Wigner correlation functional

This equation gives the dynamics of the quantum-classical system in terms of phase space variables (R, P) for the bath and the Wigner transform variables (r,p) for the quantum subsystem. This equation cannot be simulated easily but can be used when a representation in a discrete basis is not appropriate. It is easy to recover a classical description of the entire system by expanding the potential energy terms in a Taylor series to linear order in r. Such classical approximations, in conjunction with quantum equilibrium sampling, are often used to estimate quantum correlation functions and expectation values. Classical evolution in this full Wigner representation is exact for harmonic systems since the Taylor expansion truncates. 

R. M. Lynden-Bell and A. J. Stone, Reorientational correlation functions, quaternions and Wigner rotation matrices, Molec. Simul., 3 (1989), 271. 

Equilibrium time correlation function expressions for transport properties can be derived using linear response theory [3]. Linear response theory can be carried out directly on the Wigner transformed equations of motion to obtain the transport properties as correlation functions involving Wigner transformed quantities. Alternatively, we may carry out the linear response analysis in terms of abstract operators and insert the Wigner representation of operators in the final form for the correlation function. We use the latter route here. 

We see that the correlation functions have a rather complex form when expressed in terms of Wigner transformed variables, involving exponential operators of the Poisson bracket operator. 

Linearization methods start from a path integral representation of the forward and backward propagators in expressions for time correlation function, and combine them to describe the overall time evolution of the system in terms of a set of classical trajectories whose initial conditions are sampled from a quantity related to the Wigner transform of the quantum density operator. The linearized expression for a correlation function provides a powerful tool for describing systems in the condensed phase. The rapid decay of... 

To implement the linearized path integral formulation for time correlation functions the initial density operator must be Wigner transformed in the bath variables while it remains an operator in the quantum subsystem space. In the calculations presented below we assume that the system and bath do not interact initially. Consequently total probability density at t = 0 is of the form... 

As for the exchange functionals, local correlation functionals reproduce only in part the total correlation energy and GGA corrections are in order. In particular the correlation functional proposed by Lee, Yang and Parr (LYP) [71] embodies a Wigner-like functional (eqn 30) as the local contribution. The analytical expression of the LYP functional is ... 

G. J. Martyna, J. Chem. Phys. (in press, 1996). In this paper, an effective set of molecular dynamics equations are specified that provide an alternative path-integral approach to the calculation of position and velocity time correlation functions. This approach is essentially based on the Wigner phase-space function. For general nonlinear systems, the appropriate MD mass in this approach is not the physical mass, but it must instead be a position-dependent effective mass. 

The original CS correlation functional was proposed as an electron correlation correction for the Hartree-Fock method in 1975. Assuming that the volume of the region where electrons are excluded (excluded volume) is proportional to Wigner s excluded volume (Wigner 1934 Wigner and Seitz 1933), the following equation... 

Still, a Fade approximant for the gradient-corrected Wigner-type exchange-correlation functional exists and it was firstly formulated by (Rasolt Geldar, 1986) with the working form (Lee Bartolotti,... 

The variational solution of the decoupled equation of motion (Eq. Ills) for the Wigner distribution function might serve as a starting point for further studies of exchange and correlation in the dielectric function. Its connection with several other approximations has been examined, showing that many of them are particular cases or additional approximations to this variational approach. The improvement upon the RPA from dynamical exchange effects, and the fact that all checked sum rules are satisfied, gives... 

The basic problem associated with elucidating a relaxation mechanism from experimental data is most clearly seen in terms of the time-dependent conditional orientational distribution function p(0,t/l2Q,0). The quantity p(0,t/0Q,0)d0d0Q is the probability of obtaining the orientation of the body (molecule, chain segment) in the element of solid angle dO around Q at time t given its orientation was in dOo around JIq at t = 0. We may expand the distribution function in terms of the elements Dj jj(J2) of Wigner rotation matrices and the orientational time-correlation functions... 

Explicit expressions for ( )q 2 given as Cartesian tensors in [91] and in terms of Wigner rotational matrices in [90]. The former seem more appropriate for use in simulation. The isotropic (i=o) CILS spectrum is the Fourier transform of the correlation function... 

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