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.

$g_{n}=(1-\sum _{n'}|\phi _{n'}\rangle \langle \phi _{n'}|{\bf {S}})|p_{n}\rangle .$

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

$\langle b_{i}|{\bf {H}}-\epsilon {\bf {S}}|b_{j}\rangle =0,$

with the basis set

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

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