Harmonic excited state surface

Fig. 12. PET can be studied on the basis of intersecting harmonic potential-energy curves. In the approach of Marcus, the free energy of a reacting system is represented as a function of nuclear geometry on the horizontal axis. During excitation, there is a vertical transition (Franck-Condon) to a point on the excited-state surface, followed by vibrational relaxation. Electron transfer takes place at the crossing of the excited-state and ionic potential-energy curves. The transition-state energy, AGe, corresponds to the energy difference between the minimum on the excited-state surface and the point of intersection...

The quantum-mechanical description of the dynamics follows a very similar pattern. At the instant that the first photon is incident, the ground-state wavefunction makes a vertical (Franck-Condon) transition to the excited-state surface. The ground-state wavefunction is not a stationary state on the excited-state potential energy surface, so it must evolve as t increases. There are some interesting analytical properties of this time evolution if the excited-state surface is harmonic. In that case a gaussian wavepacket remains... 

Figure 15. Harmonic-excited-state Bom-Oppenheimer potential energy surface. The classical trajectory that originates at rest from the ground-state equilibrium geometry is shown superimposed. 

Figure Al.6.27. Equipotential contour plots of (a) the excited- and (b), (c) ground-state potential energy surfaces. (Here a harmonic excited state is used because that is the way the first calculations were performed.)...

The overlap for one specific normal mode (kth) has a simple form if it is assumed that (a) the force constants are the same in both ground and the excited states, (b) the potential surfaces are harmonic, (c) the transition dipole moment, H, is constant and (d) the normal coordinates are not mixed in the excited state. With these assumptions, the overlaps, <

in emission spectroscopy have the simple form... 

In the following, the way to calculate the band-shape function of femtosecond time-resolved spectra (i.e., involving the contributions of both population and coherence dynamics) for the displaced harmonic potential surfaces shall be presented. For allowed transitions a — n where n is the ground state g in the case of SE and it is a higher excited state in the case of induced absorption, Eq. (4.56) becomes... 

Apart from the heat bath mode, the harmonic potential surface model has been used for the molecular vibrations. It is possible to include the generalized harmonic potential surfaces, i.e., displaced-distorted-rotated surfaces. In this case, the mode coupling can be treated within this model. Beyond the generalized harmonic potential surface model, there is no systematic approach in constructing the generalized (multi-mode coupled) master equation that can be numerically solved. The first step to attack this problem would start with anharmonicity corrections to the harmonic potential surface model. Since anharmonicity has been recognized as an important mechanism in the vibrational dynamics in the electronically excited states, urgent realization of this work is needed. 

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. 

These equations assume that only the lowest energy state mixes with the ground state, that the potential energy surfaces are harmonic and have the same force constants, and that the displacement of the ground state potential energy surface compensates for the stabilization and destabilization of the states that mix [116]. The use of electrochemical data in the evaluation of parameters that contribute to hvmax leads to a significant correction term in strongly coupled donor-acceptor systems since the excited state species is not involved in the electrochemical processes [9, 116]. The optical and electrochemical processes are related by means of the electron-transfer equilibrium in Eq. 29. 

The theoretical background which will be needed to calculate the excited state distortions from electronic and Raman spectra is discussed in this section. We will use the time-dependent theory because it provides both a powerful quantitative calculational method and an intuitive physical picture [42,46-50]. The method shows in a simple way the inter-relationship between Raman and electronic spectroscopy. It demonstrates that the intensity of a peak in a resonance Raman spectrum provides detailed information about the displacement of the excited state potential surface along the normal mode giving rise to the peak [42,48]. It can also be used to calculate distortions from the intensities of vibronic peaks in electronic spectra [49]. For harmonic oscillators, the time-dependent theory is mathematically equivalent to the familiar Franck-Condon calculation [48]. 

If it is assumed that (a) the force constants are the same in both ground and the excited states, (b) the potential surfaces are harmonic, (c) the transition dipole moment, n, is constant and (d) the normal coordinates are not mixed in the excited state, then the overlaps, << in absorption and ( < (t)> in Raman, have simple forms (Sections III.A and III.B). None of these assumptions are requirements of the time-dependent theory. Assumption (a) introduces at most an error of 10% if the distortions are very large [25]. When the vibrational frequencies in the excited state are not known, this assumption must be used but does not introduce serious error. The harmonic approximation is used because the number of parameters to be fitted is thereby reduced and because it allows the simple expression for the overlap to be used. Because there is no distinct evidence of normal mode mixing in the molecules studied in this chapter, all of the fits were done without including... 

In Eq. (9), symmetric mode in the ground state in cm ro is the wavenumber of the inverted harmonic surface in the excited state in cm A Cj is equal to coshfnjjf) and Sj is equal to sinh(w t), and rtj is the vibrational quantum number of the yth nontotally symmetric normal mode in the ground electronic state (nj = 0,1,2, etc). To calculate the overlaps for emission or absorption spectra, Eq. (9) can be used when tij is set equal to zero. 

A direct consequence of the observation that Eqs. (12.55) provide also golden-rule expressions for transition rates between molecular electronic states in the shifted parallel harmonic potential surfaces model, is that the same theory can be applied to the calculation of optical absorption spectra. The electronic absorption lineshape expresses the photon-frequency dependent transition rate from the molecular ground state dressed by a photon, g) = g, hco ), to an electronically excited state without a photon, x). This absorption is broadened by electronic-vibrational coupling, and the resulting spectrum is sometimes referred to as the Franck-Condon envelope of the absorption lineshape. To see how this spectrum is obtained from the present formalism we start from the Hamiltonian (12.7) in which states L and R are replaced by g) and x) and Vlr becomes Pgx—the coupling between molecule and radiation field. The modes a represent intramolecular as well as intermolecular vibrational motions that couple to the electronic transition... 

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