On the form of the adiabatic and diabatic representation and the validity of the adiabatic approximation for X₃ Jahn-Teller systems

Starting with an expansion previously obtained for the adiabatic potential energy surfaces and derivative couplings for an X₃ system in the vicinity of a D_3h conical intersection, with the potential energy truncated at the second order in the distance from the intersection, and the derivative coupling carried to the highest order consistent with the existence of the "strictly diabatic" representation, we obtain the rigorous functional form to the same order of approximation for the diabatic representation of the same system. This diabatic representation is then applied to the reinterpretation of the recently observed laser-induced fluorescence spectrum of Cu₃. In a more general context, the same functional form is used to estimate quantitatively the validity of the electronic adiabatic approximation, as a function of a unitless scale parameter, over a range of this parameter corresponding to several metal trimer systems. Some of the methods and results of this study are more general than the X₃ systems to which most of the paper is addressed; we briefly indicate the possible extensions and implications for other systems.