The effective spin-orbit operator

Let’s treat the effective spin-orbit interaction operator in heavy atoms, which has the following form in coordinate representation: \[\begin{equation} \require{physics} V^c_{\text{SO}} = \sum_i^N V^c_{\text{SO}}(\vb{r}_i, m_{s \thinspace i}) \thinspace , \end{equation}\] with \[\begin{equation} V^c_{\text{SO}}(\vb{r}_i, m_{s,i}) = \xi(\vb{r}_i) \vb{l}^c(i) \vdot \vb{S}^c(i) \thinspace , \end{equation}\] in which \(\xi(\vb{r}_i)\) is a radial function, \(\vb{l}^c(i)\) is the orbital angular momentum vector operator for electron \(i\) and \(\vb{S}^c(i)\) is the spin angular momentum vector operator for electron \(i\). The spin-orbit coupling can be written as \[\begin{align} \vb{l}^c(i) \vdot \vb{S}^c(i) &= \sum_{\mu=x,y,z} l_\mu^c(\vb{r}_i) S_\mu^c(m_{s,i}) \\ &= \frac{1}{2} \qty( l^c_+(\vb{r}_i) S^c_-(m_{s,i}) + l^c_-(\vb{r}_i) S^c_+(m_{s,i}) ) + l_z^c(\vb{r}_i) S_z^c(m_{s,i}) ) \thinspace . \end{align}\]

If we define the integrals \[\begin{equation} V^\nu_{pq} = \frac{1}{2} \int \dd{\vb{r}} \phi_p^*(\vb{r}) \xi(\vb{r}) l^\nu(\vb{r}) \phi_q(\vb{r}) \thinspace , \end{equation}\] with \(\nu=+,-,z\), then we can write the second quantized form of the effective spin-orbit operator as \[\begin{equation} \hat{V}_\text{SO} = \sum_{pq}^K ( V^+_{pq} \hat{T}^{1,-1}_{pq} - V^-_{pq} \hat{T}^{1,+1}_{pq} + \sqrt{2} V^z_{pq} \hat{T}^{1,0}_{pq} ) \thinspace . \end{equation}\]