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IN THE THERMODYNAMIC PERTURBATION SCHEME

C. Aguilera-Navarro, Christina Keller, MANUEL DE LLANO, R. Guardiola

Published Jan 1, 1988• 0 citations
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Abstract

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.

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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.
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.
The authors investigate the calculational scheme for an eventual first-principles description of the ground-state energy of the many-boson Schrodinger equation. They employ a model 4 He intermolecular pair potential consisting of a hard core surrounded by an attractive rectangular well. The study is based on the well known low-density expansion for such a system, where three coefficients are known. The first corrections to the fourth coefficient are determined on the basis of the rearranged series being a thermodynamic perturbation expansion, namely one representing perturbation not about the ideal gas but about the hard-sphere fluid. Pade approximants, and variations thereof, are employed to extrapolate to physical regimes in both resulting series, in density as well as in the attractive coupling constant.
The London ground-state energy formula as a function of number density for a system of identical boson hard spheres, corrected for the reduced mass of a pair of particles in a "sphere-of-influence" picture, and generalized to fermion hard-sphere systems with two and four intrinsic degrees of freedom, has a double-pole at the ultimate regular (or periodic, e.g., face-centered-cubic) close-packing density usually associated with a crystalline branch. Improved fluid branches are constructed based upon exact, field-theoretic perturbation-theory low-density expansions for many-boson and many-fermion systems, extrapolated to intermediate densities via Padé and other approximants, but whose ultimate density is irregular or random closest close-packing as suggested in studies of a classical system of hard spheres. Results show substantially improved agreement with the best available Green-function Monte Carlo and diffusion Monte Carlo simulations for bosons, as well as with ladder, variational Fermi hypernetted chain, and so-called L -expansion data for two-component fermions.
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.