Herzberg-Teller expansion

The expressions on the right sides of Eq. (6.1-5) now involve only vibrational wavefunc-tions (vi, Vf, Ve) whereas the electronic wave functions appear in M, which is the pure electronic transition moment connecting the ground with the excited electronic state e. M is a function of nuclear coordinate and can be expanded into a Taylor series about the equilibrium position (Herzberg-Teller expansion) ... 

This is called the Herzberg-Teller expansion. The fourth term in the last line describes a vibronic coupling. The electronic part of the vibronic operator is defined... 

The interaction referred to here as well as other places in the text is between the electronic states at a reference nuclear geometry—that is, those that appear in the Herzberg-Teller expansion of the potential and which form the basis used in Eqs. (9) and (10). 

Vibrational Contributions Contribution of vibrational modes has been described for TPA [5-9, 11-17, 19, 22, 23, 31, 37, 61, 235, 309, 343-345] and for other nonlinear optical processes [346]. One classical example is the 1A j -1 B2u TP transition of benzene, the so-called green band. This electronic transition is allowed due to a vibronic coupling mechanism [346]. Semiempirical [60, 61] as well as ab initio response theory calculations using the Herzberg-Teller expansion [344] demonstrate the role of vibronic coupling. Such contributions can either enhance an allowed transition or intensify a symmetry-forbidden transition. 

We first calculate the REP of the totally symmetric fundamental OaOj -> Ojls (Zgierski, 1976). Assuming weak coupling, we carry out the usual Herzberg-Teller expansion and obtain a scattering tensor of the form... 

Chou and Jin have addressed the importance of the vibrational contributions to the polarizability and second hyperpolarizability within the two-level and the two-band models. Their study adopts the sum-over-state (SOS) expressions of the (hyper)polarizabilities expressed in terms of vibronic states and includes two states and a single vibrational normal mode. Moreover, the Herzberg-Teller expansion is applied to these SOS formulas including vibrational energy levels without employing the Plac-zek s approximation. Thus, this method includes not only the vibrational contribution from the lattice relaxation but also the contribution arising... 

The set of equations (121) can be diagonalized exactly by numerical methods. All results displayed in the figures of this and the next subsection are obtained in this way. For qualitative purposes, one can approximate these results by using perturbation expansions. For weak pseudo-Jahn-Teller (i.e., Herzberg-Teller) coupling and vanishing Renner-Jahn-Teller coupling, we have... 

Another advantage of the nuclear-ensemble approach is that it is naturally a post-Condon approximation. Because the transition moments are evaluated for geometries displaced from equilibrium position, vibronic contributions to the spectrum are computed without need of Herzberg-Teller type of expansions [5]. Thus, even dark vibronic bands are described by the simulations [15]. 

Such an expansion is, for example, at the base of the Herzberg-Teller theory of vibronic coupling we will present in Section 8.3.1.1. Unfortunately, the convergence of the expansion given in Eq. 8.11 is very slow, making its use unpractical in many... 

Within this approximation the transition occurs between vibrational states with the highest overlap. Moreover, electric dipole forbidden transitions (i.e. when /tnm(QQ " ) = 0) cannot be described within this approximation. They can be treated however by considering the expansion in Eq. (4.18) to first order, which yields the Franck-Condon-Herzberg-Teller approximation [58, 61]. 

It has been found that the above theory is quite useful in predicting the correct overall behavior of matter in the presence of electromagnetic radiation. However, in order to predict correctly Raman intensities, it has been found necessary to refine the theory by accounting for small deviations in the electronic wave functions with nuclear motion. In the framework of the Herzberg-Teller theory, it is assumed that the corrected electronic wave functions may be obtained by the use of a first-order perturbation expansion as a linear combination of the complete set of zero-order Born-Oppenheimer functions, discussed above. Since here we are mainly interested in the normal Raman effect, we shall consider only corrections to the second term in Eq. (41). If we first examine corrections to the state X, the resulting expression for the derivative of the transition moment with motion along a normal mode is ... 

Herzberg and Teller (1947) described the dependence of the electronic transition dipole moment on the nuclear geometry g by a McLaurin series expansion of the Hamiltonian... 

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