Let $X$ be a smooth projective surface and $D$ be an effective Cartier divisor (not necessarily ample) on $X$. Is there a connection between these two conditions?

$(i)$ for a large enough $n$, the linear system $|nD|$ is base point free (semiample divisor)

$(ii)$ $h^1(\mathcal O_X(D)^{\otimes t})=0$ for all $t >0$. (I couldn't find any specific name for divisors satisfying this condition)

Is the second one much more stronger and therefore one can rarely find divisor satisfying these conditions?

Is there any specific instance when these two become equivalent?

Any remark from anyone is welcome.