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Physical Review A

Some aspects of self-consistent propagator theories

M. Durga Prasad, Sourav Pal, Debashis Mukherjee

Published Mar 1, 1985• 67 citations
Publisher

Abstract

A general method for constructing bases for operator manifolds for any propagator, which satisfy ``vacuum annihilation conditions'' (VAC's) is developed. This approach is based on the observation that if the transformation of the unperturbed ground state to the correlated ground state is represented as a rotation in the Fock space, the corresponding rotation induced in the basis of the concerned operator space would generate a basis which satisfies VAC's on the correlated ground state. The associated requirements for the Hermiticity of superoperator Hamiltonian would also be met in this new basis. The proposed method is noniterative in that, once the form of the ground-state function is specified, the expansion of the operator manifold satisfying VAC's on the ground state does not require any iterative readjustment. The resultant propagators in this approach are fully linked. It is shown that this theory is equivalent to a propagator formalism in terms of hole- and particle-creation and/or annihilation operators with a modified effective Hamiltonian. The self-consistent electron and polarization propagators are considered as examples, and their underlying perturbative structures are analyzed. The role of density shift operators and higher-rank operators are discussed.

DOI
10.1103/physreva.31.1287

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A general method for constructing bases for operator manifolds for any propagator, which satisfy ``vacuum annihilation conditions'' (VAC's) is developed. This approach is based on the observation that if the transformation of the unperturbed ground state to the correlated ground state is represented as a rotation in the Fock space, the corresponding rotation induced in the basis of the concerned operator space would generate a basis which satisfies VAC's on the correlated ground state. The associated requirements for the Hermiticity of superoperator Hamiltonian would also be met in this new basis. The proposed method is noniterative in that, once the form of the ground-state function is specified, the expansion of the operator manifold satisfying VAC's on the ground state does not require any iterative readjustment. The resultant propagators in this approach are fully linked. It is shown that this theory is equivalent to a propagator formalism in terms of hole- and particle-creation and/or annihilation operators with a modified effective Hamiltonian. The self-consistent electron and polarization propagators are considered as examples, and their underlying perturbative structures are analyzed. The role of density shift operators and higher-rank operators are discussed.
Some of the difficulties encountered in the approximation of Green's functions are reviewed with particular emphasis on the self-consistent polarization propagator. Some new basic theorems are presented, involving a general procedure, which lead to both a derivation and alternate prescriptions for the computations. As demonstrated by Linderberg and Öhrn the ground state is an AGP (antisymmetrized geminal power). An energy optimized AGP ( E AGP ) in a Brueckner orbital representation can be used to generate a manifold of excited states| K ⟩ = O K + |AGP⟩, O K |AGP⟩ = 0 with the excitation operators, O + K , and killers, O K , being combinations of particle-hole and hole-particle operators. The SCPP can be expressed then as ⟨⟨ X, Y ⟩⟩ E = - ⟨AGP| XT 0 (- E + E AGP ) Y + YT 0 ( E + E AGP ) X |AGP⟩ in terms of the resolvent T 0 ( ) = P ( - PHP ) -1 with P spanned by the states| K ⟩⟨ K |, in the sense of Löwdin. Our results establish a natural relationship between hermiticity, completeness, non-redundance of operator manifolds with stationarity and killer conditions.
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.
Abstract A propagator approximation scheme is presented in the context of an abstract*‐algebra approach. The representation theory of such algebras is shown to play a crucial role in the definition of consistent approximations, i.e., approximate propagators based on model time evolutions and states. This procedure places superoperator methods of approximation on a sound Hilbert space footing. A generalization of the Fock vacuum property is introduced which leads to a simplification in the form of the model propagators. Finally a concrete example is considered that fulfills the conditions developed in this article showing that a consistent approximation to the electron propagator results in the Hartree–Fock–Boguliubov equations.
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.
Abstract We examine and compare ways of dividing into subspaces the space whose elements are density matrices or other operators for the class of model problems defined by a finite one‐particle basis set. One method of decomposition makes the significance of the subspaces apparent. We show that this decomposition is also complete, in the group‐theoretic sense, for the group of unitary transformations of the set of one‐electron basis functions. The irreducible subspaces are labeled by particle number and by an additional integer we call the reduction index. For spaces of particle‐number‐conserving operators, all subspaces with the same reduction index are isomorphic, and an analogous isomorphism exists for non‐particle‐number‐conserving cases. The general linear group also plays a key role, and we introduce the term “canonical superoperators” to characterize those superoperators which commute with this group. When an appropriate basis set is chosen for the matrix spaces, the supermatrices corresponding to these superoperators have a particularly simple form: a block structure with the only nonzero blocks being multiples of unit matrices. The superoperators of interest can be constructed in terms of two operators, , and these two have been expressed simply in terms of creation and annihilation operators. When only real orthogonal transformations of the basis are considered, a further decomposition is possible. We have introduced superoperators associated with this decomposition.