Local SIC: Difference between revisions

Revision as of 19:03, 12 August 2022

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$E^{P Z S I C - D F A} = E^{D F A} [ρ_{↑}, ρ_{↓}] - \sum_{i} {U^{} [ρ_{i σ}] + E_{X C}^{D F A} [ρ_{i σ}, 0]} .$

$E_{X C}^{L S I C - D F A} = E_{X C}^{D F A} [ρ_{↑}, ρ_{↓}] - \sum_{i, σ}^{o c c} {U^{L S I C} [ρ_{i, σ}] + E_{X C}^{L S I C} [ρ_{i, σ}, 0]}$

where

$U^{L S I C} [ρ_{i, σ}] = \frac{1}{2} \int d \vec{r} {z_{σ} (\vec{r})}^{k} ρ_{i, σ} (\vec{r}) \int d \vec{r^{'}} \frac{ρ_{i, σ} (\vec{r^{'}})}{| \vec{r} - \vec{r^{'}} |}$

$E_{X C}^{L S I C} [ρ_{i, σ}, 0] = \int d \vec{r} {z_{σ} (\vec{r})}^{k} ρ_{i, σ} (\vec{r}) ϵ_{X C}^{D F A} ([ρ_{i, σ}, 0], \vec{r}) .$

@@ Line 1: / Line 1: @@
-== Local SIC Method
+Note: this page is visible to public
+<math>
+    E^{PZSIC-DFA} = E^{DFA}[\rho_{\uparrow},\rho_{\downarrow}]- \sum_i \left \{ U^{}[\rho_{i\sigma}]+E_{XC}^{DFA}[\rho_{i\sigma},0]\right\}.
+</math>
+<math>
+    E_{XC}^{LSIC-DFA} = E_{XC}^{DFA}[\rho_{\uparrow},\rho_{\downarrow}]
+                       - \sum_{i,\sigma}^{occ}
+     \left \{ U^{LSIC}[\rho_{i,\sigma}]
+     + E_{XC}^{LSIC}[\rho_{i,\sigma},0] \right \}
+</math>
+where
+<math>
+     U^{LSIC}[\rho_{i,\sigma}] =
+       \frac{1}{2} \int d\vec{r}\,
+       \{z_{\sigma}(\vec{r})\}^k \,
+       \rho_{i,\sigma}(\vec{r})
+       \int d\vec{r'}\, \frac{\rho_{i,\sigma}(\vec{r'})}{\vert \vec{r}-\vec{r'}\vert}
+</math>
+<math>
+     E_{XC}^{LSIC}[\rho_{i,\sigma},0]
+     =  \int d\vec{r}\,  \{z_{\sigma}(\vec{r})\}^k
+     \rho_{i,\sigma} (\vec{r}) \epsilon_{XC}^{DFA}([\rho_{i,\sigma},0],\vec{r}).
+</math>

Local SIC: Difference between revisions

Revision as of 19:03, 12 August 2022

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