Exchange and correlation functionals


We used the functionals from libxc. …

Calculation of GGA potential

In libxc we have (see also “Standard subroutine calls” on ccg_dft_design) \(\sigma_0=\sigma_{\uparrow\uparrow}\), \(\sigma_1=\sigma_{\uparrow\downarrow}\) and \(\sigma_2=\sigma_{\downarrow\downarrow}\) with

\[\sigma_{ij} = \mathbf{\nabla}n_i \cdot \mathbf{\nabla}n_j\]

Uniform 3D grid

We use a finite-difference stencil to calculate the gradients:

\[\mathbf{\nabla}n_g = \sum_{g'} \mathbf{D}_{gg'} n_{g'}.\]

The \(x\)-component of \(\mathbf{D}_{gg'}\) will be non-zero only when \(g\) and \(g'\) grid points are neighbors in the \(x\)-direction, where the values will be \(1/(2h)\) when \(g'\) is to the right of \(g\) and \(-1/(2h)\) when \(g'\) is to the left of \(g\). Similar story for the \(y\) and \(z\) components.

Let’s look at the spin-\(k\) XC potential from the energy expression \(\sum_g\epsilon(\sigma_{ijg})\):

\[v_{kg} = \sum_{g'} \frac{\partial \epsilon(\sigma_{ijg'})}{\partial n_{kg}} = \sum_{g'} \frac{\partial \epsilon(\sigma_{ijg'})}{\partial \sigma_{ijg'}} \frac{\partial \sigma_{ijg'}}{\partial n_{kg}}\]

Using \(v_{ijg}=\partial \epsilon(\sigma_{ijg})/\partial \sigma_{ijg}\), \(\mathbf{D}_{gg'}=-\mathbf{D}_{g'g}\) and

\[\frac{\partial \sigma_{ijg'}}{\partial n_{kg}} = (\delta_{jk} \mathbf{D}_{g'g} \cdot \mathbf{\nabla}n_{ig'} + \delta_{ik} \mathbf{D}_{g'g} \cdot \mathbf{\nabla}n_{jg'}),\]

we get:

\[v_{kg} = -\sum_{g'} \mathbf{D}_{gg'} \cdot (v_{ijg'} [\delta_{jk} \mathbf{\nabla}n_{ig'} + \delta_{ik} \mathbf{\nabla}n_{jg'}]).\]

The potentials from the general energy expression \(\sum_g\epsilon(\sigma_{0g}, \sigma_{1g}, \sigma_{2g})\) will be:

\[v_{\uparrow g} = -\sum_{g'} \mathbf{D}_{gg'} \cdot (2v_{\uparrow\uparrow g'} \mathbf{\nabla}n_{\uparrow g'} + v_{\uparrow\downarrow g'} \mathbf{\nabla}n_{\downarrow g'})\]


\[v_{\downarrow g} = -\sum_{g'} \mathbf{D}_{gg'} \cdot (2v_{\downarrow\downarrow g'} \mathbf{\nabla}n_{\downarrow g'} + v_{\uparrow\downarrow g'} \mathbf{\nabla}n_{\uparrow g'}).\]

PAW correction

Spin-paired case:

\[\Delta E = \sum_g 4 \pi w r_g^2 \Delta r_g [\epsilon(n_g, \sigma_g) - \epsilon(\tilde n_g, \tilde\sigma_g)],\]

where \(w\) is the weight …

\[n_g = \sum_{i_ii_2} D_{i_1i_2} \phi_{j_1g} Y_{L_1} \phi_{j_2g} Y_{L_2} + n_c(r_g) = \sum_L n_{Lg} Y_L,\]


\[n_{Lg} = \sum_q D_{Lq} n_{qg} + \delta_{L,0} \sqrt{4 \pi} n_c(r_g)\]


\[D_{Lq} = \sum_p D_p G_{L_1L_2}^L \delta_{q_p,q} = \sum_p D_p B_{Lpq}.\]
\[\mathbf{\nabla} n_g = \sum_L Y_L \sum_{g'} D_{gg'} n_{Lg'} \hat{\mathbf{r}} + \sum_L \frac{n_{Lg}}{r_g} r \mathbf{\nabla} Y_L = a_g \hat{\mathbf{r}} + \mathbf{b}_g / r_g.\]

Notice that \(r \mathbf{\nabla} Y_L\) is independent of \(r\) - just as \(Y_L\) is. From the two contributions, which are orthogonal (\(\hat{\mathbf{r}} \cdot \mathbf{b}_g = 0\)), we get

\[\sigma_g = a_g^2 + \mathbf b_g \cdot \mathbf b_g / r_g^2.\]
\[\frac{\partial \Delta E}{\partial n_{Lg}} = 4 \pi w \sum_{g'} r_{g'}^2 \Delta r_{g'} \frac{\partial \epsilon}{\partial \sigma_{g'}} \frac{\partial \sigma_{g'}}{\partial n_{Lg}}.\]


\[\frac{\partial \sigma_{g'}}{\partial n_{Lg}} = 2 a_{g'} Y_L D_{g'g} + 2 \mathbf b_g \cdot (r \mathbf{\nabla} Y_L) \delta_{gg'} / r_g^2,\]

we get

\[\frac{\partial \Delta E}{\partial n_{Lg}} = 8 \pi w \sum_{g'} r_{g'}^2 \Delta r_{g'} \frac{\partial \epsilon}{\partial \sigma_{g'}} a_{g'} Y_L D_{g'g} + 8 \pi w \Delta r_g \frac{\partial \epsilon}{\partial \sigma_g} \mathbf b_g \cdot (r \mathbf{\nabla} Y_L).\]

Non-collinear case

\[\mathbf{m}_g = \sum_L \mathbf{M}_{Lg} Y_L.\]
\[n_{\alpha g} = (n_g + \alpha m_g) / 2.\]
\[2 \mathbf{\nabla} n_{\alpha g} = \mathbf{\nabla} n_g + \alpha \sum_L ( Y_L \sum_{g'} D_{gg'} \frac{\mathbf{m}_g \cdot \mathbf{M}_{Lg'}}{m_g} \hat{\mathbf{r}} + \frac{\mathbf{m}_g \cdot \mathbf{M}_{Lg}}{m_g r_g} r \mathbf{\nabla} Y_L)\]
\[= (a_g + \alpha c_g) \hat{\mathbf{r}} + (\mathbf{b}_g + \alpha \mathbf{d}_g) / r_g.\]
\[4 \sigma_{\alpha \beta g} = (a_g + \alpha c_g) (a_g + \beta c_g) + (\mathbf{b}_g + \alpha \mathbf{d}_g) \cdot (\mathbf{b}_g + \beta \mathbf{d}_g) / r_g^2.\]
\[\frac{\partial c_g}{\partial \mathbf{M}_{Lg'}} = \frac{Y_L}{m_g} ( D_{gg'} \mathbf{m}_g + \delta_{gg'} \mathbf{m}_g' - \delta_{gg'} \frac{\mathbf{m}_g \cdot \mathbf{m}_g'}{m_g^2} \mathbf{m}_g).\]
\[\frac{\partial (\mathbf{d}_g)_\gamma}{\partial \mathbf{M}_{Lg'}} = \frac{Y_L \delta_{gg'}}{m_g} ( \mathbf{m}_g r \nabla_\gamma Y_L + \sum_{L'} \mathbf{M}_{L'g} r \nabla_\gamma Y_{L'} - \frac{\mathbf{m}_g}{m_g^2} \sum_{L'} \mathbf{m}_g \cdot \mathbf{M}_{L'g} r \nabla_\gamma Y_{L'}).\]