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Arkiv Fysik

CLUSTER DEVELOPMENT OF THE DENSITY MATRICES IN THE JASTROW METHOD FOR MANY- FERMION SYSTEMS

Stefan Lundqvist

Published Jan 1, 1960• 0 citations
OpenAlex

Abstract

A method is given for the cluster development of the reduced density matrices for Jastrow wave functions. The method makes extensive use of the properties of the Dirac densitv matrices corresponding to the uncorrelated niotion, in conjunction with the method of expanding ihe distribution functions for a classical imperfect gas given by Maver and Montroll. (auth)

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A method is given for the cluster development of the reduced density matrices for Jastrow wave functions. The method makes extensive use of the properties of the Dirac densitv matrices corresponding to the uncorrelated niotion, in conjunction with the method of expanding ihe distribution functions for a classical imperfect gas given by Maver and Montroll. (auth)
A new cluster expansion technique is developed for the calculation of the energy expectation value of the many-fermion systems with singular potentials.The wave function, which is studied in this paper, is of the formwhich, in a special case, coincides with Jastrow wave function.The formula for the energy expectation value is written in terms of the complex cluster integrals.The method introduced in the derivation of this formula has an analogy to the classical cluster expansion method in the theory 0f an imperfect classical gas of multicomponents.
We study many-body correlations in the ground state of a general quantum system of bosons or fermions by including an additional Jastrow function in our recently proposed variational coupled-cluster method. Our approach combines the advantages of state-dependent correlations in the coupled-cluster theory and of the strong, short-ranged correlations of the Jastrow function. We apply a generalized linked-cluster expansion for the Jastrow wave function and provide a detailed analysis for practical evaluation of the Hamiltonian expectation value as an energy functional of the Jastrow function and the bare density-distribution functions introduced and calculated in our earlier publications; a simple, first-order energy functional is derived and detailed formulas for the higher-order contributions are provided. Our energy functional does not suffer the divergence as most coupled-cluster calculations often do when applying to Hamiltonians with hardcore potentials. We also discuss possible applications of our technique, including applications to strongly correlated fermion systems.
The reduced density matrices of quantum gases are studied by means of a Wiener integral representation described in a previous paper. They are shown to satisfy a cluster property in the form of an absolute integrability condition of the natural quantum analogues of the Ursell functions, considered as functions of the differences of their arguments. Use is made of the natural transposition to the quantum case of the algebraic formalism introduced by Ruelle in the classical case. By-products are two results on the signs of the coefficients of the Mayer expansion, in the case of Maxwell-Boltzmann and Fermi-Dirac statistics, respectively.
In order to investigate the many particle system by the variational principle, the cluster development method proposed by Jastrow is systematically formulated. It is a direct generalization of the Hartree method, in which two particle functions are included in the trial function to represent the correlations between particles. The energy expectation value is expanded in a series, each term of which is expressed in cluster integrals. It is found that the series converges rapidly when the correlations are marked only for small particle separations, so that this method is expected to be useful especially for the system of particles interacting through short range potentials with repulsive cores.