Eigenvalues of core states

Calculating eigenvalues for core states can be useful for XAS, XES and core-level shift calculations. The eigenvalue of a core state \(k\) with a wave function \(\phi_k^a(\mathbf{r})\) located on atom number \(a\), can be calculated using this formula:

\[\epsilon_k = \frac{\partial E}{\partial f_k} = \frac{\partial}{\partial f_k}(\tilde{E} - \tilde{E}^a + E^a),\]

where \(f_k\) is the occupation of the core state. When \(f_k\) is varied, \(Q_L^a\) and \(n_c^a(r)\) will also vary:

\[\frac{\partial Q_L^a}{\partial f_k} = \int d\mathbf{r} Y_{00} [\phi_k^a(\mathbf{r})]^2 \delta_{\ell,0} = Y_{00},\]

\[\frac{\partial n_c^a(r)}{\partial f_k} = [\phi_k^a(\mathbf{r})]^2.\]

Using the PAW expressions for the energy contributions, we get:

\[\frac{\partial \tilde{E}}{\partial f_k} = Y_{00} \int d\mathbf{r} \int d\mathbf{r}' \frac{\tilde{\rho}(\mathbf{r}') \hat{g}_{00}^a(\mathbf{r} - \mathbf{R}^a)} {|\mathbf{r} - \mathbf{r}'|} = Y_{00} \int d\mathbf{r} \tilde{v}_H(\mathbf{r}) \hat{g}_{00}^a(\mathbf{r} - \mathbf{R}^a),\]

\[\frac{\partial \tilde{E}^a}{\partial f_k} = Y_{00} \int_{r<r_c^a}d\mathbf{r} \int_{r'<r_c^a}d\mathbf{r}' \frac{\tilde{\rho}^a(\mathbf{r}') \hat{g}_{00}^a(\mathbf{r}) } {|\mathbf{r} - \mathbf{r}'|}\]

\[\frac{\partial E^a}{\partial f_k} = -\frac{1}{2} \int d\mathbf{r} \phi_k^a(\mathbf{r}) \nabla^2 \phi_k^a(\mathbf{r}) + \int_{r<r_c^a}d\mathbf{r} \int_{r'<r_c^a}d\mathbf{r}' \frac{\rho^a(\mathbf{r}') [\phi_k^a(\mathbf{r})]^2 } {|\mathbf{r} - \mathbf{r}'|} + \int_{r<r_c^a}d\mathbf{r} \frac{\delta E_{\text{xc}}[n(\mathbf{r})]} {\delta n} [\phi_k^a(\mathbf{r})]^2\]