Wigner-Kirkwood expansion

Quantum effects, if small, may be taken into account by expanding the functions W in powers of h. This can be accomplished by using the Wigner-Kirkwood expansion to calculate the traces which appear in Eqs. 2.35 and 2.36. In fact, if indicates the Hamiltonian of a system of... 

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 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... 

F. Barocchi, M. Neumann, and M. Zoppi, Phys. Rev. A, 36,2440 (1987). Wigner-Kirkwood Expansion Calculation of Almost Classical Static Properties of a Lennard-Jones Many-Body System. 

In particular, attention will be paid in this chapter to those based on the efficient variational approaches derived from the PI formulation, of which the first was proposed in Refs. 12 and 13. Other semiclassical approaches, such as those based on the Wigner-Kirkwood expansion, also showed early their value in this context of quantum fluids [15-17]. However, the overall performance of these latter approaches, even in sophisticated forms, is rather poor and only some comments on them will be made. 

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... 

The different techniques utilized in the non-relativistic case were applied to this problem, becoming more involved (the presence of negative energy states is one of the reasons). The most popular procedures employed are the Kirznits operator conmutator expansion [16,17], or the h expansion of the Wigner-Kirkwood density matrix [18], which is performed starting from the Dirac hamiltonian for a mean field and does not include exchange. By means of these procedures the relativistic kinetic energy density results ... 

The superscript vib is omitted with the understanding that the partition functions and particle densities correspond to vibrations. The particle density is expressed as a Wigner-Kirkwood perturbation expansion [86,87] in terms of a small parameter A ... 

Cartesian displacement for mode j and N is the number of atoms. The second-order expansion in A and fourth-order expansion in h of the Wigner-Kirkwood particle density is written as... 

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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