  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 triple-$q$-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.
