Intra-state coupling constant

In practical applications of the theory, the computational problem is simplified by restricting the electronic Hilbert space to just two or three electronic states of interest, e.g. the ground state and two excited states. A further significant simpiification arises from symmetry selection rules. It follows from the definition (10) that the first-order intra-state coupling constants can be nonzero only for totally symmetric modes. Wnn (0) is zero if and transform according to different irreducible representations. The first-order inter-state coupling constant A is nonzero only for modes which transform according to the irreducible representation Tq which fulfils... 

Let us next consider the computation of intra-state coupling constants defined in Eqs. (10) and (12) as first and second derivatives of the excitation energy with respect to ground-state normal coordinates. 

For s-wave superconductivity, in weak to intermediate coupling, one is interested in time-reversed states near the Fermi-energy. So one may confine attention to intra-band pairing. For the present case, the dimensionless electron-phonon coupling constant k has a particularly simple form, which is easily derived from the general expressions (12, 13)... 

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