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

Below are sets of HSO = ζLS eigenfunctions |j,mjexpressed in a basis |ℓ,m,msof atomic spin orbitals for a given angular momentum quantum number = 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 HαZ = Lα + 2Sα with α = 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 to m but also introduces a negative sign in the spin orbital basis if m is odd. The coefficients representing the eigenfunctions of HSO and their time-reversed counterparts reflect this. The |j,mjfunctions provided here were verified to be simultaneous eigenfunctions of HSO, j2, and j z, with phases such that j±|j,mj= α±|j,mj ± 1, with α± being a positive real number as long as mj ± 1 is in the allowed range of the projection quantum number, zero otherwise.

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### ∙ p orbitals ℓ = 1

SO Hamiltonian HSO = ζL S in the basis of = 1 spin-orbitals |l,m,ms(excluding SO coupling constant ζ) HSO Eigenfunctions |j,mjin the basis of = 1 spin-orbitals |l,m,ms
(eigenvalues: 1 for j = 12, +12 for j = 32, times ζ) Kramers-conjugates |j,mj〉′ of the HSO Eigenfunctions in the basis of = 1 spin-orbitals |l,m,ms Operator Sx in the basis of = 1 spin-orbitals |l,m,ms Operator Sy in the basis of = 1 spin-orbitals |l,m,ms Operator Sz in the basis of = 1 spin-orbitals |l,m,ms Operator Sx in the basis of = 1 SO eigenfunctions |j,mj Operator Sy in the basis of = 1 SO eigenfunctions |j,mj Operator Sz in the basis of = 1 SO eigenfunctions |j,mj Operator Lx in the basis of = 1 spin-orbitals |l,m,ms Operator Ly in the basis of = 1 spin-orbitals |l,m,ms Operator Lz in the basis of = 1 spin-orbitals |l,m,ms Operator Lx in the basis of = 1 SO eigenfunctions |j,mj Operator Ly in the basis of = 1 SO eigenfunctions |j,mj Operator Lz in the basis of = 1 SO eigenfunctions |j,mj Operator Lx + 2Sx in the basis of = 1 SO eigenfunctions |j,mj Operator Ly + 2Sy in the basis of = 1 SO eigenfunctions |j,mj Operator Lz + 2Sz in the basis of = 1 SO eigenfunctions |j,mj ### ∙ d orbitals ℓ = 2

SO Hamiltonian HSO = ζL S in the basis of = 2 spin-orbitals |l,m,ms(excluding SO coupling constant ζ) HSO Eigenfunctions in the basis of = 2 spin-orbitals |l,m,ms
(eigenvalues: 32 for j = 32, +1 for j = 52, times ζ) Kramers-conjugates |j,mj〉′ of the HSO Eigenfunctions in the basis of = 2 spin-orbitals |l,m,ms Operator Sx in the basis of = 2 spin-orbitals |l,m,ms Operator Sy in the basis of = 2 spin-orbitals |l,m,ms Operator Sz in the basis of = 2 spin-orbitals |l,m,ms Operator Sx in the basis of = 2 SO eigenfunctions |j,mj Operator Sy in the basis of = 2 SO eigenfunctions |j,mj Operator Sz in the basis of = 2 SO eigenfunctions |j,mj Operator Lx in the basis of = 2 spin-orbitals |l,m,ms Operator Ly in the basis of = 2 spin-orbitals |l,m,ms Operator Lz in the basis of = 2 spin-orbitals |l,m,ms Operator Lx in the basis of = 2 SO eigenfunctions |j,mj Operator Ly in the basis of = 2 SO eigenfunctions |j,mj Operator Lz in the basis of = 2 SO eigenfunctions |j,mj Operator Lx + 2Sx in the basis of = 2 SO eigenfunctions |j,mj Operator Ly + 2Sy in the basis of = 2 SO eigenfunctions |j,mj Operator Lz + 2Sz in the basis of = 2 SO eigenfunctions |j,mj ### ∙ f orbitals ℓ = 3

SO Hamiltonian HSO = ζL S in the basis of = 3 spin-orbitals |l,m,ms(excluding SO coupling constant ζ) HSO Eigenfunctions in the basis of = 3 spin-orbitals |l,m,ms
(eigenvalues: 2 for j = 52, +32 for j = 72, times ζ) Kramers-conjugates |j,mj〉′ of the HSO Eigenfunctions in the basis of = 3 spin-orbitals |l,m,ms Operator Sx in the basis of = 3 spin-orbitals |l,m,ms Operator Sy in the basis of = 3 spin-orbitals |l,m,ms Operator Sz in the basis of = 3 spin-orbitals |l,m,ms Operator Sx in the basis of = 3 SO eigenfunctions |j,mj Operator Sy in the basis of = 3 SO eigenfunctions |j,mj Operator Sz in the basis of = 3 SO eigenfunctions |j,mj Operator Lx in the basis of = 3 spin-orbitals |l,m,ms Operator Ly in the basis of = 3 spin-orbitals |l,m,ms Operator Lz in the basis of = 3 spin-orbitals |l,m,ms Operator Lx in the basis of = 3 SO eigenfunctions |j,mj Operator Ly in the basis of = 3 SO eigenfunctions |j,mj Operator Lz in the basis of = 3 SO eigenfunctions |j,mj Operator Lx + 2Sx in the basis of = 3 SO eigenfunctions |j,mj Operator Ly + 2Sy in the basis of = 3 SO eigenfunctions |j,mj Operator Lz + 2Sz in the basis of = 3 SO eigenfunctions |j,mj © 2018 – 2019 J. Autschbach. The material shown on this web page is based on the results of research funded by a grant from the US Department of Energy (Basic Energy Sciences, Heavy Element Chemistry program, grant DE-SC0001136). Any opinions, findings, and conclusions or recommendations expressed here are those of the author and do not necessarily reflect the views of this funding agency.