Harmonic approximation electronic spectra

Within the separable harmonic approximation, the < f i(t) > and < i i(t) > overlaps are dependent on the semi-classical force the molecule experiences along this vibrational normal mode coordinate in the excited electronic state, i.e. the slope of the excited electronic state potential energy surface along this vibrational normal mode coordinate. Thus, the resonance Raman and absorption cross-sections depend directly on the excited-state structural dynamics, but in different ways mathematically. It is this complementarity that allows us to extract the structural dynamics from a quantitative measure of the absorption spectrum and resonance Raman cross-sections. 

Figure Bl.22.7. Left resonant second-harmonic generation (SHG) spectrum from rhodamine 6G. The inset displays the resonant electronic transition induced by the two-photon absorption process at a wavelength of approximately 350 nm. Right spatially resolved image of a laser-ablated hole in a rhodamine 6G dye monolayer on fused quartz, mapped by recording the SHG signal as a function of position in the film [55] SHG can be used not only for the characterization of electronic transitions within a given substance, but also as a microscopy tool.

Figure 17 The Raman excitation spectrum for a transition to the B electronic state of iodo-benzene with one quantum of vibrational excitation in the v, vibrational mode. (Solid line) computed in the harmonic approximation for the motion in the B state. (Dotted line) The maximal entropy fit of this spectrum obtained using Eq. (97). This fit is used to determine the cross-correlation function as shown in Fig. 18. (From Ref. (102).)...

Figure 19 The Raman spectrum and time cross correlation function when the motion on the excited electronic state potential is anharmonic, compare to Figs. 17 and 18, which are for a harmonic approximation. (Top, a) Computed time correlation function using a wide window function (b) The maximal entropy representation of this function, determined from the spectrum. Note the clear separation of time scales due to the anharmonicity (cf. Fig. 20). (Bottom) The Raman excitation spectrum obtained from the computed time correlation function (a). The arrows are the sequence of computations (a) is determined from the dynamics. The spectrum is determined from (a). The maximum entropy cross-correlation function (b) uses only the spectrum as input.

A recent analysis of the electronic spectrum of aniline, however, suggests that here the harmonic oscillator approximation is incorrect, and so the calculated functions for aniline may be in error. 

As is well known, the potential energy surface of the neutral NO3 ground state is very flat in the region of the minimum (12). The harmonic approximation for the vibrational frequencies is thus expected to be rather poor. A better description of the neutral vibrational features was provided by Mayer, Cederbaum, and Koppel (12) in terms of a vibronic coupling model of interacting electronic states. As we shall later see, a similar model may be needed to properly describe the cation vibrational dynamics seen in the Wang (i) PE spectrum. 

For a spectroscopic observation to be understood, a theoretical model must exist on which the interpretation of a spectrum is based. Ideally one would like to be able to record a spectrum and then to compare it with a spectrum computed theoretically. As is shown in the next section, the model based on the harmonic oscillator approximation was developed for interpreting IR spectra. However, in order to use this model, a complete force-constant matrix is needed, involving the calculation of numerous second derivatives of the electronic energy which is a function of nuclear coordinates. This model was used extensively by spectroscopists in interpreting vibrational spectra. However, because of the inability (lack of a viable computational method) to obtain the force constants in an accurate way, the model was not initially used to directly compute IR spectra. This situation was to change because of significant advances in computational chemistry. 

This rule is only approximate because it misses out many important physical considerations. For example, e/ is taken as the field-free ionisation limit for a one-electron system, and the rule contains nothing explicit about pulse duration. Nevertheless, it has been found to work reasonably well. From this rule, we can see that a well-developed spectrum of high harmonics is only expected in species with a high ionisation threshold. 

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