New algebraic structure on electronic Fock space is studied in detail. This structure is defined in terms of a certain multiplication of many electron wave functions and has close interrelation with coupled cluster and similar approaches. Its study clarifies and simplifies the mathematical backgrounds of CC and related approaches. And even more, it leads to many relations that would be very difficult to derive using conventional technique. Formulas for action of the creation-annihilation operators on products of state vectors are derived. Explicit expressions for action of simplest particle-conserving products of the creation-annihilation operators on powers of state vectors are given. General scheme of parametrization of representable density operators of arbitrary order is presented.
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.
The quasiparticle Fock-space coupled-cluster (QFSCC) theory, introduced by us in 1985, is described. This is a theory of many-electron systems which uses the second-quantisation formalism based on the algebraic approximation: one chooses a finite spin-orbital basis, and builds a fermionic Fock space to represent all possible antisymmetric electronic states of a given system. The algebraic machinery is provided by the algebra of linear operators acting in the Fock space, generated by the fermion (creation and annihilation) operators. The Fock-space Hamiltonian operator then determines the system's stationary states and their energies. Within the QFSCC theory, the Fock space and its operator algebra are subject to a unitary transformation which effectively changes electrons into some fermionic quasiparticles. A generalisation of the coupled-cluster method is achieved by enforcing the principle of quasiparticle-number conservation. The emerging quasiparticle model of many-electron systems offers useful physical insights and computational effectiveness. The QFSCC theory requires a substantial reformulation of the traditional second-quantisation language, by making full use of the algebraic properties of the Fock space and its operator algebra. In particular, the role of operators not conserving the number of electrons (or quasiparticles) is identified.
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.
We study in this series the group theoretical structure of Fermion many-body systems arising from the canonical anticommutation relation of the annihilation-creation operators. Owing to the canonical anticommutation relation, a Fermion system with N single particle states has at least six Lie algebras of Fermion operators, a U(N), an SO(2N), an SO(2N+1), an SO(2N+2) and two U(N+1) Lie algebras. There are also two Clifford algebras of 2N and 2N+1 dimensions. The Fermion space is shown to belong to the spinor representations of the SO(2N), SO(2N+1) and SO(2N+2) groups. The canonical transformations generated by the U(N), SO(2N) and SO(2N+1) Lie algebras are characterized as the transformations to induce the linear U(N), SO(2N) and SO(2N+1) transformations for the Clifford algebras. The independent (quasi-) particle type wave functions of three kinds including the Hartree-Fock and Hartree-Bogoliubov wave functions are constructed by means of the canonical transformations and their relationship is studied. We derive three exact generator coordinate representations for state vectors in which the generator coordinates are the U(N), SO(2N) or SO(2N+1) group and the generating functions are the independent (quasi-) particle type wave functions. We characterize the structures of state vectors in the generator coordinate representations and study the relationship of the three representations.