Coulomb interaction

The standard form of Coulomb interaction in second quantization form is:

\[\begin{equation} \hat{H}_{\text{Coulomb}}= \frac{1}{2}\sum_{\sigma\sigma^{\prime}}\sum_{abcd} \left\langle a\sigma,b\sigma^{\prime}\left| \frac{1}{r_{12}} \right|c\sigma,d\sigma^{\prime}\right\rangle \hat{f}_{a\sigma}^{\dagger} \hat{f}_{b\sigma^{\prime}}^{\dagger} \hat{f}_{d\sigma^{\prime}} \hat{f}_{c\sigma}, \end{equation}\]

where $\frac{1}{r_{12}}$ is the Coulomb interaction, $r_{12}=|\vec{r}_{1}-\vec{r}_{2}|$, $a,b,c,d$ is orbital index and $\sigma,\sigma^{\prime}=\uparrow,\downarrow$ is spin index.

Slater Type Interaction

In this single particle basis, the Coulomb interaction Hamiltonian is:

\[\begin{equation} \hat{H}_{U}=\frac{1}{2}\sum_{m_{1},m_{2},m_{1}^{\prime},m_{2}^{\prime},\sigma,\sigma^{\prime}}\left\langle m_{1}\sigma,m_{2}\sigma^{\prime}\left|\frac{1}{r_{12}}\right|m_{1}^{\prime}\sigma,m_{2}^{\prime}\sigma^{\prime}\right\rangle \hat{f}_{m_{1}\sigma}^{\dagger}\hat{f}_{m_{2}\sigma^{\prime}}^{\dagger}\hat{f}_{m_{2}^{\prime}\sigma^{\prime}}\hat{f}_{m_{1}^{\prime}\sigma}. \end{equation}\]

Set

\[\alpha=m_{1}\sigma,\ \beta=m_{2}\sigma^{\prime},\ \gamma=m_{1}^{\prime}\sigma,\ \delta=m_{2}^{\prime}\sigma^{\prime},\]

thus the Coulomb U-tensor (UMAT) in the JASMINE component reads:

\[\begin{equation} \textbf{UMAT}(\alpha,\beta,\delta,\gamma)=\frac{1}{2}\delta(m_{1}+m_{2},m_{1}^{\prime}+m_{2}^{\prime})\sum_{k}c_{l}^{k}(m_{1},m_{1}^{\prime})c_{l}^{k}(m_{2}^{\prime},m_{2})F_{nl}^{k}. \end{equation}\]

Here, $F_{nl}^{k}$ is the Slater integrals:

\[\begin{equation} F_{nl}^{k}=\int_{0}^{\infty}r_{1}^{2}dr_{1}\int_{0}^{\infty}r_{2}^{2}dr_{2}R_{nl}^{2}(r_{1})R_{nl}^{2}(r_{2})\frac{r_{<}^{k}}{r_{>}^{k+1}}. \end{equation}\]

$c_{l}^{k}(m^{\prime},m^{\prime\prime})$ is related to the Gaunt coefficient for fixed $l$. It is defined as:

\[\begin{equation} c_{l}^{k}(m^{\prime},m^{\prime\prime}) = \sqrt{\frac{4\pi}{2k+1}} \int d\phi d\theta\sin(\theta) Y_{l}^{m^{\prime}*}(\theta,\phi) Y_{k}^{m^{\prime}-m^{\prime\prime}}(\theta,\phi) Y_{l}^{m^{\prime\prime}}(\theta,\phi) \end{equation}\]

Note that the Gaunt coefficient is defined as the integral over three spherical harmonics:

\[\begin{aligned} \operatorname{Gaunt}(l_1,l_2,l_3,m_1,m_2,m_3) &=\int Y_{l_1,m_1}(\Omega) Y_{l_2,m_2}(\Omega) Y_{l_3,m_3}(\Omega) \,d\Omega \\ &=\sqrt{\frac{(2l_1+1)(2l_2+1)(2l_3+1)}{4\pi}} \operatorname{Wigner3j}(l_1,l_2,l_3,0,0,0) \operatorname{Wigner3j}(l_1,l_2,l_3,m_1,m_2,m_3) \end{aligned}\]

We adopted the following Python script to generate $c_{l}^{k}(m^{\prime},m^{\prime\prime})$:

from sympy import *
from sympy.physics.wigner import gaunt

def get_gaunt(l1, l2):
    for k in range(l1 + l2 + 1):
        if not ((l1 + l2 + k) % 2 == 0 and abs(l1 - l2) <= k <= l1 + l2):
            continue
        for i1, m1 in enumerate(range(-l1, l1 + 1)):
            for i2, m2 in enumerate(range(-l2, l2 + 1)):
                x = symbols('x')
                f1 = sqrt(4*pi / (2*x + 1)) * gaunt(l1, k, l2, -m1, m1 - m2, m2)
                f2 = f1.subs(x, k)
                f3 = (-1.0)**m1
                if f2 == 0:
                    continue
                print('gaunt(', i1-l1, ',', i2-l2, ', ', k, ') = ', f2, '* (', f3, ')')

For $d$-electron system, we use get_gaunt(2,2). As for $f$-electron system, we use get_gaunt(3,3).

Kanamori Type Interaction

The Kanomori type of Coulomb interaction Hamiltonian in the JASMINE component is defined as:

\[\begin{align} \hat{H}_{U} = U^{\prime}\sum_{a\text{<}b,\sigma}\hat{f}_{a,\sigma}^{\dagger}\hat{f}_{a,\sigma}\hat{f}_{b,-\sigma}^{\dagger}\hat{f}_{b,-\sigma} + (U^{\prime}-J_{z})\sum_{a<b,\sigma}\hat{f}_{a,\sigma}^{\dagger}\hat{f}_{a,\sigma}\hat{f}_{b,\sigma}^{\dagger}\hat{f}_{b,\sigma} \\ - J_{s}\sum_{a<b,\sigma}\hat{f}_{a,\sigma}^{\dagger}\hat{f}_{a,-\sigma}\hat{f}_{b,-\sigma}^{\dagger}\hat{f}_{b,\sigma} + J_{p}\sum_{a\neq b}\hat{f}_{a,\uparrow}^{\dagger}\hat{f}_{a,\downarrow}^{\dagger}\hat{f}_{b,\downarrow}\hat{f}_{b,\uparrow} \end{align}\]

where, $a,b$ is orbital index, and $\sigma=\uparrow,\downarrow$ is spin index.