eigenvalues_of_core_states.rst 1.82 KB
 dulak committed Mar 21, 2009 1 2 3 4 5 6 .. _eigenvalues_of_core_states: ========================== Eigenvalues of core states ==========================  jensj committed Jun 20, 2013 7 8 9 10 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:  dulak committed Mar 21, 2009 11 12 13 14 15 16  .. math:: \epsilon_k = \frac{\partial E}{\partial f_k} = \frac{\partial}{\partial f_k}(\tilde{E} - \tilde{E}^a + E^a),  jensj committed Jun 20, 2013 17 18 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:  dulak committed Mar 21, 2009 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74  .. math:: \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}, .. math:: \frac{\partial n_c^a(r)}{\partial f_k} = [\phi_k^a(\mathbf{r})]^2. Using the PAW expressions for the :ref:energy contributions, we get: .. math:: \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), .. math:: \frac{\partial \tilde{E}^a}{\partial f_k} = Y_{00} \int_{r