  In this work we present a new basis set for electronic structure calculations
of crystalline solids using Density Functional theory (DFT) methods. In this
construction we take advantage of the fact that most DFT calculations use a
convergence loop in order to obtain the eigenstates of a final Khon Sham (KS)
Hamiltonian matrix whose eigenstates also give the appropriate electron density
needed to obtain the KS potential needed for that KS Hamiltonian matrix. Here
we propose that for the basis of each step of the iteration we use the previous
eigenstate basis in the interstitial region but augmented inside the MT sphere
with the solution to the spherically averaged KS Hamiltonian for the energy
window of that eigenstate. To reduce the number of times the KS potential needs
to be solved inside the MT spheres it is advantageous to use energy windows and
solve the KS Hamiltonian inside the MT region only once per window (at some
energy inside the window) so that the KS Hamiltonian needs only be solved a
small number of times per iteration, for practical applications on the order of
10 to 100 windows. This method combines the energy dependence of methods such
as Projected Augmented Wave functions (PAW) with the ability of the basis set
to adjust to the solid state (rather then atomic) environment of basis sets
such as Linearized Augmented Plane Waves (LAPW).
