  We revisit a quantum-mechanical formula of the spin magnetic quadrupole
moment (MQM) in periodic crystals. Two previous attempts were inconsistent with
each other; one is gauge dependent, and the other is gauge invariant. Here we
define the spin MQM by calculating the spin density in a nonuniform system. Our
definition is analogous to that of the charge polarization, but the result is
gauge invariant and coincides with the latter previous one. We also formulate
what we call gravitomagnetoelectric (gravito-ME) effect, in which the
magnetization is induced by a temperature gradient. Although the Kubo formula
for the gravito-ME effect provides an unphysical divergence at zero
temperature, we prove that the correct susceptibility is obtained by
subtracting the spin MQM from the Kubo formula. It vanishes at zero temperature
and is related to the ME susceptibility by the Mott relation. We explicitly
calculate the gravito-ME susceptibility in a Rashba ferromagnet and show its
experimental feasibility.
