The shifted scheme in the general-model-space diagrammatic perturbation theory

Chemical Physics 62 (19Sl) 469-479 North-Holland Publishing Company

THE SHIFFED SCHEME IN THE PERTURBATION THEORY

GENERAL-MODEL-SPACE

DIA6RAMMATK

Gabriel HOSE” and Uzi KALDOR Deportmeni of Chemistry, Tel-Avio Uniuersity, Tel Aok, 69978, Israel Received 2 March 1981 Revised manuscript received 5 July 1981

.

The shifted scheme of many-body perturbation theory is applied to open-shell states within the framework of the general-model-space theory. Rules for shifting the denominators of folded diagrams, which appear in open-shell perturbation expansions, are given. The finite-order energies in the shifted scheme obtained in two equivalent representations may differ. This happeas, for instance, in the case of triplet states. For 3L; states of the HeZ, differences up to 0.07 mhartree have been found in third order. A similar phenomenon is the size inconsistency of the shifted scheme observed by Silver in the ground state of He?. A possible advantage of the shifted scheme is its faster convergence for excited states.

1. Introduction

interactions

Two schemes corresponding to alternative

H$ = &++g

((ijiviij)-(ijlr:Iji))ala~ap;

choices of the reference hamiltonian HO are commonly used in many-body perturbation theory. The “model” scheme employs the M@ller-Plesset partitioning of the hamiltonian [l]. Ho and the perturbation V are defined as

where ho is the single-electron hamiltonian used to generate the orbitals, zl is the two-electron repulsion and f is an average eIectronic potential included in ha. The operators ~~ and a: annihilate or create an electron in the ith 0rbitaI. In the second, so-called “shifted” scheme, the reference hamiltonian is shifted with diagonal * Present address: Department of Chemistry, University Of Southern California, Los Angeles, California 90007, USA. 0301-0104/81/0000-0000/502.75

When the summation in the diagonal terms above is carried over al! orbitals one obtains the Epstein-Nesbet partitioning of the hamiltonian [2,3]. H’o is the diagonal part of the hamiItonian in the zero order basis. Partially shifted schemes are obtained when restrictions are imposed on orbital summation. In such cases Hi may have no obvious physical meaning. The shifted scheme is applied easily in the Brueckner-Goldstone expansion [4,5]. As a result it has been widely used and tested for ciosed-shell systems [6-161. It is more difficult to implement the shifted scheme to open shel!s, due to the greater complexity of the diagrams involved. In the few applicalions reported so far of open-shell theories [17-211, the model scheme is almost exclusively used. Recently we developed a diagrammatic openshell theory applicable to general model spaces, degenerate or quasi-degenerate [22]. The hole-

@ 1981 North-Holland

472

G. Hose. U. Kaldor / S/tiffed generahodel-space

theory

diagram and the other part consists off overlapping domains. The types of the folded lines are determined by the requirement that hole folded lines enter the secondary part and particle folded lines enter the other part. Line directions remain the same as in the parent diagram. Note that the secondary part must be a legitimate single-bIock diagram. Diagrams 2b2d are single-fold diagrams derived from the parent

Fig. ?. Folded diagrams (bklf) de;ived from the sin@-block diagram (a). Transirions involving at most la0 valence orbit& are considered. Diagrams !b) to (d! have a single fold. diagams Ce) to !fi have ~U’Ofolds. Dotted lines separa:e different domains.

fold diagrams with two domains each. Both domains represent an interaction between the vacuum state and another model state. The transition in the encircled domain is from the vacuum state to another model state. In the second domain, which fcrms the rest of the diagram, the transition is back to the vacuum state. Diagrams 2e-2f are double-fold diagrams having three domains. The lower encircled domain in diagram 2f and the upper one in 2e are new domain types representicg the transition between two mode1 states other than the vacuum state. All the folded diagrams with f folds may be derived by dividing :he folded diagrams with f1 folds into two overlapping parts so that the secondary part (not containing the topmost interaction line) is a legitimate single-block

diagram

2a. Subsequent

folding

is done

in the same manner while keeping the type and the direction of the previously folded lines. Thus diagrams 2e and 2f are both derived from 2d, whereas diagrams 2b and 2c do not yield further folds. Ordinary (unfolded) hole and particle lines are summed over hole and particle orbitals respectively. External open lines are summed on/y over the valence holes and particles which correspond to the transition fro-m Qi to @f. Model states may not appear as intermediate states in a domain unless overlapped by a nonmodel intermediate in another domain. The denominators of the diagrams are given by D=CE?-xEt,

(17)

where the sums are over down-going and upgoing Lines respectively, and E are the orbital energies_ The overall sign is (-l)&+lcf, where h is the number of hole (including folded hole) lines, I is the number of loops after the open incoming lines are connected to outgoing lines in ascending

order

of orbital

labels,

and f is the

number of folds. 2.3. Shifted diagrams Rules for shifting diagrams were briefly discussed previously [22] and were applied [19] in a partially shifted scheme. The rules for the application of the Epstein-Nesbet partitioning in open shells are described in detail beIow, In the Epstein-Nesbet partitioning, where all the diagonal interactions are included in H& the shifted denominator between two consecutive interaction lines in a single-block diagram is

473

G. Hose, U. Kaldor / Shiffed general-model-space t{leorY

given by

)------?

t

3*

71

i

L___

*

42 ___._-

“1

----,

i

I v

:

‘--qi 3:

f 1 v

a ;,*,

~(hh’!~lhh’)-~hh’(~[h’i2))

I

-zp,

(\‘PP’lt.lPP’)-(PP’lcIP’P)),

where (i(xli) combines the single-electron bubbie interactions of the ith orbital

(18) and

(iJxJi) =3 ((~i)vl~zi>-~~~ilu)i~>) -(ilfli),

(19)

and the summations are carried over hole (h) and particle (p) orbitals which appear in the original unshifted denominator, eq. (17). Note that the inclusion of the diagonal (ijx\i> interactions may be done simply by shifting the orbital energies. The diagonal two-electron interactions appearing in the shifted denominator (18) correspond to all possible connections between the vertical lines (hole and particle) defining the unshifted denominator (17). The description of the folded diagrams in the shifted scheme is more complicated than that of single-block diagrams because of the overlapping domain structure of the former. Since the intermediate states are defined in the individual domains, it is obvious that the diagonal twoelectron interactions which shift the denominator may connect only lines from the same domain. Inter-domain connections produce illegal diagrams when they are expanded from the denominators (fig. 3a). Another difficulty is with the diagonal interactions which connect folded lines. These lines are the “external” domain lines and are consequently in-

volved with intermediate states all over the domain. The shifted denominator in folded diagrams may therefore contain diagonal interactions connecting folded lines which do not pass between the two consecutive interaction

tines where the denominator is defined. Note that by expanding these diagonal interactions from the denominators Iegal diagrams are obtained (tig. 4~).

j

y 4&_---t

‘, \.______

-[

,y-_=“>

‘:--_.r ;

4

3

----A

:--

&

3:

I;--;

!

4,--2

y

i

3V_‘__._‘J b

Fig. 3. Forbidden shifted insertions which produce illegal high-crder diagrams. The insertion leading to diagram ca) connects two ditkrent domains. The inserted interaction in diagram (b) is illegal because it creates new folded and ordinary lines at the same time. It is clear from the discussion above that it is impossible to write a simple expression for the shifted denominator in folded diagrams, equivalent to eq. (18) in the single-block casz. To apply the shifted scheme with folded diagrams one must first determine the domain structure of the diagram, and only then work out the possible diagonal interactions which may shift the denominators in the diagram. This is best done graphically by inserting diagonal interactions between the domain lines and examining the resulting diagrams. All possible intra-domain insertions must be examined for the implementation of thz Epstein-Nesbet partitioning, but only those which generate legitimate non-cancelling higher-order diagrams shift the denominators. The shifting rules for folded diagrams are summarized below: (a) The inserted interactions may connect vertical lines belonging to the same domain; inter-domain insertions are forbidden. The inserted interactions shown in fig. 4 connect

lines From the primary domain. Fig. 3a shows an illegal inter-domain insertion. (b) The allowed insertions between ordinary Iines (hole and particle) connect lines which appear in the unshifted denominator (i.e. Iines which pass between the interaction lines where the denominator is defined). Insertions involving folded

lines may connect

foIded

lines from all

over the domain (fig. 4~). Note that insertions between folded lines may involve lines which do not appear in the unshifted denominator (17) (see Ag. 5~). (c) Each inserted interaction adds two vertical lines to the diagram, which must have the

G. How. iI. Kaidar/

4

Skifred gmeral-model-space

bid

theor!

+

-_

“4

----_-_00 --_*

I

4

I

b

2 ’ --__-_-

e

__-_--I___0 d

c

9

I

43

4 -__h

Fig. 3. Some diagonal fourth-order diagrams obtained by shifting the denominators of rhe third order folded diagram shcwn.

same character (either ordinary or folded) for the insertion to be legal. Fig. 3b shows an illegal insertion as one new line is ordinary and the other is folded. If the type (up for particle, down for hole) of the new lines coincides with their direction the iines will be ordinary, and the new inserted intzaction will be intradomain as in diagram 4a. If the type of one of the new lines disagrees with its direction, then both new lines will be folded 2nd the inserted interaction

will

form

a new

first-order

folded

domain (diagrams 4b and 4~). Note that insertions involving ordinary lines cannot produce folded lines, and only insertions between foided lines can therefore add new folds. (d) All possible legal intra-domain insertions which generate non-cancelling diagrams are

used to shift the denominators. Legal insertions generating diagrams that are cancelled by identical diagrams differing only by the number of folds should be ignored. Examples are shown in fig. 5. The insertions leading to diagrams 5a and 5c are legal, but the resulting diagrams are cancelled by diagrams Sb and 5d respectively. (e) Single-electron and bubble interactions must be inserted to all lines without restrictions. Bubble-inserted diagrams may cancel mutually, as do diagrams 4e and 4f; or they cancel other superfluous diagrams obtained by other twoelectron insertions (e.g. diagram 4h cancels 4d). Note that diagrams 4b, 4i and 4j are of equal absolute magnitude, and their SIX%gives the correct contribution to the fourth-order term.

G.

Hose.

U. ICaldor

/ Shifted

b

~eeneral-model-space

C

d

(f) Insertions which add one new hole line shift the denominator with a plus sign (diagrams

4d and 4f), the others shift with a minus sign. If the insertion adds a new fold to the diagram the sign is reversed. Thus, the insertions leading to diagrams 4c, 4e, 4i and 4j shift the denominate; with a minus sign, whereas the insertion leading to 4b has a plus sign.

3. The shifted scheme under orthogoaaf transformations Degenerate states may be described by different sets of state functions which are connected by orthogonal transformations. When Ho commutes with the symmetry operation responsible for the degeneracy, any one of the degenerate sets (symmetry adapted or not) could be chosen as the zero order states. In the model scheme this choice will not affect the results in all orders of the perturbation. The same is not true in the shifted scheme. In the EpsteinNesbet partitioning [2,3] the reference hamiltonian contains the diagonal interactions of the state functions. Under an orthogonal transformation the diagonal interactions in one degenerate set may transform into a combination of diagonal and off-diagonal interactions in the

475

second set. The shifted reference hamiltonian is therefore transformation dependent, and the finite order results in the shifted scheme depend upon the choice of state function. Note also that the shifted reference hamiltonian commutes with the symmetry operation responsible for the degeneracy only in the symmetry-adapted basis. 3.1.

Fig. 5. Superfluous legal shifting insertions. Diagrams(a) and (c) are obtained from legal insertions to diagram 3a These diagrams are cance!ledby other foldeddiagratts not accounted for by the shifting rules. Diagram (b) cancels (a) and (d) cancels Cc).

theory

Components of triplet electronic stares

Consider the triplet states having two openshell electrons. The spatial orbitals are spin restricted and correspond to pairs of degenerate spin-orbitals with r~, = ~4. All the determinants in which the two valence electrons occupy the same two spatial orbitals have the same energy. An example are the singly excited Xz states of I-Iez corresponding to the spatial configurations laf ~IT,PZW~There are four determinants belonging to each configuration: (1) (Icri, luua, nu,a) with M, = 1; (2) (lr& IS& ~,a) and (lo:, lu”cr, nc#) with MS = 0; (3) Cl,:, lcr& ncr&?) with MS = -1. The interelectron repulsion is spin independent and determinants with different MS value are therefore uncoupled. The reaction matrix for each Z: configuration is thus block-diagonalized into three. The 3Tz states of Me? may be studied in either one of the three MS spaces. All three choices are equivalent in the model scheme but not in the shifted scheme. The MS = rl determinants are symmetry adapted functions, whereas the M, = 0 determinants are not. The latter relate to the singlet and triplet functions through the orthogonal transformation

(20) According to the previous discussion it is apparent. that the shifted reference hamiitonian is different for determinants having different MS values. The exchange interaction between halffilled spatial orbitals ((n~~l~~,ll/rlzlla,,za,) for example) is diagonal for the MS = +l determinants, and is consequently included in the reference hamiltonian and summed to infinite

176

G. Hose, U. Kaldorl Shifted general-model-space tireory

order. For the &.f, = 0 determinants the interaction is off diagonal and therefore excluded from the reference hamiltonian and appearing as a perturbation. The degeneracy between the triplet components is therefore removed at finite orders of the perturbation (higher than the first). It is regained only at infinite order. This phenomenon is studied for the four lowest %; states of He2 at three internuclear separations. The basis set comprises 62 contracted gaussian functions shown in table 1. Contraction was done at each internuclear separation, using coefficients taken from the HartreeFock caIcuiation without contraction. The orbitals were calculated in the Silverstone-Yin potential [29], meaning that the lo, and 1~” orbitals are He? ground-state Hartree-Fock, and the others approximate orbitals in singly excited ZG states. The spectral distribution of the orbitals at 1.95 bohr is shown in table 2. The shifted electronic energies (Es) in table 3 were calculated in the A& = 0 two-determinant model space using the genera!-model-space theory [22]. The Brueckner-Goldstone method was applied to the A& = 1 non-degenerate case. The differences between the two sets of results, shown in table 3, are up to 240 yhartree in second order and 70 phartree in third order. Generally the differences become smaller for higher Rydberg orbitals, as expected. They show no significant dependence upon the internuclear separation. Table 1 Basis set exponents. Contraction coefficienrs ar 1.95 bohr are given in parentheses Type Exponent S

3360 (O.O0037O~+S40(0.001269jc280(0.004716) +93 (0.019312)+31.5~.0.068404)+10.5 (0.267147)+3.5(0.722346), 1.4.0.55.0.22,0.09,0.03,0.01,0.003

P.Z

5.4(0.125534)+1.08(0.964545)-+0.36 (-0.046428). 0.12.0.04,0.012.0.004

P,

30 (0.017655)+10(0.171434)+3.3(0.869321) 1.08.0.36, 0.12,0.04,0.012.0.004

d,

0.048,0.016

Table 2 Orbital energies (in hartree) at 1.95 bohr cz

cu

r;,

“g

-1.12923 -0.14914 -0.06162 -0.05603 -0.03324 -0.02885 0.00638 0.09791 0.2127s 0.73951 1.64218 2.S5506 11.69560

-0.70336 -0.09673 -0.045ss -0.02669 -0.01092 0.02069 0.07914 0.20327 0.35245 0.89876 2.7995s 3.50092 12.00026

-0.12998 -0.05698 -0.03013 -0.01872 0.06084 0.11365 0.62784 2.75558 11.20988

-0.05600 -0.03129 -0.00729 0.04917 0.14567 0.33559 0.98639 2.02564 11.51190

The higher-order corrections were estimated by the geometric approximation [6, ll]. They turn out to be much larger than the differences between the two calculations discussed above (see table 3). This indicates that the shifted scheme may be applicable to excited states. The scheme apparently converges faster than the model scheme, results for which at 1.95 bohr are included in table 3 for comparison. Silver and Wilson [ll] and Bartlett and Shavitt [13] have applied the two partitionings of the hamiltonian to closed-shell systems (Ne, Nz, H,O in their ground states) and found better convergence in the model than in the shifted scheme. Their starting point was the Hartree-Fock hamiltonian for the state under investigation, a choice inconvenient for openshell systems, particularIy excited states. Thus in contradistinction to the closed-she11 systems mentioned above, Hartree-Fock corrections do not vanish in the present case and may de significant, especially the diagonal terms. These terms may account for the faster convergence of the shifted scheme. More work on excited states, both Rydberg and valence-type, is needed before firm conclusions are drawn. 3.2. Size consistency

in thz shifted scheme

The electronic energy of a diatomic molecule is a function of the internuclear separation R.

G. Hose, (1. Kaldor/

Shifted general-mode[-space thmy

477

Table 3 Electronic energies (in hartree) of excited 32: Rydberg states.” of He?. The shifted energies (E’) shown are calculated in a model space comprising the two determinants (lo:, Iuua, ncrJ3) and (lo:, lu,& n~~~,n).Differences (in microhartree) between the Broeckner-Goldstone resuits for the (1~:. lmu,a, nog~) determinants and the two-dimensional model space energies are given in parentheses a 5,’ (2cT&

d 32: (3~~)

f 38: (4uJ

-7.979717 -7.882294 (-114) -77.984885 (~-14) -290

-7.869354 -7.957030 -7.972768 -654

-6.999859 -7.114758

-6.993128 -7.108652

h ‘2: (5crJ

R = 1.30 bohr 2; 5, 6 R=1.95

-7.992004 (-177) -8.089475 -5.091021(-+53) -25

-7.092304 -7.207319

2 E; 6

-7.191063 -7.188232 -7.194325 -113

El G E; 6

-7.849892 -7.947026 -7.952849 -729

(-58) (+7)

bohr

El EZ

R=2.?0

(-163) (-27)

(-238) (-30)

-7.095152 -7.094175 -7.100717 -345

(-90) (-24)

-7.087715 -7.088430 -7.094283 -383

-6.970557 -7.085302 (-120) (-21)

-7.065163 -7.064615 -7.071182 -403

(-5;) (-20)

bohr -6.376204 -6.478598 (-24) -6.483743 (-32) -272

-6.296651 -6.389267 (-176) -6.396389 (-70) -345

-6.292317 -6.388090 (-196) -6.395687 (-34) -358

-6.269155 -6.360063 (-99) -6.367849 (-14) -371

a1 Rydbergorbitalis shown in parentheses. ” HiDher order corrections 0

(in microhartree)

estimated by the geometric approximation

At very large R the molecular electronic energy is expected to be the sum of the separate atomic

energies. An approximate

calculation is said to be size consistent if it has this property [30-321, when the same basis set is used for the atoms and the molecule. Many-body perturbation theory is size consistent because of the linked cluster theorem [4,31]. This is true in the model scheme, which evaluates entire diagrams, but not in the shifted scheme, where just the diagonal part of certain high-order diagrams is included [26,33-351. The deviations from size consistency in the ground state of Hez are about 0.7 mhartree in third order [26]. The shifted energies also show an incorrect R-dependence at large internuclear separations [26,35,36], whereas the model energies obey the Rv6 rule. Homonuclear diatomic molecules have an inversion center and the molecular orbitals are

[6,11].

either symmetric or antisymmetric with respect to inversion. At very large R each symmetric orbital & becomes degenerate with an antisymmetric orbital 4,. Thus, at very iarge R there exists an orthogonal transformation connecting the molecular orbitals to the separate atomic orbitals: fa = 2-Wg+

4”),

fb=2-“‘~~,8-&j,

(21)

where f;l and fb are identical atomic orbitals centered on different atoms. The molecular wavefunctions at very large internuclear separation may be written in two alternative ways, using either the atomic or the molecular

orbitals. The two sets of determinants are connected by an orthogonal transformation. This transformation connects only subsets degenerate in zero order_ For the ground state of Hez, being closed-shell at all R, these subsets contain

478

G. Hose. U. Kaldor!

Shijk-d generaLmodel-space riteq

only one function each, which are therefore identical. The ground-state function in the atomic orbital basis describes two ground-state He atoms. At very large R only intra-atomic interactions do not vanish and the intermediate determinants appearing in the perturbation expansion for the ground state must also describe two He atoms. For the ground state the molecular calculation et infinite R is therefore equivalent to a separate atom calculation. The model hamiltonian HO depends upon the. ground state only and is invariant under the transformation (21). Consequently the model scheme calculation is size consistent. The shifted hamiltonian on the other hand, is transformation dependent and will be different in the two bases. Under the transformation (21) the particle-particle diagonal intra-atomic interactions included in the atomic shifted hamiltonian &I’, become

(22) where the notation is (ijjkl) = (ik(1/r&l).

(23)

The off-diagonal interactions (&$,j&~~~) appear ic the perturbation rather than in I& when the moiecular basis is used. This leads to the size inconsistency in finiie order. Note that the phenomenon is characteristic of homonuclear diatomic molecules. The molecular orbitals of heteronuclear molecules go smooth!y to atomic orbitals as R increases, and the shifted scheme is therefore expected to be size consistent. The incorrect R-dependence of the shifted energies at large R has a different otigin. When the atomic basis is used, the perturbation includes diagona! (fJalfbfb) and off-diagonal (fJi/,f&) inter-atomic Coulomb terms with an R-’ behavior. The summation of all terms gives the correct Rw6 dependence in every order. The shifted scheme takes the diagonal terms to

infinite order, thereby destroying the balance and causing an incorrect behavior at large R. The situation is even more complicated when symmetry orbitals are used for a diatomic homonuclear molecule. In this case, both the diagonal and off-diagonal terms are combinations of diagonal and off-diagonal inter-atomic Coulomb terms. For example, Silver [26] has shown that third-order diagonal and offdiagonal diagrams for Hez in the ground state display separately an R-’ behavior, whereas their sum behaves properly. Obviously an infinite-order partial summation will yield erroneous results.

4. Conclusions The shifted scheme of many-body perturbation theory has been implemented to open-shell states within the framework of the generalmodel-space theory [22]. The application of the Epstein-Nesbet partitioning to open shells is more complicated than in the closed-shell case. There are new types of diagonal interactions which must be included in the shifted denominators of folded diagrams. These diagrams appear in open-shell perturbation expansions [22-23. Rules for shifting the folded diagrams are given. The shifted zero-order hamiltonian which includes the diagonal perturbation terms may change with 2 transformation connecting two equivalent state functions. The finite-order energies in the shifted scheme may therefore depend on the representation chosen. Thus, different components of a triplet state will have different energies. For the ‘2: Rydberg states of Hez, the disagreement in third order is up to 0.07 mhartree, and is smaller than the higher order energy. Another example of the transformation-dependence of the shifted scheme is He2 at infinite internuclear separation, where size consistency is not preserved. The R-dependence of the energy at large R is also given incorrectly. A possible advantage of the shifted scheme for excited states is its faster convergence as

G. Hose. il. Kaldor / Shifted general-model-space aheory

compared to the model scheme. The zero-order functions of excited states are usualIy not the Hartree-Fock functions. The perturbation includes therefore significant Hartree-Fock corrections, which probably account for the faster convergence of the shifted scheme.

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P.S. Epstein, Phys. Rev. 28 (1926) 695. I31 R.K. Nesbet, Proc. Roy. Sot. (London) A230 (1955) 312. Phys. Rev. 100 (1955) 36. [41 K.A. Brueckner, Proc. Roy. Sot. (London) A239 (1957) 151 1. Goldstone, 267. [61 H.P. Kelly, Phys. Rev. 134 (1964) A1450: Phys. Rev. 136 (19641 B896: Advan. Chem. Phvs. 14 (1969) 129. I71 J.H. Mill& and H.P. Kelly, Phys. Rev. A4 i1970) 480. [Sl T. Lee, N.C. Dutta and T.P. Das. Phys. Rev. A4 (1970) 1410. 191 NC. Dutta and CM. Dutta, Phys. Rev. A6 (1972) 959. [lOI U. Kaldor, Phys. Rev. A7 (1973) 427. 1111 S. Wilson and D.M. Silver. Phys. Rev. Al4 (1976) 1949; J. Chem. Phys. 67 (1977) 1689. Cl21 R.J. Bartlett and D.M. Silver, .I. Chem. Phys. 64 (1976) 4578. 1131 R.J. Bartlett and I. Shavitt. Chem. Phys. Letters 50 (1977) 190; 57 (1978) 157 (E). :141 D.M. Silver and S. Wilson, J. Chem. Phys. 67 (1977) 5552.

El

479

11.51 S. Wilson, D.M. Silver and R.J. Bartlett, Mol. Phys. 33 (1977) 1177. [16] S. Wilson, Mol. Phys. 35 (1975) 1. [17] U. Kaldor, Phys. Rev. Letters 31 (1973) 1338; J. Chem. Phys. 63 (1975) 2199. [18] P.S. Stern and U. Kaldor, J. Chem. Phys. 64 (1976) 2002. [19] G. Hose and U. Kaldor, Phys. Script. 21 (19SO) 357. [20] D. Hegarty and M.A. Robb, Mol. Phys. 37 (1979) 1455. [21] S. Garpman, I. Lindgren. J. Lindgren and J. Morrison, Phys. Rev. All (1975) 758. [22] G. Hose and U. Kaldor, J. Phys. B12 (i979) 5827. [23] B.H. Brandow, Rev. Mod. Phys. 39 (1969) 771; Advan. Quantum Chem. 10 (1977) 187. [24] P.G.H. Sandaq Advan. Chem. Phys. 14 (1969) 365. [25] I. Lindgren, J. Phys. B 7 (1974) 2441. [26] D.M. Silver, Phys. Rev. A 21 (1980) 1106. [27] P.Q. Lijwdin, J. Math. Phys. 3 (1962) 963. [28] H. Feshbach. Ann. Phys. (New York) 5 (1958) 357; Ann. Phys. (New York) 19 (1962) 287. [29] H.J. Silverstone and M.L. Yin, J. Chem. Phys. 49 (1968) 2020. [30] J.A. Pople, J.S. Binkley and R. Seeger, Jnt. J. Quanhum Chem. SlQ (1956) 1. [31] R.J. Bartlett and I. Shavitt, Int. J. Quantum Chem. Sll (1977) 165; S12 (1978) 543 (E). [32] E.R. Davidson and D.W. Silver, Chem. Phys. Letters 52 (1977) 403. [33] J.P. Malrieu, J. Chem. Phys. 70 (1979) 4405. [34] R.J. Bartlett and G.D. Purvis, Phys. Script. 21 (1980) 255. [35] N.S. Ostlund and M.F. Bowen, Theoret. Chim. Acta 40 (1976) 175. [36] J.P. Malrieu and F. Spiegelman, Theoret. Chim. Acta 52 (1979) 55.