Atomic Charge Chirality from Electric Multipoles

Abstract

We establish an electric-multipole ($Q_{lm}$) criterion for chirality of local atomic charge distributions. For the 11 crystallographic chiral point groups, we identify representative lowest-rank multipole contents required to distinguish the opposite handednesses. For the axial groups $C_n$ and $D_n$ with $n=2,3,4,6$, chirality arises from the relative orientation of two azimuthal anisotropies with $m=n$, belonging to consecutive ranks $l=n$ and $n+1$. Their coexistence produces a latitude-dependent twist of the $n$-fold charge distribution. The polar component, although allowed in $C_n$, is distinct from this chiral mechanism and is not required for handedness. The cubic groups $T$ and $O$ exhibit distinct realizations, with octahedral chirality first appearing at rank nine. We further show that, within a spinless $s$-$p$-$d$-$f$ atomic basis, the scalar charge density contains multipoles only up to rank six and therefore cannot encode the handedness of $C_6$, $D_6$, or $O$. These results connect electric-multipole content and atomic orbital degrees of freedom to the microscopic encoding of crystallographic handedness.