Non-local pseudopotential energy

For clarity of notation, we first rewrite the non-local pseudopotential of equation (14) as

$$\hat{V}_{\mathrm{nl}} \equiv \sum_{I}\sum_{lm(I)} \vert\chi_{lm}^{(I)}\rangle\langle\chi_{lm}^{(I)}\vert,$$ (28)

which defines the projectors $\vert\chi_{lm}^{(I)}\rangle$.

Robust real space methods to calculate the non-local pseudopotential energy exist in the context of traditional plane-wave DFT through the work of King-Smith et al. [19]. We manage to avoid entirely the complications involved in their method through use of the FFT box.

We need to calculate matrix elements such as

$$V_{\mathrm{nl},\alpha\beta} = \langle\phi_{\alpha}\vert\hat{V}_{\mathrm{nl}}\vert\phi_{\beta}\rangle$$

$$= \sum_{I}\sum_{lm(I)} \langle\phi_{\alpha}\vert\chi_{lm}^{(I)}\rangle\langle\chi_{lm}^{(I)}\vert\phi_{\beta}\rangle,$$ (29)

which is just a matter of computing quantities like $\langle\phi_{\alpha}\vert\chi_{lm}^{(I)}\rangle$. There will only be a contribution to a particular matrix element $V_{\mathrm{nl},\alpha\beta}$ if there is at least one projector that overlaps with both $\vert\phi_{\alpha}\rangle$ and $\vert\phi_{\beta}\rangle$ (see Figure 3). We begin with the radial part of the projectors, $\zeta_{l}^{(I)}(k)$, on a reciprocal space, linear, radial grid, up to arbitrarily high wavevector. Given that the overlap condition with the NGWFs is satisfied, we then continue to calculate each overlap in turn. For example, to compute $\langle\phi_{\alpha}\vert\chi_{lm}^{(I)}\rangle$, we envisage a real space FFT box that contains the NGWF $\vert\phi_{\alpha}\rangle$ and the projector $\vert\chi_{lm}^{(I)}\rangle$. The reciprocal representation of the projector is interpolated onto the corresponding reciprocal space FFT box using

$$\chi_{lm}^{(I)}(\mathbf{k}) = e^{-i\mathbf{k}\cdot\mathbf{R}... ...} 4\pi (-i)^{l} Z_{lm}(\Omega_{\mathbf{k}})\zeta_{l}^{(I)}(k),$$ (30)

where $\mathbf{R}_{(I)}$ is the position vector of atom $I$, and $Z_{lm}$ are real spherical harmonics.

Figure 3: An example of a typical contribution to the non-local potential matrix element $V_{\mathrm{nl},\alpha\beta}$. The overlap of the projector, $\vert\chi\rangle$, with $\vert\phi_{\alpha}\rangle$ is calculated using the FFT box with solid outline, and that with $\vert\phi_{\beta}\rangle$ is done using the FFT box with dashed outline.

Performing a FFT on the projector, we obtain it in the real space FFT box, project it into the simulation cell, and take the overlap with the NGWF by summation over the grid points that lie within the localisation region of $\vert\phi_{\alpha}\rangle$. In terms of the projection operator, $\hat{P}$, this process is represented by

$$V_{\mathrm{nl}, \alpha\beta}^{\mathrm{box}} = \sum_{I}\sum_{... ...gle\langle\chi_{lm}^{(I)}\vert\hat{P}\vert\phi_{\beta}\rangle.$$ (31)

Since the FFTs are performed only on a restricted region of the simulation cell, namely the FFT box, and because the NGWFs and atom cores are strictly localised, the cost of calculation of the non-local pseudopotential matrix in this way scales as $O(N)$.