The blackbody radiation shift of the Ga$^+$ $4s^2 ^1S^e_0 to 4s4p ^3P^o_0$ clock transition is computed to be $-$$0.0140 pm 0.0048$ Hz at 300 K. The small shift is consistent with the blackbody shifts of the clock transitions of other group III ion
s which are of a similar size. The polarizabilities of the Ga$^+$ $4s^2 ^1S^e_0$, $4s4p ^3P^o_0$, and $4s4p ^1P^o_1$ states were computed using the configuration interaction method with an underlying semi-empirical core potential. A byproduct of the analysis involved large scale calculations of the low lying spectrum and oscillator strengths of the Ga$^{2+}$ ion.
A calculation of the blackbody radiation shift of the B$^+$ clock transition is performed. The polarizabilities of the B$^+$ $2s^2$ $^1$S$^e$, $2s2p$ $^1$P$^o$, and $2s2p$ $^3$P$^o$ states are computed using the configuration interaction method with
an underlying semi-empirical core potential. The recommended dipole polarizabilities are 9.64(3) $a_0^3$, 7.78(3) $a_0^3$ and 16.55(5) $a_0^3$ respectively. The derived frequency shift for the $2s^2$ $^1$S$^e$ $to$ $2s2p$ $^3$P$^o_0$ transition at 300 K is 0.0160(5) Hz. The dipole polarizabilities agree with an earlier relativistic calculation (Safronova {em et al.} Phys. Rev. Lett. {bf 107} 143006 (2011)) to better than 0.2%. Quadrupole and octupole polarizabilities and non-adiabatic multipole polarizabilities are also reported.
