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International Journal of Quantum Chemistry

Self‐energy operator and self‐energy fields in many‐body systems: Liouvillian approach

Roberto C. Bochicchio, Horacio Grinberg

Published Jan 1, 2002• 0 citations
Publisher

Abstract

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

DOI
10.1002/qua.10097

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A new type of ``intermediate'' state, \ensuremath{\Vert}\ensuremath{\Psi}${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{\mathit{J}}$〉, is introduced in the representation of many-body propagators and related resolvent matrix elements. A simple algebraic procedure is presented for constructing the unitary transformation matrix Q that relates the intermediate states and the exact energy eigenstates, \ensuremath{\Vert}${\mathrm{\ensuremath{\Psi}}}_{\mathit{n}}$〉, of the interacting system. Here the starting point is the matrix X of generalized spectroscopic amplitudes 〈${\mathrm{\ensuremath{\Psi}}}_{\mathit{n}}$\ensuremath{\Vert}C${\mathrm{^}}_{\mathit{J}}$\ensuremath{\Vert}${\mathrm{\ensuremath{\Psi}}}_{0}^{\mathit{N}}$〉, where C${\mathrm{^}}_{\mathit{J}}$ denotes a physical excitation operator and \ensuremath{\Vert}${\mathrm{\ensuremath{\Psi}}}_{0}^{\mathit{N}}$〉 is the N-electron ground state. A block QR decomposition [G. H. Golub and C. F. von Loan, MatrixB Computations (Johns Hopkins University, Baltimore, 1989)] of X according to the equation X=Q F allows one to determine explicit expressions for the subblocks of Q and the intermediate propagator representations constituted by a nondiagonal effective interaction matrix C and an effective spectroscopic matrix f. These effective quantities f and C are formulated entirely in terms of ground-state density-matrix elements and related energy expectation values. The relevance of the intermediate representations as a means for deriving computational schemes is based on the regularity and compactness of the perturbation expansions for the effective matrices f and C. These basic properties also establish that the intermediate representations are closed-form versions of the algebraic-diagrammatic construction approximation schemes derived previously as a reformulation of the original diagrammatic propagator perturbation series. The intermediate representations represent a link between algebraic and diagrammatic approaches in the field of propagator methods and are expected to be useful also in the development of nonperturbative approximations.
Abstract The methods of quantum chemistry and solid state theory to solve the many‐body problem are reviewed. We start with the definitions of reduced density matrices, their properties (contraction sum rules, spectral resolutions, cumulant expansion, and N‐representability), and their determining equations (contracted Schrödinger equations), and we summarize recent extensions and generalizations of the traditional quantum chemical methods, of the density functional theory, and of the quasiparticle theory: from finite to extended systems (incremental method), from density‐to‐density matrix (density matrix functional theory), from weak to strong correlation (dynamical mean field theory), from homogeneous (Kimball–Overhauser approach) to inhomogeneous and finite systems. Measures of the correlation strength are discussed. The cumulant two‐body reduced density matrix proves to be a key quantity. Its spectral resolution contains geminals, possibly the solutions of an approximate effective two‐body equation, and we sketch the idea of how its contraction sum rule can be used for a variational treatment. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004
Abstract Exact equations for correlation energy and kinetic energy correction functionals are derived in the time‐dependent density functional theory. Hierarchies of equations are also obtained and approximate functionals are proposed by truncation of the hierarchies at different orders. Explicit relations for the correlation energy and its kinetic component along the adiabatic path are given in terms of powers of the moments of the electron density. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003