programming. As a result, few programs have currently implemented analytic evaluation of vibronic couplings. Details about this method can be found in ref. For the implementation for SA-MCSCF and MRCI in COLUMBUS, please see ref. Crossings and avoided crossings of potential energy surfaces Vibronic coupling is large in the case of two adiabatic potential energy surfaces coming close to each other (that is, when the energy gap between them is of the order of magnitude of one oscillation quantum). This happens in the neighbourhood of an avoided crossing of potential energy surfaces corresponding to distinct electronic states of the same spin symmetry. At the vicinity of conical intersections, where the potential energy surfaces of the same spin symmetry cross, the magnitude of vibronic coupling approaches infinity. In either case the adiabatic or Born–Oppenheimer approximation fails and vibronic couplings have to be taken into account. The large magnitude of vibronic coupling near avoided crossings and conical intersections allows wave functions to propagate from one adiabatic potential energy surface to another, giving rise to nonadiabatic phenomena such as radiationless decay. The singularity of vibronic coupling at conical intersections is responsible for the existence of Geometric phase, which was discovered by Longuet-Higgins in this context. Geometric phase around the conical intersection involving the ground electronic state of the C6H3F3+ molecular ion is discussed on pages 385-386 of the textbook by Bunker and Jensen. Difficulties and alternatives Although crucial to the understanding of nonadiabatic processes, direct evaluation of vibronic couplings has been very limited. Evaluation of vibronic couplings is often associated with severe difficulties in mathematical formulation and program implementations. As a result, the algorithms to evaluate vibronic couplings are not yet implemented in many quantum chemistry program suites. The evaluation of vibronic couplings also requires correct description of at least two electronic states in regions where they are strongly coupled. This requires the use of multi-reference methods such as MCSCF and MRCI, which are computationally demanding and delicate quantum-chemical methods. This is further complicated by the fact that definition of vibronic couplings requires electronic wave functions. Unfortunately, wave function based methods are usually too expensive for larger systems and popular methods for larger systems such as density functional theory and molecular mechanics cannot generate wave function information. As a result, direct evaluation of vibronic couplings are mostly limited to very small molecules. The magnitude of vibronic coupling is often introduced as an empirical