Nondegenerate eigenfunctions

The majority of photochemistry of course deals with nondegenerate states, and here vibronic coupling effects aie also found. A classic example of non-Jahn-Teller vibronic coupling is found in the photoelection spectrum of butatiiene, formed by ejection of electrons from the electronic eigenfunctions [approximately the molecular orbitals). Bands due to the ground and first... 

There are now two possibilities the eigenvalue a, of A is either nondegenerate or degenerate. If a, is non-degenerate, then it corresponds to only one independent eigenfunction xpi, so that the function Bipi is proportional to... 

We can now show that the eigenfunctions for a molecule are bases for irreducible representations of the symmetry group to which the molecule belongs. Let us take first the simple case of nondegenerate eigenvalues. If we take the wave equation for the molecule and carry out a symmetry operation, / , upon each side, then, from 5.1-1 and 5.1-2 we have... 

We are interested in the solution of the time-independent Schrodinger equation (f is an eigenfunction and E a nondegenerate eigenvalue)... 

Thus the excitation pulse can create a superposition of i), 2) consisting of two states of different reflection symmetry. The resultant superposition possesses no symmetry properties with respect to reflection [78]. We now show that the broken symmetry created by this excitation of nondegenerate bound states translates into a nonsymmetry in the probability of populating the degenerate , n, D ), , n, L ) continuum states upon subsequent excitation. To do so we examine the properties of the bound-free transition matrix elements ( , n, q de,g Ek) that enter into the probability of dissociation. Note first that although the continuum states , n, q ) are nonsymmetric with respect to reflection, we can define symmetric and antisymmetric continuum eigenfunctions , n, s ) and , n, a ) via the relations... 

The Hermiticity of an effective Hamiltonian is first shown to be equivalent to the orthogonality of its nondegenerate eigenfunctions. This does not impose any conditions on the conservation of true eigenvector norms or of angles between degenerate model eigenfunctions, which are next demonstrated to be independent of one another. Theorem II.b then follows. 

Hence, mappings satisfying conservation B preserve norms of arbitrarily chosen degenerate eigenvectors [154], but not necessarily those of nondegenerate eigenfunctions. Thus, conservation B does not imply conservation C. The converse is also true since the condition ( <, ) = implied by conservation C for a set of model eigenfunctions that includes the <, )o, differs from Eq. (B.12). 

Thus, for example, for the PH, molecule any nondegenerate eigenfunction can only be multiplied by +1, co =... 

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