Coulomb interaction

The standard form of Coulomb interaction in second quantization form is: $$\begin{equation} \hat{H}_{U}=\frac{1}{2}\sum_{\sigma,\sigma^{\prime}}\sum_{a,b,c,d} \left\langle a\sigma,b\sigma^{\prime}\left|\frac{1}{r_{12}}\right|c\sigma,d\sigma^{\prime}\right\rangle \hat{f}_{a\sigma}^{\dagger}\hat{f}_{b\sigma^{\prime}}^{\dagger}\hat{f}_{d\sigma^{\prime}}\hat{f}_{c\sigma} \end{equation}$$ $$\begin{equation} \left\langle a\sigma,b\sigma^{\prime}\left|\frac{1}{r_{12}}\right|c\sigma,d\sigma^{\prime}\right\rangle= \int d\vec{r}_{1}d\vec{r}_{2}\phi_{\alpha\sigma}^{*}(\vec{r}_{1})\phi_{b\sigma^{\prime}}^{*}(\vec{r}_{2})\frac{1}{r_{12}}\phi_{c\sigma} (\vec{r}_{1})\phi_{d\sigma^{\prime}}(\vec{r}_{2}) \end{equation}$$ 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. In the JASMINE component, we use a array UMAT to store the $U$ tensor. BE VERY CAREFUL WITH THE ORBITAL ORDER. The indices order of UMAT is the same with that of the fermion operators, it is NOT the same with that of the $U$ tensor.

Slater Type

Expand $\frac{1}{r_{12}}$ in terms of complex spherical harmonics $Y_{l}^{m}$,

$$\begin{equation} \frac{1}{r_{12}}=\sum_{k}\frac{r_{<}^{k}}{r_{>}^{k+1}}\sum_{m}(-1)^{m}C_{m}^{k}(\theta_{1}\phi_{1})C_{-m}^{k}(\theta_{2}\phi_{2}) \end{equation}$$ where,

$$C_{m}^{k}(\theta\phi)=\sqrt{\frac{4\pi}{2k+1}}Y_{l}^{m}(\theta\phi).$$

Set $\phi(\vec{r})=R_{nl}(r)Y_{l}^{m}(\theta\phi)$, then for fixed $n,l$, we obtain,

$$\begin{eqnarray} \left\langle m_{1},m_{2}\left|\frac{1}{r_{12}}\right|m_{1}^{\prime},m_{2}^{\prime}\right\rangle & = & \sum_{km}(-1)^{m}c_{l}^{k}(m_{1},m_{1}^{\prime})c_{l}^{k}(m_{2},m_{2}^{\prime})\delta(m+m_{1}^{\prime},m_{1})\delta(-m+m_{2}^{\prime},m_{2})F_{nl}^{k}\nonumber \\ & = & \delta(m_{1}+m_{2},m_{1}^{\prime}+m_{2}^{\prime})(-1)^{m_{2}^{\prime}-m_{2}}\sum_{k}c_{l}^{k}(m_{1},m_{1}^{\prime})c_{l}^{k}(m_{2},m_{2}^{\prime})F_{nl}^{k}\nonumber \\ & = & \delta(m_{1}+m_{2},m_{1}^{\prime}+m_{2}^{\prime})\sum_{k}c_{l}^{k}(m_{1},m_{1}^{\prime})c_{l}^{k}(m_{2}^{\prime},m_{2})F_{nl}^{k} \end{eqnarray}$$

where,

$$\begin{equation} c_{l}^{k}(m^{\prime},m^{\prime\prime})=\int d\phi d\theta \sin(\theta)Y_{l}^{m^{\prime}*}(\theta\phi)C_{m^{\prime}-m^{\prime\prime}}^{k}(\theta\phi)Y_{l}^{m^{\prime\prime}}(\theta\phi) \end{equation}$$

is the Gaunt coefficient for fixed $l$, and

$$\begin{equation} F_{nl}^{k}=\int_{0}^{\infty}r_{1}^{2}dr_{1}\int_{0}^{\infty}r_{2}^{2}dr_{2}R_{nl}^{2}(r_{1})R_{nl}^{2}(r_{2})\frac{r_{<}^{k}}{r_{>}^{k+1}} \end{equation}$$ is the Slater integrals.

In this single particle basis, the Coulomb interaction Hamiltonian is: $$\begin{equation} \hat{H}_{U}=\frac{1}{2}\sum_{m_{1},m_{2},m_{1}^{\prime},m_{2}^{\prime},\sigma,\sigma^{\prime}}\left\langle m_{1}\sigma,m_{2}\sigma^{\prime}\left|\frac{1}{r_{12}}\right|m_{1}^{\prime}\sigma,m_{2}^{\prime}\sigma^{\prime}\right\rangle \hat{f}_{m_{1}\sigma}^{\dagger}\hat{f}_{m_{2}\sigma^{\prime}}^{\dagger}\hat{f}_{m_{2}^{\prime}\sigma^{\prime}}\hat{f}_{m_{1}^{\prime}\sigma} \end{equation}$$ where, $\sigma,\sigma^{\prime}$ is spin index. Set

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

thus the UMAT in the JASMINE component reads:

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

Kanamori Type

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

$$\begin{eqnarray} \hat{H}_{U} & = & U\sum_{a}\hat{f}_{a,\uparrow}^{\dagger}\hat{f}_{a,\uparrow}\hat{f}_{a,\downarrow}^{\dagger}\hat{f}_{a,\downarrow}\nonumber\\ & = & U^{\prime}\sum_{a\text{<}b,\sigma}\hat{f}_{a,\sigma}^{\dagger}\hat{f}_{a,\sigma}\hat{f}_{b,-\sigma}^{\dagger}\hat{f}_{b,-\sigma} + (U^{\prime}-J_{z})\sum_{a<b,\sigma}\hat{f}_{a,\sigma}^{\dagger}\hat{f}_{a,\sigma}\hat{f}_{b,\sigma}^{\dagger}\hat{f}_{b,\sigma}\nonumber\\ & - & J_{s}\sum_{a<b,\sigma}\hat{f}_{a,\sigma}^{\dagger}\hat{f}_{a,-\sigma}\hat{f}_{b,-\sigma}^{\dagger}\hat{f}_{b,\sigma} + J_{p}\sum_{a\neq b}\hat{f}_{a,\uparrow}^{\dagger}\hat{f}_{a,\downarrow}^{\dagger}\hat{f}_{b,\downarrow}\hat{f}_{b,\uparrow} \end{eqnarray}$$ where, $a,b$ is orbital index, and $\sigma=\uparrow,\downarrow$ is spin index.