On multipoles, their decomposition by time-reversal symmetry, and the electric toroidal monopole
The multipole decomposition of the single-site density matrix provides a symmetry-adapted representation of local electronic degrees of freedom. Conventional, so-called fixed-shell, formulations do not span the full local single-particle operator space, as only operators mapping within the same orbital $l$ are resolved. Here we construct a complete orthogonal basis of real Hermitian multipole operators for the local density matrix by extending the existing formulation to inter-shell operators. We revisit the multipole decomposition as a decomposition of the operator space by $\mathrm{SO}(3)$ by first decomposing the orbital and spin operator spaces. Then by coupling them we arrive at the spin-$\frac{1}{2}$ local single-particle operator space, staying consistent with the existing fixed-shell formulations. We then classify the multipoles by parity and time-reversal symmetry, which allows for unique identification of multipole moments of the density matrix that contribute to expectation values of fully symmetry-resolved observables. As an application, we analyze the two enantiomers of chiral trigonal tellurium by computing the electric toroidal monopole moment selected by symmetry.