Scalar coupling nonadiabatic

Here, a few comments are in order. The matrix of derivative couplings F is antihermitian. The matrix of scalar couplings G is composed of an hermitian as well as an antihermitian part. Of course, the dressed kinetic energy operator —(1/2M)(V - - F) in our basic Eq. (10) is hermitian, as is also the case for the nonadiabatic couplings A in Eq. (9a). The latter follows immediately from the relation (lie). The notation (V F) is self evident from Eq. (lid). Since F is a vector matrix, it can be written as F = (Fi, F2,..., Fjv ), where the matrices Fq, are simply defined by their... 

Thus this simple theory predicts that radiationless transitions will occur when the energy gap AE(Q) is small and the scalar product between the velocity vector and the nonadiabatic coupling Q g(Q) is large. Here Q is the nuclear coordinate vector in Eq. [11] and g(Q) is defined in Eq. [13]. 

The coupling terms Ay,- are made of two contributions, the nonadiabatic derivative couplings, Fji Q) = (Oy-(i, 2) V 0,-(r, Q)), which are vectors in the nuclear space, and the scalar nonadiabatic couphngs [second term on the right-hand side (RHS) of Eq. 8.7]. By differentiating Eq. 8.4 with respect to the normal coordinates Q, one easily obtains the following expression for the derivative couplings [17] ... 

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