A Semiclassical Approach

Methods that are not either completely classical or quantum mechanical are denoted semiclassical. In the present ch q ter we shall describe a semiclas-sical approach that is capable of reducing the numerical problem to one, that is more maneagable than that arising from a complete classical description, and that avoids introducing the partitioning in primary and secondary lattice atoms as described in the previous chapter. Furthermore, the theory works with a quantum boson description of the phonons. 

Semiclassical treatments of atom/molecule surface dynamics have been considered by several groups [99,100, 67, 104, 105]. These theories are very similar in spirit, but the method presented here has the advantage that it can in a relatively simple way account for surface temperature effects. The theory can be derived by assuming a Hartree product-type wavefunction 

In the equation for the phonon wavefunction we shall furthermore replace the expectation value by the classical limit, i.e., Vq( j(0 yiiO, Zi(t) ), 

According to eq. (8.11) the gas atoms move in an average potential (a mean field potential) of the phonons. However, in order to obtain a temperature-dependent description we need to add some corrections to the mean field theory [106]. These corrections have to do with the correction for detailed balance and are well known from gas-phase molecular dynamics [109], These corrections will be described in further detail below. 

The classical limit for the motion of the incoming atom/molecule can be obtained from eq. (8.11) by using a Gaussian wavepacket (GWP) of the form 

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The key feature to making a semiclassical approach practical is to avoid having to deal explicitly with the double-ended boundary conditions in Eq. (3.3) [16-20]. (The initial condition x(X, p( 0) = X is obviously easy to deal with.) To do this, one uses the standard coordinate space representation of Eq. (2.5),... 

L. Demeio and B. Shizgal, /. Chem. Phys., 98, 5713 (1993). Time Dependent Nucleation. II. A Semiclassical Approach. 

Our group has coupled the MST (PCM) method to Metropolis Monte Carlo sampling algorithms (MC-MST [78]). Within this approach cavitation and van der Waals terms are computed as in normal MST, while a semiclassical approach [79, 80] is used to compute the electrostatic component of solvation (see Equation (4.37)). Solute-solute energy terms are computed using a classical force field and Metropolis is then applied to the effective energy shown in Equation (4.38). 

Shin [24] has utilized a semiclassical approach to examine orientational effects on vibrational excitation. He treats the problems of (XX-A) and (XY-A) collisions, where A is an atom interacting with atoms of a diatomic molecule XX or XY through a Morse potential, and also the problems of (XX-XX) collisions and (XY-XY) collisions. The orientation dependence of the transition probabilities for Oa-Ar, 02-02, HBr-Ar, and HBr-HBr are plotted in polar diagrams. It is found, for example, that in the latter case only a very small range of angles gives the greatest contribution, namely, around that orientation in which the two H atoms lie between the Br atoms. 

The PCM has been conceived for quantum calculations, according to the scheme shown in the previous Sections. At the same time, it has been conceived as a means to extend to solvation problems a strategy based on a semiclassical approach . In past years, we found it convenient to treat on the same footing intermolecular interactions in vacuo, chemical substitution effects, internal geometry relaxation, and electronic excitation. 

V. Aquilanti, S. Cavalli, G. Grossi, and A. Lagana, A semiclassical approach to the dynamics of chemical reactions within the hyperspherical formalism. J. Mol. Struct., 93 319-323, 1983. 

This term (Equation 19.31), linear in F, is zero for a nondegenerate system with no permanent electric dipole moment, whose Hamiltonian is unaffected by the parity operation [94]. In centrosymmetric nondegenerate polymers with no permanent dipole moment, the linear Stark effect ensues from disorder [95]. In a semiclassical approach, the shift in energy caused by a permanent dipole moment my can be expressed as ... 

Eq. (3.6) is definitely not valid, since the density p is not constant but possesses local fluctuations. However, their influence is important only for low energies, or long wavelengths and in this case a semiclassical approach like that of Kane (1963) can be used. 

Quantitative spin system evolution in the presence of the interspin couplings has to be described in the language of quantum-mechanics. This is a reflection of the quantum-mechanical nature of the spins and the fact that the spin couplings via propagator operators encode NMR observables. Nevertheless, the description of nuclear magnetization relaxation processes is done by a semiclassical approach... 

To circumvent this difficulty, Luzhkov and Warshel derived a semiclassical approach in which the computed solvent spectral shift is corrected for the difference in the polarization effects in the ground and excited states ... 

The theoretical approach to the interpretation of ESR spectra is based on the solution of the SLE. This is essentially a semiclassical approach based on the Liouville equation for the magnetic probability density of the molecule augmented by a stochastic operator which describes the relevant relaxation processes that occur in the system and is responsible for the broadening of the spectral lines [2]. The SLE approach can be linked profitably to density functional theory (DFT) evaluation of geometry and magnetic parameters of the radical in its... 

G. D. Billing, On a semiclassical approach to energy transfer in polyatomic molecules, Chem. Phys. 33 227 (1978). 

Spencer, C.F., Loring, R.F. (1996). Dephasing of a solvated two-level system A semiclassical approach for parallel computing. J. Chem. Phys. 105 6596-6606 Loring, R.F, Yan, Y.Y., Mukamel, S. (1987). Time-and frequency-resolved fluorescence line shapes as a prohe of solvation dynamics. Chem. Phys. Lett. 135 23-29. 

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