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We consider a fully quadratic vibronic model Hamiltonian for studyingphotoinduced electronic transitions through conical intersections. Using asecond order perturbative approximation for diabatic couplings we derive ananalytical expression for the time evolution of electronic populations at agiven temperature. This formalism extends upon a previously developedperturbative technique for a linear vibronic coupling Hamiltonian. Theadvantage of the quadratic model Hamiltonian is that it allows one to useseparate quadratic representations for potential energy surfaces of differentelectronic states and a more flexible representation of interstate couplings.We explore features introduced by the quadratic Hamiltonian in a series of 2Dmodels, and then apply our formalism to the 2,6-bis(methylene) adamantylcation, and its dimethyl derivative. The Hamiltonian parameters for themolecular systems have been obtained from electronic structure calculationsfollowed by a diabatization procedure. The evolution of electronic populationsin the molecular systems using the perturbative formalism shows a goodagreement with that from variational quantum dynamics.

A perturbative formalism for electronic transitions through conical intersections in a fully quadratic vibronic model