Dressed kinetic energy operator

The formulation (10) demonstrates that the coupled motion of electrons and nuclei can be reduced to the study of nuclear motion in the matrix potential V. The impact of the coupling on the electronic motion has been transferred to the dressed kinetic energy operator, which now reads —(1/2M)(V +F), instead of -(1/2M)V for the bare nuclei. 

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... 

It is worthwhile to have a closer look at Eq. (19). Because of the quadratic structure of the dressed kinetic energy operator, -(1/2M) (V- -F), its block in the space spaimed by the states belonging to g cannot be written as a single quadratic term, but rather as a sum of two quadratic terms... 

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