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Physical Review

Variational Method for the Ground State of Multispecies Quantum Fluids

Kenneth R. Allen, Tucson Dunn

Published Jun 5, 1968• 4 citations
Publisher

Abstract

A series expansion for the ground-state energy of a multispecies quantum fluid is obtained. This expansion is derived by assuming that the ground-state wave function for the fluid can be approximated by the symmetrized ground-state wave function for a mixture of quantum ideal gases times a product of pair functions. The lowest-order approximation to the energy is minimized with respect to variations in these pair functions in order to obtain an approximation to the ground-state wave function for the system. In principle, this variational procedure can be carried out to any desired order in the energy expansion, and the machinery for doing so is explicitly exhibited. The results of this variation in the lowest order are applied to a two-species system consisting of electrons and nuclei. The results are consistent at high densities.

DOI
10.1103/physrev.170.293

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A series expansion for the ground-state energy of a multispecies quantum fluid is obtained. This expansion is derived by assuming that the ground-state wave function for the fluid can be approximated by the symmetrized ground-state wave function for a mixture of quantum ideal gases times a product of pair functions. The lowest-order approximation to the energy is minimized with respect to variations in these pair functions in order to obtain an approximation to the ground-state wave function for the system. In principle, this variational procedure can be carried out to any desired order in the energy expansion, and the machinery for doing so is explicitly exhibited. The results of this variation in the lowest order are applied to a two-species system consisting of electrons and nuclei. The results are consistent at high densities.
The ground state of a two-dimensional charged quantum system is studied using the method of correlated-basis functions. For particles obeying the Bose-Einstein statistics we compute the ground-state energy variationally for all particle densities using the Bijl-Dingle-Jastrow trial wave function. The Bose ground-state energy and wave function so determined are used next to study the Fermi system. We compute the Fermi ground-state energy using a cluster expansion approach, which is again valid for all particle densities, and find that the ground state is paramagnetic. While these calculations provide variational upper bounds to the ground-state energies, we also establish independent lower bounds to the ground-state energies.
The variational method for calculating energy of quantum fluids, and its applications to the Bose liquid 4He^{4}\mathrm{He}4He, Fermi neutron gas, and liquid 3He^{3}\mathrm{He}3He are discussed. The correlation functions are parametrized by their healing distance, and can depend on the states occupied by the correlated particles in the model wave function. They are calculated by constrained variation of the lowest-order contributions. The healing distance has a prescribed value in lowest-order calculations, whereas it is sufficiently large in hopefully exact energy calculations. The many-body cluster contributions in Bose fluids are summed with successive approximations of an integral equation due to van Leeuwen et al. A simple diagrammatic cluster expansion is presented for Fermi liquids, and its direct diagrams are summed with the integral equation. The contribution of exchange diagrams is shown to decrease rapidly with the number of exchanges, and their sums are truncated after the energy has converged to within a few percent.
Abstract A modification of the Rayleigh—Ritz variational principle is described which makes possible a calculation of the energy, wave function, and pair distribution function f12≡f(x1,x2) of a Bose fluid, such as liquid 4He, at absolute zero. The assumptions made are: (i) two-body interactions with potential Uij, (ii) trial wave functions of the form f12≡f(x1,x2) and (iii) the Kirkwood ‘superposition' approximation. Under these approximations, the expectation energy is E=12n2∫∫d3X1d3X2f12U12−h2m−1(∇18ϕ12)+(∇1ϕ12)2+n∫d3X2f13f23ϕ12⋅∇1ϕ13, where n ≡ N/V. It is shown here that making E stationary with respect to independent variations in f and ɸ corresponds to simultaneously applying the ordinary Rayleigh-Ritz principle and solving the Born-Green-Yvon integral equation for f. The method is illustrated by reproducing Bogolyubov’s results for the case where U is small. The case where U is large must be dealt with numerically, but transformations for simplifying the computations are given here.
Under the assumption that the ground-state wave function is expressed as a product of pair functions, the expectation value of the ground-state energy is rigorously expressed in terms of the pair distribution function. An integro-differential equation is derived for the pair distribution function from the variational principle that the expectation value of the energy is minimum. It is pointed out that the expression for the energy and the integro-differential equation become identical with those derived by Abe if the hyper-netted chain (HNC) approximation is introduced. It is proved that the HNC approximation is compatible with the virial theorem. The nature of the formulas is investigated by applying them to some cases that have been investigated by other theories. It is suggested that the present theory may in general predict the phonon-like excitations.
A new variational approach is proposed for the study of the many-body problem with the purpose of combining the advantages of perturbation-theoretic and variational approaches, while avoiding their disadvantages. A trial wave function is constructed so that a general expansion for the expectation value of an arbitrary physical quantity can be derived. Each term in the expansion can be calculated with the use of the techniques of Feynman diagrams, but in contrast with usual perturbation-theoretic approaches, every term in the expansion always converges. The procedure for applying the method to real systems is illustrated by a calculation of the ground-state energy of the electron gas at high densities.
We have used the variational and diffusion quantum Monte Carlo methods to calculate the energy, pair-correlation function, static structure factor, and momentum density of the ground state of the two-dimensional (2D) homogeneous electron gas. We have used highly accurate Slater-Jastrow-backflow trial wave functions and twist averaging to reduce finite-size effects where applicable. We compare our results with others in the literature and construct a local-density-approximation exchange-correlation functional for 2D systems.