How is Coulomb U implemented in EDMFTF code (formerly wien2k_dmft)?

Slater parametrization of the Coulomb $U$ is considered more accurate, and it takes the form:

\begin{eqnarray} \hat{U}=\frac{1}{2}\sum_{\{m\},s,s'}\left< Y_{l m_1}(r)\frac{u_l(r)}{r}Y_{l m_2}(r')\frac{u_l(r')}{r'}|V_{DMFT}(r-r')|Y_{l m_3}(r')\frac{u_l(r')}{r'}Y_{l m_4}(r)\frac{u_l(r)}{r}\right>\psi^\dagger_{m_1 s}\psi^\dagger_{m_2 s'} \psi_{m_3 s'} \psi_{m_4 s} \end{eqnarray}

In EDMFTF code (in particular when using projector 5) we use DMFT basis which has exactly this form, i.e., radial function $u_l(r)$ (which is printed in projectorw.dat) times the spherical harmonics.

As long as $V_{DMFT}(r-r')$ is a function of $r-r'$ (for example a form $$V_{DMFT}(r-r')=\frac{e^{-\lambda(r-r')}}{|r-r'|}$$ ), the above expression can be simplified by a few Slater parameters $F^k$, which we will derive below. We will then relate this parameters with other parametrizations, such as Kanamori.

The above Coulomb repulsion can always be rewritten as

\begin{eqnarray} \hat{U}=\frac{1}{2}\sum_{\{m\},s,s'} U_{m_1 m_2 m_3 m_4} \psi^\dagger_{m_1 s}\psi^\dagger_{m_2 s'} \psi_{m_3 s'} \psi_{m_4 s} \end{eqnarray}

With the Coulomb repulsion matrix elements $U_{m_1,m_2,m_3,m_4}$ having the form:

\begin{eqnarray} U_{m_1,m_2,m_3,m_4}=\sum_{m,k} \frac{4\pi\;}{2k+1} \left< Y_{l m_1}|Y_{km}|Y_{l m_4} \right> \left< Y_{l m_2}|Y_{km}^*|Y_{l m_3}\right> F^k \end{eqnarray}

If the Coulomb repulsion in real space is unscreened ($V(r)=1/r$), we can use the well known relation

\begin{eqnarray} \frac{1}{|r-r'|}=\frac{4\pi}{2k+1}\frac{r_<^k}{r_>^{k+1}}Y_{km}(\hat{r})Y_{km}^*(\hat{r}') \end{eqnarray}

to derive

\begin{eqnarray} F^k=\int dr \int dr' (u_l(r))^2 (u_l(r'))^2 \frac{r_<^k}{r_>^{k+1}} \end{eqnarray}

If the Coulomb repulsion is screened in Yukawa form

\begin{eqnarray} V_{DMFT}(r-r')=\frac{e^{-\lambda|r-r'|}}{|r-r'|} \end{eqnarray}

we can use similar expression but using Modified Bessel functions $I_k$ and $K_k$ (scipy.special.iv and scipy.special.kv), to express

\begin{eqnarray} \frac{e^{-|r-r'|}}{|r-r'|}=\sum_k \frac{4\pi I_{k+1/2}(r_<) K_{k+1/2}(r_>)}{\sqrt{r_< r_>}}\sum_m Y_{km}(\hat{r})Y_{km}^*(\hat{r}') \end{eqnarray}

The expression for Slater integrals is then a bit more involved, but still very simple to compute numerically

\begin{eqnarray} F^k =\int dr \int dr' (u_l(r))^2(u_l(r'))^2 \;(2k+1)\frac{I_{k+1/2}(\lambda r_<) K_{k+1/2}(\lambda r_>)}{\sqrt{r_< r_>}} \end{eqnarray}

Hence our implemented default form is:

\begin{eqnarray} \hat{U}=\frac{1}{2}\sum_{\{m\},s,s',k} \frac{4\pi\; F^k}{2k+1} \left< Y_{l m_1}|Y_{km}|Y_{l m_4} \right> \left< Y_{l m_2}|Y_{km}^*|Y_{l m_3}\right> \psi^\dagger_{m_1 s}\psi^\dagger_{m_2 s'} \psi_{m_3 s'} \psi_{m_4 s} \end{eqnarray}

In EDMFTF code one needs to provide the Slater parameters in "params.dat" file.

This is the convention used in "params.dat" file : For $d$-electrons, we use

\begin{eqnarray} && F^0 \equiv U_{params}\\ && F^2 \equiv 112/13\; J^{(2)}_{params}\\ && F^4 \equiv 70/13\; J^{(4)}_{params} \end{eqnarray}

and for $f$-electrons we use

\begin{eqnarray} && F^0 \equiv U_{params}\\ && F^2 \equiv 6435/(286+195*0.668+250*0.494) J^{(2)}_{params}\\ && F^4 \equiv 0.668*6435/539.76 J^{(4)}_{params}\\ && F^6 \equiv 0.494*6435/539.76 * J^{(6)}_{params} \end{eqnarray}

If a single $J_{params}$ is given, than $J^{(2)}$ and $J^{(4)}$ are both equal to $J_{params}$. If a list of $J$'s is given, one can change independently $F^2$ and $F^4$ (or $F^6$). These relations are to a very good approximation constant across the entire periodic table, although they are derived for Hydrogen atom. (You can check youself to derive them using Mathematica).

We enter $U_{params}$ & $J_{params}$ for convenience, rather than specifying $F^0$, $F^2$, $F^4$, ....

It is sometimes useful to have alternative forms of the Colulomb repulsion, and among them, the Kanamori parametrization seems most popular. The Kanamori interaction in EDMFTF is implemented in somewhat non-standard form, by making it as similar as possible to Slater parametrization.

The Kanamori parameters are specified by the following integral

\begin{eqnarray} U_K = \int d^3r \int d^3r' \phi_\alpha(r)\phi_\alpha(r')V_C(r-r')\phi_\alpha(r')\phi_\alpha(r)\\ U'_K = \int d^3r \int d^3r' \phi_\alpha(r)\phi_\beta(r')V_C(r-r')\phi_\beta(r')\phi_\alpha(r)\\ J_K = \int d^3r \int d^3r' \phi_\alpha(r)\phi_\beta(r')V_C(r-r')\phi_\alpha(r')\phi_\beta(r) \end{eqnarray}

where $U_K$ ($J_K$) is obtained when $\alpha=\beta$ ($\alpha\ne\beta$) on the right hand side.

These Kanamori parameters can also be written in terms of Slater integrals, and then we can compare their values. We can derive

\begin{eqnarray} U^K_{\alpha\beta\gamma\delta} = \sum_{k,\{ m\} } U^*_{\alpha m_1}U^*_{\beta m_2}U_{\gamma m_3} U_{\delta m_4} \frac{4\pi}{2k+1}\langle Y_{l m_1}|Y_{km}|Y_{l m_4}\rangle \langle Y_{l m_2}|Y_{k m}^*|Y_{l m_3}\rangle F^k \end{eqnarray}

Where $U_{\alpha m}$ transforms from the real harmonics $\alpha=\{xz,yz,xy\}$ to spheric harmonics $Y_{lm}$.

Using the relation between the 3j symbols and Spherical harmonics integrals, we can write

\begin{eqnarray} \langle Y_{l m_1}|Y_{km}|Y_{l m_4}\rangle = \sqrt{\frac{(2l+1)^2(2k+1)}{4\pi}}(-1)^{m_1} \begin{bmatrix} l&k&l\\ 0&0&0 \end{bmatrix} \begin{bmatrix} l&k&l\\ -m_1&m&m_4 \end{bmatrix} \end{eqnarray}

and then the Kanamori parameters are related to Slater integrals by

\begin{eqnarray} U^K_{\alpha,\beta,\gamma,\delta}=\sum_k F^k\; \sum_{\{ m\} } U^*_{\alpha m_1}U^*_{\beta m_2}U_{\gamma m_3} U_{\delta m_4} (2l+1)^2(-1)^{m_1+m_2+m} \left(\begin{bmatrix} l&k&l\\ 0&0&0 \end{bmatrix}\right)^2 \begin{bmatrix} l&k&l\\ -m_1&m&m_4 \end{bmatrix} \begin{bmatrix} l&k&l\\ -m_2&-m&m_3 \end{bmatrix} \end{eqnarray}

This can easily be evaluated in Mathematica, and it has the form

\begin{eqnarray} U^K_{\alpha\alpha\beta\beta}=U^K_{\alpha\beta\alpha\beta}= \begin{bmatrix} F^0+\frac{4}{49}(F^2+F^4)&\frac{4}{49}F^2+\frac{5}{147}F^4&\frac{1}{49}F^2+\frac{10}{147}F^4&\frac{1}{49}F^2+\frac{10}{147}F^4&\frac{4}{49}F^2+\frac{5}{147}F^4\\ \frac{4}{49}F^2+\frac{5}{147}F^4&F^0+\frac{4}{49}(F^2+F^4)&\frac{3}{49}F^2+\frac{20}{441}F^4&\frac{3}{49}F^2+\frac{20}{441}F^4&\frac{5}{63}F^4\\ \frac{1}{49}F^2+\frac{10}{147}F^4&\frac{3}{49}F^2+\frac{20}{441}F^4&F^0+\frac{4}{49}(F^2+F^4)&\frac{3}{49}F^2+\frac{20}{441}F^4&\frac{3}{49}F^2+\frac{20}{441}F^4\\ \frac{1}{49}F^2+\frac{10}{147}F^4&\frac{3}{49}F^2+\frac{20}{441}F^4&\frac{3}{49}F^2+\frac{20}{441}F^4&F^0+\frac{4}{49}(F^2+F^4)&\frac{3}{49}F^2+\frac{20}{441}F^4\\ \frac{4}{49}F^2+\frac{5}{147}F^4&\frac{5}{63}F^4&\frac{3}{49}F^2+\frac{20}{441}F^4&\frac{3}{49}F^2+\frac{20}{441}F^4&F^0+\frac{4}{49}(F^2+F^4) \end{bmatrix} \end{eqnarray}
\begin{eqnarray} U^K_{\alpha\beta\beta\alpha}= \begin{bmatrix} F^0+\frac{4}{49}(F^2+F^4)&F^0-\frac{4}{49}F^2+\frac{2}{147}F^4&F^0+\frac{2}{49}F^2-\frac{8}{147}F^4&F^0+\frac{2}{49}F^2-\frac{8}{147}F^4&F^0-\frac{4}{49}F^2+\frac{2}{147}F^4\\ F^0-\frac{4}{49}F^2+\frac{2}{147}F^4&F^0+\frac{4}{49}(F^2+F^4)&F^0-\frac{2}{49}F^2-\frac{4}{441}F^4&F^0-\frac{2}{49}F^2-\frac{4}{441}F^4&F^0+\frac{4}{49}F^2-\frac{34}{441}F^4\\ F^0+\frac{2}{49}F^2-\frac{8}{147}F^4&F^0-\frac{2}{49}F^2-\frac{4}{441}F^4&F^0+\frac{4}{49}(F^2+F^4)&F^0-\frac{2}{49}F^2-\frac{4}{441}F^4&F^0-\frac{2}{49}F^2-\frac{4}{441}F^4\\ F^0+\frac{2}{49}F^2-\frac{8}{147}F^4&F^0-\frac{2}{49}F^2-\frac{4}{441}F^4&F^0-\frac{2}{49}F^2-\frac{4}{441}F^4&F^0+\frac{4}{49}(F^2+F^4)&F^0-\frac{2}{49}F^2-\frac{4}{441}F^4\\ F^0-\frac{4}{49}F^2+F^0\frac{2}{147}F^4&F^0+\frac{4}{49}F^2-\frac{34}{441}F^4&F^0-\frac{2}{49}F^2-\frac{4}{441}F^4&F^0-\frac{2}{49}F^2F^0-\frac{4}{441}F^4&F^0+\frac{4}{49}(F^2+F^4) \end{bmatrix} \end{eqnarray}

Here the matrix is written in real harmonics, and we use the order, as in EDMFTF code, i.e.,

\begin{eqnarray} [z^2, x^2-y^2, xz, yz, xy] \end{eqnarray}

Hence the upper-left $2\times 2$ matrix corresponds to $e_g$ sector, and the lower-right to the $t_{2g}$ sector.

We can also verify that for $\alpha\ne\beta$, we have

\begin{eqnarray} U^K_{\alpha\beta\beta\alpha}-(F^0+\frac{4}{49}(F^2+F^4))=-2 U_{\alpha\alpha\beta\beta} \end{eqnarray}

which is equivalent to the equation

\begin{eqnarray} U_K-U'_K=2J_K \end{eqnarray}

Hence we see that the Kanamori parametrization becomes equivalent to Slater parametrization, when only a t2g subshell (or only eg subshell ) is correlated, and the following relation between Slater parameteres and Kanamori parameters holds (see the right-lower 3x3 submatrix)

\begin{eqnarray} &&U_K = F^0+\frac{4}{49}(F^2+F^4)\\ &&J_K^{t2g} = \frac{3}{49}F^2+\frac{20}{441}F^4\\ &&J_K^{eg}=\frac{4}{49}F^2+\frac{5}{147}F^4\\ &&{U'_K}^{t2g}=F^0-\frac{2}{49}F^2-\frac{4}{441}F^4=U_K-2\,J_K\\ &&{U'_K}^{eg}=F^0-\frac{4}{49}F^2+\frac{2}{147}F^4=U_K-2\,J_K \end{eqnarray}

Also notice that the factor, which appears in front of $N(N-1)/2$ term has the form

\begin{eqnarray} t2g: &&U_K-3J_K^{t2g}=F^0-\frac{5}{49}F^2-\frac{8}{147}F^4 \rightarrow F^0-\frac{320}{273}J_S \approx F^0-1.172 J_S\\ eg : &&U_K-3 J_K^{eg}=F^0-\frac{8}{49}F^2-\frac{1}{49}F^4 \rightarrow F^0-\frac{138}{91}J_S \approx F^0-1.517 J_S\\ average : &&U_K-3 J_K=F^0-\frac{596}{455} J_S\approx F^0-1.31 J_S \end{eqnarray}

also notice that the factor in from of the $S^2$ term is $2J_K$, which is

\begin{eqnarray} t2g : && 2 J^{t2g}_K= \frac{1264}{819}J_S \approx 1.543 J_S\\ eg : && 2 J^{eg}_K = \frac{484}{273} J_S \approx 1.773 J_S \end{eqnarray}

Also notice that Kanamori $J_K$ is not equal to Slater $J_S$, namely,

\begin{eqnarray} J_K^{t2g} = \frac{632}{819} J_S \approx 0.77 J_S\\ J_K^{eg} = \frac{242}{273} J_S \approx 0.89 J_S \end{eqnarray}

Using the standard Kanamori Hamiltonian, we can then write how the 4-dimensional tensor of $U$ should look like in Kanamori representation, to be as close as possible to the Slater representation:

\begin{eqnarray} && U_{\alpha\alpha\alpha\alpha} = F^0+\frac{4}{49}(F^2+F^4)\\ && U_{\alpha\beta\alpha\beta}=\frac{3}{49}F^2+\frac{20}{441} F^4\\ && U_{\alpha\alpha\beta\beta} = \frac{3}{49}F^2+\frac{20}{441} F^4\\ && U_{\alpha\beta\beta\alpha}=F^0-\frac{4}{49}F^2-\frac{4}{441}F^4 \end{eqnarray}

To check the results of the script, we might just check some energies in the imp.0/actqmc.cix file.

For example, the energy of the last state, which has 10 electrons in 5 orbitals (fully occupied $d$ shell) is probably the simplest.

In this case, we can apply operator $\hat{U}$ on the state $\left|22222\right>$ to obtain

\begin{eqnarray} \hat{U}\left|22222\right>= \frac{1}{2} \sum_{\alpha\beta,s,s}\left( U_{\alpha\beta\beta\alpha} n_{\alpha s}n_{\beta s'}-\delta(s,s') U_{\alpha\beta\alpha\beta} n_{\alpha s}n_{\beta s} \right) \left|22222\right> \end{eqnarray}

We can derive the following

\begin{eqnarray} && \sum_{\alpha\ne\beta}\sum_{ss'} n_{\alpha s}n_{\beta s'}\left|22222\right>=2\times 2\times 5\times 4\left|22222\right>=80\left|22222\right>\\ && \sum_{\alpha}\sum_{s\ne s'} n_{\alpha s}n_{\alpha s'}\left|22222\right>=2\times 5 \left|22222\right>=10\left|22222\right>\\ && \sum_{\alpha\ne\beta}\sum_{s} n_{\alpha s}n_{\beta s}\left|22222\right>=2\times 5\times 4\left|22222\right>=40\left|22222\right> \end{eqnarray}

Therefore we have

\begin{eqnarray} \hat{U}\left|22222\right>=\frac{1}{2} \left(10 U_{\alpha\alpha\alpha\alpha}+80 U_{\alpha\beta\beta\alpha}-40 U_{\alpha\beta\alpha\beta}\right)\left|22222\right> =(-\frac{120}{49}F^2 -\frac{380}{441}F^4)\left|22222\right> \end{eqnarray}

EDMFTF allows the input of Slater paramters $F^k$. But in literature the Hund's parameter $J$ is more common, therefore we transform $J$'s to Slater parameters. The proportionality constant can be derived, for example using Hydrogen atom wave functions. The relation is: \begin{eqnarray} && F^2 = \frac{14}{1.625} J\ && F^4 = \frac{14\times 0.625}{1.625} J \end{eqnarray}

Using Hund's $J$ parameter, we than have $$\hat{U}\left|22222\right> = -25.7387 J\left|22222\right>$$

In addition to Coulomb $U$, the orbital energies are added to the energy of each state to obtain the energy of the atomic state. We thus have: \begin{eqnarray} \hat{H}^{local}\left|22222\right> = \left(2(E_1+E_2+E_3+E_4+E_5) −25.7387J\right)\left|22222\right> \end{eqnarray}