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

\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}).

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

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

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

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

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