Spectra, nonadiabatic methods

Figure 20. The (So —> S2) absorption spectrum of pyrazine for reduced three- and four-dimensional models (left and middle panels) and for a complete 24-vibrational model (right panel). For the three- and four-dimensional models, the exact quantum mechanical results (full line) are obtained using the Fourier method [43,45]. For the 24-dimensional model (nearly converged), quantum mechanical results are obtained using version 8 of the MCTDH program [210]. For all three models, the calculations are done in the diabatic representation. In the multiple spawning calculations (dashed lines) the spawning threshold 0,o) is set to 0.05, the initial size of the basis set for the three-, four-, and 24-dimensional models is 20, 40, and 60, and the total number of basis functions is limited to 900 (i.e., regardless of the magnitude of the effective nonadiabatic coupling, we do not spawn new basis functions once the total number of basis functions reaches 900).

The failures of the Born-Oppenheimer separation of the electronic and nuclear motions show up in the spectra of molecules as homogeneous or heterogeneous perturbations in the spectra41. See, e.g. Ref. (42) for an example, a fully ab initio study of the spectrum of the calcium dimer in a coupled manifold of electronic states. Theoretical methods needed to describe the dynamics of molecules in nonadiabatic situations are being developed now. See Ref. (43) for a review. 

A semiclassical description is well established when both the Hamilton operator of the system and the quantity to be calculated have a well-defined classical analog. For example, there exist several semiclassical methods for calculating the vibrational autocorrelation function on a single excited electronic surface, the Fourier transform of which yields the Franck-Condon spectrum. ° In particular, semiclassical methods based on the initial-value representation of the semiclassical propagator,which circumvent the cumbersome root-search problem in boundary-value based semiclassical methods, have been successfully applied to a variety of systems (see, for example, the reviews Refs. 85, 86 and references therein). These methods cannot directly be applied to nonadiabatic dynamics, though, because the Hamilton operator for the vibronic coupling problem [Eq. (1)] involves discrete degrees of freedom (discrete electronic states) which do not possess an obvious classical counterpart. 

Figure 10.1 Nonadiabatic So S2/S1 absorption spectrum of pyrazine. Computational results obtained by the MCTDH method (solid line) and by the semiclassical methods (dashed lines) for the small 4-mode model (b) and the full-dimensionaUty 24-mode model (c) are compared to experimental results (a). (From ref. 97, Copyright 2004. Reproduced with permission of World Scientific Publishing Co.)...

Nonadiabatic So S2/Sj Absorption Spectrum of Pyrazine Mostofthe semiclassical theoretical models have been developed in the hmit of a dynamics taking place on a single electronic PES. In order to apply semiclassical methods to nonadiabatic dynamics, further generalization is needed and it is necessary to write down a Hamiltonian with a well-defined classical limit. In this context, the problem of the description of discrete quantum degrees of freedom (electronic states) can be tackled in two steps the first is a mapping onto continuous variables, and the second is a semiclassical treatment of the resulting problem [94,95]. These methods have been recently reviewed by Stock and Thoss [96]. 

---