kkrimp/theory.md

-###### KKRimp program: Theory {#kkrimp_program_theory}
+# KKRimp program: Theory
 
 This page should give you a brief introduction to the theoretical
 background of the impurity calculations. To find more details on the
 theory and a detailed description on the method follow the following
 links:
 
-` * PhD thesis of D. Bauer (https://publications.rwth-aachen.de/record/229375)`\
-` * Dederichs et al., MRS Proceedings 253, 185 (1991)`
-
-##### Green function based DFT calculations {#green_function_based_dft_calculations}
-
-Starting point: Schrödinger equation \$\$ \\mathcal{H}\|\\psi\\rangle =
-E\|\\psi\\rangle \$\$ This can formally be written with the Green
-function \$\\mathcal{G}\$ as \$\$ (\\epsilon
--\\mathcal{H})\\mathcal{G}(\\epsilon)=1, \\quad \\epsilon=E+i\\delta,
-\\quad \\delta\>0 \$\$ Thus by knowing the GF one can extract the
-systems properties, i.e. the expectation value \$\\langle
-\\mathcal{A}\\rangle\$ of an operator \$\\hat{\\mathcal{A}}\$: \$\$
-\\langle\\mathcal{A}\\rangle=-\\frac{1}{\\pi}\\mathrm{Im}\\int\_{-\\infty}\^{E\_F}\\mathrm{d}E\\mathrm{Tr}\\left\[
-\\hat{\\mathcal{A}}\\mathcal{G}(E) \\right\]\$\$ For example the density
-\$\\rho\$ can be calculated via
-\$\$\\rho(\\vec{x})=-\\frac{1}{\\pi}\\mathrm{Im}\\int\_{-\\infty}\^{E\_F}\\mathrm{d}E\\mathcal{G}(\\vec{x},\\vec{x};E)\$\$
+ * PhD thesis of D. Bauer (https://publications.rwth-aachen.de/record/229375)
+ * Dederichs et al., MRS Proceedings 253, 185 (1991)
+
+## Green function based DFT calculations
+
+Starting point: Schrödinger equation
+```math
+\mathcal{H}\|\psi\rangle = E\|\psi\rangle
+```
+This can formally be written with the Green
+function $`\mathcal{G}`$ as
+```math
+(\epsilon -\mathcal{H})\mathcal{G}(\epsilon)=1, \quad \epsilon=E+i\delta,
+\quad \delta\>0
+```
+Thus by knowing the GF one can extract the
+systems properties, i.e. the expectation value $`\langle
+\mathcal{A}\rangle`$ of an operator $`\hat{\mathcal{A}}`$:
+```math
+\langle\mathcal{A}\rangle=-\frac{1}{\pi}\mathrm{Im}\int_{-\infty}^{E_F}\mathrm{d}E\mathrm{Tr}\left\[
+\hat{\mathcal{A}}\mathcal{G}(E) \right\]
+```
+For example the density
+$`\rho`$ can be calculated via
+```math
+\rho(\vec{x})=-\frac{1}{\pi}\mathrm{Im}\int_{-\infty}^{E_F}\mathrm{d}E\mathcal{G}(\vec{x},\vec{x};E)
+```
 which gives the exression
-\$\$\\rho=-\\frac{1}{\\pi}\\int\\mathrm{d}\\vec{x}\\rho(\\vec{x})=-\\frac{1}{\\pi}\\sum\_{n}\\int\\mathrm{d}\\vec{r}
-\\sum\_{L=(l,m)}\\rho\^n\_L(r) Y\_L(\\hat{r})\$\$ where the second
+
+```math
+\rho=-\frac{1}{\pi}\int\mathrm{d}\vec{x}\rho(\vec{x})=-\frac{1}{\pi}\sum_{n}\int\mathrm{d}\vec{r}
+\sum_{L=(l,m)}\rho^n_L(r) Y_L(\hat{r})
+```
+where the second
 expression is the formulation in the KKR formalism where all quantities
-are expressed in atom centered (at \$\\vec{R}\^n\$) voronoi cells and
-therein expanded in real spherical harmonics \$Y\_L(\\hat{r})\$. {{
-:kkrimp/voronoi\_cells.png?600 \|Division of space in KKR formalism}}
+are expressed in atom centered (at $`\vec{R}^n`$) voronoi cells and
+therein expanded in real spherical harmonics $`Y_L(\hat{r})`$. {{
+:kkrimp/voronoi_cells.png?600 \|Division of space in KKR formalism}}
 
-##### Impurity embedding {#impurity_embedding}
+## Impurity embedding
 
 The KKR formalism relies on the multiple scattering principle. The basic
 idea is to divide the problem of calculating the Green function of a
@@ -38,57 +53,70 @@ crystal into three steps: (i) compute the scattering properties of free
 Lippmann-Schwinger equation, (ii) construct the multiple scattering via
 the structural Green function from systems geometry and (iii) construct
 the Green from the in (i) obtained wavefunctions and single-site
-\$t\$-matrix and the in thep (ii) computed structural GF.
+$`t$`-matrix and the in thep (ii) computed structural GF.
 
-#### (i) Single-site problem {#i_single_site_problem}
+### (i) Single-site problem
 
 Solving the single site problem is done by computing the solution of the
 Lippmann-Schwinger equation
-\$\$R\^n\_L(\\vec{r})=J\_L(\\vec{r})+\\sqrt{E}\\sum\_{L''}H\_{L''}(\\vec{r})t\_{L'',L}(E)\$\$
-With the singel-site \$t\$-matrix that is defined as
-\$\$\\underline{\\underline{t}}=V+V\\underline{\\underline{g}}\_{ref}\\underline{\\underline{t}}\$\$
-This contains the atomic potential \$V\$ and the analytically known
-reference GF \$g\_{ref}\$ which traditionally used to be the free space
+
+```math
+R^n_L(\vec{r})=J_L(\vec{r})+\sqrt{E}\sum_{L''}H_{L''}(\vec{r})t_{L'',L}(E)
+```
+With the singel-site $`t`$-matrix that is defined as
+
+```math
+\underline{\underline{t}}=V+V\underline{\underline{g}}_{ref}\underline{\underline{t}}
+```
+This contains the atomic potential $`V`$ and the analytically known
+reference GF $`g_{ref}`$ which traditionally used to be the free space
 green function by nowadays for numerical reasons is the Green function
 of repulsive muffin-tin potentials.
 
-#### (ii) Structural Green function {#ii_structural_green_function}
+### (ii) Structural Green function
 
 The structural Green function takes the system's geometry via the
 analytically known reference green function
-\$\\underline{\\underline{g}}\_{ref}\^{nn'}\$ and includes the multiple
+$`\underline{\underline{g}}_{ref}^{nn'}`$ and includes the multiple
 scattering properties in theis lattice by solving the Dyson equation
-\$\$\\underline{\\underline{G}}\^{nn'}=\\underline{\\underline{g}}\_{ref}\^{nn'}+\\sum\_{n''}
-\\underline{\\underline{g}}\_{ref}\^{nn''}\\underline{\\underline{t}}\^{n''}\\underline{\\underline{G}}\^{n''n'}\$\$
-Where the atomic scattering properties at site \$n''\$ enter via the
-single-site \$t\$ matrix \$\\underline{\\underline{t}}\^{n''}\$.
 
-#### (iii) Calculation of the full Green function {#iii_calculation_of_the_full_green_function}
+```math
+\underline{\underline{G}}^{nn'}=\underline{\underline{g}}_{ref}^{nn'}+\sum_{n''}
+\underline{\underline{g}}_{ref}^{nn''}\underline{\underline{t}}^{n''}\underline{\underline{G}}^{n''n'}
+```
+Where the atomic scattering properties at site $`n''`$ enter via the
+single-site $`t`$ matrix $`\underline{\underline{t}}^{n''}`$.
+
+### (iii) Calculation of the full Green function
 
 The last step is to combine the two steps that were explained above and
 compute the Green function of the system that contains a single-site
 contribution and a multiple scattering or backscattering contribution.
-Then the charge density at the atomic site \$n\$ is given by
-\$\$\\rho\_L\^{n}(r)=-\\frac{1}{\\pi}\\int\_{-\\infty}\^{E\_F}\\mathrm{d}E\\left\[
-\\underline{R}\_L\^n(r;E)\\underline{\\overline{S}}\_L\^n(r;E)+\\underline{R}\_L\^n(r;E)\\underline{\\underline{G}}\^{nn}\\underline{\\overline{G}}\_L\^n(r;E)
-\\right\]\$\$ Where the expression in square brackets is the Trace over
+Then the charge density at the atomic site $`n`$ is given by
+
+```math
+\rho_L^{n}(r)=-\frac{1}{\pi}\int_{-\infty}^{E_F}\mathrm{d}E\left\[
+\underline{R}_L^n(r;E)\underline{\overline{S}}_L^n(r;E)+\underline{R}_L^n(r;E)\underline{\underline{G}}^{nn}\underline{\overline{G}}_L^n(r;E)
+\right\]
+```
+Where the expression in square brackets is the Trace over
 the Green function and the first part is called single-site contribution
 and the latter multiple-scattering or backscattering contribution.
 
-##### Self-consistent algorithm {#self_consistent_algorithm}
+## Self-consistent algorithm
 
 To calculate the impurity Green function we make use of the Dyson
 equation. The idea is that an impurity locally changes the potential.
 This perturbation extends only over a small area that typically
 containes the impurity and the first few shells of neighbors. Then
 outside this impurity region the potentials do not change any more and
-consequently the difference in the single-site \$t\$-matrices of these
-positions vanish, i.e. \$\\Delta t=0\$. This allows us to embed the
+consequently the difference in the single-site $t$-matrices of these
+positions vanish, i.e. $`\Delta t=0`$. This allows us to embed the
 impurity in a finite real space cluster into the host system. The
 calcualtion is then done as follows.
 
-` - Compute the host systems Green function and write out the structural Green function as well as the single-site $t$ matrix of the host $t_{\mathrm{host}}$.`\
-` - From the impurity potential compute the $t$-matrix of the impurity and construct $\Delta t=t_{\mathrm{host}}-t_{\mathrm{imp}}$`\
-` - Solve the impurity Dyson equation: $\underline{\underline{G}}=\underline{\underline{G}}_{\mathrm{host}}+\underline{\underline{G}}_{\mathrm{host}}\Delta t\underline{\underline{G}}$`\
-` - Compute the new impurity potential from the Green function and update the input potential`\
-` - Repeat steps 2.-4. until the impurity potential converges`
+ - Compute the host systems Green function and write out the structural Green function as well as the single-site $`t`$ matrix of the host $`t_{\mathrm{host}}`$.
+ - From the impurity potential compute the $t$-matrix of the impurity and construct $`\Delta t=t_{\mathrm{host}}-t_{\mathrm{imp}}`$
+ - Solve the impurity Dyson equation: $`\underline{\underline{G}}=\underline{\underline{G}}_{\mathrm{host}}+\underline{\underline{G}}_{\mathrm{host}}\Delta t\underline{\underline{G}}`$
+ - Compute the new impurity potential from the Green function and update the input potential
+ - Repeat steps 2.-4. until the impurity potential converges