Semiclassical technique

Semiclassical techniques like the instanton approach [211] can be applied to tunneling splittings. Finally, one can exploit the close correspondence between the classical and the quantum treatment of a harmonic oscillator and treat the nuclear dynamics classically. From the classical trajectories, correlation functions can be extracted and transformed into spectra. The particular charm of this method rests in the option to carry out the dynamics on the fly, using Born Oppenheimer or fictitious Car Parrinello dynamics [212]. Furthermore, multiple minima on the hypersurface can be treated together as they are accessed by thermal excitation. This makes these methods particularly useful for liquid state or other thermally excited system simulations. Nevertheless, molecular dynamics and Monte Carlo simulations can also provide insights into cold gas-phase cluster formation [213], if a reliable force field is available [189]. 

Using semiclassical techniques Berry obtained the following representation for Wscar(q,p) [73] ... 

A reactionary movement started with the work of Leopold and Per-cival (1980). Using modern semiclassical techniques these authors were able to show that the old quantum mechanics was not so bad after all. Improving the old theory with the help of Maslov indices and variational techniques, Leopold and Percival showed that the old quantum theory yields results for the ground state and excited states of helium that are within the experimental accuracy achieved by the 1920s. Thus, Leopold and Percival turned the failure of the old quantum theory into a success, since the accuracy of the semiclassical theory improves with increasing quantum numbers and turns out to be a very useful tool for the computation of highly excited states. 

The most recent advance in the theory of the helium atom was the discovery of its classically chaotic nature. In connection with modern semiclassical techniques, such as Gutzwiller s periodic orbit theory and cycle expansion techniques, it was possible to obtain substantial new insight into the structure of doubly excited states of two-electron atoms and ions. This new direction in the application of chaos in atomic physics was initiated by Ezra et al. (1991), Kim and Ezra (1991), Richter (1991), and Bliimel and Reinhardt (1992). The discussion of the manifestations of chaos in the helium atom is the focus of this chapter. 

Extremely accurate results for reaction probabilities, e.g. were obtained in a few cases (e.g. I+HI and isotopic variants, [22,52] for H+MuH [22], see Figure 6), using purely semiclassical techniques for scattering phaseshifts. Formula (15) allows to predict oscillations due to interference between even and odd propagation in the energy dependence of probabilities. These oscillations have been found in numerical work [7]. As shown in Table I (from [22]) excellent agreement with exact calculation was obtained for resonances in H+MuH (see also Figure 5). 

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