Integral-equation theories and Mayer-sampling Monte Carlo: a tandem approach for computing virial coefficients of simple fluids

Mayer-sampling Monte Carlo (MSMC) has enabled computation of higher-order virial coefficients than previously possible for a variety of potential models, but it is not required for computation of the entire virial coefficient for models that are spherically symmetric: approximations that result from the hypernetted-chain (HNC) or Percus–Yevick (PY) integral-equation theories in conjunction with the compressibility equation (c) or virial equation (v) can be computed quickly by fast Fourier transforms. For the fourth and fifth virial coefficients of the Lennard–Jones potential (with parameters σ and ε), we demonstrate that the corrections to each of the four approximations (HNC(c), HNC(v), PY(c), and PY(v)) are faster to compute to a desired precision by MSMC than the full coefficient itself, with the exception of the PY(v) correction at fifth order, and that the optimal decomposition with regard to precision can be identified using a fraction of the steps required to obtain precise virial coefficients. At reduced temperatures kT/ε greater than 4, the PY(c) correction is fastest to compute by MSMC at both fourth and fifth orders. For lower temperatures, the HNC(v) decomposition is most efficient at fourth order, while the HNC(c) decomposition is most efficient at fifth order. These results are specific to the Lennard–Jones potential, but the method for determining the optimal decomposition is applicable to any spherically symmetric potential.