An algorithmic framework for approximating eigenpairs of dense, but diagonally dominant symmetric matrices is presented. It is based on a new divide-and-conquer approach for computing approximate eigenvalues and eigenvectors of a block-tridiagonal matrix. Application of this method to eigenproblems arising in Materials Science and Quantum Chemistry is discussed. It is shown that for many of these problems low rank approximations of the off-diagonal blocks as well as relaxation of deflation criteria permit the computation of approximate eigenpairs with prescribed accuracy at significantly reduced computational costs compared to standard methods as, for example, implemented in LAPACK. (Joint work with Robert C. Ward, California Institute of Technology).