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Physical Review A

Comparison within the algebraic approximation of configuration interaction and many-body perturbation theory for the Be ground state

David Martin Silver, Stephen S. Wilson, Carlos F. Bunge

Published Apr 1, 1979• 35 citations
Publisher

Abstract

Calculations of the electronic energy of the Be-atom ground state are performed using a configuration-interaction technique and a nondegenerate formulation of diagrammatic many-body perturbation theory. Both methods are applied within the algebraic approximation defined by three finite-basis sets. One of the basis sets contains only $s$ functions, and it defines an algebraic problem without degeneracies where agreement between the two methods is excellent. The other two basis sets, although more complete, both contain $p$ functions, and they define algebraic problems involving a near degeneracy where agreement between the two methods is found to be less satisfactory. It is concluded that either a degenerate formulation of perturbation theory is required, or certain diagrams in the model perturbative scheme need to be summed to high order through denominator shifts in order to properly handle the nondynamic-correlation effect found in the Be atom.

DOI
10.1103/physreva.19.1375

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Configuration interaction approach for the computation of the electronic self-energy

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In order to properly understand the utility of many-body perturbation theory as applied to finite systems, we use the configuration interaction approach to compute the electronic self-energy. The validity of the commonly used $GW$ approximation from many-body perturbation theory is tested by comparing the self-energy explicitly computed according to the diagrammatic expansions to that from the inversion of the Dyson equation. It is constructed as a functional of the interacting Green function $G$ and screened Coulomb interaction $W$. $W$ is explicitly computed through the diagonalization of the many-body Hamiltonian and, thus, takes polarization effects into account beyond the random phase approximation. The $\mathrm{Na}_{9}{}^{+}$ cluster and the dissociation of the ${\mathrm{C}}_{2}$ molecule are discussed as examples. We find that the $GW$ approximation yields accurate results for weakly correlated systems with a predominantly single-determinant ground state such as $\mathrm{Na}_{9}{}^{+}$ clusters. The ${\mathrm{C}}_{2}$ molecule is a pathological system with multideterminantal ground state where the $GW$ approach ceases to be valid.

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Introduction

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In this lecture we present many-body perturbation theory as a method to determine quasiparticle excitations in solids, especially electronic band structures, accurately from first principles.The main ingredient is the electronic self-energy that, in principle, contains all many-body exchange and correlation effects beyond the Hartree potential.As its exact mathematical expression is unknown, approximations must be used in practical calculations.The approximation is obtained using a systematic algebraic approach on the basis of Green function techniques.It constitutes an expansion of the self-energy up to linear order in the screened Coulomb potential, which describes the interaction between the quasiparticles and includes dynamic screening through the creation of exchange-correlation holes around the bare particles.The implementation of the approximation relies on a perturbative treatment starting from density functional theory.Besides a detailed mathematical discussion we focus on the underlying physical concepts and show some illustrative applications.

•Jan 1, 2006•19 citations
Calculations of the electronic energy of the Be-atom ground state are performed using a configuration-interaction technique and a nondegenerate formulation of diagrammatic many-body perturbation theory. Both methods are applied within the algebraic approximation defined by three finite-basis sets. One of the basis sets contains only sss functions, and it defines an algebraic problem without degeneracies where agreement between the two methods is excellent. The other two basis sets, although more complete, both contain ppp functions, and they define algebraic problems involving a near degeneracy where agreement between the two methods is found to be less satisfactory. It is concluded that either a degenerate formulation of perturbation theory is required, or certain diagrams in the model perturbative scheme need to be summed to high order through denominator shifts in order to properly handle the nondynamic-correlation effect found in the Be atom.
In order to properly understand the utility of many-body perturbation theory as applied to finite systems, we use the configuration interaction approach to compute the electronic self-energy. The validity of the commonly used GWGWGW approximation from many-body perturbation theory is tested by comparing the self-energy explicitly computed according to the diagrammatic expansions to that from the inversion of the Dyson equation. It is constructed as a functional of the interacting Green function GGG and screened Coulomb interaction WWW. WWW is explicitly computed through the diagonalization of the many-body Hamiltonian and, thus, takes polarization effects into account beyond the random phase approximation. The Na9+\mathrm{Na}_{9}{}^{+}Na9​+ cluster and the dissociation of the C2{\mathrm{C}}_{2}C2​ molecule are discussed as examples. We find that the GWGWGW approximation yields accurate results for weakly correlated systems with a predominantly single-determinant ground state such as Na9+\mathrm{Na}_{9}{}^{+}Na9​+ clusters. The C2{\mathrm{C}}_{2}C2​ molecule is a pathological system with multideterminantal ground state where the GWGWGW approach ceases to be valid.
Scope The book focuses primarily on many-body (or better, many-electron) methods for electron correlation. These include Rayleigh–Schrödinger perturbation theory (RSPT) , particularly in its diagrammatic representation (referred to as many-body perturbation theory , or MBPT ), and coupled-cluster (CC) theory; their relationship to configuration interaction (CI) is included. Further extensions address properties other than the energy, and also excited states and multireference CC and MBPT methods. The many-body algebraic and diagrammatic methods used in electronic structure theory have their origin in quantum field theory and in the study of nuclear matter and nuclear structure. The second-quantization formalism was first introduced in a treatment of quantized fields by Dirac (1927) and was extended to fermion systems by Jordan and Klein (1927) and by Jordan and Wigner (1928). This formalism is particularly useful in field theory, in scattering problems and in the study of infinite systems because it easily handles problems involving infinite, indefinite or variable numbers of particles. The diagrammatic approach was introduced into field theory by Feynman (1949a,b) and applied to many-body systems by Hugenholtz (1957) and by Goldstone (1957). Many-body perturbation theory and its linked-diagram formalism were first introduced by Brueckner and Levinson (1955) and by Brueckner (1955), and were formalized by Goldstone (1957). Other important contributions to the methodology, first in field theory and then in the theory of nuclear structure, are due to Dyson (1949a,b), Wick (1950), Hubbard (1957, 1958a,b) and Frantz and Mills (1960). Applications to the electronic structure of atoms and molecules began with the work of Kelly (1963, 1964a,b, 1968), and molecular applications using finite analytical basis sets appeared in the work of Bartlett and Silver (1974a, b).
The second-order multireference perturbation theory employing multiple partitioning of the many-electron Hamiltonian into a zero-order part and a perturbation is formulated in terms of many-body diagrams. The essential difference from the standard diagrammatic technique of Hose and Kaldor concerns the rules of evaluation of energy denominators which take into account the dependence of the Hamiltonian partitioning on the bra and ket determinantal vectors of a given matrix element, as well as the presence of several two-particle terms in zero-order operators. The novel formulation naturally gives rise to a “sum-over-orbital” procedure of correlation calculations on molecular electronic states, particularly efficient in treating the problems with large number of correlated electrons and extensive one-electron bases. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 395–401, 1999
We extend the self-consistent Green's functions formalism to take into account three-body interactions. We analyze the perturbative expansion in terms of Feynman diagrams and define effective one- and two-body interactions, which allows for a substantial reduction of the number of diagrams. The procedure can be taken as a generalization of the normal ordering of the Hamiltonian to fully correlated density matrices. We give examples up to third order in perturbation theory. To define nonperturbative approximations, we extend the equation-of-motion method in the presence of three-body interactions. We propose schemes that can provide nonperturbative resummation of three-body interactions. We also discuss two different extensions of the Koltun sum rule to compute the ground state of a many-body system.
Background: The Gorkov approach to self-consistent Green's function theory has been formulated by Som\`a, Duguet, and Barbieri in [Phys. Rev. C 84, 064317 (2011)]. Over the past decade, it has become a method of reference for first-principles computations of semimagic nuclear isotopes. The currently available implementation is limited to a second-order self-energy and neglects particle-number nonconserving terms arising from contracting three-particle forces with anomalous propagators. For nuclear physics applications, this is sufficient to address first-order energy differences (i.e., two neutron separation energies, excitation energies of states dominating the one-nucleon spectral function), ground-state radii and moments on an accurate enough basis. However, addressing absolute binding energies, fine spectroscopic details of $N\ifmmode\pm\else\textpm\fi{}1$ particle systems or delicate quantities such as second-order energy differences associated with pairing gaps, requires going to higher truncation orders.Purpose: The formalism is extended to third order in the algebraic diagrammatic construction (ADC) expansion with two-body Hamiltonians.Methods: The expansion of Gorkov propagators in Feynman diagrams is combined with the algebraic diagrammatic construction up to the third order as an organization scheme to generate the Gorkov self-energy.Results: Algebraic expressions for the static and dynamic contributions to the self-energy, along with equations for the matrix elements of the Gorkov eigenvalue problem, are derived. It is first done for a general basis before specifying the set of equations to the case of spherical systems displaying rotational symmetry. Workable approximations to the full self-consistency problem are also elaborated on. The formalism at third order it thus complete for a general two-body Hamiltonian.Conclusions: Working equations for the full Gorkov-ADC(3) are now available for numerical implementation.
In this lecture we present many-body perturbation theory as a method to determine quasiparticle excitations in solids, especially electronic band structures, accurately from first principles.The main ingredient is the electronic self-energy that, in principle, contains all many-body exchange and correlation effects beyond the Hartree potential.As its exact mathematical expression is unknown, approximations must be used in practical calculations.The approximation is obtained using a systematic algebraic approach on the basis of Green function techniques.It constitutes an expansion of the self-energy up to linear order in the screened Coulomb potential, which describes the interaction between the quasiparticles and includes dynamic screening through the creation of exchange-correlation holes around the bare particles.The implementation of the approximation relies on a perturbative treatment starting from density functional theory.Besides a detailed mathematical discussion we focus on the underlying physical concepts and show some illustrative applications.