Time dependent self consistent field limitations

For the case of intramolecular energy transfer from excited vibrational states, a mixed quantum-classical treatment was given by Gerber et al. already in 1982 [101]. These authors used a time-dependent self-consistent field (TDSCF) approximation. In the classical limit of TDSCF averages over wave functions are replaced by averages over bundles of trajectories, each obtained by SCF methods. 

The MFT equation of motion (25) can be derived in many ways, including the WKB approximation, the eikonal method, a (semi)classical time-dependent self-consistent field ansatz, density-matrix approaches, and the classical limit of algebraic quantization. Depending on the specific approach used, slightly different MFT schemes may result. For example, the classical force can be described either by the average of the quantum force as in Eq. (25) or by the derivative of the average quantum potential. 

However, if the correction terms are introduced as demonstrated above, it is just a way of approaching the limit in which the trial function is expressed as ff(r, t)X(R, t) with no restriction on the form of X(R, f), i.e., it is not for instance a GWP at all times We shall denote this limit the self-consistent field limit (SCF) or rather the time-dependent SCF limit (TDSCF). The reason for this designation is that the equations for the two wavefunctions are solved self-consistently, i.e., the TDSE for one mode involves the average interaction with the other and vice versa. Thus, if we insert a product type wavefunction r r, t)X R, t) in the TDSE, equation (7), using the expansion (1), multiply fi om the left with X(R, t) and integrate over R we get... 

Viscoelastic contact problems have drawn the attention of researchers for some time [2,3,104,105]. The mathematical peculiarity of these problems is their time-dependent boundaries. This has limited the ability to quantify the boundary value contact problems by the tools used in elasticity. The normal displacement (u) and pressure (p) fields in the contact region for non-adhesive contact of viscoelastic materials are obtained by a self-consistent solution to the governing singular integral equation given by [106] ... 

To make the discussion more quantitative, let us now consider the doping dependence of Rh(S,T) in terms of the f-f -/model using the saddle-point and relaxation time approximations, where FS and correlation effects are involved via the renormalized SB band (Eq. 33). As we have pointed out above, in our approach the SB quasi-particle band dispersion has to be determined in a self-consistent way at each doping level 5. This should be in contrast to the NZA SB mean-field approach to the t-f -/model of Chi and Nagi [61] where, in the 7 -> 0 limit, the calculation of transport properties is based on the simple replacement Sj — Sj = -2 (5[(cos x + cos ) + 2(r7r)cos cosAy] ofthe non-interacting band dispersion (Eq. 2). 

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