  Nuclear spin exchange occurs in ultracold collisions of fermionic
alkaline-earth-like atoms due to a difference between s- and p-wave phase
shifts. We study the use of an optical Feshbach resonance, excited on the
${}^1S_0 \to {}^3P_1$ intercombination line of ${}^{171}$Yb, to affect a large
modification of the s-wave scattering phase shift, and thereby optically
mediate nuclear exchange forces. We perform a full multichannel calculation of
the photoassociation resonances and wave functions and from these calculate the
real and imaginary parts of the scattering length. As a figure of merit of this
interaction, we estimate the fidelity to implement a $\sqrt{SWAP}$ entangling
quantum logic gate for two atoms trapped in the same well of an optical
lattice. For moderate parameters one can achieve a gate fidelity of $\sim95% $
in a time of $\sim 50 \mu$s.
