Approximations for short wavelengths

APPROXIMATIONS FOR SHORT WAVELENGTHS 3.1. Coupled quantum and classical equations 

The PWT equations can be approximated in a semiclassical limit, obtained to lowest order in hA, so that for two operators A and B, 

An equation for y(P, R, t) follows from the trace over quantum variables, 

It is possible to further simplify the equations taking advantage of the quasiclassical nature of the P and R variables, by introducing effective potentials or forces to guide their motion through phase space, by the approximation cq, Av) — V, A, with V P,R, t) an effective potential function relating to the coupling Hamiltonian of quantal and classical variables. This leads to a new potential V(P,P, t) = ViR) + V P,R, t), and a new classical Hamiltonian H i P,R,t) =H. i P,R) + V P,R,t), so that the equation for Aw becomes 

The same approximation can be made in the equations of motion for y to obtain 

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Ground-state densities from electron propagators - optimized Thomas-Eermi approximation for short wavelength modes. J. Chem. Phys. 92, 6687-6696 (1990). 

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