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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 $ p_n$ is orthonormalized to the current set of wavefunctions i.e.

$\displaystyle \vspace*{1mm} g_n =(1- \sum_{n'} \vert \phi_{n'} \rangle \langle\phi_{n'} \vert {\bf S} ) \vert p_n\rangle .$ (7.2)

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

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

with the basis set

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

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