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.
Scope The book focuses primarily on many-body (or better, many-electron) methods for electron correlation. These include Rayleigh–Schrödinger perturbation theory (RSPT) , particularly in its diagrammatic representation (referred to as many-body perturbation theory , or MBPT ), and coupled-cluster (CC) theory; their relationship to configuration interaction (CI) is included. Further extensions address properties other than the energy, and also excited states and multireference CC and MBPT methods. The many-body algebraic and diagrammatic methods used in electronic structure theory have their origin in quantum field theory and in the study of nuclear matter and nuclear structure. The second-quantization formalism was first introduced in a treatment of quantized fields by Dirac (1927) and was extended to fermion systems by Jordan and Klein (1927) and by Jordan and Wigner (1928). This formalism is particularly useful in field theory, in scattering problems and in the study of infinite systems because it easily handles problems involving infinite, indefinite or variable numbers of particles. The diagrammatic approach was introduced into field theory by Feynman (1949a,b) and applied to many-body systems by Hugenholtz (1957) and by Goldstone (1957). Many-body perturbation theory and its linked-diagram formalism were first introduced by Brueckner and Levinson (1955) and by Brueckner (1955), and were formalized by Goldstone (1957). Other important contributions to the methodology, first in field theory and then in the theory of nuclear structure, are due to Dyson (1949a,b), Wick (1950), Hubbard (1957, 1958a,b) and Frantz and Mills (1960). Applications to the electronic structure of atoms and molecules began with the work of Kelly (1963, 1964a,b, 1968), and molecular applications using finite analytical basis sets appeared in the work of Bartlett and Silver (1974a, b).
Connections between the Brueckner and Goldstone forms of the linked-cluster theorem are clarified in an elementary fashion employing a finite many-body system of electrons interacting via two-body forces as an illustrative example. It is shown that the order-by-order cancellations of Brueckner unlinked terms in the many-body wave function and energy occur in the Goldstone development upon the additions of the different time orderings of Feynman-Goldstone diagrams having disconnected vacuum parts. The factoring of the vacuum amplitude from the wave function, upon which customary presentations of the theorem focus attention, is seen to be irrelevant to the cancellations of Brueckner unlinked terms, and has to do, rather, with the presence of secular and normalization terms in the time-dependent perturbation functions for an adiabatically switched static perturbation. Similarly, the equivalence of the vacuum amplitude with the exponential of all connected vacuum diagrams, originally demonstrated by Feynman in the case of hole theory, is seen to be irrelevant to the cancellations of Brueckner unlinked terms in the energy, which occur in the Goldstone development upon the summations over all time orderings of disconnected vacuum diagrams. The distinction between Brueckner unlinked terms on the one hand, and secular and normalization terms on the other, is confused in customary presentations of the linked-cluster theorem by the use of exponential switching, following Gell-Mann and Low, and is clarified in the present development by considering the Friedrichs limit of ideal adiabatic switching.
Abstract This paper shows that the spin‐shift formalism developed in B. T. Pickup and A. Mukhopadhyay [Int. J. Quantum Chem. 26, 101 (1984)] supports a one‐component diagrammatics which has a systematics akin to that in the spin‐orbital many‐body theory. The diagrams are neither Goldstone nor Yutsis type, and characterize the chain U(2R) ⊃ U(R)⊗SU(2) on which the spin‐shift formalism is based. Accordingly, while the lines in such diagrams are labeled by the orbital indices, the diagram structure adequately reflects the irreducible representation of the group U(R). In this sense the paper presents a unitary group approach to the natural generalization of the usual many‐body theory for the spin‐adapted cases. A set of very simple rules is derived; their similarity with the corresponding rules in the ordinary many‐body theory and practical utility are discussed in connection with (a) matrix elements over many‐electron spin states and (b) closed‐ and open‐shell many‐body perturbation theory. A possibility of integral‐driven many‐body perturbation theory for open‐shells is indicated. Connections of this formalism with others are also discussed.
By use of basic results of diagrammatic perturbation theory and combinatorial arguments, a formula is derived for the number of topologically distinct, connected, time ordered Goldstone diagrams contributing to the nth order perturbation energy. A general perturbation consisting of one- and two-particle operators is considered. The result is extended to an effective one-particle operator that implicitly includes all equal-time contractions and to multiple one-particle perturbations that enter only a restricted number of times in any order; the former is important for a Hartree–Fock zero order function and the latter for a system with external or internal field perturbations. It is pointed out that the method for computing the number of disconnected diagrams is quite general and independent of the nature of the perturbation and that the connected topologies which possess unusual symmetry (i.e., those for which all sets of vertex space–time labels are not distinct) are of a very specific and easily identified type. Numerical results for the various cases considered are given through 5th order.