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The case of an orthogonal basis
In the special case of an orthogonal basis there
is no distinction between covariant and contravariant quantities with
respect to the expansion coefficients of this basis, and
as such we use Latin suffixes to denote them:
. In this case, Eq. (24)
becomes

(30) 
where we have defined
.
Let be that unitary transformation which
diagonalises the (Hermitian) matrix , i.e.,
,
where
is a matrix with eigenvalues
on its diagonal:

(31) 
Denoting transformed variables by
, we apply to Eq. (30) to
obtain

(32) 
From Eq. (31) we see that
is diagonal, hence
Eq. (32) becomes

(33) 
In other words, for the case of an orthogonal basis , the
transformed gradient is preconditioned by premultiplying by a
diagonal matrix of inverse eigenvalues
. Postmultiplication by the overlap matrix
is still present in order to account for the
nonorthogonality of the localized functions .
Next: Psinc functions
Up: Preconditioned iterative minimization
Previous: General formalism
Arash Mostofi
20031028