kkrsusc/theory.md

@@ -17,28 +17,28 @@ function (KKR-GF) method, space is partitioned into nonoverlapping
 regions surrounding the atoms, labeled $i$. These regions are taken as
 spherical in the atomic sphere approximation (ASA), and the KS potential
 is also assumed to be spherical around each atom,
-$`V\^{\text{KS}}_i(r)$`, with $`r = |\vec{r}\,|$` and $`\hat{r} =
-\vec{r}/r$`. Then the KS GF is expressed in terms of energy-dependent
+$V\^{\text{KS}}_i(r)$, with $r = |\vec{r}\,|$ and $\hat{r} =
+\vec{r}/r$. Then the KS GF is expressed in terms of energy-dependent
 scattering solutions for each atomic potential,
-$`R_{i\ell}\^\sigma(r;E)\,Y_{L}(\hat{r})$` and
-$`H_{i\ell}\^\sigma(r;E)\,Y_{L}(\hat{r})$`, which are products of
-radial functions and (real) spherical harmonics, for each spin $`\sigma
-= \;\uparrow,\,\downarrow$` and angular momentum $`L = (\ell,m)$`.
-$`R_{i\ell}\^\sigma(r;E)$` is regular at the center of the ASA
-sphere, and $`H_{i\ell}\^\sigma(r;E)$` diverges there.
+$R_{i\ell}\^\sigma(r;E)\,Y_{L}(\hat{r})$ and
+$H_{i\ell}\^\sigma(r;E)\,Y_{L}(\hat{r})$, which are products of
+radial functions and (real) spherical harmonics, for each spin $\sigma
+= \;\uparrow,\,\downarrow$ and angular momentum $L = (\ell,m)$.
+$R_{i\ell}\^\sigma(r;E)$ is regular at the center of the ASA
+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}') \; ,
 ```
-where $`r_\< = \min(r,r\')$` and $`r_\> =
-\max(r,r\')$`, and $`G\^{\sigma,\text{str}}_{iL,jL\'}(E)$` is the
+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.
 
-Near the Fermi energy ($`E_\text{F}$`) one may approximate
-$`R_{i\ell}\^\sigma(r;E) \approx
-R_{i\ell}\^\sigma(r;E_\text{F})$`.
+Near the Fermi energy ($E_\text{F}$) one may approximate
+$R_{i\ell}\^\sigma(r;E) \approx
+R_{i\ell}\^\sigma(r;E_\text{F})$.
 
 ## Susceptibility
 
@@ -51,9 +51,9 @@ The transverse magnetic Kohn-Sham susceptibility is given by
    & & \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) \, .
 ```
 
-Here, $`\chi\^{\uparrow\downarrow}$` and
-$`\chi\^{\downarrow\uparrow}$` correspond to $`\chi\^{+-}$` and
-$`\chi\^{-+}$`, respectively.
+Here, $\chi\^{\uparrow\downarrow}$ and
+$\chi\^{\downarrow\uparrow}$ correspond to $\chi\^{+-}$ and
+$\chi\^{-+}$, respectively.
 
 ### Dyson-like equation