# Spin orbit coupling operator (l.s)

The spin-orbit interaction is defined as: $$\xi \sum_i l_i \cdot s_i,$$ with $l_i$ and $s_i$ the one electron orbital and spin operators respectively and the sum over $i$ summing over all electrons. The prefactor $\xi$ is an atom dependent constant, which is to a good approximation material independent and given as: $$\xi = \left\langle R(r) \left| \frac{1}{2m^2c^2}\frac{1}{r}\frac{\mathrm{d}V(r)}{\mathrm{d} r} \right| R(r) \right\rangle.$$ The derivative of the potential multiplied by $1/r$ is only contributing close to the nucleus where electrons have relativistic speeds. We therefore can make the approximation that the potential has a spherical form and one can separate the radial and angular parts of the wave-function. Using these approximations one can derive the equation above starting from the Dirac equation and using perturbation theory.

In second quantization the spin-orbit operator becomes: \begin{eqnarray} \sum_i l_i \cdot s_i &=& \sum_i l_z^i s_z^i + \frac{1}{2} (l_i^+ s_i^- + l_i^-s_i^+)\\ \nonumber &=& \sum_{m=-l}^{m=l} \sum_{\sigma} m \sigma a^{\dagger}_{m\sigma}a^{\phantom{\dagger}}_{m\sigma} \\ \nonumber &+& \sum_{m=-l}^{m=l-1}\frac{1}{2} \sqrt{l+m+1}\sqrt{l-m}(a^{\dagger}_{m+1,\downarrow}a^{\phantom{\dagger}}_{m,\uparrow}+a^{\dagger}_{m,\uparrow}a^{\phantom{\dagger}}_{m+1,\downarrow}). \end{eqnarray} The equivalent operator in Quanty is created by:

Example.Quanty
Oppldots = NewOperator("ldots", NF, IndexUp, IndexDn)