Wigner-Kirkwood Approximations

Wigner-Kirkwood related expansions follow by taking the approximations further. We assume (11.30) as a starting point and linearize the exponential of the correction term for the potential... 

Semi-classical approximations. In classical formulae, quantum effects may be accounted for to low order. For example, the the Wigner-Kirkwood expansion of the pair distribution function may be used [136, 302],... 

In the sum over states formula, excited states for the vibrational modes need to be included up to convergence. A more convenient integral expression is provided by classical or semiclassical theories. At high temperatures and low frequencies, the vibrational motion behaves increasingly classically and the semiclassical Wigner-Kirkwood expression is an excellent approximation to the quantum partition function [78] for low-frequency vibrations at pyrolysis temperatures. The semiclassical... 

In recognition of the quantum nature of the intermolecular O H hydrogen bond, interest exists in investigating ways of including a quantum mechanical correction of the interaction potential and its effect on solvation. The numerous techniques for handling quantum effects in water simulations include explicit wavefunction techniques, the Wigner-Kirkwood expansion of the classical potential surface, and approximate corrections to... 

A first step toward quantum mechanical approximations for free energy calculations was made by Wigner and Kirkwood. A clear derivation of their method is given by Landau and Lifshitz [43]. They employ a plane-wave expansion to compute approximate canonical partition functions which then generate free energy models. The method produces an expansion of the free energy in powers of h. Here we just quote several of the results of their derivation. 

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