(68b) Improving the Efficiency of Virial-Coefficient Calculations: a Hybrid Approach Employing Integral-Equation Theories and Mayer-Sampling Monte Carlo

Mayer-sampling Monte Carlo (MSMC) has enabled calculation of high-order virial coefficients for a variety of potential models, including coefficients up to B₆ for several water models and B₈ for the Lennard-Jones (LJ) potential. A particularly appealing application of such results is the estimation of the critical point from the virial equation of state (VEOS), but virial coefficients of even higher order are required to apply the VEOS at densities approaching the critical density. Calculation of these higher-order coefficients remains impractical with the current method.

We present two improved virial-coefficient calculation methods appropriate for spherically symmetric potentials. In the first method, approximations to the virial coefficients, as defined by an integral-equation theory (IET), are computed quickly via a deterministic algorithm. Only the corrections to these approximations are computed by MSMC. For the LJ potential, we demonstrate that this hybrid approach is faster than MSMC alone for computing fourth and fifth virial coefficients.

However, the quality of the approximate virial coefficients diminishes with increasing order. In the deterministic method, an approximate virial coefficient is computed from an approximate density-expansion coefficient of the direct correlation function. In accordance with the IET, this approximate coefficient of the direct correlation function is computed from approximate coefficients of lower-order, such that the approximations compound. As the order increases, so does the magnitude of the required correction relative to that of the full virial coefficient, reducing the benefit of computing only the correction by MSMC.

We present another hybrid approach which can prevent the rapid deterioration of the quality of the approximations, and thus prove more useful for calculation of higher-order virial coefficients. In this second hybrid method, MSMC is used to correct the coefficient of the direct correlation function prior to computing the corresponding virial coefficient and to approximating a higher-order coefficient of the direct correlation function via the IET.

Extension of this hybrid method to more complicated potential models of practical interest is especially desirable as pure MSMC calculations for such models require considerably more time than those for the LJ potential. Water, for example, is often described with a multisite potential, in which interactions between sites on different molecules are spherically symmetric but the composite molecular interactions are not. Generalized integral-equation theories have been developed to compute approximate correlation functions for such potentials. We describe how we can employ these theories to generalize the hybrid method for virial-coefficient calculation.