On-the-Fly Second Virial Coefficients

A simple but approximate algorithm is described for computing second virial coefficients based on equilibrated molecular configurations that may be generated during any Monte Carlo or molecular dynamics simulation. The algorithm uses simple quadrature based on sampling every binary pair in the configuration and moving their center-center distances from zero to infinity. Comparisons are made in the literature results using more sophisticated sampling and integration for n-alkanes of ethane through n-dodecane. Accuracy is within the error bars determined by block averaging, and temperature effects can be inferred using a single configurational temperature, including perturbative virial coefficients. Predictably, the accuracy is best at the configurational temperature and when the configurational density is lowest. More notably, good accuracy is achieved from configurations at intermediate densities, and the trend at high density conveys valuable insight about conformational changes. The algorithm is simple enough to assign as a homework problem in an introductory molecular modeling course and reinforces the elementary knowledge of pairwise potentials among multisite molecules, numerical integration, and conformational averaging. The result is also quite valuable on its own merits, especially considering thermodynamic integration to compute phase equilibria. Additionally, the incidental data derived from the computation can provide simple but meaningful insights into the nature of multisite interactions, as demonstrated by relating the Mayer averaged potential to an effective Mie potential. Altogether, the argument is made that the computation of the second virial coefficient should be considered to be a routine metric of any molecular simulation, such as the radial distribution function, pressure, or energy.