Quantum representation, vibrational modes

The two main nuclear modes affecting electronic energies of the donor and acceptor are intramolecular vibrations of the molecular skeleton of the donor-acceptor complex and molecular motions of the solvent. If these two nuclear modes are uncoupled, one can arrive at a set of simple relations between the two spectral moments of absorption and/or emission transitions and the activation parameters of ET. The most transparent representation is achieved when the quantum intramolecular vibrations are represented by a single, effective vibrational mode with the frequency Vy (Einstein model). 

The coordinate q carries the symmetry of the vibrational mode and will play the same role as x in our diatomic example. Classically, this would mean that all vibrations have the irreducible representation of the pattern of atomic movements. However, referring to Equation (A6.34), the quantum picture is different. In the ground state, q will only appear as its square since the direct product of any nondegenerate irreducible representation with itself is Aj, this means that the ground state of any vibrational mode will also be Ai. The first excited state will contain q in the Hermite polynomial, and so does have the same symmetry as the irreducible representation of the classical picture of the vibration. 

It is further shown in Chapter 10 that, when each of the normal modes is in its ground state, each of the y/j is totally symmetric and hence y/v is totally symmetric. If one of the normal modes is excited by one quantum number, the corresponding it may then belong to one of the irreducible representations other than the totally symmetric one, say T, and thus the entire vibrational wave function f/Y will belong to the representation T,. Simple methods for finding the representations to which the first excited states of the normal modes belong are explained in Chapter 10. In this section we will quote without proof results obtained by these methods. 

Consider an electronic transition coupled to both the relatively sharp vibron transitions and the broad spectrum of phonons. The vibrons will split the single-electron transitions characterized by the quantum number n into a series of lines characterized by the vibrational quantum number j. Each vibron level is split further into closely spaced phonon lines, which we represent by a set of quantum numbers /, representing the phonon modes. Thus, in our simple representation a state can be given by the three quantum numbers (in order of decreasing energy) (n,j, /). However, the number of phonon modes is so large that experimentally, at high temperatures, one measures only an unresolved band. 

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