Highly Anharmonic Crystal

The free energy of the constrained hard-sphere crystal is evaluated by Hermite expansion techniques applied to the Boltzmann factors of the partition function. An independent-oscillator comparison potential generates a Hermite polynomial basis set in which exp( − 12Σjυij) is expanded. A first approximation results from retention of the first term, the Gaussian average. A second approximation results from retention and exponentiation of the succeeding two terms, and corresponds to a coupled-oscillator potential. For both approximations the pressure deviation from the molecular dynamics pressure is close to the statistical irregularities of the latter, for densities such that the unconstrained solid is thermodynamically stable. The free energy in second approximation, significantly improved over the first, is excessive by 0.2kT at high densities and by up to 0.3kT at low and intermediate densities, where the constraint is required to stabilize the lattice. The methods, readily applicable to smooth potentials (and more promising for them) are similar to those earlier applied to polymer theory. Related methods, based on expansions of the potential itself, have been devised by Koehler and others for weak or moderate crystal anharmonicity.