Adiabatic electronic energies and nonadiabatic couplings to all orders for system of three identical nuclei with conical intersectiona)

One of us has previously developed a technique for obtaining the leading terms in a perturbation expansion of the Born–Oppenheimer energies and derivative couplings for the X3 system near the conical intersection at a C3v configuration. In a preceding paper, these leading terms have been utilized to study various aspects of the adiabatic approximation for both model and real systems of this type. In the present article, this technique is generalized and extended to all orders, yielding rigorous functional forms for both energies and derivative couplings. In particular, the ‘‘nonremovable’’ part of the derivative coupling, which cannot be transformed away by going to a diabatic basis, is explicitly exhibited. Convenient approximations for both removable and nonremovable couplings are also obtained. These should facilitate the estimation of the effect of the different couplings in various situations.