  In this work we present a new basis set for electronic structures (Density
Functional Theory (DFT)) calculations. This basis set extends Soler Williams
Linearized Augmented Plane Wave (SLAPW) basis sets by allowing variable Muffin
Tin (MT) sphere radii for the different angular momentum channels and for
different magnitude wave vectors of the augmented plane waves. With the correct
choice of MT radius, this allows us to match additional derivatives of the wave
function at the MT radius without having to resort to additional higher
derivative terms as part of the wave function expansion inside the MT sphere.
This should lead to low wave vector basis set and low linearization energy
errors, arguably as good as APW basis set size and LAPW basis linearization
errors. We call these basis sets SAPWMR and SLAPWMR depending on the number of
derivative like wave functions kept inside the MT radii. Similarly local
orbital (LO and lo basis wave functions depending on the number of derivative
terms in the MT radius) are suggested with a variable radius. that reduces the
number of derivative like terms needed to make them continuous or continuously
differentiable. As such semi-core states can also be well handled by our
methods. Furthermore in the appendix Full Potential Hamiltonian calculations
(FLAPW) are extended to FSLAPWMR full potential calculations for Hamiltonian
matrix elements. In the Appendix we introduce some further ideas to improve the
speed of DFT calculations which are relevant to the basis sets presented here
and to other basis sets such as the LAPW basis set.
