kkrsusc/theory.md

@@ -30,8 +30,10 @@ sphere, and $H_{i\ell}\^\sigma(r;E)$ diverges there.
 The Korringa-Kohn-Rostoker Green function is given by
 
 ```math
- 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'} \nonumber\
- &&\hspace{-4em} + R_{i\ell}^\sigma(r;E)\,G^{\sigma,\text{str}}_{iL,jL'}(E)\,R_{j\ell'}^\sigma(r';E)\big) Y_{L'}(\hat{r}') \; ,
+\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'} \\
+&\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 $r_\> =
@@ -49,9 +51,11 @@ R_{i\ell}\^\sigma(r;E_\text{F})$.
 The transverse magnetic Kohn-Sham susceptibility is given by
 
 ```math
- \chi^{\sigma\bar{\sigma}}_{0,ij}(\vec{r}\,,\vec{r}\,';\omega)
-   &=& -\frac{1}{\pi}\!\int^{E_\text{F}}\!\!\!\!\text{d}E \nonumber \\
-   & & \hspace{-6em}\Big(G_{ij}^{\bar\sigma}(\vec{r}\,,\vec{r}\,';E+\omega+\mathrm{i} 0)\,\text{Im}\,G_{ji}^{\sigma}(\vec{r}\,',\vec{r}\,;E) + \text{Im}\,G_{ij}^{\bar\sigma}(\vec{r}\,,\vec{r}\,';E)\,G_{ji}^{\sigma}(\vec{r}\,',\vec{r}\,;E-\omega-\mathrm{i} 0)\Big) \, .
+\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)
+\end{aligned}
 ```
 
 Here, $\chi\^{\uparrow\downarrow}$ and
@@ -68,7 +72,9 @@ Let us turn our attention to the Dyson equation for the GF, including
 the self-energy describing the coupling to the magnetic excitations:
 
 ```math
- 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\
-   & & \hspace{-6em}\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) \; .
+\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)
+\end{aligned}
 ```
+