Semi-Classical Treatments

The inclusion of quantum corrections in the reaction coordinate of the TST involves some conceptual and computational difficulties. Conceptually, the precise localisation of the transition state in a PES implies an absolute uncertainty in its conjugated momentum, (eq. (1.8)). This was avoided in the classical derivation of the TST, because the length of the transition state at the top of the barrier. As in eq. (6.5), cancelled with the translational partition function along the reaction coordinate in eq. (6.12), and the actual size of the transition state is irrelevant for the calcnlations. 

Eyring adopted an ad hoc procednre that has been successful in including quantum effects in TST. First, the classical partition functions are replaced by their quantum analogues. Second, the classical rate is mnltiplied by a transmission coefficient that takes into account the qnantnm effects along the reaction coordinate. The quantum partition functions assume that the transition state can be treated as a stable system. The separate quantisation of the reaction coordinate is based on the vibrational adiabaticity assumption, that is, all the other vibrational modes of the reactive system very rapidly adjust to the reaction coordinate and maintain the continuity and smoothness of the PES. 

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

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

In this semi-classical treatment, effects of isotopic substitution on equilibria and rates in chemical reactions reflect the changes in zero-point energy differences between (i) reactants (R) and products (P) - equilibrium isotope effects (EIE), or (ii) between reactants (R) and transition structures (TS) - kinetic isotope effects (KIE). In both cases, these changes reflect changes of force constants at the isotopically substituted site. When bonds are broken and made at the isotopic atom, the effects are described as primary, otherwise they are described as secondary. 

Semi-classical treatment of the interaction of molecules with electromagnetic waves leads to equations for s and As in terms of molecular properties ... 

However, this is not so in the semi-classical treatment (see, for example, Dunham [65]) for which the zero of energy is defined by... 

This quantity is the leading term in the Dunham expansion which is obtained by higher-order quantisation than that given in equation (6.358). In the semi-classical treatment therefore,... 

See Ref. [148] for a physically appealing derivation.) Equation (12) yields an efficient, accurate and formally exact scheme to perform quantum dynamics. It involves the action of a handed, sparse and Toeplitz matrix on a vector. The propagation scheme has been shown to accurately represent [147] all quantum dynamical features including zero-point effects, tunneling as well as over-barrier reflections and in this sense differs from standard semi-classical treatments. The approach also substantially differs from other formalisms such as centroid dynamics [97,98,168-171], where the Feynman path centroid is propagated in a classical-like... 

There is a parallel between the wave packet-based semi-classical treatment of nuclear motion developed here and the Thomas-Fermi theory of an electronic system in a slowly varying vector potential. In the semi-classical electronic theory as well as here, one naturally arrives at a locally linear approximation to the scalar-potential-derived forces and a locally uniform approximation to the magnetic force derived from the vector potential. See, R. A. Harris and J. A. Cina, J. Chem. Phys. 79, 1381 (1983) C. J. Grayce and R. A. Harris, Molec. Phys. 71, 1 (1990). 

Cattaneo, P., Persico, M., Semi classical Treatment of the Photofragmentation of Azomethane, Chem. Phys. Lett. 1998, 289, 160 166. 

This result suggests the possibility of an even more extreme approach to the treatment of ion-state mobility in which the ion states are considered to move as essentially free particles which are then scattered by the lattice phonons. A semi-classical treatment along these lines by Kim Schmidt (1967) gives reasonably good agreement with experiment. It is also worth noting that, from the band structure described by (9.79), a proton should behave, for small k, like a particle of effective mass m, where m is given by the familiar formula for the electronic case... 

The density g(f) in Eqs. (7.1) and (7.2) came from a semi-classical treatment employing scattering centers. Quantum mecha,nically one frequently obtains an identical expression. In the quantum mechanical treatment one starts off with the matrix element between identical initial and final states of all the nucleons involved in the process. In those cases where the matrix element can be reduced to a sum of individual nucleon matrix elements and where plane waves can be employed for the scattered particle (Born approximation) one obtains expression (7.1) and (7.2). If the calculation are carried through considering all... 

Many theories have been published which relate the third-order susceptibility to molecular properties. These theories range from full quantum electrodynamics treatments to simple classical models. The simple semi-classical treatment given below is sufficient for our purposes since it will expose the basic physics of the coherent scattering process and will also give us an expression in terms of the conventional spontaneous Raman transition polarizabilities. 

Semi-Classical Treatments of Electron Transfer Rate from Weak to Strong Electronic Coupling Regime... 

Semi-Classical Treatments of Electron Transfer Rate... 

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