find . -name \*.txt -type f -exec /Users/ruess/Downloads/pandoc-2.7.2/bin/pandoc -f mediawiki -t markdown -o {}.md {} \;
then replace all .txt.md files by .txt
and finally run git mv on all .txt files to convert to .md files

kkrimp/theory.md

+###### 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)\$\$
+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
+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}}
+
+##### 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
+crystal into three steps: (i) compute the scattering properties of free
+(reference) electrons off an atomic potential by solving the
+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.
+
+#### (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
+green function by nowadays for numerical reasons is the Green function
+of repulsive muffin-tin potentials.
+
+#### (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
+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}
+
+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
+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}
+
+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
+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`