Normal coordinate transition probability

If the impurity-ion interaction with the host-lattice vibrations is linear with the phonon normal coordinates, the probability W for a transition of the impurity ion from electronic state i to state f with creation of a phonon is given at low temperatures by [27])... 

A general method for the calculation of the transition probability in the harmonic approximation developed in Ref. 44 enabled us to take into account, in a rigorous way, both the dependence of the tunneling of the quantum particles on the coordinates of other degrees of freedom of the system and the effects of the inertia and nonadiabaticity of the tunneling particle, taking into account the mixing of the normal coordinates of the system in the initial and... 

Below we will use Eq. (16), which, in certain models in the Born-Oppenheimer approximation, enables us to take into account both the dependence of the proton tunneling between fixed vibrational states on the coordinates of other nuclei and the contribution to the transition probability arising from the excited vibrational states of the proton. Taking into account that the proton is the easiest nucleus and that proton transfer reactions occur often between heavy donor and acceptor molecules we will not consider here the effects of the inertia, nonadiabaticity, and mixing of the normal coordinates. These effects will be considered in Section V in the discussion of the processes of the transfer of heavier atoms. 

The above method enables us to calculate the transition probability at various initial nonequilibrium conditions. As an example, we will consider the transition from the state in which the initial values of the coordinate and velocity of the reactive oscillator are equal to zero.85 In this case, the normalized distribution function has the form... 

The vibrational overlap integrals play a key role in electron transfer. A region of vibrational overlap defines values of the normal coordinate where a finite probability exists for finding coordinates appropriate for both reactants and products. The greater the overlap, the greater the transition rate. The vibrational overlap integrals can be evaluated explicitly for harmonic oscillator wavefunctions. An example is shown in equation (26) for the overlap between an initial level with vibrational quantum number v = 0 to a level v = v where the frequency (and force constant) are taken to be the same before and after electron transfer. 

In general, in the above considerations the coordinate x is presumed to describe nuclear motion normal to the intersection line L of the diabatic.potential energy surfaces of reactants and products. In particular cases, however, the coordinate x can coincide with a dynamically separable reaction coordinate. Then, the whole manydimensional problem of calculating the transition probability for any energy value is simply reduced to a one-dimensional one. Such is, for instance, the situation in a system of oscillators making harmonic vibrations with the same frequency in both the initial and final state /67/ which we considered in Sec.3.1.1. The diabatic surfaces (50.1) then represent two similar (N+1>dimensional rotational paraboloids which intersect in a N-dimensional plane S, and the intersection... 

E being the energy for u-motion measured relative to the crossing point of the electronic curves (199.II) If the reaction is adiabatic 1)f the normal coordinate can also be used to evaluate the transition probability for not very large absolute values of 8 0 by the formula (179.11) (C = 1), which involves both nuclear tunneling and reflection on the lower adiabatic surface. Otherwise, a numerical solution of the system (166.11) of differential equations, referred to the normal coordinate (x— u) is needed. 

The expression for tr pcPc depends on the characteristic intramolecular vibrations and on the model used for their description. In this section, we shall present only one example where the vibrations are harmonic and the entangling of the normal coordinates is absent. (For the general case of the harmonic approximation, see Ref. 49.) Then the expression for the transition probability takes the form ... 

The expressions for transition probability in the general case, when the frequencies of intramolecular vibrations and the systems of normal coordinates may change during the transition, are rather cumbersome and are given in Refs. [4, 5]. In the systems where the intramolecular vibrations can be subdivided into two groups - classical and quantum ones - and there is no entangling of normal coordinates and no variation of vibration frequencies, the expression for transition probability is simplified and takes the form... 

We present results of such calculations to illustrate some of the effects of normalmode mixing. There are obviously a large number of possible examples and a few of them will be included here. The calculations are performed for nti = m2 = ttij = 0, that is, for the transition from the lowest vibrational level of the optically excited initial s state into the vibrational manifold of the lower electronic state (the zero-temperature limit). The results that we shall display are the final-state vibrational distribution (as given by the multidimensional surface of 13) and the relative nomadiative decay probability, both represented as functions of the normal coordinate rotation. 

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