Electronic states diabatic nuclear motion Schrodinger

This can be used to rewrite the diabatic nuclear motion Schrodinger equation for an incomplete set of n electronic states as... 

This makes it desirable to define other representations in addition to the electronically adiabatic one [Eqs. (9)-(12)], in which the adiabatic electronic wave function basis set used in the Bom-Huang expansion (12) is replaced by another basis set of functions of the electronic coordinates. Such a different electronic basis set can be chosen so as to minimize the above mentioned gradient term. This term can initially be neglected in the solution of the / -electionic-state nuclear motion Schrodinger equation and reintroduced later using perturbative or other methods, if desired. This new basis set of electronic wave functions can also be made to depend parametrically, like their adiabatic counterparts, on the internal nuclear coordinates q that were defined after Eq. (8). This new electronic basis set is henceforth refened to as diabatic and, as is obvious, leads to an electronically diabatic representation that is not unique unlike the adiabatic one, which is unique by definition. 

In the two-adiabatic-electronic-state Bom-Huang description of the total orbital wave function, we wish to solve the corresponding nuclear motion Schrodinger equation in the diabatic representation... 

For the two-state case with real electronic wave functions, the nuclear motion Schrodinger equations are given by (74) and (106) for the adiabatic and diabatic representations, respectively. For this case, all the matrices in those equations have dimensions 2X2 and the xad(R) and xd(R) vectors have dimensions 2X1, whereas those appearing in W(1)ad and W(1)d have the dimensions of R, namely, 3(N — 1) X 1 where N is the number of nuclei in the system. Equation (69) furnishes a more explicit version of (74) and the A(q) appearing in (106) is given by (107) with (3(q) obtained from (115). These versions of (74) and (106) are rigorously equivalent, once the appropriate boundary conditions for xad(R) and xd(R) discussed in Secs. III.B.l and III.B.2 are taken into account. The main differences between and characteristics of those equations are the following ... 

In one quantum mechanical approach based on the diabatic approximation , the electron is assumed to be confined initially at one of the reactant sites and electron transfer is treated as a transition between the vibrational levels of the reactants to those of the products. The quantum mechanical treatment begins with the time dependent Schrodinger equation, Hip = -ihSiplSt, where the wavefunction tj/ is written as a sum of the initial (reactant) and final (product) states. In the limit that the Bom-Oppenheimer approximation for the separation of electronic and nuclear motion is valid, the time dependent Schrodinger equation eventually leads to the Golden Rule result in equation (25). 

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