Subgroup Techniques in Crystal and Molecular Physics

Abstract

The determination of the form of tensors and matrix elements is greatly simplified by using the smallest possible group of symmetry operations that is relevant. Whenever some indices are only permuted (aside from \ifmmode\pm\else\textpm\fi{} signs) by the symmetry operations, we show that complete symmetry information is contained in the subgroup of operations that leaves these indices invariant or exchanges them. For a particular tensor element ${T}_{123}$ this is the group of the indices 123. For a tensor function of a vector ${T}_{\ensuremath{\mu}\ensuremath{\nu}}$\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}(E) this is the group of operations that leave the vector E invariant. For a force constant matrix ${{K}_{\ensuremath{\mu}\ensuremath{\nu}}}^{\mathrm{mn}}$ this is the group of the bond $\mathrm{mn}$. For an anharmonic force constant ${{K}_{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\lambda}}}^{\mathrm{mnp}}$ this is the group of the triangle connecting the atoms $\mathrm{mnp}$. For a matrix element ${V}^{\mathrm{k}{\mathrm{k}}^{\ensuremath{'}}{\mathrm{k}}^{\ensuremath{'}\ensuremath{'}}}$ connecting three vectors k,k\ensuremath{'}k\ensuremath{''} of the Brillouin zone this is the common group of the wave vectors, i.e., the group ${G}_{s}$ of elements that leave k, k\ensuremath{'} and k\ensuremath{''} invariant (modulo a reciprocal lattice vector), in some cases augmented by the elements that permute k, k\ensuremath{'} and k\ensuremath{''}. The proper use of exchange elements is shown to be determined by the behavior of the operator $V$ under time-reversal and Hermitian conjugation. The results are valid in the presence of spin-orbit interaction.