  Let $F\subseteq\mathbb{P}^3$ be a smooth determinantal quartic surface which
is general in the N\"other-Lefschetz sense. In the present paper we give a
complete classification of locally free sheaves $\mathcal{E}$ of rank $2$ on
$F$ such that $h^1\big(F,\mathcal{E}(th)\big)=0$ for $t\in\mathbb{Z}$.
