Conical intersections in molecular photochemistry: The phase-change approach

This chapter demonstrates the approach of the crude Born-Oppenheimer approximation and shows that to carry out different orders of perturbation, the ability to calculate the matrix elements of the derivatives of Coulomb interactions with respect to nuclear coordinates is essential. Therefore we studied the case of the diatomic molecule and here we demonstrated the basic skill of computing the relevant matrix elements in Gaussian basis sets. The formulas for diatomic molecules, up to the second derivatives of the Coulomb interaction, are shown to demonstrate that some basic techniques can be developed to carry out the calculation of the matrix elements of even higher derivatives. The formulas obtained may be complicated. First, they are shown to be nonsingular. Second, the Gaussian basis set with angular momentum can be dealt with in similar ways. Third, they are expressed as multiple finite sums of certain simple functions, of order up to the angular momentum of the basis functions, and thus they can be computed efficiently and accurately. We show the application of this approach on the hydrogen molecule. The calculated equilibrium position and force constant seem to be reasonable. To obtain more reliable results, we have to employ a larger basis set to higher orders of pertubation to calculate the equilibrium geometry and wave function.