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# 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 ${\displaystyle p_{n}}$ is orthonormalized to the current set of wavefunctions i.e.
${\displaystyle g_{n}=(1-\sum _{{n'}}|\phi _{{n'}}\rangle \langle \phi _{{n'}}|{{\bf {S}}})|p_{n}\rangle .}$
Then the linear combination of this 'search direction' ${\displaystyle g_{n}}$ and the current wavefunction ${\displaystyle \phi _{n}}$ is calculated which minimizes the expectation value of the Hamiltonian. This requires to solve the ${\displaystyle 2\times 2}$ eigenvalue problem
${\displaystyle \langle b_{i}|{{\bf {H}}}-\epsilon {{\bf {S}}}|b_{j}\rangle =0,}$
${\displaystyle b_{{i,i=1,2}}=\{\phi _{{n}}/g_{{n}}\}.}$