Vibrational wave function overlap

The quantity J dr is called the vibrational overlap integral, as it is a measure of the degree to which the two vibrational wave functions overlap. Its square is known as the Franck-Condon factor to which the intensity of the vibronic transition is proportional. In carrying out the integration the requirement that r remain constant during the transition is necessarily taken into account. 

Vibrational factors involve the operation of the vibrational wave-function overlap with the radiationless transitions. 

The intensity of a vibronic transition depends upon the square of the overlap integral of the vibrational wave functions,... 

In Chapters 2 and 4, the Franck-Condon factor was used to account for the efficiency of electronic transitions resulting in absorption and radiative transitions. The efficiency of the transitions was envisaged as being related to the extent of overlap between the squares of the vibrational wave functions, /2, of the initial and final states. In a horizontal radiationless transition, the extent of overlap of the /2 functions of the initial and final states is the primary factor controlling the rate of internal conversion and intersystem crossing. 

Electronic transitions between two energy states are governed bytheFranck-Condon principle. In quantum mechanical terminology, the Franck-Condon overlap integral f xf x, dxv is important. xf and x" are, respectively, vibration wave functions for v in the final electronic state, and v" in the initial electronic state. 

Fig. 7.1 Overlap between vibrational wave functions of two electronic states.

Show that if the overlap between torsional-vibration wave functions corresponding to oscillation about different equilibrium configurations is neglected, the perturbation-theory secular equation (1.207) for internal rotation in ethane has the same form as the secular equation for the Hiickel MOs of the cyclopropenyl system, thereby justifying (5.96)-(5.98). Write down an expression (in terms of the Hamiltonian and the wave functions) for the energy splitting between sublevels of each torsional level. 

Thus we expect intersystem crossing of 1A1 methylene to 3B2 methylene. It is seen from eq. (8-5) that the transition rate is especially high when the overlap of the WlAi-v(0) and vP3Bi v(0) vibrational wave functions is a maximum this occurs when these vibrational states have vibrational kinetic... 

We notice that the electronic transition moment has been multiplied with a vibrational-overlap integral. In the solution of the vibrational problem, the vibrational wave functions will depend only upon the geometry and the force constants of the molecule. Therefore, only if all these parameters are identical in the two electronic states 1 and 2 will the two sets of vibrational wave functions be the solutions to the same Schrodinger equation. 

Energy near-resonance and favorable overlap of vibrational states are the dominant factors affecting the magnitudes of the charge-transfer cross sections in the AB + -AB systems. It was found188 that an adequate theoretical treatment of the H2+ -H2 system necessitated inclusion of the effects of vibration-rotation interaction in calculating vibrational overlaps from accurate vibrational wave functions. Charge-transfer cross sections were thus computed as a function of different vibrational and rotational levels of the incident-ion species. 

Thus the overlap between the vibrational wave functions of the tetrahedral 2 and Jahn-Teller distorted 1 is very small (weak Franck-Condon factors), and the small onset signal in the original PES measurement was simply overlooked in the significantly less sensitive PES experiments (as compared with PIMS). 

An internal conversion (IC) is observed when a molecule lying in the excited state relaxes to a lower excited state. This is a radiationless transition between two different electronic states of the same multiplicity and is possible when there is a good overlap of the vibrational wave functions (or probabilities) that are involved between the two states (beginning and final). 

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... 

It is important to distinguish between two general situations, namely crossing and not-crossing of the i//° and the p surfaces. The poor overlap of vibrational wave-functions of i//° and lowest vibrational level of ip for the noncrossing situation contrasts with the significant overlap for the crossing situation. 

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... 

Figure 23 Predissociation of the v = 2 vibrational level of the bound electronic state by a vibrational continuum wave function of the dissociative electronic state after radiative excitation (arrows) from the electronic ground state Po- The circle around the potential curve crossing point indicates an area of large overlap between the vibrational wave functions.

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. 

Quantum mechanically there is a finite probability that inversion may occur even when the vibrational energy of the molecule is lower than the potential barrier Fmax. The vibrational wave functions for the parallel vibration [Eq. (1)] in the left (pl) and right (ips) potential minima penetrate the barrier and overlap to some extent. A given vibrational state is then described by a linear combination of and tpB into a symmetrical y>+ and an antisymmetrical yi function ... 

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