Slater parametrization of the Coulomb $U$ is considered more accurate, and it takes the form:
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
With the Coulomb repulsion matrix elements $U_{m_1,m_2,m_3,m_4}$ having the form:
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}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:
This is for example also derived in: http://www.annualreviews.org/doi/pdf/10.1146/annurev-conmatphys-020911-125045
To check the results of the atom_d.py 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
We can derive the following
Therefore we have
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}