Atomic spinors (spin-orbit eigenfunctions) expressed in a basis of atomic spin orbitals, and matrix elements of the angular momentum and Zeeman operators

[This page uses MathJax and takes a while to render completely. If the matrices below don’t display properly, please load this (older, no longer maintained) version instead.]

Below are sets of ${Ĥ}^{\text{SO}}=\zeta \widehat{L}\cdot \widehat{S}$ eigenfunctions $|j,m_j⟩$ expressed in a basis $|\ell ,m_{\ell },m_s⟩$ of atomic spin orbitals for a given angular momentum quantum number $\ell =1$, 2, or 3. These functions were generated with a Mathematica notebook similar to the Mathematica notebook discussed on this page. We also give these eigenfunctions subject to time reversal, and matrix elements of the spin and orbital angular momenum operators, and of the Zeeman (magnetic field perturbation) operator ${Ĥ}_{\alpha }^{\text{Z}}={\widehat{L}}_{\alpha }+2{Ŝ}_{\alpha }$ with $\alpha =x,y,z$. The definition of the orbital angular momentum eigenfunctions in the basis uses the Condon-Shortley phase. As a consequence, time reversal changes $m_{\ell }$ to $-m_{\ell }$ but also introduces a negative sign in the spin orbital basis if $m_{\ell }$ is odd. The coefficients representing the eigenfunctions of ${Ĥ}^{\text{SO}}$ and their time-reversed counterparts reflect this. The $|j,m_j⟩$ functions provided here were verified to be simultaneous eigenfunctions of ${Ĥ}^{\text{SO}}$, ${ĵ}^2$, and ${ĵ}_z$, with phases such that ${ĵ}_{\pm }|j,m_j⟩={\alpha }_{\pm }|j,m_j\pm 1⟩$, with ${\alpha }_{\pm }$ being a positive real number as long as $m_j\pm 1$ is in the allowed range of the projection quantum number, zero otherwise.

If you use this material for research please cite our publication [211] along with this web page. Please let us know in case you spot a mistake here. Thank you.

$\bullet$ p orbitals $\ell =1$

SO Hamiltonian ${Ĥ}^{\text{SO}}=\zeta \widehat{L}\cdot \widehat{S}$ in the basis of $\ell =1$ spin-orbitals $|l,m_{\ell },m_s⟩$ (excluding SO coupling constant $\zeta$)

${Ĥ}^{\text{SO}}$ Eigenfunctions $|j,m_j⟩$ in the basis of $\ell =1$ spin-orbitals $|l,m_{\ell },m_s⟩$
(eigenvalues: $-1$ for $j=1∕2$, $+1∕2$ for $j=3∕2$, times $\zeta$)

Kramers-conjugates $|j,m_j{⟩}^{\prime }$ of the ${Ĥ}^{\text{SO}}$ Eigenfunctions in the basis of $\ell =1$ spin-orbitals $|l,m_{\ell },m_s⟩$

Operator ${Ŝ}_x$ in the basis of $\ell =1$ spin-orbitals $|l,m_{\ell },m_s⟩$

Operator ${Ŝ}_y$ in the basis of $\ell =1$ spin-orbitals $|l,m_{\ell },m_s⟩$

Operator ${Ŝ}_z$ in the basis of $\ell =1$ spin-orbitals $|l,m_{\ell },m_s⟩$

Operator ${Ŝ}_x$ in the basis of $\ell =1$ SO eigenfunctions $|j,m_j⟩$

Operator ${Ŝ}_y$ in the basis of $\ell =1$ SO eigenfunctions $|j,m_j⟩$

Operator ${Ŝ}_z$ in the basis of $\ell =1$ SO eigenfunctions $|j,m_j⟩$

Operator ${\widehat{L}}_x$ in the basis of $\ell =1$ spin-orbitals $|l,m_{\ell },m_s⟩$

Operator ${\widehat{L}}_y$ in the basis of $\ell =1$ spin-orbitals $|l,m_{\ell },m_s⟩$

Operator ${\widehat{L}}_z$ in the basis of $\ell =1$ spin-orbitals $|l,m_{\ell },m_s⟩$

Operator ${\widehat{L}}_x$ in the basis of $\ell =1$ SO eigenfunctions $|j,m_j⟩$

Operator ${\widehat{L}}_y$ in the basis of $\ell =1$ SO eigenfunctions $|j,m_j⟩$

Operator ${\widehat{L}}_z$ in the basis of $\ell =1$ SO eigenfunctions $|j,m_j⟩$

Operator ${\widehat{L}}_x+2{Ŝ}_x$ in the basis of $\ell =1$ SO eigenfunctions $|j,m_j⟩$

Operator ${\widehat{L}}_y+2{Ŝ}_y$ in the basis of $\ell =1$ SO eigenfunctions $|j,m_j⟩$

Operator ${\widehat{L}}_z+2{Ŝ}_z$ in the basis of $\ell =1$ SO eigenfunctions $|j,m_j⟩$

$\bullet$ d orbitals $\ell =2$

SO Hamiltonian ${Ĥ}^{\text{SO}}=\zeta \widehat{L}\cdot \widehat{S}$ in the basis of $\ell =2$ spin-orbitals $|l,m_{\ell },m_s⟩$ (excluding SO coupling constant $\zeta$)