Cluster integrals and virial coefficients for realistic molecular models

We present a concise, general, and efficient procedure for calculating the cluster integrals that relate thermodynamic virial coefficients to molecular interactions. The approach encompasses nonpairwise intermolecular potentials generated from quantum chemistry or other sources; a simple extension permits efficient evaluation of temperature and other derivatives of the virial coefficients. We demonstrate with a polarizable model of water. We argue that cluster-integral methods are a potent yet underutilized instrument for the development and application of first-principles molecular models and methods.

Figure 1

Graphical representation of (3c). Green lines and faces (shaded polygons, left-hand side of equation) represent $f_C$, and red (seen on right-hand side of equation) represents $f_Q$.

Figure 2

Graphs representing some of the terms that sum to give $f_B\left(8\right)$. Green regions and lines represent $f_C$ faces. The orange number above and left of each graph is its weight as given by (6). Beneath each graph is shown the functions to which it contributes (with $v$ ranges shown in square brackets, when needed).

Figure 3

Pressure-density isotherms of water for three supercritical temperatures. Approximant equation of state (AEOS) lines represent the parametric approximant [51] using values of $T_{\mathrm{c}}$ and ${\rho }_{\mathrm{c}}$ from Ref. [48]; curves are shown for A3–A6, but are not distinguishable on the plot. Pink lines labeled $\text{A}{6}^{*}$ show the A6 parametric approximant using $T_{\mathrm{c}}=647\mathrm{K}$ and ${\rho }_{\mathrm{c}}=0.37\mathrm{g}/{\mathrm{cm}}^3$. VEOS lines are shown only for the $T=800\mathrm{K}$ isotherm, and are truncated where they start to diverge; this is representative of behavior at other temperatures. Experimentally derived data are as represented by an accurate correlation [52, 53] (black dotted-dashed lines), and solid symbols are results of $NPT$ Monte Carlo simulations for GCPM water. Error bars indicate uncertainties at 68% confidence level, and for AEOS are smaller than the line thickness.