# TPSS notes¶

## Kinetic energy density¶

Inside the augmentation sphere of atom $$a$$ ($$r<r_c^a$$), we have:

$\psi_{\sigma\mathbf{k}n}(\mathbf{r}) = \sum_i \phi_i^a(\mathbf{r}) \langle\tilde{p}_i^a | \tilde{\psi}_{\sigma\mathbf{k}n} \rangle.$

The kinetic energy density from the valence electrons will be:

$\frac{1}{2} \sum_{\mathbf{k}n} f_{\sigma\mathbf{k}n} \sum_{i_1i_2} \langle \tilde{\psi}_{\sigma\mathbf{k}n} | \tilde{p}_{i_1}^a \rangle \langle \tilde{p}_{i_2}^a | \tilde{\psi}_{\sigma\mathbf{k}n} \rangle \mathbf{\nabla}\phi_{i_1}^a \cdot \mathbf{\nabla}\phi_{i_2}^a = \frac{1}{2} \sum_{i_1i_2} D_{\sigma i_1i_2}^a \mathbf{\nabla}\phi_{i_1}^a \cdot \mathbf{\nabla}\phi_{i_2}^a.$

Here, we insert $$\phi_i^a(\mathbf{r})=Y_L\phi_j^a(r)$$ and use:

$\mathbf{\nabla}\phi_i^a(\mathbf{r}) = \mathbf{\nabla}Y_L \phi_j^a(r) + Y_L \frac{d \phi_j^a}{dr} \mathbf{r} / r,$

to get:

$\mathbf{\nabla}\phi_{i_1}^a \cdot \mathbf{\nabla}\phi_{i_2}^a = \mathbf{\nabla}Y_{L_1} \cdot \mathbf{\nabla}Y_{L_2} \phi_{j_1}^a(r) \phi_{j_2}^a(r) + Y_{L_1} Y_{L_2} \frac{d \phi_{j_1}^a}{dr} \frac{d \phi_{j_2}^a}{dr}.$

Similar equations hold for the pseudo kinetic energy density.