Semi classical method

For the collision dynamics, a semi-classical method is quite accurate for intermediate energies about 50 keV. Note the use of a more recent collision program using the propagation method for the N " (ls2s) + He metastable system. 

Some authors have described the time evolution of the system by more general methods than time-dependent perturbation theory. For example, War-shel and co-workers have attempted to calculate the evolution of the function /(r, Q, t) defined by Eq. (3) by a semi-classical method [44, 96] the probability for the system to occupy state v]/, is obtained by considering the fluctuations of the energy gap between and 11, which are induced by the trajectories of all the atoms of the system. These trajectories are generated through molecular dynamics models based on classical equations of motion. This method was in particular applied to simulate the kinetics of the primary electron transfer process in the bacterial reaction center [97]. Mikkelsen and Ratner have recently proposed a very different approach to the electron transfer problem, in which the time evolution of the system is described by a time-dependent statistical density operator [98, 99]. 

Induced dipole autocorrelation functions of three-body systems have not yet been computed from first principles. Such work involves the solution of Schrodinger s equation of three interacting atoms. However, classical and semi-classical methods, especially molecular dynamics calculations, exist which offer some insight into three-body dynamics and interactions. Very useful expressions exist for the three-body spectral moments, with the lowest-order Wigner-Kirkwood quantum corrections which were discussed above. 

The anharmonicities of the potential contribute by the terms involving the constants x, g, y,. .. as well as the energy shifts AEx = 0(h2),. .. and the frequency shifts Aw, = 0(h2),. These anharmonic constants can be calculated by the Van Vleck contact transformations [20] as well as by a semi-classical method based on an h expansion around the equilibrium point [14], which confirms that the Dunham expansion (2.8) is a series in powers of h. Systematic methods have been developed to carry out the Van Vleck contact transformations, as in the algebraic quantization technique by Ezra and Fried [21]. It should be noted that the constants x and g can also be obtained from the classical-mechanical Birkhoff normal forms [22], The energy shifts AEx,... 

In this chapter we have summarized recent results obtained by applying semi-classical methods to the study of vibrational and dissociative dynamics on the femtosecond scale. 

We have presented results which extend and confirm Janev el al. recommanded data for the C6+- H(l.s) collision system. The problems raised by the choice of the intemuclear trajectory in semi-classical methods were shortly discussed. A more detailed study regarding those questions is in progress [21],... 

In 1974, Polinger [19] used the WKB method for the E e JT problem. Recently, it was also used in the explanation of the non-degenerate ground state for the same problem [4]. Further discussions were given by O Brien [11] on the semi-classical method for both one-dimensional and multi-dimensional JT problems. All these semi-classical approaches simplified the multi-dimensional cases to onedimensional problems. 

In cases where quantum calculations are too demanding because of the large number of reactant and product states that are coupled together, semi-classical methods might provide a tractable alternative. Semi-classical collision theory exploits the ease of solving Hamilton s classical equations for motion over a reaction potential energy surface to approximate the... 

Semi-classical methods of computing otu generally involve the use of equation (IS) with wave functions derived firom model considerations. The accuracy and complexity of the calculation depend greatly on the... 

To sum up, it seems clear from experiment that the polarizability of a molecule is increased by a second molecule at large intermolecular separations and decreased at small intermolecular separations. Ab initio methods reproduce this behaviour but are extremely difficult to apply evmi in the amplest of cases. Classical methods are much easim to use but are valid only in the long-range limit, failing completely at short range. Semi-classical methods seem to offer the most promise for obtaining a pl general, yet accurate representation of ofu. To date, however, this promise has not been fulfilled. 

The problems for quantum chemists in the mid-forties were how to improve the methods of describing the electronic structure of molecules, valence theory, properties of the low excited states of small molecules, particularly aromatic hydrocarbons, and the theory of reactions. It seemed that the physics needed was by then all to hand. Quantum mechanics had been applied by Heitler, London, Slater and Pauling, and by Hund, Mulliken and Hiickei and others to the electronic structure of molecules, and there was a good basis in statistical mechanics. Although quantum electrodynamics had not yet been developed in a form convenient for treating the interaction of radiation with slow moving electrons in molecules, there were semi-classical methods that were adequate in many cases. 

What is quite clear from all these studies of quadrupolar lineshapes is that molecular motion in inorganic ionic solids is relatively common. The lack of previous studies has led to the idea that motion in such systems is unlikely, yet the few studies that have now been done all report positive findings. This is clearly an area of further work, both to examine the extent of molecular motion in inorganic/ionic systems (with a view ultimately of understanding what effects such motions may have on the material properties of ionic systems) and to examine how far semi-classical methods may be employed to analyze the spectral data from such systems. 

Child, M. S. (1980b), in Semi-Classical Methods in Molecular Scattering and Spectroscopy (M.S. Child, ed.) p. 127, Reidel Publ. Dordrecht, Holland. 

Many chemical and biochemical systems of interest are too large for analysis with quantum mechanical methods, either ab initio or semiempirical. However, some of their properties may be modeled by classical or semi-classical methods. Classical computational methods do not provide electronic structure information. 

Conversely, structural and elastic properties of ionic and semi-ionic solids were studied successfully in a completely different context with semi-classical methods, " based on force-field model potentials. 

Raff, LM, Thompson DL. The classical approach to reactive scattering. In Baer M, Editor. Theory of chemical reaction dynamics, Vol. 3. Boca Raton CRC Press 1985. pp. 1-121. Truhlar, DG, Muckerman JT. Reactive scattering cross section. Ill Quasiclassical and semi-classical methods. In Bemtein RB, Editor. Atom-molecule collision theory. New York Plenum Press 1979, pp. 505 6. 

Finally, some comments on two recent studies on methods for quantum dynamics simulations. In the first study, by Mendive-Tapia et al., the convergence of non-adiabatic direct dynamics in conjunction with frozen-width variational Gaussian product basis functions is evaluated. The simulation of non-adiabatic dynamics can be subdivided into two groups semi-classical methods (like the trajectory surface hopping approach) and wavepacket methods (for example, the... 

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