Although originally invented to solve large systems of equations
iteratively, often the conjugate gradient algorithm is also
the method of choice for finding the minimum of multi-variable
functions. It requires only the knowledge of the first derivative, and,
because of its moderate storage requirements, is particularly well
suited when the dimensionality 
of the parameter space is
large. For quadratic forms, the convergence becomes super-linear when
the number of iterations approaches 
-- a condition never
realized in problems of the size considered here, where
. Nevertheless, conjugate gradients
are robust and efficient, and have been widely used to compute the
electronic structure of solids by means of a direct minimization of
the energy functional[2].
The inclusion of the orthonormality constraints, which are explicitly
present in the original DFT/LDA functional of equation (1),
requires a modification of the conjugate-gradient scheme
[2]. In contrast to this, we follow
the approach by Mauri et al[1], where a
functional 
is adopted which has the orthonormality
constraints included implicitly:
In Eq. (4), we have the matrix 
,
with 
being the identity, and 
the overlap matrix between the
wave functions. 
is the number of electrons, and
represents the ionic,
exchange-correlation, and Hartree energy of the Kohn-Sham [5]
functional, which depend
on the modified charge density
(the factors of two in front of the sums takes care of the spin
degeneracy). The wave functions 
are not
constrained to
be orthonormal. However, one can show[1] that a) the minimum 
of the unconstrained functional (4) is the same as in
Eq. (1), and b) at the minimum, the wave
functions 
become orthonormal, i.e. minimization of
(4) automatically yields a set of orthonormal wave
functions. The parameter 
in (4) has to be chosen such
that the Hessian matrix associated with the minimization problem
(4) becomes positive definite. In section 3 we
will have a closer look at the impact of 
on the performance of
the conjugate gradient algorithm used to minimize (4).
