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Journal of Physics A Mathematical and General

Quantum mechanical ground states, nonlinear Schrodinger equations and linked cluster expansions

Leo P. Kadanoff, Mahito Kohmoto

Published Jun 1, 1981• 14 citations
Publisher

Abstract

The ground state wavefunction of an infinitely large, spatially homogeneous system is evaluated in terms of wavefunctions for linked clusters of points within the system. The linked cluster wavefunction is shown to obey a nonlinear form of the Schrodinger equation, with the unlinked pieces providing the nonlinearity. A variational principle for the ground state energy is derived.

DOI
10.1088/0305-4470/14/6/009

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The ground state wavefunction of an infinitely large, spatially homogeneous system is evaluated in terms of wavefunctions for linked clusters of points within the system. The linked cluster wavefunction is shown to obey a nonlinear form of the Schrodinger equation, with the unlinked pieces providing the nonlinearity. A variational principle for the ground state energy is derived.
New forms of the linked cluster expansion for the ground-state energy of a normal spherical system are given in terms of the exact single-particle propagator and of the associated self-energy operator. These forms are shown to be stationary with respect to variations of the latter quantities. A new concise proof of the Hugenholtz-Van Hove theorem on single-particle energies is then possible. The variational principle also explains the insensitivity of recent binding energy calculations to the choice of single-particle energies. It is shown how all results may be extended to systems with anisotropic features.
The general form of the energy of the ground state of a many-fermion system is shown to be exactly of the form proposed by Brueckner and Bethe, without approximation. In a variational treatment, if the trial wave function is picked containing only pair correlations, together with all possible unlinked pairs, it is described by a two-body excitation matrix 〈m1m2∣A∣p1p2〉〈{m}_{1}{m}_{2}|A|{p}_{1}{p}_{2}〉〈m1​m2​∣A∣p1​p2​〉. Variation of this matrix in the Ritz-Rayleigh principle yields a set of integral equations of the scattering type for the matrix AAA. Hole-state energies are given self-consistently in terms of the matrix AAA, but particle-state energies are Hartree-Fock energies. This may be corrected for by widely enlarging the class of terms admitted into the wave function. If the approximation is then made of omitting a class of terms, defined as cross-linked clusters in $〈\ensuremath{\psi}|H|\ensuremath{\psi}〉$, the particle-state energies are easily renormalized. Variation then leads to an infinite hierarchy of integral equations.
In parallel with standard density-functional theory, we study the energy of the ground state of a finite many-body system as a functional of the one-particle density matrix. We show that the formulation of a variational principle that is valid beyond the Hartree-Fock limit requires that two-body correlations be included not only in the ground-state energy but also in the constraints. As an illustration, we apply a linear-response argument to derive formulas for first-order corrections to the Hartree-Fock density matrix. Further analysis suggests an approach in terms of the density matrix of an independent-particle system, which can be introduced by the application of an alternative variational principle. This approach is reminiscent of Kohn-Sham theory, but the effective external potential is not required to be local. This variational method can be implemented in a systematic fashion by means of the linked-cluster expansion. In an appendix we study a variant of the Hohenberg-Kohn theorem for nonlocal potentials.
In order to go over from a perturbation expansion of the grand partition function (the unlinked-cluster expansion) to an expansion of the thermodynamic potential (the linked-cluster expansion) in powers of the interaction, it is necessary to treat carefully those terms in which creation (or annihilation) operators for the same state occur twice or more. The unlinked- and linked-cluster expansions for a system of fermions are here shown to be equivalent by a direct comparison of the terms which occur in each. The relation between the two expansions is illustrated by the example of a system of fermions interacting only with a single-particle potential.