The electronic adiabatic-diabatic transformation matrix: A theoretical and numerical study of a three-state system

In this work, we consider a diabatic 3 x 3 potential matrix which is used to study the three adiabatic-diabatic transformation angles that form the corresponding 3 x 3 adiabatic-diabatic transformation matrix. The three angles are known to be solutions of three coupled first-order differential equations (Top, Z. H.; Baer, M. J. Chem. Phys. 1977, 66, 1363). These equations are solved here for the first time and are shown to be stable and to yield meaningful solutions. Since many sets of equations can be formed for this purpose efforts were made to classify the various sets of equations, with the aim of gaining more physical content for the calculated angles. The numerical treatment was applied to a three-state diabatic potential matrix devised for the Nas excited states (Cocchini, F.; Upton, T. H.; Andreoni, W. J. Chem. Phys. 1988, 88, 6068). A comparison between two-state and three-state angles reveals that, in certain cases, the two-state angles contain information regarding the interaction of the lower state with the upper states. However in general the two-state treatment may fail in yielding the correct topological features of the system. One of the main results of this study is that the adiabatic-diabatic transformation matrix, upon completion of a cycle, becomes diagonal again with the numbers ii in its diagonal.