# Exchange and correlation functionals¶

## Libxc¶

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:

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

and

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

## 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 …

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

where

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

and

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

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

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

Inserting

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

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¶

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