Multi-vibrational electron transitions

The theory of the multi-vibrational electron transitions based on the adiabatic representation for the wave functions of the initial and final states is the subject of this chapter. Then, the matrix element for radiationless multi-vibrational electron transition is the product of the electron matrix element and the nuclear element. The presented theory is devoted to the calculation of the nuclear component of the transition probability. The different calculation methods developed in the pioneer works of S.I. Pekar, Huang Kun and A. Rhys, M. Lax, R. Kubo and Y. Toyozawa will be described including the operator method, the method of the moments, and density matrix method. In the description of the high-temperature limit of the general formula for the rate constant, specifically Marcus s formula, the concept of reorganization energy is introduced. The application of the theory to electron transfer reactions in polar media is described. Finally, the adiabatic transitions are discussed. 

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... 

The theory of multi-phonon electron transitions due to non-adiabatic interaction between the electron terms is expounded in this chapter. The participation of the large number of the vibration degrees of freedom in the transition is caused by the non-orthogonality of the oscillator wave function in the initial and final states. This non-orthogonality may be connected with the following factors the shift of the equilibrium positions of the oscillations, the change of the frequencies in the transition, and the change of the set of normal coordinates of the vibration system. 

The nuclear wave functions h/v)- corresponding to different electronic terms, i.e. for different /u are not orthogonal with each other, (Xnjln ) 0 at ji v for any ns and. It is this non-orthogonality that is the principal reason for the population change in many modes of the vibration system (multi-phonon transition), as it was first noticed by Frenkel [3]. 

Formation of Intermediate N20 ( E+) Complex in Electronically Non-Adiabatic Channel of NO Synthesis. Estimate the maximum gas temperature To, when the formation of the intermediate N20 ( E+) complex proceeds as a multi-quantum transition and is determined by the vibrational energy of N2 molecules. Use relations (6-25) and (6-26) and estimate the probability of the multi-quantum transition. Explain why this mechanism is a preferred one in cold plasma-chemical systems. 

The potential energy (PE) of multi-atomic species varies with intermolecular distances. Fig. 1.16 shows the generic one-dimensional PE cmve for a diatomic molecule. A triatomic molecule would require a two-dimensional surface, and polyatomics a multi-dimensional surface but the key features can be illustrated by reference to the diatomic PE curve. Vibrational energy spacings are in the order of 100-4000 cm , much smaller than electronic energy spacings, and transitions between vibrational levels can be induced by IR and NIR photons. Strong bonds... 

Figure 15.1 Jablonski diagram illustrating the transitions between the electronic states in a polyatomic molecule, in a condensed phase. The vibrational levels in each electronic state are only schematically represented, because there are likely to be a very large number of vibrational modes that exist in a multi-dimensional surface. The wavy hnes represent the vibrational relaxation within the electronic states.

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