kkrsusc/theory.md

-# 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)
- * [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
-
-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
-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
-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
-```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
-$`R_{i\ell}\^\sigma(r;E) \approx
-R_{i\ell}\^\sigma(r;E_\text{F})$`.
-
-## Susceptibility
-
-### Kohn-Sham susceptibility
-
-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) \, .
-```
-
-Here, $`\chi\^{\uparrow\downarrow}$` and
-$`\chi\^{\downarrow\uparrow}$` correspond to $`\chi\^{+-}$` and
-$`\chi\^{-+}$`, respectively.
-
-### Dyson-like equation
-
-#### Electron Self-energy
-
-#### 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:
-```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) \; .
-```
+# 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)
+ * [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
+
+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
+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
+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
+```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
+$`R_{i\ell}\^\sigma(r;E) \approx
+R_{i\ell}\^\sigma(r;E_\text{F})$`.
+
+## Susceptibility
+
+### Kohn-Sham susceptibility
+
+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) \, .
+```
+
+Here, $`\chi\^{\uparrow\downarrow}$` and
+$`\chi\^{\downarrow\uparrow}$` correspond to $`\chi\^{+-}$` and
+$`\chi\^{-+}$`, respectively.
+
+### Dyson-like equation
+
+#### Electron Self-energy
+
+#### 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:
+```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) \; .
+```