  A systematic study is performed by the {\it ab-initio} density functional
theory of the anisotropy of the orbital moments $<l_z>$ and the magnetic dipole
term $<T_z>$ in bulk ${\rm CrO_2}$. Two different band-structure techniques are
used (FLAPW and LMTO-ASA), and the electronic correlations are treated by the
local-spin-density approximation (LSDA), the LSDA+ orbital polarization method,
and the LSDA+$U$ method. The calculated anisotropies of $<l_z>$ and $<T_z>$ are
very large compared to Fe, Ni and Co but still a factor of 5 and 2 smaller than
the anisotropies obtained from a recently suggested analysis of the X-ray
magnetic circular dichroism spectra for a thick layer of ${\rm CrO_2}$.
