Basis set

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:

(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.