Nonadiabatic coupling first derivative

An early step in bringing analytic gradient techniques to nonadiabatic quantum chemistry came in 1984 with the introduction of an algorithm for evaluating the first derivative coupling for electronic states i(x X) and, (x X)... 

Thus the first derivative nonadiabatic coupling matrix element consists of two terms... 

We now consider the CSF contribution to the first derivative nonadiabatic coupling matrix element. A CSF, 06, is an antisymmetric, symmetry-adapted product of molecular orbitals, defined in Eq. (5b) so that its derivative has the form... 

The singularity of the nonadiabatic couplings at the locus of intersection derives from general principles discussed in other places in this book (see, for example, Chapters 1 and 7). The vibronic secular problem corresponding to the Hamiltonian (3) has been solved, and electronic absorption spectra for an optical transition from a nondegenerate state into the coupled surfaces of Fig. 1 been computed numerically by Longuet-Higgins et al. This represents the first study of its kind in the literature. 

The second derivative terms 3 2 of the nonadiabatic coupling are mostly much smaller than the first derivative terms and hence are usually believed to be negligible. However, their omission will lead to a non-hermitian Hamiltonian due to a non-hermitian coupling... 

Despite this strong mixing the nonadiabatic coupling between these states vanishes, since, in a two-state model the nonadiabatic first-derivative coupling matrix element is given by [58]... 

As usual, the bracket notation ( ) implies integration over the electronic coordinates only, while F/j(R ) and G/j(R ) are the first- and second-order nonadiabatic matrix coupling elements, respectively. By using a derivation similar to that employed for the Hellmann-Feynman theorem, it may be shown that... 

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