The energy-wavenumber characteristic

Given the band energies to second order, the total energy is given by  

The evaluation presents some problems of book keeping, which are simplified by temporarily introducing the short hand notation  

Remembering that the electron distribution consists of three terms (a uniform distribution no = Z fQo, orthogonalisation holes which net negative contribution — Z — Z) at each ion position, and induced density oscillations), the Coulomb interaction term (n)(n) can be decomposed into  

The integral over the kinetic term gives jNZEp. For the contribution 

This argument is rather heuristic. A full derivation in detail is quite involved, but can e.g. be found in Harrison. 

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Fig. 6.6 The wave-vector dependence of the energy-wavenumber characteristic, ( ) which has a node at q0 and a weak logarithmic singularity in its slope at q = 2kF. Also shown are a set of degenerate cubic reciprocal lattice vectors that are centred on q0. A tetragonal distortion would lift their degeneracy away from the node at q0 as shown, thereby lowering the band-structure energy. (After Heine and Weaire (1970).)...

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