Semiclassical procedure

The continuation of the strategy presents at this point a bifurcation. The solute M may be described with a semiclassical procedure similar to that used for solvent molecules, or with a QM approach. The first method is often called classical (or semiclassical) MM description [3], the second a combined QM/MM approach [4], The physics of the first method is rather elementary, but notwithstanding this it opened the doors to our present understanding of the solvation of molecules. The second method is markedly more accurate, because the QM description of the solute has the potential of taking into account subtler solvent effects, such as the solvent polarization of the solute electronic polarization and the changes in geometry within M. 

Semiclassical procedures which give results in quantitative agreement with the exact quantum results over the whole angular range have been published only recently (Berry, 1966 Miller, 1968 Berry, 1969 Mullen and Thomas, 1973 Mount, 1973). Therefore we start with a description of such a uniform semiclassical approximation (Section III.A). Next, all cross section features are summarized. Their dependence on the reduced energy and reduced wave number as well as their sensitivity to the potential are discussed in Section III. B. Finally, the influence of other than spherically symmetric interactions such as anisotropic potentials occurring in atom-molecule scattering or any kind of inelasticities and chemical reactions are studied (Section III.C). 

Semiclassical Procedures for the Determination of the Potential The angular momentum dependence Starting with the expression of the JWKB-phase shift (9) and using the transformation (Sabatier, 1965 Vollmer and Kruger, 1968 Vollmer, 1969)... 

In some cases P/j may be obtained experimentally from low-density gas-phase determinations of vibrational relaxation rates (although sometimes it is necessary to extrapolate downward to liquid state temperatures). At low densities (t ) can be obtained from gas kinetic theory collisional cross-sections once a radius has been chosen, while t, can be found experimentally. Their ratio gives P j, the transition probabihty per collision. Purely theoretical calculations of P j can be carried out in principle using quantum scattering theory, although this becomes difficult for molecules of even moderate size. A number of semiclassical procedures have been proposed among the most popular is the SSH model, which gives ... 

In the case of coherent laser light, the pulses are characterized by well-defined phase relationships and slowly varying amplitudes (Haken, 1970). Such quasi-classical light pulses have spectral and temporal distributions that are also strictly related by a Fourier transformation, and are hence usually refered to as Fourier-transform-limited. They are required in the typical experiments of coherent optical spectroscopy, such as optical nutation, free induction decay, or photon echoes (Brewer, 1977). Here, the theoretical treatments generally adopt a semiclassical procedure, using a density matrix or Bloch formalism to describe the molecular system subject to a pulsed or continuous classical optical field, which generates a macroscopic sample polarization. In principle, a fully quantal description is possible if one represents the state of the field by the coherent or quasi-classical state vectors (Glauber, 1965 Freed and Villaeys, 1978). For our purpose, however. 

We are able to define a multipole susceptibility A m,j) of particle in a similar manner to the semiclassical treatment presented in Chapter 4. On substituting the quantum mechanical susceptibilities (7.48) for the respective classical terms, we can find the dispersion energy between particles 1 and 2 on strictly similar lines to the semiclassical procedure. 

Approximations have been reviewed in the case of short deBroglie wavelengths for the nuclei to derive coupled quantal-semiclassical computational procedures, by choosing different types of many-electron wavefunctions. Time-dependent Hartree-Fock and time-dependent multiconfiguration Hartree-Fock formulations are possible, and lead to the Eik/TDHF and Eik/TDMCHF approximations, respectively. More generally, these can be considered special cases of an Eik/TDDM approach, in terms of a general density matrix for many-electron systems. 

Another very good example of the mapping procedure, which can be practiced to design a semiclassical intramolecular logic gate molecular circuit, is the nontrivial... 

In the previous discussion the semiclassical separation of particles and antiparticles employed projection operators and the associated subspaces of the Hilbert space. By suitable choices of bases such a separation can also be constructed with the help of unitary operators rotating the Hamiltonian into a block-diagonal form. Such a procedure is closely analogous to the Foldy-Wouthuysen transformation that provides a similar separation in a non-relati-vistic limit. A (unitary) semiclassical Foldy-Wouthuysen transformation Usc rotates the Dirac-Hamiltonian Hd into... 

The process of formation of a bubble having a critical radius, can be computed using a semiclassical approximation. The procedure is rather straightforward. First one computes, using the well known Wentzel-Kramers-Brillouin (WKB) approximation, the ground state energy Eq and the oscillation frequency //() of the virtual QM drop in the potential well U JV). Then it is possible to calculate in a relativistic framework the probability of tunneling as (Iida Sato 1997)... 

A semiclassical description is well established when both the Hamilton operator of the system and the quantity to be calculated have a well-defined classical analog. For example, there exist several semiclassical methods for calculating the vibrational autocorrelation function on a single excited electronic surface, the Fourier transform of which yields the Franck-Condon spectmm [108, 109, 150, 244]. In particular, semiclassical methods based on the initial-value representation of the semiclassical propagator [104-111, 245-248], which circumvent the cumbersome root-search problem in boundary-value-based semiclassical methods, have been successfully applied to a variety of systems (see, for example, Refs. 110, 111, 161, and 249 and references therein). The mapping procedure introduced in Section VI results in a quantum-mechanical Hamiltonian with a well-defined classical limit, and therefore it... 

It should be noted, however, that the limit 0 is only a formal procedure, which does not necessarily lead to a unique or correct semiclassical limit. In the case of the mapping formulation, this is because of the following reasons (i) For a given molecule, the frequencies f)mi(x) will in general also depend in a nontrivial way on h. (ii) A slowly varying term may as well be included in the stationary phase treatment [147]. (iii) As indicated by the term resulting from the commutator = 8 , the effective action constant ... 

We want to mention here that the application of the near-nuclear corrections could have been performed with the complete relativistic functional, and we have utilized the semi-relativistic expressions just for simplicity and for testing them. For not large Z, the remaining errors above mentioned should be addressed to limitations of the semiclassical approach and of the procedure utilized for the near-nuclear corrections, rather than to the truncation of the expansion in powers of... 

Results of similar accuracy as relativistic TFDW are found with a simple procedure based on near-nuclear correction which leave space for further improvements. For the reasons mentioned at the end of previous section the direct way to improve the present approach seems to be the refinement of the near nuclear corrections, a problem that we have just tackled with success in the non-relativistic framework [31,32]. The aim was to describe the near-nuclear region accurately by means of using the quantum mechanical exact asymptotic expression up to of the different ns eigenstates of Schodinger equation with a fit of the semiclassical potential at short distancies to the exact asymptotic behaviour (with four terms) of the potential near the nucleus. The result is that the density below Tq becomes very close to Hartree-Fock values and the improvement of the energy values is large (as an example, the energy of Cs+ is improved from the Ashby-Holzman result of-189.5 keV up to -205.6, very close to the HF value of -204.6 keV). This result makes us expect that a similar procedure in the relativistic framework may provide results comparable to Dirac-Fock ones. 

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