Projection operators

We must also introduce an operator, $\hat{P}(\alpha\beta)$, that is required to project NGWFs between their representation in terms of the plane-waves of the entire simulation cell and those of the FFT box. This operator has arguments $\alpha$ and $\beta$ as it is dependent upon the exact location of the FFT box within the simulation cell, which in turn depends upon the particular matrix element that is being calculated. We thus define for the pair of NGWFs $\phi_{\alpha}$ and $\phi_{\beta}$:

$$\hat{P}(\alpha\beta) = \frac{1}{\Omega}\sum_{k=0}^{n_{1}-1}\... ...+K_{\alpha\beta})(l+L_{\alpha\beta})(m+M_{\alpha\beta})}\vert,$$ (23)

where $K_{\alpha\beta}$, $L_{\alpha\beta}$ and $M_{\alpha\beta}$ are integers denoting the grid point of the simulation cell at which the origin of the FFT box ($k=l=m=0$) is located, and $\Omega$ is the volume per grid point.

When this operator acts upon a function with the periodicity of the simulation cell, i.e., a function given by equation (3), it maps it onto the `same' function with the periodicity of the FFT box:

$$\hat{P}(\alpha\beta)\phi_{\alpha}(\mathbf{r}) = \sum_{k=0}^{... ...{2}-1}\sum_{m=0}^{n_{3}-1} c_{klm,\alpha} d_{klm}(\mathbf{r}),$$ (24)

where $c_{klm,\alpha} \equiv C_{(k+K_{\alpha\beta})(l+L_{\alpha\beta})(m+M_{\alpha\beta}),\alpha}$.

Similarly, we define a supplementary operator, $\hat{Q}(\alpha\beta)$, that performs the same task, but for the fine grid representation:

$$\hat{Q}(\alpha\beta) = \frac{8}{\Omega} \sum_{x=0}^{2n_{1}-1... ..._{\alpha\beta})(y+2L_{\alpha\beta})(z+2M_{\alpha\beta})}\vert.$$ (25)

The operators $\hat{P}^{\dagger}(\alpha\beta)$ and $\hat{Q}^{\dagger}(\alpha\beta)$ map functions from the FFT box back to the simulation cell on the standard and fine grids respectively.