Different characterization of $\Delta_2^P$
That set is BH2, and BH is a subset of PNP. ​ ​ ​ If ​ BH ⊆ BH2 ​ "then the polynomial hierarchy collapses to" BH3(2), where BH3(2) is the 2nd "level of the Boolean hierarchy over $\Sigma_2^P$."

(So, to answer your question: ​ No, I can't give such a description.)