The approach to calculate improved, two-state, adiabatic-to-diabatic transformation angles (also known as mixing angles), presented before (see Das et al., J Chem Phys 2010, 133, 084107), was used here while studying the F + H 2 system. However, this study is characterized by two new features: (a) it is the first of its kind in which is studied the interplay between Renner-Teller (RT) and Jahn-Teller (JT) nonadiabatic coupling terms (NACT); (b) it is the first of its kind in which is reported the effect of an upper singular RT-NACT on a lower two-state (JT) mixing angle. The fact that the upper NACT is singular (it is shown to be a quasi-Dirac δ-function) enables a semi-analytical solution for this perturbed mixing angle. The present treatment, performed for the F + H 2 system, revealed the existence of a novel parameter, η, the Jahn-Renner coupling parameter (JRCP), which yields, in an unambiguous way, the right intensity of the RT coupling (as resembled, in this case, by the quasi-Dirac δ-function) responsible for the fact that the final end-of-the contour angle (identified with the Berry phase) is properly quantized. This study implies that the numerical value of this parameter is a pure number (independent of the molecular system): η = 2√2/π (= 0.9003) and that there is a good possibility that this value is a novel characteristic molecular constant for a certain class of tri-atomic systems. Copyright © 2011 Wiley Periodicals, Inc.