kkrsusc/theory.md

-###### KKRsusc program: Theory {#kkrsusc_program_theory}
+# KKRsusc program: Theory
 
 This page should give you a brief introduction to the theoretical
 background of the KKRsusc program. To find more details on the theory
 and a detailed description on the method follow these links:
 
-` * `[`S.`` ``Lounis`` ``et`` ``al.,`` ``Phys.`` ``Rev.`` ``Lett.`` ``105,`` ``187205`` ``(2010)`](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.105.187205 "wikilink")\
-` * `[`S.`` ``Lounis`` ``et`` ``al.,`` ``Phys.`` ``Rev.`` ``B`` ``83,`` ``035109`` ``(2011)`](http://journals.aps.org/prb/abstract/10.1103/PhysRevB.83.035109 "wikilink")\
-` * `[`B.`` ``Schweflinghaus`` ``et`` ``al.,`` ``Phys.`` ``Rev.`` ``B`` ``89,`` ``235439`` ``(2014)`](http://journals.aps.org/prb/abstract/10.1103/PhysRevB.89.235439 "wikilink")
+ * [S. Lounis et al., Phys. Rev. Lett. 105, 187205 (2010)](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.105.187205)
+ * [S. Lounis et al., Phys. Rev. B 83, 035109 (2011)](http://journals.aps.org/prb/abstract/10.1103/PhysRevB.83.035109)
+ * [B. Schweflinghaus et al., Phys. Rev. B 89, 235439 (2014)](http://journals.aps.org/prb/abstract/10.1103/PhysRevB.89.235439)
 
-##### Projected Green function {#projected_green_function}
+## Projected Green function
 
 The Kohn-Sham (KS) Green function (GF) is the resolvent of the
-corresponding Hamiltonian, \$G\_{\\text{KS}}(E) = (E -
-\\mathcal{H}\_{\\text{KS}})\^{-1}\$. In the Korringa-Kohn-Rostoker Green
+corresponding Hamiltonian, $`G_{\text{KS}}(E) = (E -
+\mathcal{H}_{\text{KS}})\^{-1}$`. In the Korringa-Kohn-Rostoker Green
 function (KKR-GF) method, space is partitioned into nonoverlapping
-regions surrounding the atoms, labeled \$i\$. These regions are taken as
+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.
-
-The Korringa-Kohn-Rostoker Green function is given by \\begin{eqnarray}
-
-` 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}') \; ,`
-
-\\end{eqnarray} where \$r\_\< = \\min(r,r\')\$ and \$r\_\> =
-\\max(r,r\')\$, and \$G\^{\\sigma,\\text{str}}\_{iL,jL\'}(E)\$ is the
+$`R_{i\ell}\^\sigma(r;E)\,Y_{L}(\hat{r})$` 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
 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
+## Susceptibility
 
-#### Kohn-Sham susceptibility {#kohn_sham_susceptibility}
+### Kohn-Sham susceptibility
 
 The transverse magnetic Kohn-Sham susceptibility is given by
-\\begin{eqnarray}
-
-` \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) \, .`
+```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) \, .
+```
 
-\\end{eqnarray}
+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
 
-#### Dyson-like equation {#dyson_like_equation}
+#### Electron Self-energy
 
-##### Electron Self-energy {#electron_self_energy}
-
-#### Dyson equation {#dyson_equation}
+#### Dyson equation
 
 Let us turn our attention to the Dyson equation for the GF, including
 the self-energy describing the coupling to the magnetic excitations:
-\\begin{eqnarray}
-
-` 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) \; .`
-
-\\end{eqnarray}
+```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) \; .
+```