Wavepackets cross-correlation function

The cross-correlation functions Cj/ are the links between the motion of the wavepacket in the upper state, on one hand, and the Raman spectrum on the other hand. Its behavior in time controls the fluorescence intensities into the vibrational states of the electronic ground state. 

The nonresonant term may be obtained from the resonant term by the replacement cj, - tus, and, henceforth, will be neglected. Equation (2.9) states that the scattering amplitude is the half-Fourier transform of the overlap of the time-evolving wavepacket with the final state of interest (multiplied by the transition moment). Equation (2.9) bears a close resemblance to Eq. (2.3) for the absorption cross section, but there are three differences to note (1) the cross-correlation function of the moving wavepacket with the final vibrational state of interest is required, rather than the autocorrelation function (2) an integral over the range [0, oo], not [-00,00], is required for the Raman amplitude (3) The cross-section / " (to) is proportional to the absolute value squared of a ... 

Figure Al.6.14. Schematic diagram showing the promotion of the initial wavepacket to the excited electronic state, followed by free evolution. Cross-correlation functions with the excited vibrational states of the ground-state surface (shown in the inset) determine the resonance Raman amplitude to those final states (adapted from [14].

Resonance Raman (RR) spectra yield valuable additional information on the excited-state dynamics and allow for a further test of the theoretical model. RR excitation profiles and spectra can be computed via a straightforward generalization of the Lanczos algorithm to allow for several initial states (namely, all those states where RR transitions are sought for). This amounts to a block-Lanczos or band-Lanczos procedure, both of which are well established in the literature. Alternatively, RR amplitudes may be obtained through Fourier transformation of appropriate cross-correlation functions which can be directly extracted from the time-dependent wavepacket 4 (t). di4... 

Figure B3.4.7. Schematic example of potential energy curves for photo-absorption for a ID problem (i.e. for diatomics). On the lower surface the nuclear wavepacket is in the ground state. Once this wavepacket has been excited to the upper surface, which has a different shape, it will propagate. The photoabsorption cross section is obtained by the Fourier transfonn of the correlation function of the initial wavefimction on tlie excited surface with the propagated wavepacket.

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