We consider a unit cell (which we shall refer to as the simulation cell) with primitive lattice vectors
, volume
, and
grid points along direction
, where the
are integers. We define our basis functions to be the cell periodic, bandwidth limited Dirac delta functions (Figure 1) given by,
where and
are integers, and the
are the reciprocal lattice vectors:
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(2) |
The NGWFs
are expanded in terms of our basis
,
where the
are the expansion coefficients of
in the basis
and the sum is over all the grid points of the simulation cell,
![]() |
(4) |
where , and
are integers.
There is one basis function centered on each grid point of the simulation cell. They have the property that they are non-zero at the grid point on which they are centered and zero at all other grid points (48). This basis spans the same Hilbert space as the basis of plane-waves that can be represented by the real space grid of our simulation cell: a unitary transformation relates the two. Further properties are derived in Appendix A.
Due to the localisation of the NGWFs, the expansion coefficient,
, of a particular NGWF,
, is equal to zero if the grid point
does not fall within the LR of
. Consequently, because the size of each LR is independent of system size, each NGWF is expanded in terms of a number of basis functions that is independent of system size.