Electron classical model

Bartell and co-workers have made significant progress by combining electron diffraction studies from beams of molecular clusters with molecular dynamics simulations [14, 51, 52]. Due to their small volumes, deep supercoolings can be attained in cluster beams however, the temperature is not easily controlled. The rapid nucleation that ensues can produce new phases not observed in the bulk [14]. Despite the concern about the appropriateness of the classic model for small clusters, its application appears to be valid in several cases [51]. 

The two-center two-electron repulsion integrals ( AV Arr) represents the energy of interaction between the charge distributions at atom Aand at atom B. Classically, they are equal to the sum over all interactions between the multipole moments of the two charge contributions, where the subscripts I and m specify the order and orientation of the multipole. MNDO uses the classical model in calculating these two-center two-electron interactions. 

To give a simple classical model for frequency-dependent polarizabilities, let me return to Figure 17.1 and now consider the positive charge as a point nucleus and the negative sphere as an electron cloud. In the static case, the restoring force on the displaced nucleus is d)/ AtteQO ) which corresponds to a simple harmonic oscillator with force constant... 

From this point of view it is of interest to examine the consequences of full ther-malization of the classical Drude oscillators on the properties of the system. This is particularly important given the fact that any classical fluctuations of the Drude oscillators are a priori unphysical according to the Bom-Oppenheimer approximation upon which electronic induction models are based. It has been shown [12] that under the influence of thermalized (hot) fluctuating Drude oscillators the corrected effective energy of the system, truncated to two-body interactions is... 

In the (semi-)classical models of ETR (Marcus the Russian school), redox orbitals of reactants overlap at a close separation, followed by swift electron transfer. The activated complex, considered in equilibrium with the reactants, consists of these overlapping orbitals. In the tunneling model, the electron penetrates... 

Schmidt (1976) has given a classical model for the field dependence of quasi-free electron mobility that predicts p(E) in the high-field limit. At any... 

In a study by da Silva et al. (1988), the hydrogen was assumed to be in the X—AB position. They constructed a spring model of this structure and fit the spring constants to demonstrate that experimentally measured frequencies could be produced for H—B, H—Al, and H—Ga pairs in the X—AB configuration. Although their original study described the electronic structure in terms of SW-Xa-cluster calculations, these vibrational fits were produced from a classical model. 

Spin-orbit coupling problems are of a genuine quantum nature since a priori spin is a quantity that only occurs in quantum mechanics. However, already Thomas (Thomas, 1927) had introduced a classical model for spin precession. Later, Rubinow and Keller (Rubinow and Keller, 1963) derived the Thomas precession from a WKB-like approach to the Dirac equation. They found that although the spin motion only occurs in the first semiclassical correction to the relativistic classical electron motion, it can be expressed in merely classical terms. 

Figure 1. Potential energy plot of the reactants (precursor complex) and products (successor complex) as a function of nuclear configuration Eth is the barrier for the thermal electron transfer, Eop is the energy for the light-induced electron transfer, and 2HAB is equal to the splitting at the intersection of the surfaces, where HAB is the electronic coupling matrix element. Note that HAB << Eth in the classical model. The circles indicate the relative nuclear configurations of the two reactants of charges +2 and +5 in the precursor complex, optically excited precursor complex, activated complex, and successor complex.

The value of log rn for the Fe(H20) 2+ - Fe(H20)6 + exchange (which features a relatively large inner-sphere barrier) is plotted as a function of 1/T in Figure 5. The nuclear tunneling factors are close to unity at room temperature but become very large at low temperatures. As a consequence of nuclear tunneling, the electron transfer rates at low temperatures will be much faster than those calculated from the classical model. 

The calculation of the transmission coefficient for adiabatic electron transfer modeled by the classical Hamiltonian Hajis based on a similar procedure developed for simulations of general chemical reactions in solution. The basic idea is to start the dynamic trajectory from an equilibrium ensemble constrained to the transition state. By following each trajectory until its fate is determined (reactive or nonreactive), it is possible to determine k. A large number of trajectories are needed to sample the ensemble and to provide an accurate value of k. More details... 

The most simple, but general, model to describe the interaction of optical radiation with solids is a classical model, due to Lorentz, in which it is assumed that the valence electrons are bound to specific atoms in the solid by harmonic forces. These harmonic forces are the Coulomb forces that tend to restore the valence electrons into specific orbits around the atomic nuclei. Therefore, the solid is considered as a collection of atomic oscillators, each one with its characteristic natural frequency. We presume that if we excite one of these atomic oscillators with its natural frequency (the resonance frequency), a resonant process will be produced. From the quantum viewpoint, these frequencies correspond to those needed to produce valence band to conduction band transitions. In the first approach we consider only a unique resonant frequency, >o in other words, the solid consists of a collection of equivalent atomic oscillators. In this approach, coq would correspond to the gap frequency. 

On a purely classical level, it has been shown that the mapping formalism recovers the classical electron analog model of Meyer and Miller [89]. The Langer-hke modifications [101] that were empirically introduced in this model could be identified as a zero-point energy term that accounts for quantum fluctuations in the electronic DoF [102, 103]. This in practice quite important... 

Considering the semiclassical description of nonadiabatic dynamics, only the mapping approach [99, 100] and the equivalent formulation that is obtained by requantizing the classical electron analog model of Meyer and Miller [112] appear to be amenable to a numerical treatment via an initial-value representation [114, 116, 117, 121, 122]. Other semiclassical formulations such as Pechukas path-integral formulation [45] and the various connection... 

Figure 22 shows the same quantities for the intramolecular electron-transfer Model IVb. Similar to what occurs in the pyrazine model, the classical level density obtained with y = 1 overestimates the total and state-specific level density while for y = 0 the classical level densities are too small. Employing a ZPE correction of y = 0.8 results in a very good agreement with the total quantum mechanical level density, while the criterion to reproduce the state-specific level density results in a ZPE correction of y = 0.6. 

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