We present a generalized projection-based order-$N$ method which is applicable within nonorthogonal basis sets of spatially localized orbitals. The projection to the occupied subspace of a Hamiltonian, performed by means of a Chebyshev-polynomial representation of the density operator, allows the nonvariational computation of band-structure energies, density matrices, and forces for systems with nonvanishing gaps. Furthermore, the explicit application of the density operator to local basis functions gives a powerful method for the calculation of Wannier-like functions without using eigenstates. In this paper, we investigate such functions within models of diamond and fourfold-coordinated amorphous carbon starting from bonding pairs of hybrid orbitals. The resulting Wannier states are exponentially localized and show an ellipsoidal spatial dependence. These results are used to maximize the efficiency of a linear-scaling orthonormalization scheme for truncated Wannier functions.

• Received 10 March 1997

DOI:https://doi.org/10.1103/PhysRevB.57.6391