Projected Augmented Waves (PAW): extended resolution of unity method

The Projected Augmented Waves (PAW) method is based on a linear transformation between the pseudo wavefunctions and the all electron wavefunctions. To obtain high accuracy with this method, it is important that the local part of the linear transform (inside each atomic sphere) be defined over a complete basis set (with deviations from completeness leading to corrections to the total energy not computed within current implementations of PAW). Here we show how to make this basis much closer to complete without significant additional computational work and without modifying the transformation in any significant way thereby making the modifications we propose easy to implement in current electronic structure codes for PAW.. This is done by extending the resolution of unity used for the transform to include more smooth wavefunctions (which have nothing to do with the atomic problem) and having them linearly transform via the identity.