Formulation of the problem

where is the single-particle Hamiltonian of the system, with energy eigenvalues and corresponding eigenstates . The eigenstates satisfy the orthogonality constraints given by

for all and . For instance, within the Kohn-Sham scheme of density-functional theory [22,23,24], is the Kohn-Sham Hamiltonian and is the effective potential.

The total band-structure energy is given by

In the case of linear-scaling calculations, the lowest
extended eigenstates
(
) are expressed in terms of a set of
localized functions
(
) that are generally
nonorthogonal:

and on substitution of Eq. (4) into the orthogonality relation given by Eq. (2) we find that satisfies

where a distinction has been made between contravariant and covariant quantities [26,27] through the use of superscript and subscript Greek suffixes, respectively.

Substituting Eq. (4) into the energy expression of
Eq. (3), and defining

the band-structure energy becomes

where is referred to as the

We consider the localized functions
to
be represented in terms of a basis
as follows:

Defining

Suffixes and run over the localized functions , and run over the basis functions and runs over the extended orthogonal orbitals . We have adopted the Einstein summation convention for all repeated Greek suffixes, and continue to do so from here on.

It is both convenient and physically meaningful to perform the
minimization of the energy functional in two nested loops, as in the
ensemble density-functional method of Marzari *et
al.* [29]: in the inner loop
we minimize the energy with respect to the elements of the density
kernel
using one of a number of
methods [30,31,32] to impose the
constraint that the ground state density matrix be idempotent and give
the correct number of electrons; in the outer loop we optimize the
localized functions
with respect to their
coefficients
in the basis
[16].