Quantum dynamics through conical intersections: combining effective modes and quadratic couplings.

We present a detailed study for the short-time dynamics through conical intersections in molecular systems related to the quadratic vibronic coupling (QVC) Hamiltonian [Müller, H.; Köppel, H.; Cederbaum, L. S. New J. Chem. 1993, 17, 7-29] and the effective-mode formalism [Cederbaum, L. S.; Gindensperger, E.; Burghardt, I. Phys. Rev. Lett. 2005, 94, 113003]. Our approach is based on splitting the nuclear degrees of freedom of the whole system into system modes and environment modes. It was found that only three-effective environmental modes together with the system's modes are needed to describe the short-time dynamics of the complex system correctly. In addition, a detailed mathematical proof is given in the appendix to demonstrate that the exact cumulants are recovered up to the second order within the cumulant expansion of the autocorrelation function. The butatriene molecule is studied as an explicit showcase example to stress the viability of our proposed scheme and to compare with other systems.