General Expressions for the Transition Probability

If the interaction between the reactants leading to the reaction is weak enough (nonadiabatic processes), the transition probability per unit time may be calculated using the formula of the first order in quantum mechanical 

It should be noted that Eq. (14) contains only the nondiagonal part of the interaction operator between the reactants, Its diagonal part, causing the distortion of the reactant states in each channel, is included in the Hamiltonian Hi and Hf. 

At first, we shall calculate the transition probability for fixed quantum numbers of the electrons, a and jS. The subscripts a and (3 will, however, be temporarily omitted. To separate the electronic and nuclear movements in the channels of the reaction, we shall use the Born-Oppenheimer approximation. Introducing the integral representation of the 8 function, Eq. (14) may be transformed to 

in the transitional configuration (at = O ), the quantity 2ifj is equal to the resonance splitting of the potential energy surfaces, A e( 0 ), in the following, we shall use for if, the notation, = A e( 0 )/2 also at arbitrary values of the heavy particle coordinates, C . In the two-level 

Equation (17) shows that, in the approximation used, the nondiagonal part of the interaction operator, has the formt 

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In this section we will derive two general expressions for the transition probability one for regular perpendicular-mode EPR (Bl 1B) and one for parallel-mode EPR CBj IIB). The two expressions are related in the sense that they also provide the correct ratio of intensities (perpendicular over parallel) for data obtained with a single, dual mode resonator. The expressions are derived here, and not just given, because all expressions thus far published in the EPR literature contain small inconsistencies and/or errors. 

Using the general expressions for the transition probability presented in Sections 3 and 4, one may obtain for the rate of the reaction (66) the expression... 

In order to describe now a real photoemission process, one has to calculate the matrix elements resulting from the presence of a perturbing radiation field A. The most important term which has to be considered in the new single-particle Hamiltonian is h = e/mc)A r)p. For a particular transition in which an electron occupying the orbital 0,- is emitted into a continuum state 0 of energy a, we shall retain from the general expression for the transition probability (Martin and Shirley 1976) only the leading term ... 

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