The concept of an effective Hamiltonian in Fock space is introduced. It is based on the division of the entire one-particle space into subspaces of ‘‘active’’ and ‘‘inactive’’ orbitals. The effective Fock space Hamiltonian has—for active model states—the same eigenvalues as the full Hamiltonian. The theory outlined in this context differs from that of paper I mainly in a different definition of the ‘‘diagonal part’’ of an operator, and in the fact that the ‘‘quasidegenerate case’’ applies throughout. The separation theorem, and as a consequence the connected diagram theorem, is shown to hold, in a more limited sense though, even for those normalizations where it did not in the context of universal wave and energy operators. Unlike in the theory of the ‘‘universal’’ operators of paper I the Fock space and n-particle Hilbert space approaches with analogous normalizations are no longer equivalent. In particular, the Primas normalization with a fully Lie-algebraic structure does not lead to a connected diagram expansion if it is formulated in n-particle Hilbert space, only so in a Fock space formulation. In n-particle Hilbert space with the present definition of the diagonal part of an operator the normalizations b (‘‘canonical’’) and c (‘‘Primas’’) happen to agree. As an alternative to the construction of the wave and energy operator W and L by perturbation theory the nonperturbative approach is presented as a generalization of the coupled-cluster method, in detail both in the intermediate and in the unitary normalization. In the unitary variant only a linear system for σ (the logarithm of the wave operator) has to be solved in order to get L correct through fifth order in perturbation theory with important contributions of higher orders included. A generalization of the Hartree–Fock method to Fock space theory is outlined, which guarantees stationarity of all (active) eigenstates with respect to one-particle transformations. A generalized electron pair theory is also defined. An analysis of the necessary computational steps shows that the nonperturbative approaches do not require significantly more computational effort than perturbation theory to the corresponding order. As a numerical example the H2 molecule in a small basis is discussed.
The concept of an effective Hamiltonian in Fock space is introduced. It is based on the division of the entire one-particle space into subspaces of ‘‘active’’ and ‘‘inactive’’ orbitals. The effective Fock space Hamiltonian has—for active model states—the same eigenvalues as the full Hamiltonian. The theory outlined in this context differs from that of paper I mainly in a different definition of the ‘‘diagonal part’’ of an operator, and in the fact that the ‘‘quasidegenerate case’’ applies throughout. The separation theorem, and as a consequence the connected diagram theorem, is shown to hold, in a more limited sense though, even for those normalizations where it did not in the context of universal wave and energy operators. Unlike in the theory of the ‘‘universal’’ operators of paper I the Fock space and n-particle Hilbert space approaches with analogous normalizations are no longer equivalent. In particular, the Primas normalization with a fully Lie-algebraic structure does not lead to a connected diagram expansion if it is formulated in n-particle Hilbert space, only so in a Fock space formulation. In n-particle Hilbert space with the present definition of the diagonal part of an operator the normalizations b (‘‘canonical’’) and c (‘‘Primas’’) happen to agree. As an alternative to the construction of the wave and energy operator W and L by perturbation theory the nonperturbative approach is presented as a generalization of the coupled-cluster method, in detail both in the intermediate and in the unitary normalization. In the unitary variant only a linear system for σ (the logarithm of the wave operator) has to be solved in order to get L correct through fifth order in perturbation theory with important contributions of higher orders included. A generalization of the Hartree–Fock method to Fock space theory is outlined, which guarantees stationarity of all (active) eigenstates with respect to one-particle transformations. A generalized electron pair theory is also defined. An analysis of the necessary computational steps shows that the nonperturbative approaches do not require significantly more computational effort than perturbation theory to the corresponding order. As a numerical example the H2 molecule in a small basis is discussed.
The Fock space Hamiltonian H has a simpler structure than its projection Hn to n-particle Hilbert space. It is therefore recommended to diagonalize H—to the extent that this is possible—before one specifies n. Diagonalization of H is possible if one defines the diagonal and nondiagonal parts of an operator appropriately. It is shown that the diagonalized Fock space Hamiltonian L, called an energy operator, contains all information about the eigenvalues of H in a simply coded form. For a spinfree Hamiltonian, spin can be completely eliminated and all interesting quantities are expressible in terms of spinfree excitation operators (generators of the unitary group). The construction of the wave operator W and the energy operator L is formulated in terms of perturbation theory both in the strictly degenerate and the quasidegenerate version. Three variants are discussed that differ in the normalization of W, namely, a intermediate normalization, b and c unitary normalization with two additional conditions, b: WD=WD+, c:σD=(ln W)D=0. The transformation of operator products to normal product form is achieved by means of a generalization of Wick’s theorem to spinfree quantum chemistry. The contributions to the various orders of L, W, and σ are illustrated by means of diagrams. Only variant c guarantees a connected-diagram expansion of the Fock space energy operator L. None of the three variants satisfies a generalization of the Wigner (2n+1) rule, but with a slight change in the normalization from b to b′ or c to c′ (that does, in the case of c′, not affect the connectedness of the diagram expansion) L(2n) and L(2n+1) are expressible in terms of the W(k), k⩽n. A generalization of the variation principle for the Fock space energy operator is given and its consequences are discussed. Generalized Fock space Brillouin conditions are derived and a variational scheme for the construction of the energy operator is proposed. We finally compare the present Fock space approach with the traditional degenerate perturbation theory in n-particle Hilbert space. If the diagonal part ΩD of an operator Ω is defined in the spirit of the theory of the universal wave and energy operators, then the two approaches lead to the same results (for the same normalization), but the Fock space approach looks simpler and more transparent.
A particle-hole formalism is defined which is consistent with the tensor notation (and the index pairing) of the generators of the unitary group and their many-particle generalization. The corresponding generalized Wick theorem is formulated. The classification of basis operators as ‘‘closed,’’ ‘‘open,’’ ‘‘closed from above,’’ and ‘‘closed from below’’ given previously is extended to the present context. Both the perturbative and nonperturbative construction of effective Fock space Hamiltonians is discussed in the particle-hole formalism, especially for the case that the ‘‘physical vacuum’’ satisfies a Hartree–Fock condition. For the calculations of the correlation energy of the physical vacuum Cizek’s CP-MET is compared formally with the unitary variant, the UNICEP (unitary coupled electron pair) approximation. A simple derivation of Goldstone’s ‘‘linked-cluster expansion’’ is also given.
The problem of finding the spectrum of the Fock-space Hamiltonian \ifmmode \hat{H}\else \^{H}\fi{} for a system of many fermions is analyzed. The quasiparticle formalism is employed, with ``holes'' and ``particles'' defined with respect to some N-particle determinantal wave function (the model vacuum). The basic idea behind the proposed approach is to perform a similarity transformation of Hamiltonian \ifmmode \hat{H}\else \^{H}\fi{}, such that the resulting effective Hamiltonian \ifmmode \hat{G}\else \^{G}\fi{} is, unlike \ifmmode \hat{H}\else \^{H}\fi{}, a quasiparticle-number--conserving operator. It is shown that eigenvalues of \ifmmode \hat{G}\else \^{G}\fi{}, corresponding to small numbers of quasiparticles (0,1,2) can be easily calculated. This is equivalent to finding eigenvalues of \ifmmode \hat{H}\else \^{H}\fi{} for certain states of N, N\ifmmode\pm\else\textpm\fi{}1, and N\ifmmode\pm\else\textpm\fi{}2 particles. The construction of the operator transforming \ifmmode \hat{H}\else \^{H}\fi{} into \ifmmode \hat{G}\else \^{G}\fi{} (the wave operator) stems from an analysis of the structure of the algebra of linear operators acting in a (finite-dimensional) Fock space. The exponential Ansatz for the wave operator is used, resulting in a generalization of the coupled-cluster (CC) method of Coester [Nucl. Phys. 7, 421 (1958)]. The generalized CC equations determining the wave operator, and equations determining the effective Hamiltonian \ifmmode \hat{G}\else \^{G}\fi{}, are presented in a diagrammatic form. An effort has been made to obtain a concise notation for expressing these equations in an algebraic form. Approximation schemes, necessary for practical applications of the proposed method, are also studied.
We generalize here the formalism of the preceeding paper to encompass the case of the general incomplete model space. The classification of operators as diagonal or nondiagonal depends in this case upon the specific m-valence model space. It is stressed that even then one has to work in Fock space in order to get connected-diagram expansions, since connectedness is a Fock space property. Two choices of separable normalization of the wave operator W leading to a connected Heff are discussed. It is shown that the intermediate normalization is not separable in general and hence not compatible with a connected-diagram expansion. We also discuss how to generate ‘‘subduced’’ incomplete model spaces of lower particle rank such that Heff remains a valid effective Hamiltonian for these subduced model spaces as well. We discuss the nature of the various disconnected diagrams encountered in many-body formalisms and point out which of these are really worth worrying about. We finally comment on the question of the separability of the wave function into proper fragments.
A detailed and systematic presentation of algebraic and diagrammatic techniques introduced in paper I [L. Z. Stolarczyk and H. J. Monkhorst, Phys. Rev. A 32, 725 (1985)] is given. Performing a similarity transformation of an operator in Fock space, e.g., a Hamiltonian in the generalized coupled-cluster method (see paper I), leads to lengthy algebraic formulas for the linear parameters (amplitudes) of the transformed operator. These formulas can be expressed in a compact form by means of the so-called reduced diagrams. Precise rules are given for constructing such diagrams (related to the so-called directed graphs of the graph theory) and the algebraic expressions corresponding to them. In particular, rules for generating a connected-diagram expansion of the amplitudes of the transformed operator are formulated. Several examples illustrating the use of this diagrammatic-algebraic approach are considered.
Abstract In a previous paper the theory of perturbations was discussed for system which possess only discrete spectra. The theory is now extended to systems with both discrete and continuous spectra, and in addition a slightly more general method of perturbation is considered. The Hamiltonian H of the perturbed systems is split up into three parts H0, H1, H2 (t), two of which can be chosen arbitrarily. The solutions of the perturbed problem can then be worked out in terms of the characteristic functions corresponding to the Hamiltonian H0 + H2 (T), where T is any arbitrary fixed time. In general, however, the series of perturbations does not converge, and different modes of splitting up the Hamiltonian will lead to different formal results. The perturbation theory does usually give the correct physical results, and it is therefore necessary to give some explanation of how this happens. In (I) the asymptotic nature of certain solutions was emphasised, but this does not extend to the more general systems treated here. It is suggested that the usual boundary conditions do not give an adequate description of any but the simplest atomic problems, and that more detailed restrictions, determined by the experimental conditions, ought to be substituted. As this is impossible in practice, an alternative method is to regard the motion as a perturbation on that Hamiltonian H0 + H2 (T) which most nearly corresponds to the experimental conditions. (For example, we may regard an electron as free or bound according as the experiment determines the number of free or bound electrons respectively, whereas in the usual theory the initial Hamiltonian is taken as fundamental.) This criterion, together with the initial conditions, suffices to fix the division of H into the three parts H0, H1, H2 (t) which at first were chosen arbitrarily. The formal solution can be worked out on this basis, and it is then assumed that this solution is asymptotic to the one which would be obtained if the full boundary conditions were used. This suggestion is in accordance with the fact that a hydrogen atom in an electric field gives rise not only to the usual Stark effect, but also to an ionisation effect. In § 2 the perturbation equation are derived in a slightly more general form than has previously been given. The theories of Born,* Dirac and Oppenheimer are particular cases of that given here, and are obtained by particular choices of H0, H1, H2 (t). To obtain Born’s theory we take H1 = 0, and for Oppenheimer’s we take H2 (t) = H2 (T) and T = 0. In Dirac’s theory we have H1 = 0, H2 (0) = 0 and T = 0. Each of these theories is appropriate for the solution of special problems, but the more general method of this paper is required for the full discussion of the perturbation theory.