kkrsusc/theory.md

 ---
 title: 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:
+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)
+ \* [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 Kohn-Sham (KS) Green function (GF) is the resolvent of the corresponding Hamiltonian, <span dir="">G\_{\\text{KS}}(E) = (E - \\mathcal{H}\_{\\text{KS}})^{-1}</span>. In the Korringa-Kohn-Rostoker Green function (KKR-GF) method, space is partitioned into nonoverlapping regions surrounding the atoms, labeled <span dir="">i</span>. 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, <span dir="">V\\^{\\text{KS}}\_i(r)</span>, with <span dir="">r = |\\vec{r}\\,|</span> and <span dir="">\\hat{r} = \\vec{r}/r</span>. Then the KS GF is expressed in terms of energy-dependent scattering solutions for each atomic potential, <span dir="">R\_{i\\ell}\\^\\sigma(r;E)\\,Y\_{L}(\\hat{r})</span> and <span dir="">H\_{i\\ell}\\^\\sigma(r;E)\\,Y\_{L}(\\hat{r})</span>, which are products of radial functions and (real) spherical harmonics, for each spin <span dir="">\\sigma = \\;\\uparrow,\\,\\downarrow</span> and angular momentum <span dir="">L = (\\ell,m)</span>. <span dir="">R\_{i\\ell}\\^\\sigma(r;E)</span> is regular at the center of the ASA sphere, and <span dir="">H\_{i\\ell}\\^\\sigma(r;E)</span> diverges there.
 
 The Korringa-Kohn-Rostoker Green function is given by
 
@@ -41,11 +20,24 @@ G_{ij}^\sigma(\vec{r},\vec{r}';E) &= \sum_{LL'}Y_L(\hat{r}) \big(\sqrt{E}\,R_{i\
 \end{aligned}
 ```
 
-where $r_{\text{min}} = \min(r,r')$ and $r_{\text{max}} = \max(r,r')$, and $G^{\sigma,\text{str}}_{iL,jL'}(E)$ is the structural GF, describing backscattering effects.
+where <span dir="">r\_{\\text{min}} = \\min(r,r')</span> and <span dir="">r\_{\\text{max}} = \\max(r,r')</span>, and <span dir="">G^{\\sigma,\\text{str}}\_{iL,jL'}(E)</span> is the structural GF, describing backscattering effects.
+
+Near the Fermi energy (<span dir="">E\_\\text{F}</span>) one may approximate <span dir="">R\_{i\\ell}\\^\\sigma(r;E) \\approx R\_{i\\ell}\\^\\sigma(r;E\_\\text{F})</span>.
+
+We now notice that the quantities present in the Green's function (The regular <span dir="">R(\\mathbf{r};E)</span>, and irregular <span dir="">H(\\mathbf{r};E)</span> component of the radial wave function) depend on both the energy and the position. This would make many calculations unfeasable, thus a separation of the energetic and spatial components is used.
 
-Near the Fermi energy ($E_\text{F}$) one may approximate
-$R_{i\ell}\^\sigma(r;E) \approx
-R_{i\ell}\^\sigma(r;E_\text{F})$.
+Using a new basis set <span dir="">\\ket{\\phi\_{ilb}}</span>, where <span dir="">i</span> accounts for the lattice site, <span dir="">l</span> for the orbital, and <span dir="">b</span> for the energy. One can disentangle the spatial dependence using
+
+```math
+
+G^{proj}_{iLb,jL^'b^'}(E) = \int_{0}^{r_{MT}}drr^{2}\int_{0}^{r_{MT}}dr^{'}r^{'2} Y_{L}(\hat{r})\phi_{ilb}(r)G_{iL;jL^{'}}(r,r^{'};E)Y_{L^{'}}(\hat{r}^{'})\phi_{jl^{'}b^{'}}(r^{'})
+```
+The basis function used for the projection is constructed as
+
+```math
+
+\phi_{ilb} = \frac{R_{il}(R;E_{b}}{\sqrt{\int_{0}^{r_{MT}}dr^{'}r^{'2}(R_{il}(r^{'};E_{b})^{2}}
+```
 
 ## Susceptibility
 
@@ -61,10 +53,7 @@ The transverse magnetic Kohn-Sham susceptibility is given by
 \end{aligned}
 ```
 
-
-Here, $\chi_{\uparrow\downarrow}$ and
-$\chi_{\downarrow\uparrow}$ correspond to $\chi_{+-}$ and
-$\chi_{-+}$, respectively.
+Here, <span dir="">\\chi\_{\\uparrow\\downarrow}</span> and <span dir="">\\chi\_{\\downarrow\\uparrow}</span> correspond to <span dir="">\\chi\_{+-}</span> and <span dir="">\\chi\_{-+}</span>, respectively.
 
 ### Dyson-like equation
 
@@ -72,8 +61,7 @@ $\chi_{-+}$, respectively.
 
 #### 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:
+Let us turn our attention to the Dyson equation for the GF, including the self-energy describing the coupling to the magnetic excitations:
 
 ```math
 \begin{aligned}
@@ -81,5 +69,3 @@ G_{ij}^\sigma(\vec{r},\vec{r}';E) &= G_{0,ij}^\sigma(\vec{r},\vec{r}';E) + \sum_
 &\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}
 ```
\ No newline at end of file
-
-