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Physical Review A

Energy of a Boson Fluid with Lennard-Jones Potentials

Malvin H. Kalos

Published Jul 1, 1970• 158 citations
Publisher

Abstract

A more powerful version of a numerical method published before is given which permits the integration of the Schr\"odinger equation for a large ensemble of particles. A new method for obtaining quantum expectation values is outlined. Numerical results are tabulated for the energy of the ground state of a system of 32 bosons interacting by Lennard-Jones potentials and with periodic boundary conditions. This is a commonly used model of liquid ${\mathrm{He}}^{4}$. The present results are substantially lower and closer to experiment than variational estimates based on a Jastrow-type wave function.

DOI
10.1103/physreva.2.250

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The thermodynamic perturbation scheme for the very successful construction of classical liquid equations of state is applied in the quantum regime to the ground-state energy of a many-boson system interacting pairwise via a Lennard-Jones inter­ molecular potential. The ensuing density and attractive-coupling double expansion for the energy is series-analyzed with Pade techniques, as well as generalizations thereof. Results through sixth order compare with the Green Function Monte Carlo simula­ tions for the same system at the one-percent level of accuracy.

•Jan 1, 1988•0 citations
A more powerful version of a numerical method published before is given which permits the integration of the Schr"odinger equation for a large ensemble of particles. A new method for obtaining quantum expectation values is outlined. Numerical results are tabulated for the energy of the ground state of a system of 32 bosons interacting by Lennard-Jones potentials and with periodic boundary conditions. This is a commonly used model of liquid He4{\mathrm{He}}^{4}He4. The present results are substantially lower and closer to experiment than variational estimates based on a Jastrow-type wave function.
The perturbation theoretic procedure is developed for a system of strongly interacting particles, by replacing the singular potential with an equivalent operator. As an application of the method, an expression for the ground state energy of a Bose system, which corresponds to Lenz's formula for the hard-sphere system, is obtained for the Lennard-Jones potential. The cohesive energy of liquid He4 is calculated and is shown to be about twice as large as the experimentally observed one. Some difficulties arising from the negative value of the scattering length are discussed, and it is suggested that the assumption, usually taken for Bose system, that nearly all the particles are in the single particle state with momentum 0 should be altered for the actual liquid He4.
A trial wave function describing the ground state of a quantum system of NNN interacting bosons is written in the Jastrow form, a product of pair functions. With the interaction potential chosen to represent liquid He4{\mathrm{He}}^{4}He4, and with the parametrized form of the pair function chosen to include a long-range term which has been found necessary to represent the zero-point motion of the long-wavelength density oscillations, a variational calculation has been performed using a new approximate integral equation for the pair distribution function. This equation, which can also be used for classical fluids, is found to be more accurate for repulsive potentials than the Percus-Yevick equation and comparable to (but much simpler than) the Percus-Yevick 2 equation. The essential results are that including the zero-point motion in the wave function tends to lower the energy, raises the equilibrium density, corrects the behavior of the structure function and the momentum distribution of the particles in the low-wave-number region, and slightly decreases the Bose-Einstein condensate fraction. The value of the lower limit on the wavelength of the density oscillations was determined variationally to be about three interparticle spacings.
The exact, low-density many-boson expansion for the ground-state energy is rearranged in a quantum-thermodynamic perturbation series. Extrapolation to physical densities is implemented order by order via Pad\'e-type approximants. Several ways of splitting the pair Lennard-Jones (LJ) or Aziz potentials are considered, including the Barker-Henderson (BH) and Weeks-Chandler-Andersen (WCA) schemes familiar from classical fluid studies. Results obtained using the LJ interaction with WCA splitting, when compared with benchmark simulation data, agree excellently in energy and density, but only moderately so in sound velocity. For the Aziz potential, however---perhaps because of a softer core and/or its ability to support a two-particle bound state---, neither potential splitting (BH or WCA) gave results of the LJ quality, but a method intermediate between BH and WCA proves to be superior. Our results are also compared with other recent alternative calculations, both variational and perturbative.
"A theory is developed to allow the systematic calculation of the properties of quantum fluids given the microscopic two-body potential. Physical arguments are used to construct a correlation operator that approximately transforms the states of a non-interacting system into the corresponding states of the interacting-system. The theory can be regarded as a microscopic realization of Landau's assumption of a one-to-one correspondence between the non-interacting and interacting states. It is variational since the correlation operator is determined by minimizing E(p) the expectation value of the ground-state energy. We first give a survey of various other many-body theories including the simpler Jastrow variational method. Differential equations are derived to calculate the correlation operator and the related Jastrow correlation"" function. A comparison of results in several Bose systems with various methods is given. The Fermi hypernetted chain integral equations are generalized to allow the calculation of E(p) in Fermi systems, using the correlation operator wave function. The method is applied to liquid 3He and the v2-model of nuclear matter. In 3He the correlation operator gives results in much closer agreement with experiment than that with the Jastrow wave function. A new set of hypernetted chain equations are then derived to allow the calculation of the excited st;ate energies in Bose systems. The theory is then applied to the calculation of the single quasi-particle excitation spectrum of liquid 4He. The spectrum is found to be a stringent test on the form of the correlation operator, deficiencies in the correlation operator giving corresponding errors in the spectrum. Good results are obtained for the density dependence of the roton energy. A variational principle for the free energy is used to generalize the correlation operator technique to finite temperatures. Results are given for the finite temperature properties of the v8-model of nuclear matter."
The thermodynamic perturbation scheme for the very successful construction of classical liquid equations of state is applied in the quantum regime to the ground-state energy of a many-boson system interacting pairwise via a Lennard-Jones inter­ molecular potential. The ensuing density and attractive-coupling double expansion for the energy is series-analyzed with Pade techniques, as well as generalizations thereof. Results through sixth order compare with the Green Function Monte Carlo simula­ tions for the same system at the one-percent level of accuracy.