  Let -\Delta denote the Dirichlet Laplace operator on a bounded open set in
\mathbb{R}^d. We study the sum of the negative eigenvalues of the operator -h^2
\Delta - 1 in the semiclassical limit h \to 0+. We give a new proof that yields
not only the first term of the asymptotic formula but also the second term
involving the surface area of the boundary of the set. The proof is valid under
weak smoothness assumptions on the boundary.
