Semiclassical polarizability theory

A multitude of semiempirical and semiclassical theories have been developed to calculate electron impact ionization cross sections of atoms and atomic ions, with relatively few for the more complicated case of molecular electron impact ionization cross sections. One of the earlier treatments of molecular targets was that of Jain and Khare.38 Two of the more successful recent approaches are the method proposed by Deutsch and Mark and coworkers12-14 and the binary-encounter Bethe method developed by Kim and Rudd.15,16 The observation of a strong correlation between the maximum in the ionization efficiency curve and the polarizability of the target resulted in the semiempirical polarizability model which depends only on the polarizability, ionization potential, and maximum electron impact ionization cross section of the target molecule.39,40 These and other methods will be considered in detail below. 

The first theoretical model of optical activity was proposed by Drude in 1896. It postulates that charged particles (i.e., electrons), if present in a dissymmetric environment, are constrained to move in a helical path. Optical activity was a physical consequence of the interaction between electromagnetic radiation and the helical electronic field. Early theoretical attempts to combine molecular geometric models, such as the tetrahedral carbon atom, with the physical model of Drude were based on the use of coupled oscillators and molecular polarizabilities to explain optical activity. All subsequent quantum mechanical approaches were, and still are, based on perturbation theory. Most theoretical treatments are really semiclassical because quantum theories require so many simplifications and assumptions that their practical applications are limited to the point that there is still no comprehensive theory that allows for the predetermination of the sign and magnitude of molecular optical activity. 

The second method follows the semiclassical perturbation approach to dispersion theory. However, it recognizes from the outset that the polarizabilities must be complex quantities and that electronic absorption bands are not shaped like 6 functions. This second approach has been carried through with an attempt at making allowances for the fact that E =t= E. However, as might be anticipated, the most general form of the expressions achieved by this method cannot be utilized until some specific assumption is made about E. ... 

Waals energy formulae. Our procedure produces an explicit model, derived from the wave function of the system, for the dynamical polarizabilities associated with the building blocks, which are bonds, lone pairs and in general localized electron pairs. Thus one arrives to the quantum chemical analogs of the quantum harmonic oscillators (QHO) appearing in the semiclassical theory of dispersion forces, elaborated within a RPA framework by Tkatchenko et al. [72, 73]. 

The microscopic (hyper)polarizabilities are studied by means of the so-called theory of the response functions which is of importance for aU molecular and cluster entities (Roman et al. 2006). The most commonly used approach in studying the linear and nonlinear optical properties of clusters is the so-caUed semiclassical one. According to this approach a classical treatment is used to describe the response of the cluster to an external field (radiation) while the system itself is treated using the lows and techniques of quantum mechanics. This is done by using a Hamiltonian which combines both of the above treatments ... 

Conversely, for slow collisions the combined system of incoming electron and target molecule has to be considered, leading in the exit channel to a full three-body problem. Quantum-mechanical (approximate) calculations are difficult and have been carried out only for a few selected examples. Therefore, other methods have been developed with the goal of obtaining reasonably accurate cross sections using either classical or semiclassical theories and by devising semiempirical formulae. Some of these concepts are based on the Born-Bethe formula [22] and on the observation that the ejection of an atomic electron with quantum numbers (n,J) is approximately proportional to the mean-square radius of the electron shell (n,J). This leads also to proposed correlations of the ionization cross section with polarizability, dia-... 

Abe " has developed an alternative semiclassical theory of the solvent effects on electronic spectra. This theory is based on the averaging of the intermolecular interaction energy over all solute-solvent configurations within the approximation of pair interactions. The theory involves the dipole moments and polarizabilities of the solute molecule and takes into account the temperature dependence arising from the Boltzmarm factor. 

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