  Crystal-field (CF) effects on the rare-earth (RE) ions in ferrimagnetic
intermetallics NdCo$_5$ and TbCo$_5$ are evaluated using an ab initio density
functional + dynamical mean-field theory approach in conjunction with a
quasi-atomic approximation for on-site electronic correlations on the localized
4$f$ shell. The study reveals an important role of the high-order sectoral
harmonic component of the CF in the magnetism of RECo$_5$ intermetallics. An
unexpectedly large value is computed in the both systems for the corresponding
crystal-field parameter (CFP) $A_6^6 \langle r^6 \rangle$, far beyond what one
would expect from only electrostatic contributions. It allows solving the
enigma of the non-saturation of zero-temperature Nd magnetic moments in
NdCo$_5$ along its easy axis in the Co exchange field. This unsaturated state
had been previously found out from magnetization distribution probed by
polarised neutron elastic scattering but had so far remained theoretically
unexplained. The easy plane magnetic anisotropy of Nd in NdCo$_5$ is strongly
enhanced by the large value of $A_6^6\langle r^6 \rangle$. Counter-intuitively,
the polar dependence of anisotropy energy within the easy plane remains rather
small. The easy plane magnetic anisotropy of Nd is reinforced up to high
temperatures, which is explained through $J$-mixing effects. The calculated ab
initio anisotropy constants of NdCo$_5$ and their temperature dependence are in
quantitative agreement with experiment. Unlike NdCo$_5$, the $A_6^6 \langle r^6
\rangle$ CFP has negligible effects on the Tb magnetism in TbCo$_5$ suggesting
that its impact on the RE magnetism is ion-specific across the RECo$_5$ series.
The origin of its large value is the hybridization of RE and Co states in a
hexagonally coordinated local environment of the RE ion in RECo$_5$
intermetallics.
