Title: Multi-radius Soler-Williams Augmented Plane Waves (SAPWMR), Multi-Radius Soler-Williams Linearized Augmented Plane Waves (SLAPWMR) and extensions

Abstract: 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.