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Progress of Theoretical Physics

Cluster Expansion Technique in the Many-Fermion Systems. I

Ki-ichi Nakamura

Published Dec 1, 1960• 5 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.24.1195

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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.
Cluster development may furnish a powerful device for the calculation of the expectation values of the observables of a many-fermion system with respect to dynamically correlated state vectors. The generalized normalization integral, a generating function for the required expectation values, is defined, and four of the many possible decompositions of this function into cluster integrals are explored. Two of these decompositions are slight extensions of the conventional ones of Iwamoto and Yamada and Aviles, Hartogh, and Tolhoek. The other two are product decompositions, leading to new ``factor-cluster'' formalisms. A factor-cluster expansion is applied to the evaluation of the n-particle spatial distribution function.
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 cluster variation method, which was originally devised for quantum-mechanical many-body systems at zero temperature, is extended to treat finite-temperature systems. The new method is based on a variational principle that the partition function is the maximum of a certain functional, APF (approximate partition function). This method includes the conventional cluster variation method at zero temperature as a limiting case. A procedure is given for calculating APF by cluster expansion for fermion systems. In this procedure, use is made of single-particle energies which should be determined self-consistently. The non-orthogonality problem which comes from the presence of correlations among particles is solved by the use of linear combinations of (generalized) Jastrow wave functions. Further development of this method is briefly discussed.
A simple alternative derivation of conventional cluster expansion formulas of Iwamoto and Yamada and others for the expectation value ≪H > of the energy of a many-Fermion system is offered. The assumed trial function is of the general form FΦp/(FΦp, FΦp)1/2, where Φp is a Slater determinant of appropriate single-particle functions and the factor F is attached to account for the possibility of strong interparticle correlations. The proposed scheme is applicable to finite as well as infinite systems and is easily generalized to the calculation of non-diagonal as well as diagonal matrix elements, with respect to the members of the set {FΦp/(FΦp, FΦp)1/2}. Explicit results for ≪H > are displayed to second cluster order, contributions being classified according to their assigned order of magnitude in a correlation parameter ξ=|Nω/Ω| of the sort introduced by Iwamoto and Yamada and Clark and Westhaus.
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.