Kinetic energy

We want to calculate matrix elements of the form

$$T_{\alpha\beta} = \langle\phi_{\alpha}\vert\hat{T}\vert\phi_{\beta}\rangle.$$ (26)

The NGWFs are localised in real space, so we need only consider those elements $T_{\alpha\beta}$ for which the localisation regions of $\phi_{\alpha}$ and $\phi_{\beta}$ overlap as other contributions will be effectively zero. Once we have established that there is an overlap, we imagine the pair of NGWFs as being enclosed within the FFT box that has been defined for the calculation, as shown in Figure 2. We then apply the operator $\hat{P}(\alpha\beta)$ to $\vert\phi_{\beta}\rangle$ to give it the periodicity of the FFT box. $\hat{P}(\alpha\beta) \vert\phi_{\beta}\rangle$ may be then fast Fourier transformed to reciprocal space with a computational cost that scales as $O(N_{\mathrm{box}}\log N_{\mathrm{box}})$, where $N_{\mathrm{box}}$ is the number of grid points in the FFT box and is independent of system-size. $\hat{T}$ is applied to $\hat{P}(\alpha\beta) \vert\phi_{\beta}\rangle$ in reciprocal space by multiplying by $\vert\mathbf{k}\vert^{2}/2$ at each reciprocal lattice point, $\mathbf{k}$, in the FFT box. Performing another FFT, we obtain $\hat{T}\hat{P}(\alpha\beta)\vert\phi_{\beta}\rangle$ in the real space FFT box, again with a cost of $O(N_{\mathrm{box}}\log N_{\mathrm{box}})$. We may then apply $\hat{P}^{\dagger}(\alpha\beta)$ to this to map it back into the simulation cell where the matrix element $T_{\alpha\beta}^{\mathrm{box}}$, given by

$$T_{\alpha\beta}^{\mathrm{box}} = \langle\phi_{\alpha}\vert\hat{P}^{\dagger}(\alpha\beta)\hat{T}\hat{P}(\alpha\beta)\vert\phi_{\beta}\rangle$$

$$= \Omega \sum_{K,L,M \in \mathrm{LR}^{\alpha}} C_{KLM,\alpha} [... ...\dagger}(\alpha\beta)\hat{T}\hat{P}(\alpha\beta)\phi_{\beta}](\mathbf{r}_{KLM})$$ (27)

is calculated by summation over the grid points that lie within the localisation region of $\phi_{\alpha}$ ( $\mathrm{LR}^{\alpha}$). The superscript `box' signifies that a quantity has been calculated using the FFT box technique. We will show in section 6 that $T_{\alpha\beta}^{\mathrm{box}}$ is an accurate approximation to $T_{\alpha\beta}$ . This is because the FFT box technique is essentially a method of coarse-sampling the frequency components of the NGWFs in reciprocal space. As the NGWFs are truly localised in real space, we expect them to be very smooth in reciprocal space, and thus amenable to coarse-sampling.

For a single matrix element, the FFT box method makes the cost of calculation independent of system size. Thus, for all non-zero matrix elements the cost scales as $O(N)$.