Projected random vectors and the recursion method in the electronic-structure problem

We develop a technique to determine the occupied eigenstates in the matrix formulation of the electronic-structure problem. The theory uses a random vector projected onto the electron occupied subspace by use of a Fermi-Dirac projection operator. This random starting vector is inserted into the recursion scheme to generate all occupied eigenenergies and eigenvectors of the system. The method produces a tridiagonal Hamiltonian matrix, which unlike the original Hamiltonian matrix, can be diagonalized even for a very large system. Hellmann-Feynman forces are readily obtained because the eigenvectors can be efficiently computed. Care must be taken to correct for instabilities in the three-term recurrence which gives rise to spurious solutions.