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arXiv:physics/9710033 · Physical Review A

First-order static excitation potential: Scheme for excitation energies and transition moments

Joachim Brand, Lorenz S. Cederbaum

Published Jun 1, 1998• 9 citations
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Abstract

We present an approximation scheme for the calculation of the principal excitation energies and transition moments of finite many-body systems. The scheme is derived from a first-order approximation to the self-energy of a recently proposed extended particle-hole Green's function. A Hermitian eigenvalue problem is encountered of the same size as the well-known random phase approximation (RPA). We find that it yields a size consistent description of the excitation properties and removes an inconsistent treatment of the ground-state correlation by the RPA. By presenting a Hermitian eigenvalue problem the new scheme avoids the instabilities of the RPA and should be well suited for large-scale numerical calculations. These and additional properties of the new approximation scheme are illuminated by a very simple exactly solvable model.

DOI
10.1103/physreva.57.4311

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We present an approximation scheme for the calculation of the principal excitation energies and transition moments of finite many-body systems. The scheme is derived from a first-order approximation to the self-energy of a recently proposed extended particle-hole Green's function. A Hermitian eigenvalue problem is encountered of the same size as the well-known random phase approximation (RPA). We find that it yields a size consistent description of the excitation properties and removes an inconsistent treatment of the ground-state correlation by the RPA. By presenting a Hermitian eigenvalue problem the new scheme avoids the instabilities of the RPA and should be well suited for large-scale numerical calculations. These and additional properties of the new approximation scheme are illuminated by a very simple exactly solvable model.
We present a method for the treatment of correlations in finite Fermi systems in a more consistent way than the random phase approximation (RPA). The main differences with respect to previous approaches are underlined and discussed. In particular, no use of renormalized operators is made. By means of the method of linearization of equations of motion, we derive a set of RPA-like equations depending only on the one-body density matrix. The latter is no more assumed to be diagonal and it is expressed in terms of the XXX and YYY amplitudes of the particle-hole phonon operators. This set of nonlinear equations is solved via an iterative procedure, which allows us to calculate the energies and the wave functions of the excited states. After presenting our approach in its general formulation, we test its quality by using metal clusters as a good test laboratory for a generic many-body system. Comparison to RPA shows significant improvements. We discuss also how the present approach can be further extended beyond the particle-hole configuration space, getting an approximation scheme in which energy weighted sum rules are exactly preserved, thus solving a problem common to all extension of RPA proposed until now. The implementation of such approach will be afforded in a future work.
An approach for particle-hole correlation functions, based on the so-called self-consistent random-phase approximation is developed. This leads to a fully self-consistent RPA-like theory that satisfies the fff-sum rule and several other theorems. As a first step, a simpler self-consistent approach, the renormalized RPA, is solved numerically in the one-dimensional Hubbard model. The charge and the longitudinal spin susceptibility, the momentum distribution, and several ground-state properties are calculated and compared with the exact results. Especially at half-filling, our approach provides quite promising results and matches the exact behavior apart from a general prefactor. The strong-coupling limit of our approach can be described analytically.
We present a new extension of the random-phase approximation method: the quasiboson approximation is avoided and correlations are included in the ground state without resorting to renormalized operators or renormalized matrix elements; the configuration space is enlarged by considering also elementary excitations corresponding to the annihilation of a particle (hole) and the creation of another particle (hole) on the correlated ground state, together with the particle-hole ones. Two new and relevant advantages of this method with respect to the existing extensions of random-phase approximation are highlighted: (i) the energy weighted sum rules are completely satisfied; (ii) the problem of the existence of nonphysical states in the spectrum, related to the inclusion of particle-particle and hole-hole configurations, is solved: a way to unambiguously disentangle physical from nonphysical states in the excitation spectrum is presented. The method is applied here to a three-level Lipkin model where its quality can be judged by comparing with the exact results. Both advantages (i) and (ii) shall lead to feasible future applications of this extended RPA to several realistic cases.
A self-consistent RPA method is presented for N-particle Fermi systems with two- and three-body forces. The RPA equations for the generalised 'particle-hole' and 'particle-particle' problem, respectively, are derived by minimising the mean excitation energies defined analogously to mean single-particle energies. This corresponds to the equation-of-motion method. For the generalised 'one-particle/two-particle-one-hole problem' the Green-function scheme is used in order to establish a connection between two possible ways of treating the problem. It is shown that the RPA Hamiltonians have almost the same structure as in the case of pure two-body forces, and that Rowe's general RPA equations are contained as a special case of the treatment.
We derive the explicit expression of the three self-energies that one encounters in many-body perturbation theory: the well-known GW self-energy, as well as the particle-particle and electron-hole T-matrix self-energies. Each of these can be easily computed via the eigenvalues and eigenvectors of a different random-phase approximation linear eigenvalue problem that completely defines their corresponding response function. For illustrative and comparative purposes, we report the principal ionization potentials of a set of small molecules computed at each level of theory. The performance of these schemes on strongly correlated systems (B2 and C2) is also discussed.
Within the framework of the many-body Green's-function method we present a new approach to the polarization propagator for finite Fermi systems. This approach makes explicit use of the diagrammatic perturbation expansion for the polarization propagator, and reformulates the exact summation in terms of a simple algebraic scheme, referred to as the algebraic diagrammatic construction (ADC). The ADC defines in a natural way a set of approximation schemes (nnnth-order ADC schemes) which represent infinite partial summations exact up to nnnth order of perturbation theory. In contrast to the random-phase-approximation (RPA)-like schemes, the corresponding mathematical procedures are essentially Hermitian eigenvalue problems in limited configuration spaces of unperturbed excited configurations. Explicit equations for the first- and second-order ADC schemes are derived. These schemes are thoroughly discussed and compared with the Tamm-Dancoff approximation and RPA schemes.