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International Journal of Quantum Chemistry

The linked-diagram expansion of the ground state of a many-electron system: A time-independent derivation

Rolf Manne

Published Jun 18, 2009• 6 citations
Publisher

Abstract

A time-independent derivation of the linked-diagram expansion of the ground-state energy of a many-electron system is given for the case of a nondegenerate unperturbed system. The starting point is the Rayleigh-Schrödinger perturbation theory, and the derivation follows in many respects suggestions by Čížek and Paldus, including the use of the Frantz-Mills factorization theorem for the cancellation of unlinked and renormalization diagrams.

DOI
10.1002/qua.560120822

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A time-independent derivation of the linked-diagram expansion of the ground-state energy of a many-electron system is given for the case of a nondegenerate unperturbed system. The starting point is the Rayleigh-Schrödinger perturbation theory, and the derivation follows in many respects suggestions by Čížek and Paldus, including the use of the Frantz-Mills factorization theorem for the cancellation of unlinked and renormalization diagrams.
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A new approach to the diagrammatic formulation of many-body perturbation theory for open-shell systems is presented. The formalism is based on a generalized form of the Bloch equation that also generates the Rayleigh--Schroedinger perturbation expansion for a system with several open shells. Second quantization in the particle-hole formulation is used together with Wick's theorem in order to derive graphical rules in the usual way. The linked-diagram property of the wave operator and of the effective interaction is shown by expanding the wave operator in terms of normal products of connected diagram clusters, in analogy with the exp(S) formalism of Coester and Kuemmel for closed-shell systems. Self-consistent coupled-cluster equations are derived for the open-shell case from the generalized Bloch equation by a straightforward extension of the procedure of Cizek and Paldus. The application of such equations for investigating different properties of open-shell atoms is discussed. 56 references.
A nondegenerate perturbation theory is studied using the adiabatic-theorem formalism. A new factorization of the adiabatic time-evolution operator analogous to that obtained by Morita in his derivation of the linked-cluster theorem is given by carrying out a series of intermediate representations. This leads to a new expansion for the perturbed eigenvectors \ensuremath{\Vert}\ensuremath{\alpha}〉 and energies ${E}_{\ensuremath{\alpha}}$ which are analytically expressed with respect to two matrix series scrV(n){\mathrm{scrV}}^{(n)}scrV(n) and E(n){E}^{(n)}E(n). The nth term of each series (scrV(n){\mathrm{scrV}}^{(n)}scrV(n) and E(n){E}^{(n)}E(n)) includes perturbational contributions of power p, p\ensuremath{\ge}n. An explicit recurrence, scrV(n){\mathrm{scrV}}^{(n)}scrV(n)=V(${\mathrm{scrV}}^{n\mathrm{\ensuremath{-}}1)}$, ${E}^{(n\mathrm{\ensuremath{-}}1)}$), E(n){E}^{(n)}E(n)=E(${\mathrm{scrV}}^{(n\mathrm{\ensuremath{-}}1)}$, ${E}^{(n\mathrm{\ensuremath{-}}1)}$) is derived and allows us to calculate these matrices to an arbitrary iteration order (n). This is a central feature of our approach since the eigenvectors and eigenvalues can be generated, to this arbitrary order, by a simple numerical program.
The equations of motion for the general many-time causal Green's functions for a fermion system are iterated and are shown not to lead to unlinked graphs, which is a general proof of the linked cluster theorem. An explicit expression is obtained for the perturbation expansion of an arbitrary Green's functions which is applied to the one- and two-particle Green's functions. By connecting lines systematically in a set of diagrams obtained from the equations of motion, the usual topologically different linked graphs and rules are generated.
A perturbation theory is described which leads to two alternative methods for calculating the eigenvalues and normalized eigenstates of a perturbed Hamiltonian, the results being given to all orders in the perturbation. The expansions obtained are formulated in terms of two sets of partial sums which have simple, easily characterized forms, and they combine some of the advantages of both the Rayleigh-Schr\"odinger and Brillouin-Wigner perturbation methods while avoiding some of the difficulties of each. In particular, one form of the perturbation expansions given here will in most cases of interest have a larger radius of convergence than the Rayleigh-Schr\"odinger series. As is usual for infinite-order treatments, the explicit form of the final result applies only to nondegenerate levels. However, some of the intermediate relations obtained in the derivation are of general validity. Since certain classes of unlinked diagrams appear in the expansions, the method described here is not suitable for application to systems containing a large number of particles.