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# 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
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
```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}}
## 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
Solving the single site problem is done by computing the solution of the
Lippmann-Schwinger equation
```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
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
```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
```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
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
contains 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
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
# 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
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
```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 $`Y_L(\hat{r})`$.
![Division of space in KKR formalism](uploads/c14e758bc8d197dc5f6c369e845e72b4/voronoi_cells.png)
## 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
Solving the single site problem is done by computing the solution of the
Lippmann-Schwinger equation
```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
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
```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
```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
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
contains 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
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