Eckart wavefunction

Katriel s trick [64] gives rise to the generalization of the relation (5.4) for any excited state, while Eckart s estimate [80] means that analogous relations hold for the wavefunctions. 

Matrix elements between two n-electron wavefunctions can be factored into spin and spatial (including all information about orbital angular momenta) parts. It has been shown frequently that the Wigner-Eckart theorem can be applied as follows (Langhoff and Kern, 1977 McWeeny, 1965 Cooper and Musher, 1972) ... 

In the following, we assume that the BO approximation is valid and the Eckart conditions are met. Additionally, for the sake of simplifying the notation, all transitions are supposed to involve a single initial and a single final electronic state. Consequently, the summation will be done over the initial vibrational states (also noted i) from now on, with the vibrational wavefunction simply noted as %. The subscript //will also be dropped in the transition dipole moment = df). In these conditions, the Taylor expansion given in Eq. 8.35 can be noted in the condensed form... 

By 1930, Eckart had established that if the errors in the energy values are small, the corresponding errors in the wavefunction will be on the order of the square of the energy errors. See [Eckart, 1930]. 

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