  The energy minimization involved in density functional calculations of
electronic systems can be carried out using an exponential transformation that
preserves the orthonormality of the orbitals. The energy of the system is then
represented as a function of the elements of a skew-Hermitian matrix that can
be optimized directly using unconstrained minimization methods. An
implementation based on the limited memory Broyden-Fletcher-Goldfarb-Shanno
approach with inexact line search and a preconditioner is presented and the
performance compared with that of the commonly used self-consistent field
approach. Results are presented for the G2 set of 148 molecules, liquid water
configurations with up to 576 molecules and some insulating crystals. A general
preconditioner is presented that is applicable to systems with fractional
orbital occupation as is, for example, needed in the k-point sampling for
periodic systems. This exponential transformation direct minimization approach
is found to outperform the standard implementation of the self-consistent field
approach in that all the calculations converge with the same set of parameter
values and it requires less computational effort on average. The formulation of
the exponential transformation and the gradients of the energy presented here
are quite general and can be applied to energy functionals that are not unitary
invariant such as self-interaction corrected functionals.
