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systems properties, i.e. the expectation value $`\langle
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\mathcal{A}\rangle`$ of an operator $`\hat{\mathcal{A}}`$:
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```math
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\langle\mathcal{A}\rangle=-\frac{1}{\pi}\mathrm{Im}\int_{-\infty}^{E_F}\mathrm{d}E\mathrm{Tr}\left\[
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\hat{\mathcal{A}}\mathcal{G}(E) \right\]
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\langle\mathcal{A}\rangle=-\frac{1}{\pi}\mathrm{Im}\int_{-\infty}^{E_F}\mathrm{d}E\mathrm{Tr}\left[
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\hat{\mathcal{A}}\mathcal{G}(E) \right]
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```
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For example the density
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$`\rho`$ can be calculated via
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| ... | ... | @@ -41,8 +41,8 @@ which gives the exression |
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where the second
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expression is the formulation in the KKR formalism where all quantities
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are expressed in atom centered (at $`\vec{R}^n`$) voronoi cells and
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therein expanded in real spherical harmonics $`Y_L(\hat{r})`$. {{
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:kkrimp/voronoi_cells.png?600 \|Division of space in KKR formalism}}
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therein expanded in real spherical harmonics $`Y_L(\hat{r})`$.
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## Impurity embedding
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| ... | ... | @@ -95,9 +95,9 @@ contribution and a multiple scattering or backscattering contribution. |
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Then the charge density at the atomic site $`n`$ is given by
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```math
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\rho_L^{n}(r)=-\frac{1}{\pi}\int_{-\infty}^{E_F}\mathrm{d}E\left\[
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\rho_L^{n}(r)=-\frac{1}{\pi}\int_{-\infty}^{E_F}\mathrm{d}E\left[
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\underline{R}_L^n(r;E)\underline{\overline{S}}_L^n(r;E)+\underline{R}_L^n(r;E)\underline{\underline{G}}^{nn}\underline{\overline{G}}_L^n(r;E)
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\right\]
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\right]
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```
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Where the expression in square brackets is the Trace over
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the Green function and the first part is called single-site contribution
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| ... | ... | @@ -115,8 +115,8 @@ positions vanish, i.e. $`\Delta t=0`$. This allows us to embed the |
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impurity in a finite real space cluster into the host system. The
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calculation is then done as follows.
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* Compute the host systems Green function and write out the structural Green function as well as the single-site $`t`$ matrix of the host $`t_{\mathrm{host}}`$.
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* From the impurity potential compute the $`t`$-matrix of the impurity and construct $`\Delta t=t_{\mathrm{host}}-t_{\mathrm{imp}}`$
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* Solve the impurity Dyson equation: $`\underline{\underline{G}}=\underline{\underline{G}}_{\mathrm{host}}+\underline{\underline{G}}_{\mathrm{host}}\Delta t\underline{\underline{G}}`$
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* Compute the new impurity potential from the Green function and update the input potential
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* Repeat steps 2.-4. until the impurity potential converges |
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- Compute the host systems Green function and write out the structural Green function as well as the single-site $`t`$ matrix of the host $`t_{\mathrm{host}}`$.
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- From the impurity potential compute the $`t`$-matrix of the impurity and construct $`\Delta t=t_{\mathrm{host}}-t_{\mathrm{imp}}`$
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- Solve the impurity Dyson equation: $`\underline{\underline{G}}=\underline{\underline{G}}_{\mathrm{host}}+\underline{\underline{G}}_{\mathrm{host}}\Delta t\underline{\underline{G}}`$
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- Compute the new impurity potential from the Green function and update the input potential
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- Repeat steps 2.-4. until the impurity potential converges |
