization of the reference space Hamiltonian, H,, whereas the QDVARPT method allows the reference space vector and energy to relax through coupling with the configurations in C,. This relaxation reflects the quasi-degenerate nature of the method.

In this paper, we propose a correction to the QDVARPT method that is related to the MRACPF method [3], where Ex in Eq. (1) is given by

(2)

where n is the number of correlated electrons. We will refer to this corrected method as QDVARPT with average pair correction (QDVARPT + APC).

Another way of unifying different MR methods has been discussed by Gdanitz and Ahlrichs [3]. Their approach considers the correlation energy functional

Ex = CPTH,,Cp + (2/n) * ( E - C,TH,,C,),

where qo is fixed and is the zeroth-order reference wave function obtained by di- agonalizing the reference space Hamiltonian. q, spans that part of the reference space that is orthogonal to qo, and qq contains all remaining single and double excitations out of the reference space. Eo is fixed and is the zeroth-order reference space energy. g, and g, are constants that are different for different MR

methods: g, = g, = 1 recovers the MRSDCI energy expression; g, = g, = 0 is MRCEPA(O); and g, = 1, g, = 2/n gives MRACPF (n is the number of correlated electrons).

The expression for the correlation energy is minimized with respect to changes in q, and qq, and this is equivalent to solving the following matrix

CALCULATIONS ON MODEL SYSTEMS 275

equation :

H W - E &a IIa7

&a

[ 2 &a - Eo - gaEc Rq ][:I = 0 . (4) &q - EO - gqEc

A comparison of Eqs. (1) and (4) reveals the similarity between the MRACPF

and the QDVARPT + APC methods. The main difference is the choice of Eo, with the QDVARPT method allowing the reference space energy to increase as the vector, C,, changes through interactions with single and double excitations.

The theoretical justification for the choice Ex in Eq. (2) can be illustrated by a single-reference treatment of a system of N noninteracting helium atoms [3]. Consider a minimal basis set description of He where only two orbitals are used and the full CI wave function will be a mixture of a determinant, A, where the lowest orbital is doubly occupied, and a determinant, B, where the highest orbital is doubly occupied. The solution for the full CI on the N monomer system will be a linear combination of products of determinants in which particular helium atoms are either in the A or B configuration. If the matrix is truncated at the doubles level and account is taken of the symmetry of the system, then the following two-by-two matrix results:

For the CI problem, Ex equals E, , the correlation energy. a is the matrix element between A and B, and A is the difference between the energy of determinant B and determinant A. If instead of solving this SDCI problem for the N atom system one chooses Ex so that E , reproduces the full CI answer, one ends up with

Ex = (l/N) . E , . (6) It is the generalization of this that produces the pair correction term in Eq. (2).

Two problems with this analysis arise: First, Eq. (6) could be said to be derived for noninteracting pairs of electrons.

This is slightly misleading because the partitioning of the correlation energy between interpair and intrapair contributions depends on any unitary transformation among the occupied orbitals. We define the total interpair energy of lowest magnitude that can be achieved by such a transformation as the irreducible interpair energy. Clearly, Eq. (6) should not be applicable to systems with nonzero irreducible interpair energies. In particular, one might expect its performance to be especially bad for systems where the irreducible interpair energy was significant in comparison to the corresponding intrapair energy.

Second, Eq. (6) is derived for a single-reference system, and it is not immedi- ately obvious to what extent it is applicable to arbitrary multireference problems. This question can be examined by investigating a multireference approach to the calculation of the correlation energy of N noninteracting helium atoms. If we build up the reference space from all double excitations and truncate the full CI

to include all doubles out of this space, the following three-by-three matrix re-

276 MURRAY, RACINE, AND DAVIDSON

sults after account is made of symmetry:

- E , V%LX 0 VR(Y A - E , d 2 ( N - l)a][::] = 0 . ( 7 )

0 WCW 2 A - Ex

The choice of Ex that reproduces the full CI answer is

Ex = ( 2 / N ) * E , , (8)

where the correlation energy is measured with respect to the SCF energy rather than to the reference space energy. This different answer shows that Eq. (6) cannot be rigorously applied to multireference systems where the reference space is selected on the grounds of importance. For this dilute helium gas model, the only size-consistent reference space after the SCF is the full CI, and for most large systems, restriction of the MRSDCI to size-consistent reference spaces is impractical.

To solve Eq. (l), it is necessary that the matrix (H, - Ex) be positive definite [S]; otherwise, intruder states may be obtained. Similar conditions apply to solving Eq. (4), and it can be seen that the choice of go = 0 in Eq. (4) makes the MR- CEPA(O) method particularly vulnerable to intruder states from other functions in the reference space.

Another approach to the production of size-consistent multireference energies is through correction of the MRSDCI answer. The multireference generalization of the Davidson correction [9] (MRSDCI + D) is an example of this approach, and the Pople correction [lo] can be generalized in a similar way. The Pople correction to the SDCI energy can be written as

where C is the square of the coefficient of the Hartree-Fock configuration in the SDCI wave function and n is the number of correlated electrons. We have defined the MRSDCI + P energy similarly with C now given by the sum of the squares of the coefficients of the reference space functions in the MRSDCI wave function. It has been pointed out [3] that if the MRSDCI wave function is placed into the MRACPF energy functional, then an expression like the MRSDCI + P energy is pro- duced, so it is to be expected that for small systems the Pople correction will give similar results to the MRACPF and QDVARPT + APC methods. However, for large systems, where size-consistency effects are expected to severely affect the quality of the MRSDCI wave function, the Pople correction cannot be expected to give re- liable answers.

111. Application to Small Molecular Systems

Calculations were performed on some small molecules with small basis sets for which the full CI results were known or could easily be performed. Three geometries along the Czv insertion of beryllium into Hz were studied. Two reference configurations and MCSCF orbitals were used in the calculation. Some


TABLE I . Energy differences in mHartree from the full CI value with various multireference methods for three geometries along the Czu insertion of beryllium into Hz; ten Cartesian basis functions were used with two reference configura-

tions:A = (la,)’ ( 2 ~ ~ ) ’ (3~1)’ and B = (la,)’ (2~1)’ (16’)’.

Method Geometry 1“ Geometry zb Geometry 3‘

MCSCF

MRSDCI MRSDCI + D

MRCISD + P

MRCEPA(O)~

MRLCC

QDVARPT

MRACPF‘

QDVARPT + APCe

MRLCC + APCe

53.31 0.84

-2.32 -0.82 -3.28 -2.62 -3.01 -0.90 -0.91 -0.61

64.35 2.01

-3.18 -0.88 +4.30 - 2.40 -5.03 -0.90 -1.03 +0.90

66.68 3.01

-3.24 -0.44 -5.50 -5.50 -5.50 -0.53 -0.53 -0.48

%eometry 1 r(H-H) = 2.78; r(Be-Hz) = 2.5. bGeometry 2 r(H-H) = 2.55; r(Be-Hz) = 2.75. ‘Geometry 3 r(H-H) = 2.32; r(Be-Hz) = 3.0. dTaken from Ref. 3. eAs in Ref. 3, the number of electron pairs is taken to be 2 since there are no

functions in the basis set to correlate the 1s’ electrons.

information concerning the geometries, basis sets, and reference configurations are given in Table I, and more details can be found in [2]. It can be seen that the pair correction improves the results for both QDVARPT and MRLCC. As expected, MRACPF, MRSDCI + P, and QDVARPT + APC all give similar numbers, and they offer a general improvement over the MRSDCI method. The corrected LCC method gives good answers also, but over the three geometries, the potential energy curve is less parallel to the full CI results than with the other methods, casting doubts over its reliability. The second geometry studied is interesting because the MCSCF method does not give a good description of the wave function. The ratio of the coefficients of the two reference functions using the different methods is -0.57 for MCSCF, -0.85 for full CI, -0.82 for MRSDCI, -0.89 for QDVARPT, and -0.85 for QDVARPT + APC. This indicates that the pair correction gives improved wave functions as well as energies.

Calculations were also performed on the lowest singlet state of CH2 using four different reference spaces, and the results are shown in Table 11. The geometry, basis set, and first three reference spaces were taken from [ll]. The fourth reference space contains the 38 configurations with the largest weight in an SDCI calculation and was designed to be of a similar size to the third space, which comes from a CASSCF. The best numbers again come from MRACPF, QDVARPT + APC, and MRSDCI + P, where a systematic improvement is observed as the reference space improves. For the best reference space, the MRLCC gives considerably worse results than does MRSDCI, and when the APC correction is used, the answers get worse. One way of looking at this is that the pair correction will always decrease the correlation energy obtained from QDVARPT or MRLCC, and MRLCC cannot be


TABLE 11. Energy differences in mHartree from the full CI value using various multireference methods with different reference spaces for the lowest 'A, state

of CHI; the la, orbital was frozen in all calculations.

No. CSFS in reference space

Method 1 2 56 57

CI in ref. space MRSDCI

MRSDCI + D

MRSDCI + P

MRLCC

QDVARPT

M R A C P F ~

QDVARPT + AF'C

MRLCC + APC

140.89 +8.90

+2.38 -0.04

-2.95 -2.95 f1.90 +1.90 +1.90

119.56 +5.09 -0.49 + 1.08 - 1.27 -1.27 +1.19 + 1.00 +1.01

81.71 +1.49 -1.18 -0.52 -0.55 -1.56 -0.40 -0.53 +0.42

38.64 +0.23 -0.32 -0.12 +0.65 -0.38

-0.17 +0.82

"Taken from Ref. 3.

relied upon to overestimate the correlation energy for good reference spaces. It can also be seen that the relative improvement of QDVARPT + APC compared to MRSDCI seems to decrease as large reference spaces are used.

The average pair correction is worrying from a theoretical point of view because it is derived from the standpoint that the interpair correlation is very small relative to the intrapair correlation. It is interesting to see what the correction does for systems where this is not true. In neon, only 40% of the correlation energy comes-from intrapair correlation [12], so it is to be expected that the model of noninteracting helium monomers would give a very poor description of this system. We have calculated the deviations from full CI for neon using several theoretical methods and the results are given in Table 111. A Dunning split valence basis set was used and the 1s orbital was frozen. Single-reference calculations

TABLE 111. Energy differences in mHartree from the full CI value of -128.51755 Hartree using various multireference methods with different reference spaces for the ground state of the neon atom; a Dunning split valence basis set was used

and the Is orbital was frozen in all calculations.


Method 1 7

CI in ref. space +157.22 +100.76 MRSDCI +4.55 +0.98 MRSDCI f D +1.35 -0.55 MRSDCI + P +2.15 -0.25 MRLCC +1.26 -0.17 QDVARF'T +1.26 -0.64 MRLCC + AFC +2.11 +0.24 QDVARF'T + AF'C +2.11 -0.23


were performed, together with calculations using a reference space of the four configurations with the greatest weight in an SDCI calculation. Good answers are again obtained using QDVARPT + APC and MRSDCI + P for the largest reference space. For the single-reference case where LCC and QDVARPT are equivalent methods, the APC correction makes the answer worse because the uncorrected result has not overestimated the correlation energy. This brings into question the reliability of the correction when applied to single-reference systems.

IK Application to a Metallic Model System

So far we have tested the method on small systems with small basis sets and generally found it to give significant improvement over MRSDCI and QDVARPT. The underlying assumption is that good results for these systems will translate into good results for large systems. This assumption is especially suspect in this case because the theory behind the APC is not rigorous.

An alternative is to build other model systems that might be better prototypes for larger systems. We have tried to model a metallic environment by placing eight electrons in a sphere of radius 8.0 au so as to approximately match the valence electron density in sodium metal. The uniform positive background is modeled by a cage consisting of 935 points of positive charge arranged in cubic close packing, each with equal charge. This gives a nuclear-nuclear repulsion of about 4.70041 au, which compares favorably with the limit of 4.8 au as the number of equally spaced charges tends to infinity. Basis functions are placed with similar packing at centers in the sphere. Three basis sets were used. In the smallest, an s function with exponent of 0.0382 was placed at 19 sites. Another basis was generated by augmenting this set with a p function of exponent 0.0370 at each site. The largest basis set had these s andp functions at 43 sites in the sphere. The exponents were obtained by optimizing the SCF energy.

Table IV gives the results for the small basis set with a frozen core approxima- tion, relative to a full CI calculation. Table V gives the results of when all eight electrons are correlated relative to a near full CI answer. The numbers are some- what erratic, but it can be seen that QDVARPT + APC and MRSDCI + P always give considerably better answers than does MRSDCI. However, the pair-corrected QDVARPT does not always give improvement over the uncorrected method.

Tables VI and VII give correlation energies in the medium and large basis sets, respectively. Although the calculated SCF energy does not change much, the calculated correlation energy increases considerably as the basis set increases from (19s) to (19s, 19p). Further increasing the basis set to (43s, 43p) produces a much smaller change, indicating that the correlation energy is nearly converged. The QDVARPT + APC correlation energy of -152 mHartrees with the largest basis set should be within 5 mH of the exact result. Because of the increasing interest in density functional theory, we have also tried to estimate the correlation energy of the system using this method. Table VIII shows the correlation energy calculated using various density functional schemes [13-221 that have been im- plemented [23] in our suite of quantum chemistry programs. The functionals are evaluated using the SCF density from the largest basis-set calculation. The an-


TABLE IV. Energy differences in mHartree from the full c i value of -0.51154 Hartree using various multireference methods with different reference spaces for the metallic model system; the SCF energy is -0.44413 hartree and the basis

set is (19s).


Method 1 30


MRSDCI + D

MRSDCI f P

MRLCC

QDVARPT

MRLCC + APC

QDVARPT f APC

+67.41 +11.47 +3.44 +4.84 -0.36 -0.36 +4.73 +4.73

+18.56 + 1.06 +0.41 +0.44 +0.79 +0.08 +1.12 +0.43

TABLE V. Similar calculations to Table IV except now all eight electrons are correlated; differences are in mHartree from a near (all excitations up to

sextuples) full CI value of -0.53315 Hartree.


Method 1 11 31


MRSDCI + D

MRSDCI + P

MRLCC

QDVARPT

MRLCC + APC

QDVARPT 4- APC

+89.02 +18.17 +5.75 +5.55 - 1.92 - 1.92 +5.37 +5.37

+66.00 +7.07 -3.25 -3.35 -8.87

-10.31 -3.87 -2.97

+39.18 +2.85 -0.75 -0.35 -0.24 -2.08 f1.07 -0.56

TABLE VI. Correlation energies in mHartree using various multireference methods with different reference spaces for the metallic model system; the SCF

energy is -0.44612 Hartree and the basis set is (19s, 19p).


Method 1 27


MRSDCI 4- D

MRSDCI + P

MRLCC

QDVARPT

MRLCC + APC

QDVARPT + APC

0.0 -110.64 - 132.58 - 133.68 - 149.66 - 149.66 -134.48 - 134.48

-38.13 - 127.56 -143.28 -143.28 -152.64 - 152.95 -143.61 -143.94


TABLE VII. Correlation energies in mHartree using various multireference methods with different reference spaces for the metallic model system; the SCF

energy is -0.44834 Hartree and the basis set is (43s, 43p).


Method 1 17

ci in ref. space MRSLXI

MRSDCI + D MRSDCI + P

MRLCC

QDVARPT

MRLCC + APC

QDVARPT f APC

0.0 -118.09 -140.96 -141.96 - 158.95 -158.95 - 142.97 - 142.97

-25.54 -131.02 -150.96 -151.66 - 164.85 - 166.72 - 150.63 -152.31

TABLE VIII. Correlation energies in mHartree for the metallic model system using different density functional methods based on the SCF density; our best

ab initio estimate is included for comparison.

Method Correlation energy

QDVARPT + APC

Colle-Salvetti Carravetta-Clementi/Colle-Salvetti Lie-Clementi Stoll-Pavlidou-Preuss/Gunnarson-Lundqvist Stoll-Pavlidou-Preuss/Vosko-Wilk-Nusair Perdew/Perdew-Zunger Becke Lee-Yang-Parr

- 152.3 -107.6 -90.2 -62.8 -94.6

-118.8 - 198.7 - 113.8 -107.2

swers from the smaller basis sets are almost the same, as are the SCF energies, indicating that we are reasonably close to the Hartree-Fock limit. It appears that the density functional calculations are very poor, since most methods badly underestimate the correlation energy (as judged from our best variational calculation in Table VII). The Perdew/Perdew-Zunger functional [19,20] is the only one that does not underestimate and it appears to give a 20% overestimate.

It is of interest to consider why density functional theory performs so badly on this system where one might expect it to do well. First, our model system may not be a very good model for a metal. We have plotted the density of the orbitals and the total density along a plane that goes through the center of the sphere. The plot for the lowest-energy orbital, which has the shape of an s orbital, is shown in Figure 1. Figure 2 has one of the triply degenerate HOMOS that have the expected shape of hydrogenicp functions. Figure 3 shows the total density that is approximately spherically symmetric. All plots shown are from the SCF density with the largest basis set. The smaller basis sets give similar results. The plots show that the density in the sphere, while not constant, changes smoothly and gradually

282

C"""""""""" ' ' ' ' ' 1 ' ~ ' ' ' ' ~ ' ' i ' ' ' '

t8 a 6 I I I I I % I I I I I I I I I I I I I 8 1 I I I I I 8 8 I 8 8 a 8 ,-I

. .

from one region to another. This system is therefore a reasonable model for the metallic environment.

We have also plotted the total density reconstructed from the natural orbitals of an SDCI calculation, and this plot was very similar, with the only difference being that the two inner contours shifted slightly. Our conclusion from this is that the use of the SCF density, rather than the exact density, should not have changed the density functional answers very much.

The remaining explanation for the poor performance of the density functional methods is that they are not good at estimating the correlation energy in metallic systems. If these functionals could get the true metallic environment right, then they should be able to get the model system right, since the problems associated

' with edge effects and nonconstant density are not so great. The reason is that most of these methods have been parameterized for atoms and molecules where the form of the density is very different. Some of them [16-181 were originally fit to free electron gas results, but then a self-interaction correction was used that in this system changes the answers dramatically. Without a self-interaction correction, the correlation energy for an eight-electron free electron gas system at this density is about -250 mHartrees [24]. It could be that the correction term is still not fully taking into account the subtle effects in moving from an infinite num-

CALCULATIONS ON MODEL SYSTEMS

C " " ' " " ' " ' " " " " " " " " " " " ' " "

283

THE BZU W I T A L XY PLW

Figure 2. The density of one electron in the BzU orbital of the metallic model system. Contours a to i represent densities of 0.0001 to 0.0009 in increments of 0.0001.

No contour existed for a density of 0.0010. Tick marks are 0.5 au apart.

ber of electrons to only eight; however, such a difficulty should arise in dealing with small atoms and molecules where the methods can be relied upon to give a good estimate of the correlation energy. These results bring into question the general applicability of the density functional correlation potentials used here.

IV. Conclusions

The average pair correction proposed here gives improved results over the QDVARPT method, at least for the majority of the small systems measured here. It also appears to give an improved wave function. The Pople correction to MRSDCI gives similar energies for small systems, but it is expected that this method will become more unreliable for large systems. The pair correction does not have any rigorous theoretical backing but should probably be employed on the basis that empirically it gives better results.

Acknowledgment

This work was supported by grant CHF-8503415 from the National Science Foundation. We thank S. Chakravorty for his help with the density functional calculations.

284

L

. . . . . . . . . . . . . . . . . . . . . . 1 1 1 1 1 1 1 1 1 1 ~ 1 1 J ' ~ ~ ~ I

THE TOTRL DENSITY ( X Y PLFwuEl Figure 3. The total density of the metallic model system. Contours a t o e represent densities of 0.0008 to 0.0040 in increments of 0.0008. No contour existed for a density of 0.0048. If the density was completely uniform and strictly confined in a sphere of radius 8.0 au, then the density would be 0.0036; this corresponds to an r,

value of 4.0 au. Tick marks are 0.5 au apart.

Bibliography

[I ] R. J. Cave and E. R. Davidson, J. Chem. Phys. 89, 6798 (1988). [2] W. D. Laidig and R. J. Bartlett, Chem. Phys. Lett. 104, 424 (1984). [3] J. Gdanitz and R. Ahlrichs, Chem. Phys. Lett. 143, 413 (1988). [4] R. J. Cave and E.R. Davidson, J. Chem. Phys. 88,5770 (1988). [5] B. Jeziorski and J. Paldus, J. Chem. Phys. 88, 5673 (1988). [6] B. Jeziorski and H. J. Monkhorst, Phys. Rev. A 24, 1668 (1981). [7] R. J. Bartlett, Annu. Rev. Phys. Chem. 32, 359 (1981). [8] C.W. Murray, S.C. Racine, and E .R . Davidson, submitted. [9] S.R. Langhoff and E.R. Davidson, Int. J. Quantum Chem. 8, 61 (1974); P. Bruna, S.D.

Peyerimhoff, and R. J. Buenker, Chem. Phys. Lett. 72, 278 (1980). [lo] J. A. Pople, R. Seeger, and R. Krishnan, Int. J. Quantum Chem. Symp. 11, 149 (1977). [ l l ] C.W. Bauschlicher Jr. and P. R. Taylor, J. Chem. Phys. 85, 6510 (1986). [12] T. L. Barr and E. R. Davidson, Phys. Rev. A 1, 644 (1970). [13] R. Colle and 0. Salvetti, Theor. Chim. Acta 37, 55 (1975); Ibid., Theor. Chim. Acta 53, 55

[14] V. Carravetta and E. Clementi, J. Chem. Phys. 81, 2646 (1984). [15] G . C. Lie and E. Clementi, J. Chem. Phys. 60, 1275 (1974); Ibid., 60, 1288 (1974). [16] H. Stoll, C. M. E. Pavlidou, and H. Preuss, Theor. Chim. Acta 49, 143 (1978); A. Savin, H.

(1979); hid. , J. Chem. Phys. 79, 1404 (1983).

Stoll, and H. Preuss, Theor. Chim. Acta 70, 407 (1986).


[17] 0. Gunnarson and B.I. Lindqvist, Phys. Rev. B 13, 4274 (1976). [18] S. H. Vosko, L . Wilk, and M. Nusair, Can. J. Phys. 58, 1200 (1980); J. B. Lagowski and S. H.

[19] J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981). [20] J. P. Perdew, Phys. Rev. B 33, 8822 (1986). [21] A. D. Becke, J. Chem. Phys. 88, 1053 (1988). [22] C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B 37, 785 (1988). [23] S. Chakravorty and E. R. Davidson, to be published. [24] D. Ceperley, Phys. Rev. B 18, 3126 (1978).

Vosko, J. Phys. B 21, 203 (1988).

Received May 7, 1991 Accepted for publication August 26, 1991

Documents

Calculations on model systems using quasi-degenerate variational perturbation theory with an average pair correction