Semi-classical theory

The theory coimecting transport coefficients with the intemiolecular potential is much more complicated for polyatomic molecules because the internal states of the molecules must be accounted for. Both quantum mechanical and semi-classical theories have been developed. McCourt and his coworkers [113. 114] have brought these theories to computational fruition and transport properties now constitute a valuable test of proposed potential energy surfaces that... 

We have shown in this chapter how some experiments made it necessary in some cases to use a quantum description of light instead of the standard semi-classical theory where only the atomic part is quantized. A brief description of different helds in terms of their statistical properties was also given. This description makes it possible to discriminate between the different sources using the intensity autocorrelation function (r). 

W. H. Miller, Semi-classical theory for non-separable systems construction of good action-angle variables for reaction rate constants, Faraday Disc. Chem. Soc. 62, 40 (1977). 

Nonadiabatic Processes in Condensed Matter Semi-classical Theory and Implementation. 

Historically, the intervention of tunneling has usually been invoked when the observed KIE exceeded limits set by semi-classical theory. A recent example is the hydrogen atom transfer step in methylmalonyl-CoA mutase (MCM) catalyzed... 

The accurate quantum mechanical first-principles description of all interactions within a transition-metal cluster represented as a collection of electrons and atomic nuclei is a prerequisite for understanding and predicting such properties. The standard semi-classical theory of the quantum mechanics of electrons and atomic nuclei interacting via electromagnetic waves, i.e., described by Maxwell electrodynamics, turns out to be the theory sufficient to describe all such interactions (21). In semi-classical theory, the motion of the elementary particles of chemistry, i.e., of electrons and nuclei, is described quantum mechanically, while their electromagnetic interactions are described by classical electric and magnetic fields, E and B, often represented in terms of the non-redundant four components of the 4-potential, namely the scalar potential and the vector potential A. 

In this review we shall first establish the theoretical foundations of the semi-classical theory that eventually lead to the formulation of the Breit-Pauli Hamiltonian. The latter is an approximation suited to make the connection to phenomenological model Hamiltonians like the Heisenberg Hamiltonian for the description of electronic spin-spin interactions. The complete derivations have been given in detail in Ref. (21), but turn out to be very involved and are thus scattered over many pages in Ref. (21). For this reason, we aim here at a summary that is as brief and concise as possible so that all relevant connections between different levels of approximation are evident. This allows us to connect present-day quantum chemical methods to phenomenological Hamiltonians and hence to establish and review the current status of these first-principles methods applied to transition-metal clusters. 

Since the Marcus model was initially a classical or later a semi-classical theory, the introduction of quantum effects was considered to account for these observations. In particular, tunnelling pathways of e.t. would increase the rates of some reactions. A closer analysis has, however, led to the conclusion that this could not be a general explanation [80]. 

Whilst semi-classical theories have been successful in describing non-reactive processes such as elastic and inelastic scattering, they have had less impact on calculations for reactions. The majority of calculations have been for simple reactions such as H + H2, F + H2 or H + Cl2 which have been treated collinearly for the most part. The most successful use of semi-classical theory is for treating tunnelling and it has found use extending the scope of purely classical calculations by describing classically forbidden processes and threshold effects [156]. 

The reader is referred to the reviews of Child [157] and Connor [143] for detailed discussion and description of semi-classical theories and for references to earlier works on the subject. 

Miklavc, A. and Fischer, S.F. (1978) Semi-classical theory of collision-induced vibrational-rotational transitions. Apphcation to methyl halides. J. Chem. Phys. 69, 281-287. 

Miklavc, A. (1980) On the semi-classical theory of coUision-induced vibrational-rotational transitions in molecules. J.Chem. Phys. 72, 3805-3808. 

Miklavc, A. (1980) Semi-classical theory of vibrationaL rotational and translational energy exchange in collisions of polyatomic molecules. Mo/. Phys. 39, 855-864. 

As discussed in Section III.B, quantum interference structure tends to be quenched by these sums, and if the transition is classically allowed, the semi-classical theory then effectively degenerates to a completely classical result. 

The role of X2 vibrational quantization in the M + X2 electron jump mechanism has also received recent theoretical attention. It has been suggested179 that this may be treated classically to a good approximation and criteria for the validity of this have been determined180 from semi-classical theory. Potential energy surfaces have been calculated semi-empirically181 for K + Cl2 and ab initio182 for Li + F2. 

Classical and semi-classical theories of electron transfer provide quantitative models for determining the reaction pathway. Of particular importance is the theory of nonequilibrium solvent polarization based on the dielectric continuum model.5 From these theories Eqs. 

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