Quasidegenerate Perturbation Theories. a Canonical Van Vleck Formalism and Its Relationship to Other Approaches Isaiah Shavitt and Lynn T

Quasidegenerate perturbation theories. A canonical van Vleck formalism and its relationship to other approaches Isaiah Shavitt and Lynn T. Redmon

Citation: J. Chem. Phys. 73, 5711 (1980); doi: 10.1063/1.440050 View online: http://dx.doi.org/10.1063/1.440050 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v73/i11 Published by the American Institute of Physics.

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Downloaded 19 Dec 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions Quasidegenerate perturbation theories. A canonical van Vleck formalism and its relationship to other approaches

Isaiah Shavitt Battelle Columbus Laboratories. Columbus, Ohio 43201 and Department of Chemistry. Ohio State University. Columbus. Ohio 43210

Lynn T. Redmon

Battelle Columbus Laboratories. Columbus. Ohio 43201 (Received 11 June 1980; accepted 22 August 1980)

Three forms of quasidegenerate perturbation theory are discussed and compared in terms of a common general formulation based ~n a similarity transformation which decouples the model space and complementary space components of the Hamiltonian. The discussion is limited to formal, rather than many body (diagrammatic), aspects. Particular attention is focused on a "canonical" form of van Vleck perturbation theory, for which new and highly compact formulas are obtained. Detailed comparisons are made with the Kirtman-Certain-Hirschfe1der form of the van Vleck approach and with the approach based on intermediate normalization which has been used as the basis for most of the diagrammatic formulations of quasidegenerate perturbation theory.

I. INTRODUCTION call "canonical" VVPT. (Klein, as well as Primas, 38,39 treated the case of exactly degenerate, rather than Quasidegenerate perturbation theory (QDPT) has been quasidegenerate, zero-order subspaces.) Combining receiving increasing attention in recent years (for sev some elements from the treatments of Primas38•39 and eral excellent discussions and reviews see Refs. 1-10). of J,5rgensen, 40.41 we shall present a simple derivation It provides the perturbation theory analog of "multiref of the canonical VVPT formalism, and obtain highly erence" configuration interaction (Cn techniques, 11.12 compact expressions for the decoupling operator and which have proved effective in the treatment of potential the resulting effective Hamiltonian. We shall compare surfaces and excited states of molecular systems. this formalism with another version of VVPT discussed "One-dimensional" perturbation theory (based on a sin by Kirtman43•44 and Certain and Hirschfelder, 45,46 and gle-configuration zero-order function) has been quite shall present simple derivations of explicit equations successful in applications to many near-equilibrium connecting canonical VVPT with the more common ground state systems and in some other cases in which QDPT formulation based on intermediate normaliza a Single configuration provides a reasonably adequate tion and a non-Hermitian model Hamiltonian. 2- 10 The starting point (for some examples see Refs. 13-21). different quasidegenerate approaches will be discussed Practically all such applications have used the many in terms of a common general formulation (see also body form of Rayleigh-Schrodinger perturbation theory Klein5 and B~andow4) which clearly shows their relation (RSPT) (for reviews see, for example, Refs. 22 and ships. Only the formal aspects of the QDPT expansions 23). This approach has Significant advantages over CI will be discussed here. We expect to discuss their techniques because of its use of the linked cluster ex many-body realization in terms of diagrammatic ex pansion24• 25 and its "extensivity" property, 2.18.18,26.27 pansions in future contributions. i. e., its correct scaling with the size of the system. 28 While coupled cluster techniques appear capable of The notation to be used and the common framework extending the range of usefulness of single-configura for the treatments of the different formalisms are pre tion based treatments considerably, 27.29 including some sented in Sec. II. Canonical VVPT is derived in Sec. nearly degenerate or even fully degenerate cases, 30.31 III. The Kirtman-Certain-Hirschfelder (KCH) form of it still appears desirable to have practical computa VVPT and the intermediate normalization form of QDPT tional techniques based on a multidimensional "model are discussed in Secs. IV and V, respectively. The re space" as the zero-order approximation. (An alternative sults are discussed in Sec. VI. strategy is to use unrestricted Hartree-Fock zero-order functions, 18.19 but this appears to have some serious dis advantages. 27) In fact, QDPT may be expected to pro II. COMMON FRAMEWORK FOR aUASIDEGENERATE vide faster convergence and more general applicability PERTURBATION FORMALISMS than one-dimensional perturbation expanSions, and A. Notation should become an increaSingly important tool for the calculation of highly correlated electronic wave func The Hamiltonian H is partitioned into a zero-order tions and energies of atoms and molecules. (A coupled part and a perturbation cluster analog of QDPT has also been formulated. 7.32) H=Ho+ V • (1) In the present contribution we are concerned primari The eigenfunctions of Ho will be written in the form It), ly with a particular form of van Vleck perturbation the with eigenvalues (0 as ory4.5·33-42 (VVPT) which, in the spirit of Klein's treat ment5 (see also Jprgensen42 and Brandow'), we shall (2)

Downloaded 19 Dec 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 5712 I. Shavitt and l. T. Redmon: Quasidegenerate perturbation theories and the set of these eigensolutions will be partitioned (14) into two subsets

{t, u, ... } ={a, 13, 0 •• } U {i,j, .•. } (3) where defining the "model space" {a,j3, ••• } and its orthogonal (15) complement. The projection operator into the model space is are the "bonne functions" referred to by Bloch48 and by Jprgensen and Pedersen. 40.41 P=L la)(al , (4) " Rewriting Eq. (11) in the form and its orthogonal complement is UX=HU, (16)

Q=I-P=L li)(il • (5) and splitting it into diagonal and off-diagonal blocks us i ing Eqs. (8)-(10), we find that the condition (12) leads Any operator A can be partitioned47 into a block diag to the following implicit equations: onal part Av and a block off-diagonal part Ax HvUx=-VxUv+UxX, (17) (6) or, using the definition of W in Eq. (12), (7) Av=PAP+QAQ, Ax=PAQ+QAP • [Ho, Ux]=-VxUv-VvUx+UxW, (18) For a product of two operators we have and

(AB)D=AvBv+AxBx, (AB)x=AvBx+AxBv· (8) (19) For the Hamiltonian we note that, since Ho is diagonal, Expanding the operators in orders of the perturbation

Hv=Ho+Vv, (9) ~ ~ ~ U=Lu(n), X=LX(n)=Ho+LW(n) , (20) Hx = Vx • (10) n=O n=O n=1 with B. Quasi degenerate perturbation theory W U(O) =1, X ) =Ho , X(n) =WIn) (n> 0) , (21) The essential feature of the various QDPT formalisms explicit recursive equations are obtained for n> 0: is a similarity transformation which block diagonalizes n-l the Hamiltonian H U(n)]- - V U(n-ll V U(n-ll '"" U(m) W(n-m) [ 0, X - X V - V X + L...J x , X=U-l HU (11) m=l (22) with and n-l X=Xv=Ho+W, J + Vv U~"-l) - L U~m) W("-m) . This is not always the form in which the formalisms are m=l (23) presented (particularly for the intermediately normal l 10 ized form - ), but it provides a common and simple However, the decoupling operator U is not fully deter starting point for straightforward derivation of the equa mined by the condition (12) or, equivalently, by Eqs. tions for all of them. The decoupling operator U is (22) and (23) (note that no equation for U~") has been ob unitary in the van Vleck formalisms and produces a tained). In fact, multiplication of U by any block diag Hermitian effective (or "model") Hamiltonian PXP. It onal operator does not destroy the decoupling of X. is nonunitary in the intermediate normalization approach Different supplementary conditions on U 0. e., specifi (where it is referred to as the "wave operator") and cation of Uv) then lead to the different QDPT formalisms leads to a non-Hermitian X (which can easily be trans discussed in the next three sections. 4 formed to a Hermitian form, if desired ). The operator Different choices of U produce different model Ham W defined in Eq. (12) is referred to as the "level shift" iltonians, and, while they give the same infinite order operator, particularly in the exactly degenerate case. eigenvalues and eigenfunctions [1/1"" Eq. (14)], their Obviously, X has the same eigenvalues as H, so that truncated (finite-order) results are not generally equiv diagonalization of the model Hamiltonian P3CP provides alent. Thus, the choice of subsidiary conditions to com a subset of the eigenvalues of H. The perturbed model plete the specification of U may affect the rate of con functions are vergence. It is well known, of course, that once U v has been U I a) = Lit) (t I ul a) , (13) t specified, Eq. (22) does provide an explicit equation for Ui") through the resolvent operator and the overlap matrix between them is PUtUP (which is a unit matrix in the van Vleck case). The correspond (24) ing eigenfunctions 1/1" of H are obtained by transforming the perturbed model functions (13) with the matrix C of since eigenvectors (right eigenvectors in the non-Hermitian case) of.PJeP: (25)

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III. CANONICAL VAN VLECK PERTURBATION Bernoulli numbers B2n (see, for example, Abramowitz 51 THEORY and Stegun ): The unitarity of the decoupUng operator U in VVPT can 22n (39) be ensured by expressing it in exponential form3s •49 cn = (2n)! B2n (26) (Other notations for the Bernoulli numbers are also in use; see, for example, Jolley. 52) The first few coeffi with G an anti-Hermitian operator cients in the series are G=-G' • (27) The canonical form of VVPT (compare Klein,s J0rgen (40) sen,42 and Brandow4) is obtained by completing the spe Equation (38) can be expanded order by order: cification of U through [Ho,GU)]=-Vx , (28) [Ho, C (2)] = - [V D' CU)] , The decoupling operator and the transformed Hamilto U U [Ho, C(3)] = - [VD' C(2)] - t [[vx , C )], G )] , nian obtained from this condition will be denoted by Ue U and:fCe =Ho+ We, respectively. [Ho, G(4)] = - [V D, G(S)]_ H[[vx , C )], c(2)] In order to obtain a compact formalism we shall use a +[[vX,G(2)],C(1)]} , superoperator notation,38.39 in which with any operator [Ho, c(S)]= - [VD' c(U] - H[[v , C U)], GIS)] A we associate a superoperator A defined by50 x + [[vx, G (2)], c(a)] + [[v , GIS)], c(U]} AX=[X,A]=XA -AX (29) x + 4\-[([[v , C U )], CU)], c(1)], C U )] , (41) (where X is any operator). Positive powers of A pro x duce repeated commutators etc., and converted into explicit equations for the C (n) using the resolvent formalism [Eqs. (24) and (25)]. A 2X=[[X,A], A] , (30) (Note that G(O) = 0.) etc., and the zero power is the identity superoperator The transformed Hamiltonian 3Cc = (Jec)D can also be AOX=X • (31) obtained compactly in terms of the hyperbolic functions of the superoperator As noted by Primas, 36.S9 this allows a compact repre sentation of the Baker-Campbell-Hausdorff expansion Jec = coshG H D + sinhG Hx coshG -1 [ ] ~ =HD + G H D , G + sinhCHx

=H - (coshG -1) cothGH + sinhGH (32) D x x 2 2 =HD - cschG(cosh G - coshG - sinh G) Hx It is convenient to partition this expansion into even and odd functions of G: =HD - cschG(1- coshG) Hx eO = coshG + sinhG . (33) =HD+tanh(~G)Hx , (42) Then, if G satisfies Eq. (28), we find that or We = V + tanh{~ Vx (:fCe)D = (eO H)D = coshG HD + sinhG Hx , (34) D G)

(3Cdx = (eO H)x =coshG Hx + sinhG HD • (35) (43) The decoupling condition (12) can now be written in the where the power series coefficients tn are also related form to the Bernoulli numbers

(36) 2:1oot+2(2:1oot+2 _ 1) tn = (2n + 2)1 B:1oot+2 • (44) (note that the superoperator function G- 1 sinhG involves The first few coefficients in this series are no inverse powers of G in its power series expansion). We thus obtain a commutation relation for G: to=l, t1=-t, t2=ft-, ts=-Hs, t4=~'''' (45) [HD,G]=-CcothGHx , (37) The order-by-order computation of We (and of Jec) easily follows: or, noting Eqs. (9) and (10),

wP)=VD , (Ho, G] = - [VD' G] - G cothG Vx

~ (2) -.!. [V G(1)] We -2 x, , =-[V ,G]-Lc C:1ootV , (38) D n x (S) -.!. [V G(2)] n-O We -2 x, ,

w(4) =.!. U where the power series coefficients cn are related to the e 2 [Vx, G(3)] _.L[[[V24 X, GUl) , GIll] , G ») ,

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W~5) =t [Vx , G(41] -k{[[[vx , G(t)], GUll, G(21] By repeated use of this relation, together with Eq. (49), it is possible to convert Eq. (23) into a form in + [[[V ', G(l)), G(2l), G(1I)+ [[[V , G(ZI), G(ll], GUll} , 2n 1l x x which Wi2n1 and wi + are expressed in terms of uir'), (46) m =1,2, ... , n. The resulting equations can be written etc. We observe that W6"1 depends on G(m!, m =1,2, ... , in the formS. n -1. The operator U , and thus the perturbed model c 2n functions (13), can be obtained from Eq. (26) to the same wi ) =i (U%,-m VU%,) + U%')t VUin-I»D order as G. n [m/2) - ~ f.; (1- 4 00.k - 4 02k.m){w~ml(U%,-m+klt U%,-k»D It is clear that the canonical VVPT formalism is en tirely expressible within the domain of a Lie algebra, + (U 0 fined by can be stated in the form (54)

• This operator is block diagonal (Mx =0) and anti-Her L U(mlt U(·-ml =0 (n>O) . (47) m=O mitian, and tediOUS order-by-order comparison of the equations for UK and U shows that The sum in Eq. (47) can be split into two parts c M(OI =M(l) =M(21 =0, • U(wdt U(n-ml = [~2) (1-1. 0 } U(mlt U(n-ml 2 2m,n M(3) =4 [G(l), G(2)1 , m~ m' M(4) =4 [G(!), G(3)] , (48) M(5) =4 [Go!, G(4)] +4 [G(21, G(31] where [x} denotes the integral part of x. The KCH ortho +i [G(2), (G(!»3]-i G(!I[G(!), G(21]G(1l , (55) normalization condition for the perturbed model func etc. No simple, order-independent equation for M could tions (13) can be obtained by requiring that the diagonal blocks vanish for each of the two sums on the rhs of be found. Eq. (48) separately. Denoting the resulting decoupling A recent diagrammatic treatment44 of the KCH formal operator and transformed Hamiltonian by UK and JCK ism finds that the expressions for W;) can be written in 46 =Ho+ WK , respectively, we get fully linked form through fourth order, but that some unlinked diagrams remain in fifth order. It appears (nl) - [~2) (1 1. 0 ) (U(mlt u0) (UK D - - - 22m.. K K D • (49) likely that this is related to the lack of an order-inde m= pendent equation for UK (or, equivalently, for K or M), Together with the decoupling condition (12), or with Eqs. since it can be shown that any formalism in which JC can (22) and (23), this completely specifies UK' It is pos be expressed in the exponential form (32) satiSfies the 2s sible to write UK in exponential form separability theorem ,s and is thus fully linked, order by order, provided general order-independent equations (50) for the exponential operator exist. with K = - K t , but there appears to be no advantage for this form in the KCH treatment. 43-46 V. INTERMEDIATELY NORMALIZED aDPT Using Eqs. (16) and (20), we can derive the recursion The most widely used form of QDPT is based on in relation54 termediate normalization and leads to a non-Hermitian m+1 model Hamiltonian. 2- 10 We shall denote the decoupling U(mlt VU(n1 = U(m+llt VU(n-1l + L JC(kl U(m-k+llt U(nl operator (wave operator) and the transformed Hamilto k=O nian obtained from the intermediate normalization con n - L U(m+l>t U(n-kIJC(k) • (51) dition by Ur and Xl =Ho+ WI> respectively. This condi k=O tion takes the simple form

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Gt G (UI)D =1 . (56) JCc =e H e 56 l 2 It is convenient to define an operator '" (UJ UI t / U;HUI(Uj UI ,-I/2 l l (57) '" (Uj UI r /2 UJ UI JCI(U; UIt /2

Substituting Eqs. (56) and (57) into Eq. (19) and then =(Uj UI)I/ 2JCI (UJ UI)-I/Z Eq. (18) provides a very simple direct derivation of the =(1_X2)1/2JC (1_X2)-1/2, (68) equations determining X and WI: I

WI ",VD+VXX , (58) JCI ", coshG 3ec sechG . (69) (59) [HD'X]=-Vx+XVxX' An equation for the direct determination of the Hermitian We also find from Eq. (59) that X is anti-Hermitian: model Hamiltonian 3ec in terms of X instead of G is given in the Appendix. X= _xt . (60) Order-by-order equations for X and WI follow directly VI. DISCUSSION from Eqs. (58) and (59): Three forms of quasidegenerate perturbation theory [Ho,X Cl)] '" - Vx , have been examined here in terms of a common general formulation based on a similarity transformation of the [Ho,X CZ )] = - [V ,X(1)] , D Hamiltonian. This approach, together with the sym n-Z Cn In ll Cm Cn metric treatment of the P and Q subs paces represented [Ho,X )] =- [VD,X - ]+ L X ) Vx X -m-ll m=l by the D-X (block-diagonal! off-diagonal) partitioning of operators, has enabled simple and direct derivations of n-Z Cn ll Cm n m the relevant equations and has clearly brought out the = - [VD ,X - ]+ L X ) w: - ) (n> 2) , (61) m=1 relationships between the different QDPT forms. This derivation bypasses the complications of the usual de wj1)",V , D velopment of the intermediate normalization formalism Cn ll win) =Vx x - (n> 1) • (62) which often involves iterative removal of energy depen dence from the denominators and treatment of zero Comparing the recursion relations (61) for X with order energy differences as perturbations (see, for ex those for G [Eq. (41)], we find that ample, Kvasnicka,8 but see also his more direct "al GCl) =XUl, GCZ ) ",X CZ ) • (63) gebraic theory" in Sec. III C). The use of the D-X par titioning also allows us to avoid the extensive use of pro A more tedious comparison shows that jection operators, which tends to obscure the formalism G(3 ) =X(3 ) + t (X(1l)3 , (see, for example, Klein5).

G C4l ",X (4 ) + H(XCll)ZX CZ ) + X CllXC2lXCll +XC2l(XCll)2} . Except in the Kirtman-Certain-Hirschfelder formal ism,43-46 in which order-independent formulation is not (64) possible, the equations for the wave (or decoupling) op In fact, these are special cases of the general relation erator and for the transformed Hamiltonian have been Ship between G and X, proved in the Appendix, obtained initially in order-independent (implicit) form, from which recursive order-by-order equations for all G '" arctanhX '" t _1_ X2n +1 , (65) orders easily follow. The use of a superoperator nota n=O 2n + 1 tion and hyperbOlic functions of the superoperator has or allowed a very compact and simple derivation of the general equations for the canonical VVPT formalism. 2n 1 X = tanhG '" tn G + (66) f.; The infinite-order model Hamiltonians of the different formalisms are related by similarity transformations, [see Eqs. (44) and (45)]. Since arctanhx =1ln[(1 + x)1 but when these model Hamiltonians are truncated at a 57 (1 - x)], we have finite order, the Similarity transformation relationships are no longer exact. Thus, they would generally pro G l+X)I/Z Uc=e ", -- ( I-X duce different results order by order, with possibly dif ferent convergence characteristics. The selection of the method to be used would thus be governed by con vergence behavior as well as by computational consid (67) erations. 4 In this regard the KCH formalism has the advantage of the (2n+ 1) rule,59 while the other forms As seen in Sec. II, uj U is the overlap matrix for the I benefit from having fully linked diagrammatic expan perturbed functions U I t). Thus, as noted by Brandow4 I sions in all orders. The intermediate normalization (see also des Cloizeaux,37 Klein, 5 Kvasnicka, 8 and form (with or without Hermitization) appears to be the LevyIO), the canonical van Vleck functions Uc I a) may be viewed as the result of the symmetric orthonormal most convenient for applications involving infinite order partial summations. 4 ization58 of the Uri a) functions. The connection between the corresponding model Hamiltonians is then Equation (A5) of the Appendix provides a hybrid ap-

J. Chern. Phys., Vol. 73, No. 11, 1 December 1980

Downloaded 19 Dec 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 5716 I. Shavitt and l. T. Redmon: Quasidegenerate perturbation theories proach for QDPT calculations, using the intermediate where normalization formalism for the determination of the 2n wave operator U/ =1 +X, but obtaining the Hermitian qn = 2- (2:) (A6) model Hamiltonian Xc =Ho+ We. This type of approach has been advocated by Brandow. 4 is the coefficient of x" in the binomial power series for (1 - xrl/2. The first two terms in these expressions AC KNOWLEDGMENTS represent the Hermitian average The authors are grateful to Dr. R. J. Bartlett, Dr. B. (A7) H. Brandow and Dr. B. Kirtman for useful discussions. This research was supported by the Office of Naval Re The additional terms contribute to We beginning in fourth search under Contract N00014-79-C-0821 and by the order. The third order form of this model Hamiltonian National Science Foundation under Grant CHE-7825191. has been used by Freed and co-workers6o in their work APPENDIX on the effective valence shell Hamiltonian. In this appendix we derive Eq. (66), connecting the canonical van Vleck and the intermediately normalized forms of the wave operator. We also present an equa . Ip. O. Lawdin, J. Math. Phys. 3, 969 (1962); P. O. Lawdin tion expressing Xc -HD (or, equivalently, We - VD) in and O. Goscinski, Int. J. Quantum Chem. Symp. 5, 685 terms of V x and X directly. (1971). 2B. H. Brandow, Rev. Mod. Phys. 39, 771 (1967). From Eqs. (11) and (26) we have 3B. H. Brandow, Adv. Quantum Chem. 10, 187 (1977). 4B. H. Brandow, Int. J. Quantum Chem. 15, 207 (1979). H eG:ICc e-G = 5D. J. Klein, J. Chem. Phys. 61, 786 (1974). =(coshG + sinhG) :ICc (coshG - sinhG) (AI) 61. Lindgren, J. Phys. B 7, 2441 (1974). 71. Lindgren and J. Morrison, Atomic Many-Body Theory Taking the block-diagonal and block off-diagonal parts (Springer, Berlin, in press). of this equation separately, we find 8V . Kvasnicka, Adv. Chem. Phys. 36, 345 (1977). 9G. Hose and U. Kaldor, J. Phys. B 12, 3827 (1979); Phys. HD =coshG:ICc coshG- sinhG:ICc sinhG , (A2) Scr. 21, 357 (1980). lOB. Levy, in "Proceedings of the Fourth Seminar on Compu Vx=-coshG:lCcsinhG+sinhG:lCccoshG. (A3) tational Methods in Quantum Chemistry," Orenas, Sweden, Therefore, September 1978 (Report MPI-PAE/Astro In, Max-Planck Institut fUr Physik und Astrophysik, Munich, 1978), p. 149. [H D , tanhG) =(coshG:ICc coshG - sinhGJCe sinhG) tanhG III. Shavitt, in Methods of Electronic Structure Theory (Modern - tanhG(coshGJC coshG - sinhG:ICc sinhG) Theoretical Chemistry), edited by H. F. Schaefer (Plenum, e New York, 1977), Vol. 3, p. 189. =coshG Xc s inhG - s inhG Xc coshG 12R. J. Buenker, S. D. Peyerimhoff, and W. Butscher, Mol. Phys. 35, 771 (1978). - tanhG(coshGXc sinhG - sinhG:ICc coshG) tanhG 13R. J. Bartlett and D. M. Silver, J. Chem. Phys. 62, 3258 (1975); 64, 1260 (1976). = - Vx + tanhG Vx tanhG . (A4) 14L. T. Redmon, G. D. Purvis, and R. J. Bartlett, J. Am. Thus, tanhG satisfies the same commutation equation as Chem. Soc. 101, 2856 (1979); J. Chern. Phys. 72, 986 (1980) . X [Eq. (59)). Since also (tanhG) D =0 [this follows from Eq. (28) and the fact that tanhG is an odd function of G), 15G. D. PurvisandR. J. Bartlett, J. Chem. Phys. 68, 2114 (1978). tanhG must be identical to the X operator. 16R. J. Bartlett, I. Shavitt, and G. D. Purvis, J. Chem. Phys. 71, 281 (1979). The equation referred to above for We - V D is obtained by a lengthy and tedious sequence of algebraic manipula 17G. F. Adams, G. D. Bent, G. D. PurViS, and R. J. Bartlett, J. Chem. Phys. 71, 3697 (1979). tions, beginning with Eq. (68) and using Eqs. (58) and 18J. A. Pople, J. S. Binkley, and R. Seeger, Int. J. Quantum (59) and the properties of the binomial expansions of Chem. Symp. 10, 1 (1976). (1 - X)~1/2. The result can be expressed in any of the 19R. Krishnan and J. A. Pople, Int. J. Quantum Chem. 14, 91 forms (1978). 1 20S. Wilson and D. M. Silver, J. Chem. Phys. 66, 5400 we=vD+i[vx,x) +-2 tt ( 2)(n 2 1) qmqn+m (1977); 67, 1689 (1977). m.O n=1 n+ m n+ m+ 21V. Kella, M. Urban, I. Hubac, and P. C~rsky, Chem. Phys. 2m 2n 1 Lett. 58, 83 (1978). X {X2m[Vx ,X2n+l )X +X2m+l [Vx , X - ] X2m+l} 22 H. P. Kelly, Adv. Chem. Phys. 14, 129 (1969). =vD+Hvx,x) 23J • Paldus and J. Cfzek, Adv. Quantum Chem. 9, 105 (1975). 24K. A. Bruckner, Phys. Rev. 97, 1353; 100, 36 (1955). X2m 25J . Goldstone, Proc. R. Soc. (London) Ser. A 239, 267 +1. (n+ 2m)(n+2m + 1) qmqn+m 2 ttm.O n=1 (1957). 26H. Primas, in Modern Quantum Chemistry, edited by O. Sina x [VxX +XVx,x2n]x2m noglu (Academic, New York, 1965), Vol. 2, p. 45. 27R. J. Bartlett and G. D. Purvis, Phys. Scr. 21, 255 (1980). =VD+i[VX,X] 28 For a recent review see J. Cflek and J. Paldus, Phys. Scr. 1~~ n-m 2m ) 2n 21, 251 (1980). +-/-i/-i ( )( 1) qm X (VxX+XVx qnX , 2 n=O moO n + m n + m + 29R. J. Bartlett and G. D. Purvis, Int. J. Quantum Chem. 14, !n+m>0) (A5) 561 (1978).

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30B. G. Adams, K. Jankowski, and J. Paldus, Chem. Phys. 49The operator G defined here is the negative of the G used by Lett. 67, 144 (1979). Primas38,39 and by Klein. 5 The present notation is considered 31K. Jankowski and J. Paldus, Int. J. Quantum Chem. (in more convenient, since it gives the corresponding wave opera G press). tor as U=e • 32 1• Lindgren, Int. J. Quantum Chem. Symp. 12, 33 (1978). 50This is the negative of the superoperator defined by Primas. 38 33J. H. van Vleck, Phys. Rev. 33, 467 (1929). However, we shall not use his inverse superoperator (l/k) (A), 340. M. Jordahl, Phys. Rev. 45, 87 (1934). but employ the resolvent notation of Eqs. (24) and (25) instead. 35E. C. Kemble, The Fundamental Principles oj Quantum 51M. Abramowitz and 1. A. Stegun, Handbook oj Mathematical Mechanics (McGraw-Hill, New York, 1937). Functions (Dover, New York, 1965). 36 L. L. Foldy and S. A. Wouthuysen, Phys. Rev. 78, 29 (1950). 52 L. B. W. Jolley, Summation oj Series (Dover, New York, 37J. des Cloizeaux, Nucl. Phys. 20, 321 (1960). 1961), second edition. 38 H• Primas, Helv. Phys. Acta 34, 331 (1961). 53D. Herschbach (unpublished notes). 39H. Primas, Rev. Mod. Phys. 35, 710 (1963). 54This relation is equivalent to the equation given in the foot 4°F. Jorgensen and T. Pedersen, Mol. Phys. 27, 33, 959 note on p. 3 of Hirschfelder. 46 (1974); 28, 599 (1974). 55Equations (52) and (53) are a more compact representation 41 F • Jorgensen, Mol. Phys. 29, 1137 (1975). of Eqs. (18) and (19) of Certain and Hirschfelder45 [but note 42 F • Jorgensen, J. Chem. Phys. 68, 3952 (1978). that the last two sums in their Eq. (19) have the wrong signs). 43B. Kirtman, J. Chem. Phys. 49, 3890. (1968). 56The operator X has been called the "correlation operator" HB. Kirtman, J. Chem. Phys. (to be published). (denoted by X) by Lindgren and Morrison. 7 (It is denoted by 45p. R. Certain and J. O. Hirschfelder, J. Chem. Phys. 52, U in Levy's treatment. 10) 5977 (1970); 53, 2992 (1970). 57The operator X2 is the negative of Brandow's4 e operator. 46J. O. Hirschfelder, Chem. Phys. Lett. 54, 1 (1978). Also, because of Eq. (66), 1-X2 = sech2 G. The radicals in 47This is the same as the "even" and "odd" partitioning of Eq. (67) are to be understood in terms of the binominal ex Jorgensen and Pedersen. 40,41 In the notation of Primas, 38,39 pansions of (1 +X)I/2, etc. AD= (A) and Ax= (l/k)([K,A]), where K=Ho, but his decom 58p. O. Lowdin, Adv. Quantum Chem. 5, 185 (1970). position of the operators is in terms of multiple exactly 59See also the recent discussion of the (2n + 1) rule in one degenerate subspaces, instead of the two nondegenerate P dimensional MBPT by V. Kvasnicka, V. Laurinc, and S. Bis and Q subspaces used here. Although in some cases we kupic, Mol. Phys. 39, 143 (1980). For the applicability of shall only be interested in the PAP part of an AD and the this rule for the diagonal elements of the model Hamiltonian QAP part of an Ax (e.g., QUP of Ux ), the symmetric treat in the multidimensional case see J. O. Hirschfelder, Int. J. ment of the P and Q subspaces implied by the D and X parti Quantum Chem. 3, 731 (1969). tioning leads to simpler and more compact derivations. 60 M. G. Sheppard, K. F. Freed, M. F. Herman, and D. L. 48C. Bloch, Nucl. Phys. 6, 329 (1958). Yeager, Chem. Phys. Lett. 61, 577 (1979).

J. Chern. Phys., Vol. 73, No. 11, 1 December 1980

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