Origin of the basis set

The spherical-wave basis functions[13] used in this
work are eigenfunctions of the Helmholtz equation

where are spherical polar coordinates with origin at the center of the sphere, is a non-negative integer and is an integer satisfying . is the spherical Bessel function of order , and is a spherical harmonic. The eigenvalue is determined from the th zero of where

(2) |

The real spherical-wave basis functions used in this work are

where are the associated coefficients. For most systems tested in this work, we have used one basis sphere per atom, where the basis spheres are centered on the atoms. For some systems we have increased the number of basis spheres by placing basis spheres between the atoms. In principle it is possible to use two or more basis spheres of different radii centered on the same atom, but this arrangement has not been studied in this work. We note that even though the basis functions belonging to different basis spheres are generally nonorthogonal, one of the main advantages of this basis set is that it is possible to analytically evaluate[13] the overlap matrix elements

and kinetic energy matrix elements

We also note that the matrix elements for the nonlocal pseudopotentials in the Kleinman-Bylander[19] form can also be evaluated analytically by first expanding the projectors in the spherical-wave basis set.