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Below are sets of H^{SO} = ζL⋅S eigenfunctions |j,m_{j}〉 expressed in a basis
|ℓ,m_{ℓ},m_{s}〉 of atomic spin orbitals for a given angular momentum quantum
number ℓ = 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 H_{α}^{Z} = L_{α} + 2S_{α} 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_{ℓ} to −m_{ℓ} but
also introduces a negative sign in the spin orbital basis if m_{ℓ} is odd. The
coefficients representing the eigenfunctions of H^{SO} and their time-reversed
counterparts reflect this. The |j,m_{j}〉 functions provided here were verified to be
simultaneous eigenfunctions of H^{SO}, j^{2}, and j_{
z}, with phases such that
j_{±}|j,m_{j}〉 = α_{±}|j,m_{j} ± 1〉, with α_{±} being a positive real number as long as
m_{j} ± 1 is in the allowed range of the projection quantum number, zero
otherwise.

If you use this material for research please cite our publication [210] and this web page. And if you spot a mistake here, please let us know. Thank you.

SO Hamiltonian H^{SO} = ζL ⋅ S in the basis of ℓ = 1 spin-orbitals
|l,m_{ℓ},m_{s}〉 (excluding SO coupling constant ζ)

H^{SO} Eigenfunctions |j,m_{j}〉 in the basis of ℓ = 1 spin-orbitals
|l,m_{ℓ},m_{s}〉

(eigenvalues: −1 for j = 1∕2, +1∕2 for j = 3∕2, times ζ)

Kramers-conjugates |j,m_{j}〉′ of the H^{SO} Eigenfunctions in the basis of
ℓ = 1 spin-orbitals |l,m_{ℓ},m_{s}〉

Operator S_{x} in the basis of ℓ = 1 spin-orbitals |l,m_{ℓ},m_{s}〉

Operator S_{y} in the basis of ℓ = 1 spin-orbitals |l,m_{ℓ},m_{s}〉

Operator S_{z} in the basis of ℓ = 1 spin-orbitals |l,m_{ℓ},m_{s}〉

Operator S_{x} in the basis of ℓ = 1 SO eigenfunctions |j,m_{j}〉

Operator S_{y} in the basis of ℓ = 1 SO eigenfunctions |j,m_{j}〉

Operator S_{z} in the basis of ℓ = 1 SO eigenfunctions |j,m_{j}〉

Operator L_{x} in the basis of ℓ = 1 spin-orbitals |l,m_{ℓ},m_{s}〉

Operator L_{y} in the basis of ℓ = 1 spin-orbitals |l,m_{ℓ},m_{s}〉

Operator L_{z} in the basis of ℓ = 1 spin-orbitals |l,m_{ℓ},m_{s}〉

Operator L_{x} in the basis of ℓ = 1 SO eigenfunctions |j,m_{j}〉

Operator L_{y} in the basis of ℓ = 1 SO eigenfunctions |j,m_{j}〉

Operator L_{z} in the basis of ℓ = 1 SO eigenfunctions |j,m_{j}〉

Operator L_{x} + 2S_{x} in the basis of ℓ = 1 SO eigenfunctions |j,m_{j}〉

Operator L_{y} + 2S_{y} in the basis of ℓ = 1 SO eigenfunctions |j,m_{j}〉

Operator L_{z} + 2S_{z} in the basis of ℓ = 1 SO eigenfunctions |j,m_{j}〉

SO Hamiltonian H^{SO} = ζL ⋅ S in the basis of ℓ = 2 spin-orbitals
|l,m_{ℓ},m_{s}〉 (excluding SO coupling constant ζ)

H^{SO} Eigenfunctions in the basis of ℓ = 2 spin-orbitals |l,m_{ℓ},m_{s}〉

(eigenvalues: −3∕2 for j = 3∕2, +1 for j = 5∕2, times ζ)

Kramers-conjugates |j,m_{j}〉′ of the H^{SO} Eigenfunctions in the basis of
ℓ = 2 spin-orbitals |l,m_{ℓ},m_{s}〉

Operator S_{x} in the basis of ℓ = 2 spin-orbitals |l,m_{ℓ},m_{s}〉

Operator S_{y} in the basis of ℓ = 2 spin-orbitals |l,m_{ℓ},m_{s}〉

Operator S_{z} in the basis of ℓ = 2 spin-orbitals |l,m_{ℓ},m_{s}〉

Operator S_{x} in the basis of ℓ = 2 SO eigenfunctions |j,m_{j}〉

Operator S_{y} in the basis of ℓ = 2 SO eigenfunctions |j,m_{j}〉

Operator S_{z} in the basis of ℓ = 2 SO eigenfunctions |j,m_{j}〉

Operator L_{x} in the basis of ℓ = 2 spin-orbitals |l,m_{ℓ},m_{s}〉

Operator L_{y} in the basis of ℓ = 2 spin-orbitals |l,m_{ℓ},m_{s}〉

Operator L_{z} in the basis of ℓ = 2 spin-orbitals |l,m_{ℓ},m_{s}〉

Operator L_{x} in the basis of ℓ = 2 SO eigenfunctions |j,m_{j}〉

Operator L_{y} in the basis of ℓ = 2 SO eigenfunctions |j,m_{j}〉

Operator L_{z} in the basis of ℓ = 2 SO eigenfunctions |j,m_{j}〉

Operator L_{x} + 2S_{x} in the basis of ℓ = 2 SO eigenfunctions |j,m_{j}〉

Operator L_{y} + 2S_{y} in the basis of ℓ = 2 SO eigenfunctions |j,m_{j}〉

Operator L_{z} + 2S_{z} in the basis of ℓ = 2 SO eigenfunctions |j,m_{j}〉

SO Hamiltonian H^{SO} = ζL ⋅ S in the basis of ℓ = 3 spin-orbitals
|l,m_{ℓ},m_{s}〉 (excluding SO coupling constant ζ)

H^{SO} Eigenfunctions in the basis of ℓ = 3 spin-orbitals |l,m_{ℓ},m_{s}〉

(eigenvalues: −2 for j = 5∕2, +3∕2 for j = 7∕2, times ζ)

Kramers-conjugates |j,m_{j}〉′ of the H^{SO} Eigenfunctions in the basis of
ℓ = 3 spin-orbitals |l,m_{ℓ},m_{s}〉

Operator S_{x} in the basis of ℓ = 3 spin-orbitals |l,m_{ℓ},m_{s}〉

Operator S_{y} in the basis of ℓ = 3 spin-orbitals |l,m_{ℓ},m_{s}〉

Operator S_{z} in the basis of ℓ = 3 spin-orbitals |l,m_{ℓ},m_{s}〉

Operator S_{x} in the basis of ℓ = 3 SO eigenfunctions |j,m_{j}〉

Operator S_{y} in the basis of ℓ = 3 SO eigenfunctions |j,m_{j}〉

Operator S_{z} in the basis of ℓ = 3 SO eigenfunctions |j,m_{j}〉

Operator L_{x} in the basis of ℓ = 3 spin-orbitals |l,m_{ℓ},m_{s}〉

Operator L_{y} in the basis of ℓ = 3 spin-orbitals |l,m_{ℓ},m_{s}〉

Operator L_{z} in the basis of ℓ = 3 spin-orbitals |l,m_{ℓ},m_{s}〉

Operator L_{x} in the basis of ℓ = 3 SO eigenfunctions |j,m_{j}〉

Operator L_{y} in the basis of ℓ = 3 SO eigenfunctions |j,m_{j}〉

Operator L_{z} in the basis of ℓ = 3 SO eigenfunctions |j,m_{j}〉

Operator L_{x} + 2S_{x} in the basis of ℓ = 3 SO eigenfunctions |j,m_{j}〉

Operator L_{y} + 2S_{y} in the basis of ℓ = 3 SO eigenfunctions |j,m_{j}〉

Operator L_{z} + 2S_{z} in the basis of ℓ = 3 SO eigenfunctions |j,m_{j}〉

© 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.

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