kkrimp/theory.md

@@ -12,13 +12,13 @@ links:
 
 Starting point: Schrödinger equation
 ```math
-\mathcal{H}\|\psi\rangle = E\|\psi\rangle
+\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
+\quad \delta>0
 ```
 Thus by knowing the GF one can extract the
 systems properties, i.e. the expectation value $`\langle
@@ -53,7 +53,7 @@ crystal into three steps: (i) compute the scattering properties of free
 Lippmann-Schwinger equation, (ii) construct the multiple scattering via
 the structural Green function from systems geometry and (iii) construct
 the Green from the in (i) obtained wavefunctions and single-site
-$`t$`-matrix and the in thep (ii) computed structural GF.
+$`t`$-matrix and the in thep (ii) computed structural GF.
 
 ### (i) Single-site problem
 
@@ -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