Mean field approximation advantage

It may seem that the prospeets are bleak for the GvdW approach to electrolytes but, in fact, the reverse is the ease. We need only follow Debye and Hiickel [18] into their analysis of the sereening meehanism, almost as successful as the van der Waals analysis of short-range fluids, to see that the mean-field approximation can be applied to the correlation mechanism with great advantage. In fact, we can then add finite ion size effects to the analysis and thereby unify these two most successful traditional theories. 

In the traditional theory of the cooperative JT effect, its significant part is one-center JT problem in a low-symmetry mean field (see the last paragraph of Sect. 2.2). In particular, it includes the eigenvalue problem for the Hamiltonian, similar to (7), operating in an infinite manifold of vibrational one-center states. Compared to this relatively complex step, in the OOA, the mean-field approximation is much simpler. In the OOA, one has to solve just a finite-size matrix (2 x 2 in this case) or, for other JT cases, a somewhat larger matrix but finite anyway. In the theory of the cooperative JT effect, this important advantage of the OOA allows to proceed farther than... 

The result of this approximation is that each mode is subject to an effective average potential created by all the expectation values of the other modes. Usually the modes are propagated self-consistently. The effective potentials governing die evolution of the mean-field modes will change in time as the system evolves. The advantage of this method is that a multi-dimensional problem is reduced to several one-dimensional problems. 

The concentration of salt in physiological systems is on the order of 150 mM, which corresponds to approximately 350 water molecules for each cation-anion pair. Eor this reason, investigations of salt effects in biological systems using detailed atomic models and molecular dynamic simulations become rapidly prohibitive, and mean-field treatments based on continuum electrostatics are advantageous. Such approximations, which were pioneered by Debye and Huckel [11], are valid at moderately low ionic concentration when core-core interactions between the mobile ions can be neglected. Briefly, the spatial density throughout the solvent is assumed to depend only on the local electrostatic poten-... 

The mapping approach outlined above has been designed to furnish a well-defined classical limit of nonadiabatic quantum dynamics. The formalism applies in the same way at the quantum-mechanical, semiclassical (see Section VIII), and quasiclassical level, respectively. Most important, no additional assumptions but the standard semiclassical and quasi-classical approximations are needed to get from one level to another. Most of the established mixed quantum-classical methods such as the mean-field-trajectory method or the surface-hopping approach do invoke additional assumptions. The comparison of the mapping approach to these formulations may therefore (i) provide insight into the nature of these additional approximation and (ii) indicate whether the conceptual virtues of the mapping approach may be expected to result in practical advantages. 

The mean-polarizability approximation, discussed in detail by Agranovitch,16 presents the same advantages (simplicity, arbitrary concentrations, etc.), and the same limitations as the average-locator approximation in particular, this theory provides two bands of persistence behavior for all values of the parameters. This may be checked on the example of a cubic crystal, where the local field has a very simple form The modes of the mixed crystal are given by... 

A proof of Eq. (14) is given in several places [27, 66-68]. The advantage of this equation is that it can be used to systematically improve the approximate Hamiltonian and the free energy F by optimizing a set of variable parameters or functions contained in Hq. Specific forms chosen for lead to the Self-Consistent Phonon (SCP) method, the Mean-Field (MF) method and the Time-Dependent Hartree (TDH) or Random-Phase Approximation (RPA). 

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