An effective potential for calculating free energies. I. General concepts and approximations

We present a new analytical method to calculate free energies of molecules based on a high temperature expansion of an effective potential which is a function of the mean position and fluctuation of the coordinates of the molecule. We first introduce an effective potential Veff(x̄,β) which is a convex function of the mean position x̄ and whose sole minimum gives the free energy. Then, we define a convex effective potential Veff(x̄,Δ,β) which after variation over the mean fluctuation Δ yields Veff(x̄,β). We expand Veff(x̄,Δ,β) in a high temperature series. The first two terms of the series yield an effective diffused potential method to calculate free energies. The diffusion times are calculated via a variational principle. Besides free energies, the method yields an analytical annealing method for energy minimization.