Energy bands in the free-electron approximation symmorphic space groups

5 Energy bands in the free-electron approximation symmorphic space groups 

When calculating FE energy states along particular directions in the BZ it is often convenient to work in Cartesian coordinates, that is to use the (e basis rather than the (b basis. The matrix representation of a reciprocal lattice vector bm is 

In applying the projection operator method for the calculation of symmetrized linear combinations of eigenfunctions, we shall need the effect of a space-group function operator (R ) on the FE eigenfunction mk(r), which is 

for symmorphic space groups r and v are vectors in the space of the crystal lattice and are measured in units of a lattice constant a of that lattice, so that r/a =xei+ye2+ze3 = [x y z], where [x y z] means a vector whose Cartesian components x, y, and z are dimensionless numbers. Similarly, when v = w 0, [w W2 W3] = wid + W2e2 + means the vector whose Cartesian components are wq, w2, and w3, in units of a. 

We now use the projection operator method for finding the linear combinations of the degenerate eigenfunctions 0O(X), 1 0](X) that form bases for the IRs of P(k) = D4h. For 

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