We provide a first-principle, materials-specific theory of multipolar order and superexchange in NpO$_2$ by means of a non-collinear local-density approximation +$U$ (LDA+$U$) method. Our calculations offer a precise microscopic description of the tr
iple-$q$-antiferro ordered phase in the absence of any dipolar moment. We find that, while the most common non-dipolar degrees of freedom (e.g., electric quadrupoles and magnetic octupoles) are active in the ordered phase, both the usually neglected higher-order multipoles (electric hexadecapoles and magnetic triakontadipoles) have at least an equally significant effect.
