  A first principles study of the dynamics of $^6$Li($^{2}$S) +
$^6$Li$^{174}$Yb($^2\Sigma^+$)$ \to ^6$Li$_2(^1\Sigma^+$) + $^{174}$Yb($^1$S)
reaction is presented at cold and ultracold temperatures. The computations
involve determination and analytic fitting of a three-dimensional potential
energy surface for the Li$_2$Yb system and quantum dynamics calculations of
varying complexities, ranging from exact quantum dynamics within the
close-coupling scheme, to statistical quantum treatment, and universal models.
It is demonstrated that the two simplified methods yield zero-temperature
limiting reaction rate coefficients in reasonable agreement with the full
close-coupling calculations. The effect of the three-body term in the
interaction potential is explored by comparing quantum dynamics results from a
pairwise potential that neglects the three-body term to that derived from the
full interaction potential. Inclusion of the three-body term in the
close-coupling calculations was found to reduce the limiting rate coefficients
by a factor of two. The reaction exoergicity populates vibrational levels as
high as $v=19$ of the $^6$Li$_2$ molecule in the limit of zero collision
energy. Product vibrational distributions from the close-coupling calculations
reveal sensitivity to inclusion of three-body forces in the interaction
potential. Overall, the results indicate that a simplified model based on the
long-range potential is able to yield reliable values of the total reaction
rate coefficient in the ultracold limit but a more rigorous approach based on
statistical quantum or quantum close-coupling methods is desirable when product
rovibrational distribution is required.
