Absorption cross section autocorrelation function

This equation expresses the total absorption cross section as the half-Fourier transform of an autocorrelation function, (>l)(f = 0) >( )), where the angular brackets indicate integration over all spatial coordinates. 

In deriving this relation we used (4.4) and (4.5). S(t) on the left-hand side of (4.6) is the autocorrelation function. It is the central quantity in the time-dependent formulation of spectroscopy and photodissociation. Multiplying (4.6) by etEft/nj integrating over t, and using Equations (2.56), (2.57), and (2.67) yields the final expression for the total absorption cross section,... 

The total absorption cross section as a function of energy is proportional to the Fourier transformation of the time-dependent autocorrelation function S(t). 

In the time-independent approach one has to calculate all partial cross sections before the total cross section can be evaluated. The partial photodissociation cross sections contain all the desired information and the total cross section can be considered as a less interesting by-product. In the time-dependent approach, on the other hand, one usually first calculates the absorption spectrum by means of the Fourier transformation of the autocorrelation function. The final state distributions for any energy are, in principle, contained in the wavepacket and can be extracted if desired. The time-independent theory favors the state-resolved partial cross sections whereas the time-dependent theory emphasizes the spectrum, i.e., the total absorption cross section. If the spectrum is the main observable, the time-dependent technique is certainly the method of choice. 

A much simpler picture emerges in the time domain. The corresponding autocorrelation function, depicted in Figure 8.4, exhibits three well resolved recurrences with very small amplitudes. The recurrence times 2"i, T2, and T3 are incommensurable which indicates that they reflect different types of molecular motion. Since the recurrences are well separated we can write S(t) as a sum S(t) = Si(t), i = 0,..., 3, with So representing the main peak at t = 0. The Fourier transformation is linear so that the absorption cross section also splits into four individual terms,... 

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 6 Illustration of the relation between the time-dependence of the autocorrelation function 5(t) (left-hand side) and the energy dependence of the absorption cross section am(E) (right-hand side), (a) and (b) direct dissociation (c) and (d) indirect dissociation...

Figure 8 Absorption cross sections calculated from the autocorrelation functions displayed in Figure 7. n and k denote the NO stretching and the XNO bending quantum numbers, respectively. Reprinted, with permission of the American Chemical Society, from Ref. 45...

The evolution of the wave packet is governed by the TDSE. The cross section for absorption onto the /th electronic surface is given by the Fourier transform of the autocorrelation function ... 

Raman emission spectra can be calculated in a manner similar to Eq. (6) for the absorption spectrum. The main difference is the replacement of the autocorrelation function by the so-called cross-correlation function, which also can be obtained from the propagation of a single wave packet, respectively two when two interacting states are involved. Finally, also the partial dissociation cross sections for producing the fragments in a particular set of quantum states, denoted by a, can be calculated from the evolving wave packet according to ... 

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