  We consider long-range interactions between two alkali-metal atoms in their
respective ground states. We extend the previous relativistic many-body
calculations of C_6 dispersion coefficients [Phys.Rev. Lett. {\bf 82}, 3589
(1999)] to higher-multipole coefficients C_8 and C_10. A special attention is
paid to usually omitted contribution of core-excited states. We calculate this
contribution within relativistic random-phase approximation and demonstrate
that for heavy atoms core excitations contribute as much as 10% to the
dispersion coefficients. We tabulate results for both homonuclear and
heteronuclear dimers and estimate theoretical uncertainties. The estimated
uncertainties for C_8 coefficients range from 0.5% for Li_2 to 4% for Cs_2.
