Single band steepest descent scheme: Difference between revisions
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to the current set of wavefunctions i.e. | to the current set of wavefunctions i.e. | ||
<math> | ::<math> | ||
g_n =(1- \sum_{n'} | \phi_{n'} \rangle \langle\phi_{n'} | {\bf S} ) | p_n\rangle . | g_n =(1- \sum_{n'} | \phi_{n'} \rangle \langle\phi_{n'} | {\bf S} ) | p_n\rangle . | ||
</math> | </math> | ||
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This requires to solve the <math>2 \times 2</math> eigenvalue problem | This requires to solve the <math>2 \times 2</math> eigenvalue problem | ||
<math> | ::<math> | ||
\langle b_i | {\bf H} - \epsilon {\bf S} | b_j \rangle = 0, | \langle b_i | {\bf H} - \epsilon {\bf S} | b_j \rangle = 0, | ||
</math> | </math> | ||
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with the basis set | with the basis set | ||
<math> | ::<math> | ||
b_{i,i=1,2} = \{ \phi_{n} / g_{n} \}. | b_{i,i=1,2} = \{ \phi_{n} / g_{n} \}. | ||
</math> | </math> | ||
---- | ---- | ||
[[Category:Electronic | [[Category:Electronic minimization]][[Category:Theory]] |
Latest revision as of 15:58, 6 April 2022
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 is orthonormalized to the current set of wavefunctions i.e.
Then the linear combination of this 'search direction' and the current wavefunction is calculated which minimizes the expectation value of the Hamiltonian. This requires to solve the eigenvalue problem
with the basis set