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International Journal of Quantum Chemistry

Superoperator approach to propagator approximations

Brian Weiner, Osvaldo Goscinski

Published Feb 1, 1982• 12 citations
Publisher

Abstract

Abstract The superoperator formalism for matrix propagators is reviewed, with special attention to the validity of superoperator resolution of the identity. The equation of motion for the particle‐hole propagator is discussed. Decoupling procedures based on inner projections and partitioning are examined showing the desirability of self‐consistent approximations. The conditions under which a state can be a vacuum for a given linear manifold of operators are analyzed.

DOI
10.1002/qua.560210204

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The Journal of Chemical Physics•Oct 15, 1972•2 citations
Abstract The superoperator formalism for matrix propagators is reviewed, with special attention to the validity of superoperator resolution of the identity. The equation of motion for the particle‐hole propagator is discussed. Decoupling procedures based on inner projections and partitioning are examined showing the desirability of self‐consistent approximations. The conditions under which a state can be a vacuum for a given linear manifold of operators are analyzed.
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.
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.
In the quantum theory of matter, the wavefunctions Ψ form a linear space {Ψ} which serves as a carrier space for the operators F corresponding to physical observables as well as for the system operatorsΓ associated with physical ensembles. The mappings of the operators T in turn are called superoperators, and, of particular importance are the Liouvillian and the superevolution operator used in describing the time dependence of the system operators Γ(t). The theory of such superoperators is reviewed in some detail. By studying the time dependence of expectation values (F) and time-correlation functions (F(t)G(t')), the connection between the superoperator approach and the conventional propagator method is explored. The eigenvalue problem of the superoperators is treated, and the principle importance of the superresolvents and the practical usefulness of the partitioning technique are employed. The “inner projections” of superoperators with respect to both orthogonal projectors and skew projectors are studied, and it is shown that they render “rational approximations” of the quantities involved which are always convergent. It is finally shown that the partitioning technique may be used also in solving the time-dependent Liouville equation.
Abstract An explicit definition for the self‐energy field operator and the self‐energy fields obtained from an average of the associated operator within the algebraic formalism of superoperators is presented. It stems from the formal expansion of the many‐body propagator equations of motion hierachy in quantum many‐body systems within the scenario of the Liouvillian decoupling scheme developed in previous works. An essential theoretical property of such fields for the complete expansion of the propagator to any order in the interaction potential is shown. This states that the interaction potential to a given order only depends on one q‐reduced density matrix. The contraction order q of the density matrix depends on both the nature of the operators defining the propagator and the actual order of the expansion. This result is rigorous regarding infinite summation of the irreducible terms of the self‐energy fields and provides a direct way to estimate the extent to which many‐body effects are involved in successive approximations, i.e., truncation of the excitation level given by the reference state throughout the reduced density matrix and the expansion of the propagator. © 2002 John Wiley & Sons, Inc. Int J Quantum Chem, 2002
Abstract Master equations for propagators in quantum open systems and their spectral resolutions are derived. The Zwanzig partitioning scheme along the superoperator algebra are used to derive equations of motion for partitioned operators in a Liouville space. The reservoir influence on the dynamical evolution of operators is shown to lead explicitly to dissipative effects arising from memory terms in the evolution equations of such operators. It is also shown that spectral representations may be written in a self‐consistent analytic way by means of the self‐energy fields for transition energies of the system by taking into account the lack of the complete knowledge about the reservoir. A kinematic fluid interpretation of the resultant equations is given and an explicit form of the “collision” superoperator is obtained. Finally, a simple example to illustrate the determination of self‐energy fields for the system–reservoir interaction corrections is given. © 1995 John Wiley & Sons, Inc.
Moment-conserving decoupling procedures for the one-electron Green's function (electron propagator) are investigated. By analysis of one particular system (Hubbard model) it becomes clear that the moment-conserving techniques so far proposed for interacting systems are all special cases of [1, 0], [2, 1], or [3, 2] Padé approximants differing only in which terms they omit, how they define self-consistency, and which operator algebra they utilize. The number of formally conserved moments appears not to be a particularly useful criterion of the accuracy of a decoupling procedure, since a two-moment scheme (Hartree-Fock) may be numerically closer to the exact answer than some three-moment procedures. Use of the complete Padé approximant, as suggested by Lukman and Goscinski for the particle-hole propagator, appears to be the most consistent method.