Vibration boson operator

For these vibrations, the quantization scheme of Section 4.2 can be carried over without any modification (Iachello and Oss, 1991a). The potentials in each stretching coordinate 5 are in an anharmonic force field approximation represented by Morse potentials. The boson operators (Ot,xt) correspond to the quantization of anharmonic Morse oscillators, with classical Hamiltonian... 

The boson operator b is called the vibration (or fluctuation) boson since, when applied to the ground state, it generates the vibrational modes... 

AE(t) represents the energy gap between reactant and product states which depends strongly on the thermal motion of the protein. In the spin-boson model, the two states of electron transfer are described as the spin operator, while thermal vibrations of the protein are accounted for by the boson operators. 

Here, 1 > and 2 > are the diabatic electronic states (exciton and CT states, respectively), J their electronic coupling parameter, uj the vibrational frequency, 6 and h the boson creation and annihilation operators, and g and g2 the equilibrium position shifts in the excited states 1 > and 2 >, and AE the zeroth order splitting between the two electronic states. The zero energy is set to ( AE - Eg (jS) where ft = 1. and Eg is the ground state electronic energy of a Born-Oppenheimer model. 

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