Renner–Teller intersections along the collinear axes of polyatomic molecules: $H2CN$ as a case study

The tetra-atomic $C2H2+$ cation is known to form Renner–Teller-type intersections along its collinear axis. Not too long ago, we studied the nonadiabatic coupling terms (NACTs) of this molecule [G. J. Halász et al., J. Chem. Phys. 126, 154309 (2007)] and revealed two kinds of intersections. (i) By employing one of the hydrogens as a test particle, we revealed the fact that indeed the corresponding (angular) NACTs produce topological (Berry) phases that are equal to $2π$⁠, which is a result anticipated in the case of Renner–Teller intersections. (ii) However, to our big surprise, repeating this study when one of the atoms (in this case a hydrogen) is shifted from the collinear arrangement yields for the corresponding topological phase a value that equals $π$ (and not $2π$⁠). In other words, shifting (even slightly) one of the atoms from the collinear arrangement causes the intersection to change its character and become a Jahn–Teller intersection. This somewhat unexpected novel result was later further analyzed and confirmed by other groups [e.g., T. Vertesi and R. Englman, J. Phys. B 41, 025102 (2008)]. The present article is devoted to another tetra-atomic molecule, namely, the $H2CN$ molecule, which just like the $C2H2+$ ion, is characterized by Renner–Teller intersections along its collinear axis. Indeed, we revealed the existence of Renner–Teller intersections along the collinear axis, but in contrast to the $C2H2+$ case a shift of one atom from the collinear arrangement did not form Jahn–Teller intersections. What we found instead is that the noncollinear molecule was not affected by the shift and kept its Renner–Teller character. Another issue treated in this article is the extension of (the two-state) Berry (topological) phase to situations with numerous strongly interacting states. So far the relevance of the Berry phase was tested for systems characterized by two isolated interacting states, although it is defined for any number of interacting states [M. V. Berry, Proc. R. Soc. London, Ser. A 392, 45 (1984)]. We intend to show how to overcome this limitation and get a valid, fully justified definition of a Berry phase for an isolated system of any number of interacting states (as is expected).