During the last decade the study of topological effects formed by molecular systems became a routine but it was always carried out for configurations that were limited by symmetry conditions. To be more specific this applied to the Jahn-Teller (JT) effect formed by molecular configurations of planar symmetry [see, e.g., Baer et al., Faraday Discuss. 127, 337 (2004)] and the Renner-Teller effect formed by configurations of axial symmetry [see, e.g., Halasz et al., J. Chem. Phys. 126, 154309 (2007)]. In this article we consider for the first time molecular configurations that avoid any symmetry conditions or, in other words., are characterized by the C_1 point group. We report on a detailed study of topological effects formed by such a molecular system. The study concentrates on both, the two-state (Abelian) case and the multistate (non-Abelian) case. It is shown that the theory that was originally developed to treat topological effects due the JT intersection and also applies for the study of topological effects in the most general case. The study is accompanied with numerical results.
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