Hierarchical construction of finite diabatic sets by Mathieu functions

An extension is given for the standard two component model of adiabatic, Born-Oppenheimer (BO) electronic states in a polyatonic molecule, by use of Mathieu functions of arbitrary order. The curl or compatibility conditions for the construction of a diabatic set of states based on a finite-dimensional subset of BO states are not satisfied exactly. It is shown, however, that, by successively adding higher order Mathieu functions to the BO set, the compatibility conditions are satisfied with increasingly better accuracy. We then generalize to situations in which the nonadiabatic couplings (the dynamic corrections to the BO approximation) are small (though not necessarily zero) between a finite-dimensional BO subset and the rest of the BO states. We prove that approximate diabatic sets exist, with an error that is of the order of the square of the neglected nonadiabatic couplings.