Symmetry

A calculation at a "sensible" choice of k-points is entirely analagous to a <math>\Gamma</math>-point calculation in a non-primitive cell. This leads us to our concept of effective supercells. We can efficiently apply symmetry to <math>\Gamma</math>-point wavefunctions but not (typically) k-point wavefunctions (ref: Tinkham). The question is, can we use our effective supercell concept to apply space-group symmetry operations. I think the answer is yes.

Consider a point <math>r</math> and an operation <math>{R|t}</math>. (Bold type left out, but take them as vectors and matrices as appropriate.)

<math>r'={R|t}</math>

Find the equivalent point inside our effective supercell:

<math>r=r'+L_s={R|t}r'+L_s</math>

where <math>L_s</math> is a lattice vector of the effective supercell. Finally, we can map this point back into the primitive cell.

<math>r={R|t}r+L_s+L_p</math>

where <math>L_p</math> is a primitive lattice vector.

Now, consider the effect of a symmetry operation on the i-th wavefunction at the k-th k-point:

<math>{R^{-1}|-R^{-1}t}\psi_{k,i}(r)=\psi_{k,i}(r)</math>

<math>=\psi_{k,i}(r')</math>

<math>=\psi_{k,i}(r)</math> due to the long-range periodicity of the wavefunction.

<math>=e^{ik.r}u_{k,i}(r)</math>

<math>=e^{ik.(r-L_p)}u_{k,i}(r)</math> due to the periodicity of the Bloch function.

<math>=e^{-ik.L_p}psi_{k,i}(r)</math>

Haven't got time to finish typing this now.