kkrimp/theory.md

@@ -24,8 +24,8 @@ 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\[
-\hat{\mathcal{A}}\mathcal{G}(E) \right\]
+\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
@@ -41,8 +41,8 @@ which gives the exression
 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}}
+therein expanded in real spherical harmonics $`Y_L(\hat{r})`$. 
+![Division of space in KKR formalism](uploads/c14e758bc8d197dc5f6c369e845e72b4/voronoi_cells.png)
 
 ## Impurity embedding
 
@@ -95,9 +95,9 @@ contribution and a multiple scattering or backscattering contribution.
 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\[
+\rho_L^{n}(r)=-\frac{1}{\pi}\int_{-\infty}^{E_F}\mathrm{d}E\left[
 \underline{R}_L^n(r;E)\underline{\overline{S}}_L^n(r;E)+\underline{R}_L^n(r;E)\underline{\underline{G}}^{nn}\underline{\overline{G}}_L^n(r;E)
-\right\]
+\right]
 ```
 Where the expression in square brackets is the Trace over
 the Green function and the first part is called single-site contribution
@@ -115,8 +115,8 @@ 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
 calculation 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