Update theory authored by Johannes Wasmer's avatar Johannes Wasmer
...@@ -45,6 +45,7 @@ R_{i\ell}\^\sigma(r;E_\text{F})$. ...@@ -45,6 +45,7 @@ R_{i\ell}\^\sigma(r;E_\text{F})$.
### Kohn-Sham susceptibility ### Kohn-Sham susceptibility
The transverse magnetic Kohn-Sham susceptibility is given by The transverse magnetic Kohn-Sham susceptibility is given by
```math ```math
\chi^{\sigma\bar{\sigma}}_{0,ij}(\vec{r}\,,\vec{r}\,';\omega) \chi^{\sigma\bar{\sigma}}_{0,ij}(\vec{r}\,,\vec{r}\,';\omega)
&=& -\frac{1}{\pi}\!\int^{E_\text{F}}\!\!\!\!\text{d}E \nonumber \\ &=& -\frac{1}{\pi}\!\int^{E_\text{F}}\!\!\!\!\text{d}E \nonumber \\
...@@ -63,6 +64,7 @@ $\chi\^{-+}$, respectively. ...@@ -63,6 +64,7 @@ $\chi\^{-+}$, respectively.
Let us turn our attention to the Dyson equation for the GF, including Let us turn our attention to the Dyson equation for the GF, including
the self-energy describing the coupling to the magnetic excitations: the self-energy describing the coupling to the magnetic excitations:
```math ```math
G_{ij}^\sigma(\vec{r}\,,\vec{r}\,';E) G_{ij}^\sigma(\vec{r}\,,\vec{r}\,';E)
&=& G_{0,ij}^\sigma(\vec{r}\,,\vec{r}\,';E) + \sum_{pq}\!\int\!\!\text{d}\vec{r}_1\!\int\!\!\text{d}\vec{r}_2\; \times \nonumber\ &=& G_{0,ij}^\sigma(\vec{r}\,,\vec{r}\,';E) + \sum_{pq}\!\int\!\!\text{d}\vec{r}_1\!\int\!\!\text{d}\vec{r}_2\; \times \nonumber\
... ...
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