  Appraising the projected $10^{-18}$ fractional uncertainty in the optical
frequency standards using singly ionized ions, we estimate the black-body
radiation (BBR) shifts due to the magnetic dipole (M1) and electric quadrupole
(E2) multipoles of the magnetic and electric fields, respectively. Multipolar
scalar polarizabilities are determined for the singly ionized calcium (Ca$^+$)
and strontium (Sr$^+$) ions using the relativistic coupled-cluster method;
though the theory can be exercised for any single ion clock proposal. The
expected energy shifts for the respective clock transitions are estimated to be
$4.38(3) \times 10^{-4}$ Hz for Ca$^+$ and $9.50(7) \times 10^{-5}$ Hz for
Sr$^+$. These shifts are large enough and may be prerequisite for the frequency
standards to achieve the foreseen $10^{-18}$ precision goal.
