Nonadiabatic derivative couplings

W. H. Miller In treating electronically nonadiabatic processes one often introduces (usually on physical grounds) a diabatic model, which has a nondiagonal electronic potential matrix, and then neglects any remaining derivative coupling. The total (vibronic) wave function is... 

It follows directly from this derivation that the adiabatic BO functions are coupled nonadiabatically by off-diagonal matrix elements of the form... 

These elements split into a derivative operator and a c-number. The nonadiabatic derivative couplings are given by... 

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... 

In this section we examine conical intersections, the derivative coupling, the geometric phase, and how they affect nonadiabatic processes. We therefore begin with an overview of these concepts and the Born-Huang approach from which they originate. 

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)... 

The most rigorous description of nonadiabatic effects in molecules is provided by the adiabatic representation. However, that requires knowledge of all derivative coupling matrix elements. 

With increasing system size, the implementation of ab initio electron wavepacket dynamics, such as the semiclassical Ehrenfest theory, using nuclear derivative coupling tends to be computationally more demanding because of the necessity of solving coupled perturbed equations. We therefore propose a useful treatment of nonadiabatic coupling, in which one can avoid the tedious coupled perturbed equations for the nuclear derivative of molecular orbitals and CSFs. 

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]... 

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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