Semi-classical

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... 

The calculation of the time evolution operator in multidimensional systems is a fomiidable task and some results will be discussed in this section. An alternative approach is the calculation of semi-classical dynamics as demonstrated, among others, by Heller [86, 87 and 88], Marcus [89, 90], Taylor [91, 92], Metiu [93, 94] and coworkers (see also [83] as well as the review by Miller [95] for more general aspects of semiclassical dynamics). This method basically consists of replacing the 5-fimction distribution in the true classical calculation by a Gaussian distribution in coordinate space. It allows for a simulation of the vibrational... 

Equation (A3.13.54) legitimates the use of this semi-classical approximation of the molecule-field interaction in the low-pressure regime. Since /7j(t) is explicitly time dependent, the time evolution operator is more... 

The flux-flux expression and its extensions have been used to calculate reaction probabilities for several important reactions, including H2+02 H + H2O, by explicit calculation of the action of G in a grid representation with absorbmg potentials. The main power of the flux-flux fomuila over the long mn will be the natural way in which approximations and semi-classical expressions can be inserted into it to treat larger systems. 

Finally, semi-classical approaches to non-adiabatic dynamics have also been fomuilated and siiccessfLilly applied [167. 181]. In an especially transparent version of these approaches [167], one employs a mathematical trick which converts the non-adiabatic surfaces to a set of coupled oscillators the number of oscillators is the same as the number of electronic states. This mediod is also quite accurate, except drat the number of required trajectories grows with time, as in any semi-classical approach. 

Markovic N and Billing G D 1997 Semi-classical treatment of chemical reactions extension to 3D wave packets Chem. Phys. 224 53... 

V. P. Maslov and M. V. Fedoriuk, Semi-classical Approximations in Quantum Mechanics, Reidel, Boston, 1981. 

Maslov, V. R, Fedoriuk, M. V. Semi-Classical Approximation in Quantum Mechanics. D. Reidel Publishing Company, Dordrecht, Boston, London (1981)... 

In the light of the path-integral representation, the density matrix p Q-,Q-,p) may be semi-classically represented as oc exp[ —Si(Q )], where Si(Q ) is the Eucledian action on the -periodic trajectory that starts and ends at the point Q and visits the potential minimum Q = 0 for r = 0. The one-dimensional tunneling rate, in turn, is proportional to exp[ —S2(Q-)], where S2 is the action in the barrier for the closed straight trajectory which goes along the line with constant Q. The integral in (4.32) may be evaluated by the method of steepest descents, which leads to an optimum value of Q- = Q. This amounts to minimization of the total action Si -i- S2 over the positions of the bend point Q. ... 

At first sight, the easiest approach is to fit a set of points near the saddle point to some analytical expression. Derivatives of the fitted function can then be used to locate the saddle point. This method has been well used for small molecules (see Sana, 1981). An accurate fit to a large portion of the potential energy surface is also needed for the study of reaction dynamics by classical or semi-classical trajectory methods. 

Maxwellian distribution 129 infinite-order sudden (IOS) approximation 155-6 semi-classical calculation 136-7 Sack s model rotational relaxation 19 strong collision model 219 scattering see isotropic scattering spectra ... 

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). 

Semi-Classical Pictures of Non-Adiabatic Induced Electron Ejection in Molecular Anions... 

It is instructive to examine further the approximate semi-classical form for R7 shown above because, when viewed as a rate of transition between two intersecting energy sur ces, one anticipates that connection can be made with the well known Landau-Zener theory (10). For a non-linear molecule with N atoms, the potentials (Q) depend on 3N-6 internal degrees of fi eedom (for a linear molecule, Vj f depend on 3N-5 internal coordinates). The subspace S... 

That is, the semi-classical approximation to the photon absorption rate is equivalent to a Landau-Zener treatment of the probability of hopping from Vj -i-hco to Vf induced by the electronic coupling perturbation p, f (s,0,Q). 

The semi-classical expression shown in Eq. (54) for the rate of ejection of electrons from a specified initial vibration-rotation state Xi (Q) induced by non BO coupling to all accessible neutral-molecule-plus-free-electron final states (labeled f) gives this rate as ... 

By introducing the simplest semi-classical approximation to the propagators, in which the nuclear motion kinetic energy is assumed to commute with the anion and neutral potential energy functions and with the non BO coupling operators, one obtains... 

Improving on the semi-classical treatment of the vibration-rotation motion only slightly allows Rt to be recast in a form... 

Finally, by using what is known about the geometry dependence of the mi f functions (i.e., that m, f is strongly peaked near geometries Qo where the anion and neutral surfaces approach most closely), it is possible to further simplify the semi-classical equation for Rx... 

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