(30) |

The pseudopotential components are themselves short-ranged in real-space, and vanish beyond the core radius . Therefore the action of the non-local pseudopotential depends only upon the form of the wave functions within this core region. We require the matrix elements of the non-local pseudopotential between localised basis functions which are not necessarily centred on the ion.

We therefore need to find an expansion of the basis functions in
terms of functions localised within the pseudopotential core. Since
the basis functions are all solutions of the Helmholtz equation, we
invoke the uniqueness theorem which states that the expansion we seek
is uniquely determined by the boundary conditions on the surface of
the core region and solve the Helmholtz equation subject to these
inhomogeneous boundary conditions by the standard method using the
formal expansion of the Green's function. The result is

(31) |

The coefficients
and
are defined by:

The are chosen by and play the same role as the in the expansion of the wave functions. The integral in equation (33) is straightforward to evaluate for given .

The surface integral in equation (32) is evaluated
by first rotating the coordinate system so that the new -axis is
parallel to
, thus mixing
the spherical harmonics [5]. The elements of the
orthogonal spherical harmonic mixing matrices
are
defined by the elements of the rotation matrix for the coordinate
system. In the new coordinate system, the surface integral is written
in terms of a one-dimensional integral

in which the dimensionless variable is introduced. denotes an associated Legendre polynomial, and these integrals can all be calculated indefinitely using elementary methods once the integrand is expanded into trigonometric functions.

The final result for
is then

(33) |

(34) |

The non-local pseudopotential data is therefore stored in terms of the core matrix elements defined in equation (37). In figure 2 we plot the non-local pseudopotential energy against the number of core Bessel functions for an s-local silicon pseudopotential generated according to the scheme of Troullier and Martins [3]. We see that the energy converges rapidly with the number of core Bessel functions used (the dashed line is the energy calculated with fifty core functions.) Increasing the number of core functions only increases the number of coefficients required, and the separable nature of the calculation means that even using fifty core functions requires very little computational effort.