Single band, steepest descent scheme

The Davidson iteration scheme optimizes all bands simultaneously. Optimizing a single band at a time would save the storage necessary for the NBANDS gradients. In a simple steepest descent scheme the preconditioned residual vector $p_n$ is orthonormalized to the current set of wavefunctions i.e.

$$\vspace*{1mm} g_n =(1- \sum_{n'} \vert \phi_{n'} \rangle \langle\phi_{n'} \vert {\bf S} ) \vert p_n\rangle .$$ (45)

Then the linear combination of this 'search direction' $g_n$ and the current wavefunction $\phi_n$ is calculated which minimizes the expectation value of the Hamiltonian. This requires to solve the $2 \times 2$ eigenvalue problem

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

with the basis set

$$b_{i,i=1,2} = \{ \phi_{n} / g_{n} \}.$$