  We consider a fully quadratic vibronic model Hamiltonian for studying
photoinduced electronic transitions through conical intersections. Using a
second order perturbative approximation for diabatic couplings we derive an
analytical expression for the time evolution of electronic populations at a
given temperature. This formalism extends upon a previously developed
perturbative technique for a linear vibronic coupling Hamiltonian. The
advantage of the quadratic model Hamiltonian is that it allows one to use
separate quadratic representations for potential energy surfaces of different
electronic states and a more flexible representation of interstate couplings.
We explore features introduced by the quadratic Hamiltonian in a series of 2D
models, and then apply our formalism to the 2,6-bis(methylene) adamantyl
cation, and its dimethyl derivative. The Hamiltonian parameters for the
molecular systems have been obtained from electronic structure calculations
followed by a diabatization procedure. The evolution of electronic populations
in the molecular systems using the perturbative formalism shows a good
agreement with that from variational quantum dynamics.
