Free Energy Perturbation Theory at Low Temperature

The perturbative expansion introduced by Zwanzig [R. W. Zwanzig, J. Chem. Phys. {\bf 22}, 1420 (1954)] expresses the difference in Helmholtz free energy between a system of interest and that of a reference system as series of cumulants $\kappa_n$ of the potential energy difference between the two systems. This expansion has attractive features as a method for obtaining absolute free energies for {\it ab initio} potential energy surfaces. The series is formally a power series in $\beta=1/T$, suggesting that its usefulness may be limited to high temperature. However, for smooth reference potentials, the $T$-dependence of the $\kappa_n$ contributes to the convergence. A closed form expression is derived for the $\kappa_n$ to all orders for the case that both the system and reference potentials are harmonic. In this case $\kappa_n \propto T^n$ for $n \ge 2$ and the convergence of the series is independent of temperature. More realistic cases of liquid Cu and solid Al, with a $1/r^{12}$ and harmonic reference potential, respectively, are investigated numerically by evaluating the cumulants to third order using Monte Carlo integration. In all cases, the ratio of the third order to the second order term in the expansion is found to be $\sim 0.1$, indicating good convergence. Third order contributions to the free energy are typically a few meV/atom, and comparable to their statistical errors. The statistical error in the second order free energy of Al is 0.4 meV/atom with only 100 evaluations of the {\it ab initio} energy. These results suggest that the perturbation series allows for efficient and accurate evaluation of the free energy for condensed phases.