Charge density and total electronic energy with Non-orthogonal Generalised Wannier functions

Linear-scaling DFT calculations are aimed at large systems, and in particular, large unit cells. Thus in this work we will be concerned with calculations only at the -point, i.e. . This means that the Bloch bands and therefore the NGWFs can be chosen to be real. We can also drop the dependence of the NGWFs on , so that .

Our basis set is the set of periodic bandwidth limited delta functions
that are centred on the points
of a regular
real-space grid:

where is the volume per grid point and is the result of bandwidth limiting the function to the same plane-wave components as in (5).

We represent the NGWFs in the delta function basis by

where it is straightforward to show that the amplitudes are the result of a discrete Fourier transform on the delta function expansion coefficients .

In (7) the sum over the , and indices
formally goes over the grid points of a regular grid
that extends over the *whole* simulation cell.
From now on however, we will restrict all NGWFs to have contributions
*only* from delta functions centred inside a predefined
spherical region. This spherical region is in general different
for each NGWF.
Thus we impose on (7) the condition:

(9) |

The charge density of equation (2) with our
NGWFs becomes (from now on we will use the summation convention
for repeated Greek indices)

which involves the fine grid delta functions that are defined in a similar way to the of equation (5) but include up to twice the maximum wavevector of in every reciprocal lattice vector direction (see also Appendix A). This is necessary because a product of two delta functions is a linear combination of fine grid delta functions , a result reminiscent of the Gaussian function product rule [30].

The expressions for the various contributions to the total electronic energy
with the NGWFs are simple to derive from
(10). The total energy is the sum of the kinetic energy ,
the Hartree energy , the local pseudopotential energy ,
the non-local pseudopotential energy and the
exchange and correlation energy

(11) |

To compute these matrix elements we can apply to the plane-wave representation (8) of and then evaluate the integral in real-space where it is equal to a discrete sum over grid points where obviously plays the role of of equation (6).

Calculation of the Hartree energy requires first the Hartree
potential. From equation (10) we see that the charge
density is a fine grid delta function expansion, thus the same should
be true for the Hartree potential, which is a convolution of the
charge density with the Coulomb potential. Therefore,
can be written as a linear combination of fine grid delta functions
and extends over the whole simulation cell:

(13) |

This quantity can be calculated as a discrete summation on the fine grid of the product of with or equivalently as a trace of the product of the density kernel and the potential matrix elements. The local potential matrix elements are integrals that are identically equal to discrete sums on the regular grid provided of course that is first put on the regular grid.

The local pseudopotential energy is calculated in an entirely analogous manner to the Hartree energy and can be represented by equation (14) if we put in place of and multiply it by a factor of 2 to take into account the lack of self-interaction in this case.

The non-local pseudopotential energy is the expectation
value of the non-local potential operator
in the Kleinman-Bylander form [31]:

(15) |

(16) |

The exchange-correlation energy is obtained by
approximating the exchange-correlation functional expression
as a direct summation on the fine grid, which first involves
the evaluation of a function
whose particular
form depends on our choice of
exchange-correlation functional[2]:

(17) |