kkrsusc/theory.md

@@ -31,11 +31,12 @@ The Korringa-Kohn-Rostoker Green function is given by
 
 ```math
 \begin{aligned}
-G_{ij}^\sigma(\vec{r}\,,\vec{r}\,';E) &= \sum_{LL'}Y_L(\hat{r}) \big(\sqrt{E}\,R_{i\ell}^\sigma(r_<;E)\,H_{i\ell}^\sigma(r_>;E)\delta_{ij}\delta_{LL'} \\
+G_{ij}^\sigma(\vec{r},\vec{r}';E) &= \sum_{LL'}Y_L(\hat{r}) \big(\sqrt{E}\,R_{i\ell}^\sigma(r_{\text{min}};E)\,H_{i\ell}^\sigma(r_{\text{max}};E)\delta_{ij}\delta_{LL'} \\
 &\quad + R_{i\ell}^\sigma(r;E)\,G^{\sigma,\text{str}}_{iL,jL'}(E)\,R_{j\ell'}^\sigma(r';E)\big) Y_{L'}(\hat{r}')
 \end{aligned}
 ```
 
+
 where $r_\< = \min(r,r\')$ and $r_\> =
 \max(r,r\')$, and $G\^{\sigma,\text{str}}_{iL,jL\'}(E)$ is the
 structural GF, describing backscattering effects.
@@ -52,12 +53,13 @@ The transverse magnetic Kohn-Sham susceptibility is given by
 
 ```math
 \begin{aligned}
-\chi^{\sigma\bar{\sigma}}_{0,ij}(\vec{r}\,,\vec{r}\,';\omega) &= -\frac{1}{\pi}\!\int^{E_\text{F}}\!\!\!\!\text{d}E \\
-&\quad \Big(G_{ij}^{\bar\sigma}(\vec{r}\,,\vec{r}\,';E+\omega+\mathrm{i} 0)\,\text{Im}\,G_{ji}^{\sigma}(\vec{r}\,',\vec{r}\,;E) \\
-&\quad\quad + \text{Im}\,G_{ij}^{\bar\sigma}(\vec{r}\,,\vec{r}\,';E)\,G_{ji}^{\sigma}(\vec{r}\,',\vec{r}\,;E-\omega-\mathrm{i} 0)\Big)
+\chi^{\sigma\bar{\sigma}}_{0,ij}(\vec{r},\vec{r}';\omega) &= -\frac{1}{\pi}\int^{E_{\text{F}}}\text{d}E \\
+&\quad \Big(G_{ij}^{\bar\sigma}(\vec{r},\vec{r}';E+\omega+\mathrm{i}0)\,\text{Im}\,G_{ji}^{\sigma}(\vec{r}',\vec{r};E) \\
+&\quad\quad + \text{Im}\,G_{ij}^{\bar\sigma}(\vec{r},\vec{r}';E)\,G_{ji}^{\sigma}(\vec{r}',\vec{r};E-\omega-\mathrm{i}0)\Big)
 \end{aligned}
 ```
 
+
 Here, $\chi\^{\uparrow\downarrow}$ and
 $\chi\^{\downarrow\uparrow}$ correspond to $\chi\^{+-}$ and
 $\chi\^{-+}$, respectively.
@@ -73,8 +75,9 @@ the self-energy describing the coupling to the magnetic excitations:
 
 ```math
 \begin{aligned}
-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 \\
-&\quad \times G_{0,ip}^\sigma(\vec{r}\,,\vec{r}_1;E)\,\Sigma_{pq}^\sigma(\vec{r}_1,\vec{r}_2;E)\,G_{qj}^\sigma(\vec{r}_2,\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 \\
+&\quad \times G_{0,ip}^\sigma(\vec{r},\vec{r}_1;E)\,\Sigma_{pq}^\sigma(\vec{r}_1,\vec{r}_2;E)\,G_{qj}^\sigma(\vec{r}_2,\vec{r}';E)
 \end{aligned}
 ```
 
+