Journal of Physics B Atomic and Molecular Physics

Equations-of-motion formulation of many-body perturbation theory

Esper Dalgaard, Jack Simons

Published Oct 1, 1977• 14 citations

Abstract

A formulation of many-body perturbation theory starting from the operator equation (H,Q + )= omega Q + is presented. The method of solution is based on an operator scalar product, (X/Y)=Tr(X + Y), which allows the use of resolvent and partitioning techniques to establish Rayleigh-Schrodinger or Brillouin-Wigner perturbation theory for the excitation energy and excitation operator, Q + . The excitation operator contains all information about the two states involved in the transition. Specific results are given for removal or addition of an electron and for excitations of particle-hole type and comparison with the Green's function methods is made.

DOI: 10.1088/0022-3700/10/14/012

Related papers

Resolvent Operator Formulation of Stationary State Perturbation Theory

Robert Yaris

By starting with an exact operator equation and using different methods of expanding the resolvent operator, the Schrödinger, Wigner—Brilloin, similarity transformation, gauge transformation, and first-order perturbation iteration method, perturbation expansions are generated in a rigorous and straightforward manner. It is also shown how additional perturbation and perturbation iteration methods can be generated.

The Journal of Chemical PhysicsApr 1, 19649 citations

Critical Comparison Between Equation of Motion-Green's Function Methods and Configuration Interaction Methods: Analysis of Methods and Applications

Karl F. Freed, Michael F. Herman, Danny L. Yeager

A description is provided of the common conceptual origins of many-body equations of motion and Green's function methods in Liouville operator formulations of the quantum mechanics of atomic and molecular electronic structure. Numerical evidence is provided to show the inadequacies of the traditional strictly perturbative approaches to these methods. Nonperturbative methods are introduced by analogy with techniques developed for handling large configuration interaction calculations and by evaluating individual matrix elements to higher accuracy. The important role of higher excitations is exhibited by the numerical calculations, and explicit comparisons are made between converged equations of motion and configuration interaction calculations for systems where a fundamental theorem requires the equality of the energy differences produced by these different approaches.

Physica ScriptaJan 1, 19806 citations

Many-body perturbation theory: The GW approximation

Christoph Friedrich, Arno Schindlmayr

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, 200619 citations

Resolvent operator approach to many-body perturbation theory. I. Closed shells

Ajit Banerjee, Debashis Mukherjee, Jack Simons

In this paper, we develop a time-dependent approach to many-body perturbation theory for closed shells based on the resolvent of the Schrödinger equation. We introduce a quantity S(t) = i<φ‖exp [−i(H−E0)t]‖ψ≳/<φ‖ψ≳, where φ and ψ are, respectively, the unperturbed and exact wave functions for the system and E0 is the unperturbed energy. The Fourier transform of S(t), S(ω) = <φ‖(ω+E0− H)−1‖ψ≳/ <φ‖ψ≳, is a matrix element of the resolvent containing the exact function ψ and, thus, has a pole at ω = (E−E0), the correlation energy. Starting from a time-dependent perturbation expansion of S(t) via the Gellman–Low adiabatic theorem, we have obtained a Dyson-like equation: S−1(ω) = S−10(ω)+Σ<φ‖ψ≳ for S(ω). Such a derivation requires judicious grouping of terms of the perturbation series for S(t). It has been shown that specific regroupings of the terms of S(t) into appropriate ’’top’’ and ’’bottom’’ parts and corresponding time-integration procedures yield a Σ which generates the Brillouin–Wigner (BW) or Rayleigh–Schrödinger (RS) energy series. Σ for both the cases are identified as the ’’top’’ parts. The characteristic features of the BW series, namely, the appearance of disconnected and ω-dependent diagrams is to be contrasted with that of the RS series, which contains connected and ω-independent diagrams only, and both emerge naturally as a consequence of the dissection procedure into appropriate top and bottom parts.

The Journal of Chemical PhysicsFeb 15, 19829 citations

Functionals in many-particle physics

Richard M. Martin, Lucia Reining, David M. Ceperley

To avoid confusion, it is necessary to be quite explicit about what is assumed and what is to be proved. A. Klein, Phys. Rev. 121, 950 (1961) Summary The topic of this chapter is functionals that provide a concise formulation for thermodynamic quantities and Green's function methods in interacting many-body systems. Building upon the instructive examples of density functional theory and the Hartree– Fock approximation, functionals of the Green's function are developed to provide a framework for practical methods used extensively in many-body perturbation theory (Chs. 9–15), dynamical mean-field theory (Chs. 16–20), and their combination in Ch. 21. Functional concepts give a firm foundation for those methods, for example, they give the conditions for approximations to obey conservation laws. Additional background and specific aspects are in App. H. Many physical quantities can be expressed as functionals , i.e., they depend on an entire function.1 For example, in the quantum variational method to determine a wavefunction, the total energy of a system of many electrons is a functional of a many-body trial wavefunction ( r 1 , r 2 , …): it is the expectation value of the hamiltonian. At the variational minimum, assuming the trial wavefunction is completely flexible, one finds the many-body Schrödinger equation for the ground state. Many of the properties we are interested in are, formally, simple functionals of the ground- or excited-state wavefunctions. In the search for feasible ways to calculate properties such as excitation spectra, we are led to different types of functional. The idea is to deal only with a physical, measurable quantity Q of interest and a variational functional E [ Q ] of this quantity. The variational character of E allows one to derive equations that determine Q 0 , the value of Q for the system in a desired state, typically the ground state or equilibrium state at T ≠ 0. Other properties should be different functionals F m [ Q ] that can then be evaluated at Q = Q 0 . This strategy is outlined in Fig. 8.1. To appreciate the power of such approaches, one can consider the impact of density functional theory, which is formulated as a functional of the density n ( r ) of electrons; it is defined for a range of densities and the variational minimum for the energy functional E [ n ] leads to equations to determine the actual ground-state density n 0 ( r ) and total energy E 0 .

Cambridge University Press eBooksMay 31, 20160 citations

Equations of Motion with Particle–Particle Interactions and Approximations

Norman J. M. Horing

Abstract Starting with the equation of motion for the field operator ψ(x,t) of an interacting many-particle system, the n-particle Green’s function (Gn) equation of motion is developed, with interparticle interactions generating an infinite chain of equations coupling it to (n+1)- and (n−1)-particle Green’s functions (Gn+1 and Gn−1, respectively). Particularly important are the one-particle Green’s function equation with its coupling to the two-particle Green’s function and the two-particle Green’s function equation with its coupling to the three-particle Green’s function. To develop solutions, it is necessary to introduce non-correlation decoupling procedures involving the Hartree and Hartree-Fock approximations for G2 in the G1 equation; and a similar factorization “ansatz” for G3 in the G2 equation, resulting in the Sum of Ladder Diagrams integral equation for G2, with multiple Born iterates and finite collisional lifetimes. Similar treatment of the G11-equation for the joint propagation of one-electron and one-hole subject to mutual Coulomb attraction leads to bound electron-hole exciton states having a discrete hydrogen like spectrum of energy eigenstates. Its role in single-particle propagation is also discussed in terms of one-electron self-energy Σ and the T-matrix

Jul 27, 20170 citations