I googled the title on the internet, and arrived at the following result


$$HH\_k(D)\cong H\_{DR}^{2n-k}(M).$$


Here $M$ is a smooth manifold of dimension $n$, and $D$ is the ring of differential operators on $M$. My first question is


(A): is there a known chain map realizing the above isomorphism?


I think the above isomorphism remains true if we replace $D$ by the ring of differential operators on $M$ with coefficients in a vector bundle $E$. But I am not so sure. If this is indeed the case, my second question is


(B): question (A) with coefficients in $E$?



Hochschild (co)homology of differential operators
It is not literally what you want, but very close: check Proposition 2.3 and related results in "A Riemann-Roch-Hirzebruch formula for traces of differential operators" by Markus Engeli and Giovanni Felder (<http://arxiv.org/abs/math/0702461>).


