  An algorithm for first-principles electronic structure calculations having a
computational cost which scales linearly with the system size is presented. Our
method exploits the real-space localization of the density matrix, and in this
respect it is related to the technique of Li, Nunes and Vanderbilt. The density
matrix is expressed in terms of localized support functions, and a matrix of
variational parameters, L, having a finite spatial range. The total energy is
minimized with respect to both the support functions and the elements of the L
matrix. The method is variational, and becomes exact as the ranges of the
support functions and the L matrix are increased. We have tested the method on
crystalline silicon systems containing up to 216 atoms, and we discuss some of
these results.
