Herzberg-Teller approximation

Franck-Condon and Herzberg Teller Approximations of Electronic Transition AmpUtudes... 

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]. 

The correction of the CA approximation performed above is known as vibronic coupling and the wavefunction (1.24a) is sometimes designated as the Herzberg-Teller approximation. In this approximation, the corrected molecular eigenfunction... 

Burland and Robinson 41) have presented semi-quantitative arguments (in the case of internal conversion) to determine the ratios of the electronic matrix elements 5 in what they call the Herzberg-Teller scheme and the ABO approximation. Their arguments may be qualtitativeley correct only if the CBO approximation, is employed instead of the Herzberg-Teller scheme15). 

In those cases where the adiabatic approximation can be assumed to hold, the influence of vibronic coupling on the spectroscopic properties of a system can be treated within the Herzberg-Teller (perturbative) formalism (20) for vibronic interactions. This formalism is applicable when the vibronic interaction energies are small compared to the energy spacings between the... 

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... 

Since vibronic coupling effects in the adiabatic approximation are generally small compared to energy differences between electronic levels a first order perturbation can be applied. The perturbation operator contains all terms depending on the nuclear coordinates Si in the Herzberg-Teller series of Eq. (1) acting on the system with fixed nuclei which is represented by the zero order term which in ligand field theory is... 

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... 

If / vanishes in the fixed nuclei approximation, the electronic jump is said to be electronically forbidden if it vanishes also in the moving nuclei approximation, the spectral transition is said to be vibronically (i.e., vibrationally-electronically) forbidden. Vibronically allowed emissions or absorptions are called Herzberg-Teller bands. 

This description accounts for transition dipole moments induced by displacements along the normal modes during the electronic transition, that is, a simultaneous excitation of vibrational modes. This approximation is also denoted as double-harmonic. For electric dipole forbidden transitions, the first term in Eq. [24] vanishes and the resulting expression is denoted as the Herzberg-Teller (HT) approximation. 

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]. 

The FC approximation corresponds to the first term on the RHS of Eq. 8.35, the Herzberg-Teller (HT) one to the second one, and the acronym FCHT will refer to both terms taken simultaneously. The remaining terms in Eq. 8.35 will not be taken into account in the following discussion. 

Such an alternative derivation is interesting since it allows us to connect the approximations behind the derivation of Eq. 10.83 with the dynamics on the excited state. First, it highlights that in cases where Herzberg-Teller effects are important, Eq. 10.83 is expected to work worse than in FC transitions, where fig/Q) can be considered constant and hence [r, /ije(Q)] 0. Second, expanding the exponentials in Eq. 10.84 in powers of t, we have... 

Additionally, various approximations on the transition dipole moment, namely Franck-Condon (FC), Herzberg-Teller (HT), or Franck-Condon Herzberg-Teller (FCHT) can be appfied. From the computational perspective, the state-of-the-art methods rooted into TD-DFT are nowadays the most effective routes to treat medium-to-large molecular systems. Since analytical second... 

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