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

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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. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors 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) J. Chern. Phys. 73(11), 1 Dec. 1980 0021·9606/80/235711·07$01.00 © 1980 American Institute of Physics 5711 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<X=O. (12) Win) = [Ho, U~n)]+ Vx UJt-l> + 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) J.
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