  Given a vector bundle $A$ over a smooth manifold $M$ such that the square
root $\mathcal{L}$ of the line bundle $\wedge^{\mathrm{top}}A^\ast \otimes
\wedge^{\mathrm{top}}T^\ast M$ exists, the Clifford bundle associated to the
split pseudo-Euclidean vector bundle $(E = A \oplus A^\ast, \langle \cdot,
\cdot \rangle)$, admits a spinor bundle $\wedge^\bullet A \otimes \mathcal{L}$,
whose section space can be thought of as that of Berezinian half-densities of
the graded manifold $A^\ast[1]$. We give an explicit construction of Dirac
generating operators of split Courant algebroid (or proto-bialgebroid)
structures on $A \oplus A^\ast$ introduced by Alekseev and Xu. We also prove
that the square of the Dirac generating operator gives rise to an invariant of
the split Courant algebroid.
