Estimation of the absolute internal-rotation entropy of molecules with two torsional degrees of freedom from stochastic simulations

A method of statistical estimation is applied to the problem of evaluating the absolute entropy of internal rotation in a molecule with two torsional degrees of freedom. The configurational part of the entropy is obtained as that of the joint probability density of an arbitrary form represented by a two-dimensional Fourier series, the coefficients of which are statistically estimated using a sample of the torsional angles of the molecule obtained by a stochastic simulation. The internal rotors in the molecule are assumed to be attached to a common frame, and their reduced moments of inertia are initially calculated as functions of the two torsional angles, but averaged over all the remaining internal degrees of freedom using the stochastic-simulation sample of the atomic configurations of the molecule. The torsional-angle dependence of the reduced moments of inertia can be also averaged out, and the absolute internal-rotation entropy of the molecule is obtained in a good approximation as the sum of the configurational entropy and a kinetic contribution fully determined by the averaged reduced moments of inertia. The method is illustrated using Monte Carlo simulations of isomers of stilbene and halogenated derivatives of propane. The two torsional angles in cis-stilbene are found to be much more strongly correlated than those in trans-stilbene, while the degree of the angular correlation in propane increases strongly on substitution of hydrogen atoms with chlorine.