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Residual minimization scheme, direct inversion in the
iterative subspace (RMM-DIIS)

The schemes 7.1.3-7.1.5 try to optimize the expectation value
of the Hamiltonian for each wavefunction using an increasing trial basis-set.
Instead of minimizing the expectation value it is also possible to
minimize the norm of the residual vector. This leads to a similar iteration scheme as
described in section 7.1.4, but a different eigenvalue problem
has to be solved (see Ref. [19,26]).

There is a significant difference
between optimizing the eigenvalue and the norm of the residual vector.
The norm of the
residual vector is given by

and possesses a *quadratic unrestricted* minimum at the each
eigenfunction . If you have a good starting guess for
the eigenfunction it is possible to use this algorithm without
the knowledge of other wavefunctions, and therefore
without the explicit orthogonalization of the preconditioned residual vector
(eq. 7.2).
In this case after a sweep over all bands a Gram-Schmidt orthogonalization is necessary
to obtain a new orthogonal trial-basis set.
Without the explicit orthogonalization to the current set of trial wavefunctions
all other algorithms tend to converge to the lowest band, no matter
from which band they are start.

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