This article discusses two subjects which are shown to be closely connected. The first subject concerns the existence of pure diabatic states and the derivation of an adiabatic-to-diabatic matrix within a sub-Hirbert space. If certain requirements are fulfilled this sub-space behaves, for all practical purposes, like a full Hilbert space. The second concerns the necessary condition for having uniquely defined diabatic potentials. Fulfillment of this condition has led to the introduction of a matrix - the Topological matrix - which was found to contain the topological features of a given system. In the present Letter, the study of the topological matrix is extended to reveal new features.