  In this paper we study the second Hochschild cohomology group
${HH}^2(\Lambda)$ of a finite dimensional algebra $\Lambda$. In particular, we
determine ${HH}^2(\Lambda)$ where $\Lambda$ is a finite dimensional
self-injective algebra of finite representation type over an algebraically
closed field $K$ and show that this group is zero for most such $\Lambda$; we
give a basis for ${HH}^2(\Lambda)$ in the few cases where it is not zero.
