Fully Quadratic Vibronic Model of Electronic Transitions through Conical Intersections

We introduce a perturbative method for studying photoinduced electronic transitions through conical intersections. Our approach uses a quadratic vibronic coupling Hamiltonian and second order cumulant approximation for the diabatic coupling to derive an analytical expression for time evolution of electronic populations at given temperatures. The formalism is an extension of a previous method called the non-equilibrium Fermi golden rule approach which used the linear vibronic coupling Hamiltonian with the same cumulant treatment for diabatic coupling. The advantage of the quadratic Hamiltonian is that it can include electronic states with different frequencies and normal modes. We explore these advantages with 2D models showing the improved accuracy of the new quadratic method over the linear method. We then apply our formalism to some real molecules, 2,6-bis(methylene) adamantyl cation, and its dimethyl derivative, with parameters obtained from electronic structure calculations followed by diabatization. The results show good agreement with quantum dynamics techniques.MAS