Complex-valued Hamiltonian route to probability densities in molecular solvents

For a pairwise additive energy variable <(Phi)over cap> in a molecular fluid, the associated probability density p(Phi)(beta phi)= [delta(beta phi-beta Phi)] has a Fourier transform that may be interpreted as the ratio of two partition functions, one of which is defined in terms of a complex-valued (c upsilon) Hamiltonian. As first shown by Loring, this relation may be recast in terms of a generalized c upsilon free energy A(Phi)(alpha) that is a function of a coupling parameter alpha. We formulate a c upsilon version of the XRISM integral equation to calculate A(Phi)(alpha). We apply the method to calculate the probability densities of the intermolecular binding energy for models of tetrahydrofuran and methanol. We also calculate the diabatic free energy profiles of symmetric and asymmetric model intramolecular charge transfer reactions in diatomic solutes dissolved in methyl chloride. Finally we consider the free energy profiles for a model of the ferrocene-ferrocenium electron exchange reaction in acetonitrile. For the models studied, we find that the c upsilon-XRISM results for the diabatic free energy profiles are in close agreement with the recently developed Surrogate Hamiltonian-Renormalized Dielectric Theory (SH-RDT), which is an integral equation based on a renormalized linear response formulation. The accuracy of these methods of calculating probability densities is limited by the use of the XRISM integral equation to calculate the correlation functions of the models. However, where possible, we find quite satisfactory agreement of the c upsilon-XRISM results with simulation results reported in the literature for the same models. In terms of the computing resources required for a given calculation, the integral equation methods (c upsilon-XRISM and SH-RDT) are considerably less demanding and faster than computer simulations.