Vibrational overlap matrix element

Fig. 4.1. The four significant induced dipole components of H2-He pairs ground state vibrational average (left panel) and vibrational transition matrix element v = 0 —> v = 1 (center panel) and of the ground state vibrational average of H2-Ar pairs (right panel). Overlap components are dotted multipole-induced components are represented by solid lines dashed lines indicate the associated classical multipole component after [279, 151, 280],...

The acetylene A <- X electronic transition is a bent <- linear transition that would be electronically forbidden ( - ) at the linear structure. The usual approximation is to ignore the possibility that the electronic part of the transition moment depends on nuclear configuration and to calculate the relative strengths of vibrational bands as the square of the vibrational overlap between the initial and final vibrational states (Franck-Condon factor). A slightly more accurate picture would be to express the electronic transition moment as a linear function of Q l (the fra/w-bending normal coordinate on the linear X1 state) in such a treatment, the transition moment would be zero at the linear structure and the vibrational overlap factors would be replaced by matrix elements of Qfl- Nevertheless, as long as one makes use of low vibrational levels of the A state, neglect of the nuclear coordinate dependence of the electronic excitation function is unlikely to affect the predicted dynamics or to complicate any proposed control scheme. 

The theory of multi-oscillator electron transitions developed in the works [1, 2, 5-7] is based on the Born-Oppenheimer s adiabatic approach where the electron and nuclear variables are divided. Therefore, the matrix element describing the transition is a product of the electron and oscillator matrix elements. The oscillator matrix element depends only on overlapping of the initial and final vibration wave functions and does not depend on the electron transition type. The basic assumptions of the adiabatic approach and the approximate oscillator terms of the nuclear subsystem are considered in the following section. Then, in the subsequent sections, it will be shown that many vibrations take part in the transition due to relative change of the vibration system in the initial and final states. This change is defined by the following factors the displacement of the equilibrium positions in the... 

From Eq. [239], it is apparent that the size of a particular is not only determined by the magnitude of the electronic coupling matrix element but also by the overlap of the vibrational wave functions v,- and i/. Squared overlap integrals of the type (Xi/, (Q) IXt/ (Q))q 2 are frequently called Franck-Con-don (FC) factors. In contrast to radiative processes, FC factors for nonradiative transitions become particularly unfavorable if two states differing considerably in their electronic energies exhibit similar shapes and equilibrium coordinates of their potential curves. Due to the near-degeneracy requirement, an upper state vibrational wave function, with just a few nodes... 

Rates for nonradiative spin-forbidden transitions depend on the electronic spin-orbit interaction matrix element as well as on the overlap between the vibrational wave functions of the molecule. Close to intersections between potential energy surfaces of different space or spin symmetries, the overlap requirement is mostly fulfilled, and the intersystem crossing is effective. Interaction with vibrationally unbound states may lead to predissociation. 

Sg and the matrix elements of the respective transitions (or overlap integrals of the vibrational states). 

In the case of internal conversion between states of equal multiplicity A = is the kinetic energy operator of the nuclei. In the case of weak coupling the matrix element of the perturbation operator can be split into an electronic part and a contribution due to the vibrational terms which, with the help of further simplifying assumptions, can be written as the Franck-Condon overlap integral ... 

In the vibrational parts of the diagrams, vertical lines between potential curves represent overlap integrals, which are diagonal in the vibrational quantum number for unshifted harmonic potentials of the same fundamental frequency. An inclined line with a circle represents a matrix element of the coordinate operator, Q, which changes the vibrational quantum number by 1. 

---