Efficient single band eigenvalue-minimization

A very efficient scheme for the calculation of the lowest eigenvalues, might be obtained by increasing the basis set mentioned in the previous section in each iteration step, i.e.: At the step N solve the eigenvalue problem

$$\langle b_i \vert {\bf H} - \epsilon {\bf S} \vert b_j \rangle = 0$$

with the basis set

$$b_{i,i=1,N-1} = \{ \phi_{n} / g^1_{n} / g^2_{n} / g^3_{n} / ...\}.$$

The lowest eigenvector of the eigenvalue problem is used to calculate a new (possibly preconditioned) search vector $g^N_{n}$.