  We consider computing the $k$-th eigenvalue and its corresponding eigenvector
of a generalized Hermitian eigenvalue problem of $n\times n$ large sparse
matrices. In electronic structure calculations, several properties of
materials, such as those of optoelectronic device materials, are governed by
the eigenpair with a material-specific index $k.$ We present a three-stage
algorithm for computing the $k$-th eigenpair with validation of its index. In
the first stage of the algorithm, we propose an efficient way of finding an
interval containing the $k$-th eigenvalue $(1 \ll k \ll n)$ with a non-standard
application of the Lanczos method. In the second stage, spectral bisection for
large-scale problems is realized using a sparse direct linear solver to narrow
down the interval of the $k$-th eigenvalue. In the third stage, we switch to a
modified shift-and-invert Lanczos method to reduce bisection iterations and
compute the $k$-th eigenpair with validation. Numerical results with problem
sizes up to 1.5 million are reported, and the results demonstrate the accuracy
and efficiency of the three-stage algorithm.
