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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 If the matrices below don’t display properly, please load this (older, no longer maintained) version instead.]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 ĤSO = ζL^ S^ eigenfunctions |j,mj expressed in a basis |,m,ms of 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 ĤαZ = L^ α + 2Ŝα 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 ĤSO and their time-reversed counterparts reflect this. The |j,mj functions provided here were verified to be simultaneous eigenfunctions of ĤSO, ĵ2, and ĵz, with phases such that ĵ±|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.

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.

p orbitals = 1

SO Hamiltonian ĤSO = ζL^ S^ in the basis of = 1 spin-orbitals |l,m,ms (excluding SO coupling constant ζ)

|1, 1, 1 2 |1, 1,1 2 |1, 0, 1 2 |1, 0,1 2 |1,1, 1 2 |1,1,1 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ |1, 1, 1 2 1 2 0 0 0 0 0 |1, 1,1 2 0 1 2 1 2 0 0 0 |1, 0, 1 2 0 1 2 0 0 0 0 |1, 0,1 2 0 0 0 0 1 2 0 |1,1, 1 2 0 0 0 1 2 1 2 0 |1,1,1 2 0 0 0 0 0 1 2

ĤSO Eigenfunctions |j,mj in the basis of = 1 spin-orbitals |l,m,ms
(eigenvalues: 1 for j = 12, + 12 for j = 32, times ζ)

|1 2,1 2 |1 2, 1 2 |3 2,3 2 |3 2,1 2 |3 2, 1 2 |3 2, 3 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ |1, 1, 1 2 0 0 0 0 0 1 |1, 1,1 2 0 2 3 0 0 1 3 0 |1, 0, 1 2 0 1 3 0 0 2 3 0 |1, 0,1 2 1 3 0 0 2 3 0 0 |1,1, 1 2 2 3 0 0 1 3 0 0 |1,1,1 2 0 0 1 0 0 0

Kramers-conjugates |j,mj of the ĤSO Eigenfunctions in the basis of = 1 spin-orbitals |l,m,ms

|1 2,1 2 |1 2, 1 2 |3 2,3 2 |3 2,1 2 |3 2, 1 2 |3 2, 3 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ |1, 1, 1 2 0 0 1 0 0 0 |1, 1,1 2 2 3 0 0 1 3 0 0 |1, 0, 1 2 1 3 0 0 2 3 0 0 |1, 0,1 2 0 1 3 0 0 2 3 0 |1,1, 1 2 0 2 3 0 0 1 3 0 |1,1,1 2 0 0 0 0 0 1

Operator Ŝx in the basis of = 1 spin-orbitals |l,m,ms

|1, 1, 1 2 |1, 1,1 2 |1, 0, 1 2 |1, 0,1 2 |1,1, 1 2 |1,1,1 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ |1, 1, 1 2 0 1 2 0 0 0 0 |1, 1,1 2 1 2 0 0 0 0 0 |1, 0, 1 2 0 0 0 1 2 0 0 |1, 0,1 2 0 0 1 2 0 0 0 |1,1, 1 2 0 0 0 0 0 1 2 |1,1,1 2 0 0 0 0 1 2 0

Operator Ŝy in the basis of = 1 spin-orbitals |l,m,ms

|1, 1, 1 2 |1, 1,1 2 |1, 0, 1 2 |1, 0,1 2 |1,1, 1 2 |1,1,1 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ |1, 1, 1 2 0 i 2 0 0 0 0 |1, 1,1 2 i 2 0 0 0 0 0 |1, 0, 1 2 0 0 0 i 2 0 0 |1, 0,1 2 0 0 i 2 0 0 0 |1,1, 1 2 0 0 0 0 0 i 2 |1,1,1 2 0 0 0 0 i 2 0

Operator Ŝz in the basis of = 1 spin-orbitals |l,m,ms

|1, 1, 1 2 |1, 1,1 2 |1, 0, 1 2 |1, 0,1 2 |1,1, 1 2 |1,1,1 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ |1, 1, 1 2 1 2 0 0 0 0 0 |1, 1,1 2 0 1 2 0 0 0 0 |1, 0, 1 2 0 0 1 2 0 0 0 |1, 0,1 2 0 0 0 1 2 0 0 |1,1, 1 2 0 0 0 0 1 2 0 |1,1,1 2 0 0 0 0 0 1 2

Operator Ŝx in the basis of = 1 SO eigenfunctions |j,mj

|1 2,1 2 |1 2, 1 2 |3 2,3 2 |3 2,1 2 |3 2, 1 2 |3 2, 3 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ |1 2,1 2 0 1 6 1 6 0 1 32 0 |1 2, 1 2 1 6 0 0 1 32 0 1 6 |3 2,3 2 1 6 0 0 1 23 0 0 |3 2,1 2 0 1 32 1 23 0 1 3 0 |3 2, 1 2 1 32 0 0 1 3 0 1 23 |3 2, 3 2 0 1 6 0 0 1 23 0

Operator Ŝy in the basis of = 1 SO eigenfunctions |j,mj

|1 2,1 2 |1 2, 1 2 |3 2,3 2 |3 2,1 2 |3 2, 1 2 |3 2, 3 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ |1 2,1 2 0 i 6 i 6 0 i 32 0 |1 2, 1 2 i 6 0 0 i 32 0 i 6 |3 2,3 2 i 6 0 0 i 23 0 0 |3 2,1 2 0 i 32 i 23 0 i 3 0 |3 2, 1 2 i 32 0 0 i 3 0 i 23 |3 2, 3 2 0 i 6 0 0 i 23 0

Operator Ŝz in the basis of = 1 SO eigenfunctions |j,mj

|1 2,1 2 |1 2, 1 2 |3 2,3 2 |3 2,1 2 |3 2, 1 2 |3 2, 3 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ |1 2,1 2 1 6 0 0 2 3 0 0 |1 2, 1 2 0 1 6 0 0 2 3 0 |3 2,3 2 0 0 1 2 0 0 0 |3 2,1 2 2 3 0 0 1 6 0 0 |3 2, 1 2 0 2 3 0 0 1 6 0 |3 2, 3 2 0 0 0 0 0 1 2

Operator L^x in the basis of = 1 spin-orbitals |l,m,ms

|1, 1, 1 2 |1, 1,1 2 |1, 0, 1 2 |1, 0,1 2 |1,1, 1 2 |1,1,1 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ |1, 1, 1 2 0 0 1 2 0 0 0 |1, 1,1 2 0 0 0 1 2 0 0 |1, 0, 1 2 1 2 0 0 0 1 2 0 |1, 0,1 2 0 1 2 0 0 0 1 2 |1,1, 1 2 0 0 1 2 0 0 0 |1,1,1 2 0 0 0 1 2 0 0

Operator L^y in the basis of = 1 spin-orbitals |l,m,ms

|1, 1, 1 2 |1, 1,1 2 |1, 0, 1 2 |1, 0,1 2 |1,1, 1 2 |1,1,1 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ |1, 1, 1 2 0 0 i 2 0 0 0 |1, 1,1 2 0 0 0 i 2 0 0 |1, 0, 1 2 i 2 0 0 0 i 2 0 |1, 0,1 2 0 i 2 0 0 0 i 2 |1,1, 1 2 0 0 i 2 0 0 0 |1,1,1 2 0 0 0 i 2 0 0

Operator L^z in the basis of = 1 spin-orbitals |l,m,ms

|1, 1, 1 2 |1, 1,1 2 |1, 0, 1 2 |1, 0,1 2 |1,1, 1 2 |1,1,1 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ |1, 1, 1 2 1 0 0 0 0 0 |1, 1,1 2 0 1 0 0 0 0 |1, 0, 1 2 0 0 0 0 0 0 |1, 0,1 2 0 0 0 0 0 0 |1,1, 1 2 0 0 0 0 1 0 |1,1,1 2 0 0 0 0 0 1

Operator L^x in the basis of = 1 SO eigenfunctions |j,mj

|1 2,1 2 |1 2, 1 2 |3 2,3 2 |3 2,1 2 |3 2, 1 2 |3 2, 3 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ |1 2,1 2 0 2 3 1 6 0 1 32 0 |1 2, 1 2 2 3 0 0 1 32 0 1 6 |3 2,3 2 1 6 0 0 1 3 0 0 |3 2,1 2 0 1 32 1 3 0 2 3 0 |3 2, 1 2 1 32 0 0 2 3 0 1 3 |3 2, 3 2 0 1 6 0 0 1 3 0

Operator L^y in the basis of = 1 SO eigenfunctions |j,mj

|1 2,1 2 |1 2, 1 2 |3 2,3 2 |3 2,1 2 |3 2, 1 2 |3 2, 3 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ |1 2,1 2 0 2i 3 i 6 0 i 32 0 |1 2, 1 2 2i 3 0 0 i 32 0 i 6 |3 2,3 2 i 6 0 0 i 3 0 0 |3 2,1 2 0 i 32 i 3 0 2i 3 0 |3 2, 1 2 i 32 0 0 2i 3 0 i 3 |3 2, 3 2 0 i 6 0 0 i 3 0

Operator L^z in the basis of = 1 SO eigenfunctions |j,mj

|1 2,1 2 |1 2, 1 2 |3 2,3 2 |3 2,1 2 |3 2, 1 2 |3 2, 3 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ |1 2,1 2 2 3 0 0 2 3 0 0 |1 2, 1 2 0 2 3 0 0 2 3 0 |3 2,3 2 0 0 1 0 0 0 |3 2,1 2 2 3 0 0 1 3 0 0 |3 2, 1 2 0 2 3 0 0 1 3 0 |3 2, 3 2 0 0 0 0 0 1

Operator L^x + 2Ŝx in the basis of = 1 SO eigenfunctions |j,mj

|1 2,1 2 |1 2, 1 2 |3 2,3 2 |3 2,1 2 |3 2, 1 2 |3 2, 3 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ |1 2,1 2 0 1 3 1 6 0 1 32 0 |1 2, 1 2 1 3 0 0 1 32 0 1 6 |3 2,3 2 1 6 0 0 2 3 0 0 |3 2,1 2 0 1 32 2 3 0 4 3 0 |3 2, 1 2 1 32 0 0 4 3 0 2 3 |3 2, 3 2 0 1 6 0 0 2 3 0

Operator L^y + 2Ŝy in the basis of = 1 SO eigenfunctions |j,mj

|1 2,1 2 |1 2, 1 2 |3 2,3 2 |3 2,1 2 |3 2, 1 2 |3 2, 3 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ |1 2,1 2 0 i 3 i 6 0 i 32 0 |1 2, 1 2 i 3 0 0 i 32 0 i 6 |3 2,3 2 i 6 0 0 2i 3 0 0 |3 2,1 2 0 i 32 2i 3 0 4i 3 0 |3 2, 1 2 i 32 0 0 4i 3 0 2i 3 |3 2, 3 2 0 i 6 0 0 2i 3 0

Operator L^z + 2Ŝz in the basis of = 1 SO eigenfunctions |j,mj

|1 2,1 2 |1 2, 1 2 |3 2,3 2 |3 2,1 2 |3 2, 1 2 |3 2, 3 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ |1 2,1 2 1 3 0 0 2 3 0 0 |1 2, 1 2 0 1 3 0 0 2 3 0 |3 2,3 2 0 0 2 0 0 0 |3 2,1 2 2 3 0 0 2 3 0 0 |3 2, 1 2 0 2 3 0 0 2 3 0 |3 2, 3 2 0 0 0 0 0 2

d orbitals = 2

SO Hamiltonian ĤSO = ζL^ S^ in the basis of = 2 spin-orbitals |l,m,ms (excluding SO coupling constant ζ)

|2, 2, 1 2 |2, 2,1 2 |2, 1, 1 2 |2, 1,1 2 |2, 0, 1 2 |2, 0,1 2 |2,1, 1 2 |2,1,1 2 |2,2, 1 2 |2,2,1 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |2, 2, 1 2 1 0 0 0 0 0 0 0 0 0 |2, 2,1 2 0 1 1 0 0 0 0 0 0 0 |2, 1, 1 2 0 1 1 2 0 0 0 0 0 0 0 |2, 1,1 2 0 0 0 1 2 3 2 0 0 0 0 0 |2, 0, 1 2 0 0 0 3 2 0 0 0 0 0 0 |2, 0,1 2 0 0 0 0 0 0 3 2 0 0 0 |2,1, 1 2 0 0 0 0 0 3 2 1 2 0 0 0 |2,1,1 2 0 0 0 0 0 0 0 1 2 1 0 |2,2, 1 2 0 0 0 0 0 0 0 1 1 0 |2,2,1 2 0 0 0 0 0 0 0 0 0 1

ĤSO Eigenfunctions in the basis of = 2 spin-orbitals |l,m,ms
(eigenvalues: 32 for j = 32, + 1 for j = 52, times ζ)

|3 2,3 2 |3 2,1 2 |3 2, 1 2 |3 2, 3 2 |5 2,5 2 |5 2,3 2 |5 2,1 2 |5 2, 1 2 |5 2, 3 2 |5 2, 5 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |2, 2, 1 2 0 0 0 0 0 0 0 0 0 1 |2, 2,1 2 0 0 0 2 5 0 0 0 0 1 5 0 |2, 1, 1 2 0 0 0 1 5 0 0 0 0 2 5 0 |2, 1,1 2 0 0 3 5 0 0 0 0 2 5 0 0 |2, 0, 1 2 0 0 2 5 0 0 0 0 3 5 0 0 |2, 0,1 2 0 2 5 0 0 0 0 3 5 0 0 0 |2,1, 1 2 0 3 5 0 0 0 0 2 5 0 0 0 |2,1,1 2 1 5 0 0 0 0 2 5 0 0 0 0 |2,2, 1 2 2 5 0 0 0 0 1 5 0 0 0 0 |2,2,1 2 0 0 0 0 1 0 0 0 0 0

Kramers-conjugates |j,mj of the ĤSO Eigenfunctions in the basis of = 2 spin-orbitals |l,m,ms

|3 2,3 2 |3 2,1 2 |3 2, 1 2 |3 2, 3 2 |5 2,5 2 |5 2,3 2 |5 2,1 2 |5 2, 1 2 |5 2, 3 2 |5 2, 5 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |2, 2, 1 2 0 0 0 0 1 0 0 0 0 0 |2, 2,1 2 2 5 0 0 0 0 1 5 0 0 0 0 |2, 1, 1 2 1 5 0 0 0 0 2 5 0 0 0 0 |2, 1,1 2 0 3 5 0 0 0 0 2 5 0 0 0 |2, 0, 1 2 0 2 5 0 0 0 0 3 5 0 0 0 |2, 0,1 2 0 0 2 5 0 0 0 0 3 5 0 0 |2,1, 1 2 0 0 3 5 0 0 0 0 2 5 0 0 |2,1,1 2 0 0 0 1 5 0 0 0 0 2 5 0 |2,2, 1 2 0 0 0 2 5 0 0 0 0 1 5 0 |2,2,1 2 0 0 0 0 0 0 0 0 0 1

Operator Ŝx in the basis of = 2 spin-orbitals |l,m,ms

|2, 2, 1 2 |2, 2,1 2 |2, 1, 1 2 |2, 1,1 2 |2, 0, 1 2 |2, 0,1 2 |2,1, 1 2 |2,1,1 2 |2,2, 1 2 |2,2,1 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |2, 2, 1 2 0 1 2 0 0 0 0 0 0 0 0 |2, 2,1 2 1 2 0 0 0 0 0 0 0 0 0 |2, 1, 1 2 0 0 0 1 2 0 0 0 0 0 0 |2, 1,1 2 0 0 1 2 0 0 0 0 0 0 0 |2, 0, 1 2 0 0 0 0 0 1 2 0 0 0 0 |2, 0,1 2 0 0 0 0 1 2 0 0 0 0 0 |2,1, 1 2 0 0 0 0 0 0 0 1 2 0 0 |2,1,1 2 0 0 0 0 0 0 1 2 0 0 0 |2,2, 1 2 0 0 0 0 0 0 0 0 0 1 2 |2,2,1 2 0 0 0 0 0 0 0 0 1 2 0

Operator Ŝy in the basis of = 2 spin-orbitals |l,m,ms

|2, 2, 1 2 |2, 2,1 2 |2, 1, 1 2 |2, 1,1 2 |2, 0, 1 2 |2, 0,1 2 |2,1, 1 2 |2,1,1 2 |2,2, 1 2 |2,2,1 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |2, 2, 1 2 0 i 2 0 0 0 0 0 0 0 0 |2, 2,1 2 i 2 0 0 0 0 0 0 0 0 0 |2, 1, 1 2 0 0 0 i 2 0 0 0 0 0 0 |2, 1,1 2 0 0 i 2 0 0 0 0 0 0 0 |2, 0, 1 2 0 0 0 0 0 i 2 0 0 0 0 |2, 0,1 2 0 0 0 0 i 2 0 0 0 0 0 |2,1, 1 2 0 0 0 0 0 0 0 i 2 0 0 |2,1,1 2 0 0 0 0 0 0 i 2 0 0 0 |2,2, 1 2 0 0 0 0 0 0 0 0 0 i 2 |2,2,1 2 0 0 0 0 0 0 0 0 i 2 0

Operator Ŝz in the basis of = 2 spin-orbitals |l,m,ms

|2, 2, 1 2 |2, 2,1 2 |2, 1, 1 2 |2, 1,1 2 |2, 0, 1 2 |2, 0,1 2 |2,1, 1 2 |2,1,1 2 |2,2, 1 2 |2,2,1 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |2, 2, 1 2 1 2 0 0 0 0 0 0 0 0 0 |2, 2,1 2 0 1 2 0 0 0 0 0 0 0 0 |2, 1, 1 2 0 0 1 2 0 0 0 0 0 0 0 |2, 1,1 2 0 0 0 1 2 0 0 0 0 0 0 |2, 0, 1 2 0 0 0 0 1 2 0 0 0 0 0 |2, 0,1 2 0 0 0 0 0 1 2 0 0 0 0 |2,1, 1 2 0 0 0 0 0 0 1 2 0 0 0 |2,1,1 2 0 0 0 0 0 0 0 1 2 0 0 |2,2, 1 2 0 0 0 0 0 0 0 0 1 2 0 |2,2,1 2 0 0 0 0 0 0 0 0 0 1 2

Operator Ŝx in the basis of = 2 SO eigenfunctions |j,mj

|3 2,3 2 |3 2,1 2 |3 2, 1 2 |3 2, 3 2 |5 2,5 2 |5 2,3 2 |5 2,1 2 |5 2, 1 2 |5 2, 3 2 |5 2, 5 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |3 2,3 2 0 3 10 0 0 1 5 0 1 52 0 0 0 |3 2,1 2 3 10 0 1 5 0 0 3 5 0 3 2 5 0 0 |3 2, 1 2 0 1 5 0 3 10 0 0 3 2 5 0 3 5 0 |3 2, 3 2 0 0 3 10 0 0 0 0 1 52 0 1 5 |5 2,5 2 1 5 0 0 0 0 1 25 0 0 0 0 |5 2,3 2 0 3 5 0 0 1 25 0 2 5 0 0 0 |5 2,1 2 1 52 0 3 2 5 0 0 2 5 0 3 10 0 0 |5 2, 1 2 0 3 2 5 0 1 52 0 0 3 10 0 2 5 0 |5 2, 3 2 0 0 3 5 0 0 0 0 2 5 0 1 25 |5 2, 5 2 0 0 0 1 5 0 0 0 0 1 25 0

Operator Ŝy in the basis of = 2 SO eigenfunctions |j,mj

|3 2,3 2 |3 2,1 2 |3 2, 1 2 |3 2, 3 2 |5 2,5 2 |5 2,3 2 |5 2,1 2 |5 2, 1 2 |5 2, 3 2 |5 2, 5 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |3 2,3 2 0 i3 10 0 0 i 5 0 i 52 0 0 0 |3 2,1 2 i3 10 0 i 5 0 0 i3 5 0 1 5i3 2 0 0 |3 2, 1 2 0 i 5 0 i3 10 0 0 1 5i3 2 0 i3 5 0 |3 2, 3 2 0 0 i3 10 0 0 0 0 i 52 0 i 5 |5 2,5 2 i 5 0 0 0 0 i 25 0 0 0 0 |5 2,3 2 0 i3 5 0 0 i 25 0 i2 5 0 0 0 |5 2,1 2 i 52 0 1 5i3 2 0 0 i2 5 0 3i 10 0 0 |5 2, 1 2 0 1 5i3 2 0 i 52 0 0 3i 10 0 i2 5 0 |5 2, 3 2 0 0 i3 5 0 0 0 0 i2 5 0 i 25 |5 2, 5 2 0 0 0 i 5 0 0 0 0 i 25 0

Operator Ŝz in the basis of = 2 SO eigenfunctions |j,mj

|3 2,3 2 |3 2,1 2 |3 2, 1 2 |3 2, 3 2 |5 2,5 2 |5 2,3 2 |5 2,1 2 |5 2, 1 2 |5 2, 3 2 |5 2, 5 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |3 2,3 2 3 10 0 0 0 0 2 5 0 0 0 0 |3 2,1 2 0 1 10 0 0 0 0 6 5 0 0 0 |3 2, 1 2 0 0 1 10 0 0 0 0 6 5 0 0 |3 2, 3 2 0 0 0 3 10 0 0 0 0 2 5 0 |5 2,5 2 0 0 0 0 1 2 0 0 0 0 0 |5 2,3 2 2 5 0 0 0 0 3 10 0 0 0 0 |5 2,1 2 0 6 5 0 0 0 0 1 10 0 0 0 |5 2, 1 2 0 0 6 5 0 0 0 0 1 10 0 0 |5 2, 3 2 0 0 0 2 5 0 0 0 0 3 10 0 |5 2, 5 2 0 0 0 0 0 0 0 0 0 1 2

Operator L^x in the basis of = 2 spin-orbitals |l,m,ms

|2, 2, 1 2 |2, 2,1 2 |2, 1, 1 2 |2, 1,1 2 |2, 0, 1 2 |2, 0,1 2 |2,1, 1 2 |2,1,1 2 |2,2, 1 2 |2,2,1 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |2, 2, 1 2 0 0 1 0 0 0 0 0 0 0 |2, 2,1 2 0 0 0 1 0 0 0 0 0 0 |2, 1, 1 2 1 0 0 0 3 2 0 0 0 0 0 |2, 1,1 2 0 1 0 0 0 3 2 0 0 0 0 |2, 0, 1 2 0 0 3 2 0 0 0 3 2 0 0 0 |2, 0,1 2 0 0 0 3 2 0 0 0 3 2 0 0 |2,1, 1 2 0 0 0 0 3 2 0 0 0 1 0 |2,1,1 2 0 0 0 0 0 3 2 0 0 0 1 |2,2, 1 2 0 0 0 0 0 0 1 0 0 0 |2,2,1 2 0 0 0 0 0 0 0 1 0 0

Operator L^y in the basis of = 2 spin-orbitals |l,m,ms

|2, 2, 1 2 |2, 2,1 2 |2, 1, 1 2 |2, 1,1 2 |2, 0, 1 2 |2, 0,1 2 |2,1, 1 2 |2,1,1 2 |2,2, 1 2 |2,2,1 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |2, 2, 1 2 0 0 i 0 0 0 0 0 0 0 |2, 2,1 2 0 0 0 i 0 0 0 0 0 0 |2, 1, 1 2 i 0 0 0 i3 2 0 0 0 0 0 |2, 1,1 2 0 i 0 0 0 i3 2 0 0 0 0 |2, 0, 1 2 0 0 i3 2 0 0 0 i3 2 0 0 0 |2, 0,1 2 0 0 0 i3 2 0 0 0 i3 2 0 0 |2,1, 1 2 0 0 0 0 i3 2 0 0 0 i 0 |2,1,1 2 0 0 0 0 0 i3 2 0 0 0 i |2,2, 1 2 0 0 0 0 0 0 i 0 0 0 |2,2,1 2 0 0 0 0 0 0 0 i 0 0

Operator L^z in the basis of = 2 spin-orbitals |l,m,ms

|2, 2, 1 2 |2, 2,1 2 |2, 1, 1 2 |2, 1,1 2 |2, 0, 1 2 |2, 0,1 2 |2,1, 1 2 |2,1,1 2 |2,2, 1 2 |2,2,1 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |2, 2, 1 2 2 0 0 0 0 0 0 0 0 0 |2, 2,1 2 0 2 0 0 0 0 0 0 0 0 |2, 1, 1 2 0 0 1 0 0 0 0 0 0 0 |2, 1,1 2 0 0 0 1 0 0 0 0 0 0 |2, 0, 1 2 0 0 0 0 0 0 0 0 0 0 |2, 0,1 2 0 0 0 0 0 0 0 0 0 0 |2,1, 1 2 0 0 0 0 0 0 1 0 0 0 |2,1,1 2 0 0 0 0 0 0 0 1 0 0 |2,2, 1 2 0 0 0 0 0 0 0 0 2 0 |2,2,1 2 0 0 0 0 0 0 0 0 0 2

Operator L^x in the basis of = 2 SO eigenfunctions |j,mj

|3 2,3 2 |3 2,1 2 |3 2, 1 2 |3 2, 3 2 |5 2,5 2 |5 2,3 2 |5 2,1 2 |5 2, 1 2 |5 2, 3 2 |5 2, 5 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |3 2,3 2 0 33 5 0 0 1 5 0 1 52 0 0 0 |3 2,1 2 33 5 0 6 5 0 0 3 5 0 3 2 5 0 0 |3 2, 1 2 0 6 5 0 33 5 0 0 3 2 5 0 3 5 0 |3 2, 3 2 0 0 33 5 0 0 0 0 1 52 0 1 5 |5 2,5 2 1 5 0 0 0 0 2 5 0 0 0 0 |5 2,3 2 0 3 5 0 0 2 5 0 42 5 0 0 0 |5 2,1 2 1 52 0 3 2 5 0 0 42 5 0 6 5 0 0 |5 2, 1 2 0 3 2 5 0 1 52 0 0 6 5 0 42 5 0 |5 2, 3 2 0 0 3 5 0 0 0 0 42 5 0 2 5 |5 2, 5 2 0 0 0 1 5 0 0 0 0 2 5 0

Operator L^y in the basis of = 2 SO eigenfunctions |j,mj

|3 2,3 2 |3 2,1 2 |3 2, 1 2 |3 2, 3 2 |5 2,5 2 |5 2,3 2 |5 2,1 2 |5 2, 1 2 |5 2, 3 2 |5 2, 5 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |3 2,3 2 0 3i3 5 0 0 i 5 0 i 52 0 0 0 |3 2,1 2 3i3 5 0 6i 5 0 0 i3 5 0 1 5i3 2 0 0 |3 2, 1 2 0 6i 5 0 3i3 5 0 0 1 5i3 2 0 i3 5 0 |3 2, 3 2 0 0 3i3 5 0 0 0 0 i 52 0 i 5 |5 2,5 2 i 5 0 0 0 0 2i 5 0 0 0 0 |5 2,3 2 0 i3 5 0 0 2i 5 0 4i2 5 0 0 0 |5 2,1 2 i 52 0 1 5i3 2 0 0 4i2 5 0 6i 5 0 0 |5 2, 1 2 0 1 5i3 2 0 i 52 0 0 6i 5 0 4i2 5 0 |5 2, 3 2 0 0 i3 5 0 0 0 0 4i2 5 0 2i 5 |5 2, 5 2 0 0 0 i 5 0 0 0 0 2i 5 0

Operator L^z in the basis of = 2 SO eigenfunctions |j,mj

|3 2,3 2 |3 2,1 2 |3 2, 1 2 |3 2, 3 2 |5 2,5 2 |5 2,3 2 |5 2,1 2 |5 2, 1 2 |5 2, 3 2 |5 2, 5 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |3 2,3 2 9 5 0 0 0 0 2 5 0 0 0 0 |3 2,1 2 0 3 5 0 0 0 0 6 5 0 0 0 |3 2, 1 2 0 0 3 5 0 0 0 0 6 5 0 0 |3 2, 3 2 0 0 0 9 5 0 0 0 0 2 5 0 |5 2,5 2 0 0 0 0 2 0 0 0 0 0 |5 2,3 2 2 5 0 0 0 0 6 5 0 0 0 0 |5 2,1 2 0 6 5 0 0 0 0 2 5 0 0 0 |5 2, 1 2 0 0 6 5 0 0 0 0 2 5 0 0 |5 2, 3 2 0 0 0 2 5 0 0 0 0 6 5 0 |5 2, 5 2 0 0 0 0 0 0 0 0 0 2

Operator L^x + 2Ŝx in the basis of = 2 SO eigenfunctions |j,mj

|3 2,3 2 |3 2,1 2 |3 2, 1 2 |3 2, 3 2 |5 2,5 2 |5 2,3 2 |5 2,1 2 |5 2, 1 2 |5 2, 3 2 |5 2, 5 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |3 2,3 2 0 23 5 0 0 1 5 0 1 52 0 0 0 |3 2,1 2 23 5 0 4 5 0 0 3 5 0 3 2 5 0 0 |3 2, 1 2 0 4 5 0 23 5 0 0 3 2 5 0 3 5 0 |3 2, 3 2 0 0 23 5 0 0 0 0 1 52 0 1 5 |5 2,5 2 1 5 0 0 0 0 3 5 0 0 0 0 |5 2,3 2 0 3 5 0 0 3 5 0 62 5 0 0 0 |5 2,1 2 1 52 0 3 2 5 0 0 62 5 0 9 5 0 0 |5 2, 1 2 0 3 2 5 0 1 52 0 0 9 5 0 62 5 0 |5 2, 3 2 0 0 3 5 0 0 0 0 62 5 0 3 5 |5 2, 5 2 0 0 0 1 5 0 0 0 0 3 5 0

Operator L^y + 2Ŝy in the basis of = 2 SO eigenfunctions |j,mj

|3 2,3 2 |3 2,1 2 |3 2, 1 2 |3 2, 3 2 |5 2,5 2 |5 2,3 2 |5 2,1 2 |5 2, 1 2 |5 2, 3 2 |5 2, 5 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |3 2,3 2 0 2i3 5 0 0 i 5 0 i 52 0 0 0 |3 2,1 2 2i3 5 0 4i 5 0 0 i3 5 0 1 5i3 2 0 0 |3 2, 1 2 0 4i 5 0 2i3 5 0 0 1 5i3 2 0 i3 5 0 |3 2, 3 2 0 0 2i3 5 0 0 0 0 i 52 0 i 5 |5 2,5 2 i 5 0 0 0 0 3i 5 0 0 0 0 |5 2,3 2 0 i3 5 0 0 3i 5 0 6i2 5 0 0 0 |5 2,1 2 i 52 0 1 5i3 2 0 0 6i2 5 0 9i 5 0 0 |5 2, 1 2 0 1 5i3 2 0 i 52 0 0 9i 5 0 6i2 5 0 |5 2, 3 2 0 0 i3 5 0 0 0 0 6i2 5 0 3i 5 |5 2, 5 2 0 0 0 i 5 0 0 0 0 3i 5 0

Operator L^z + 2Ŝz in the basis of = 2 SO eigenfunctions |j,mj

|3 2,3 2 |3 2,1 2 |3 2, 1 2 |3 2, 3 2 |5 2,5 2 |5 2,3 2 |5 2,1 2 |5 2, 1 2 |5 2, 3 2 |5 2, 5 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |3 2,3 2 6 5 0 0 0 0 2 5 0 0 0 0 |3 2,1 2 0 2 5 0 0 0 0 6 5 0 0 0 |3 2, 1 2 0 0 2 5 0 0 0 0 6 5 0 0 |3 2, 3 2 0 0 0 6 5 0 0 0 0 2 5 0 |5 2,5 2 0 0 0 0 3 0 0 0 0 0 |5 2,3 2 2 5 0 0 0 0 9 5 0 0 0 0 |5 2,1 2 0 6 5 0 0 0 0 3 5 0 0 0 |5 2, 1 2 0 0 6 5 0 0 0 0 3 5 0 0 |5 2, 3 2 0 0 0 2 5 0 0 0 0 9 5 0 |5 2, 5 2 0 0 0 0 0 0 0 0 0 3

f orbitals = 3

SO Hamiltonian ĤSO = ζL^ S^ in the basis of = 3 spin-orbitals |l,m,ms (excluding SO coupling constant ζ)

|3, 3, 1 2 |3, 3,1 2 |3, 2, 1 2 |3, 2,1 2 |3, 1, 1 2 |3, 1,1 2 |3, 0, 1 2 |3, 0,1 2 |3,1, 1 2 |3,1,1 2 |3,2, 1 2 |3,2,1 2 |3,3, 1 2 |3,3,1 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |3, 3, 1 2 3 2 0 0 0 0 0 0 0 0 0 0 0 0 0 |3, 3,1 2 0 3 2 3 2 0 0 0 0 0 0 0 0 0 0 0 |3, 2, 1 2 0 3 2 1 0 0 0 0 0 0 0 0 0 0 0 |3, 2,1 2 0 0 0 1 5 2 0 0 0 0 0 0 0 0 0 |3, 1, 1 2 0 0 0 5 2 1 2 0 0 0 0 0 0 0 0 0 |3, 1,1 2 0 0 0 0 0 1 2 3 0 0 0 0 0 0 0 |3, 0, 1 2 0 0 0 0 0 3 0 0 0 0 0 0 0 0 |3, 0,1 2 0 0 0 0 0 0 0 0 3 0 0 0 0 0 |3,1, 1 2 0 0 0 0 0 0 0 3 1 2 0 0 0 0 0 |3,1,1 2 0 0 0 0 0 0 0 0 0 1 2 5 2 0 0 0 |3,2, 1 2 0 0 0 0 0 0 0 0 0 5 2 1 0 0 0 |3,2,1 2 0 0 0 0 0 0 0 0 0 0 0 1 3 2 0 |3,3, 1 2 0 0 0 0 0 0 0 0 0 0 0 3 2 3 2 0 |3,3,1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 3 2

ĤSO Eigenfunctions in the basis of = 3 spin-orbitals |l,m,ms
(eigenvalues: 2 for j = 52, + 32 for j = 72, times ζ)

|5 2,5 2 |5 2,3 2 |5 2,1 2 |5 2, 1 2 |5 2, 3 2 |5 2, 5 2 |7 2,7 2 |7 2,5 2 |7 2,3 2 |7 2,1 2 |7 2, 1 2 |7 2, 3 2 |7 2, 5 2 |7 2, 7 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |3, 3, 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |3, 3,1 2 0 0 0 0 0 6 7 0 0 0 0 0 0 1 7 0 |3, 2, 1 2 0 0 0 0 0 1 7 0 0 0 0 0 0 6 7 0 |3, 2,1 2 0 0 0 0 5 7 0 0 0 0 0 0 2 7 0 0 |3, 1, 1 2 0 0 0 0 2 7 0 0 0 0 0 0 5 7 0 0 |3, 1,1 2 0 0 0 2 7 0 0 0 0 0 0 3 7 0 0 0 |3, 0, 1 2 0 0 0 3 7 0 0 0 0 0 0 2 7 0 0 0 |3, 0,1 2 0 0 3 7 0 0 0 0 0 0 2 7 0 0 0 0 |3,1, 1 2 0 0 2 7 0 0 0 0 0 0 3 7 0 0 0 0 |3,1,1 2 0 2 7 0 0 0 0 0 0 5 7 0 0 0 0 0 |3,2, 1 2 0 5 7 0 0 0 0 0 0 2 7 0 0 0 0 0 |3,2,1 2 1 7 0 0 0 0 0 0 6 7 0 0 0 0 0 0 |3,3, 1 2 6 7 0 0 0 0 0 0 1 7 0 0 0 0 0 0 |3,3,1 2 0 0 0 0 0 0 1 0 0 0 0 0 0 0

Kramers-conjugates |j,mj of the ĤSO Eigenfunctions in the basis of = 3 spin-orbitals |l,m,ms

|5 2,5 2 |5 2,3 2 |5 2,1 2 |5 2, 1 2 |5 2, 3 2 |5 2, 5 2 |7 2,7 2 |7 2,5 2 |7 2,3 2 |7 2,1 2 |7 2, 1 2 |7 2, 3 2 |7 2, 5 2 |7 2, 7 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |3, 3, 1 2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |3, 3,1 2 6 7 0 0 0 0 0 0 1 7 0 0 0 0 0 0 |3, 2, 1 2 1 7 0 0 0 0 0 0 6 7 0 0 0 0 0 0 |3, 2,1 2 0 5 7 0 0 0 0 0 0 2 7 0 0 0 0 0 |3, 1, 1 2 0 2 7 0 0 0 0 0 0 5 7 0 0 0 0 0 |3, 1,1 2 0 0 2 7 0 0 0 0 0 0 3 7 0 0 0 0 |3, 0, 1 2 0 0 3 7 0 0 0 0 0 0 2 7 0 0 0 0 |3, 0,1 2 0 0 0 3 7 0 0 0 0 0 0 2 7 0 0 0 |3,1, 1 2 0 0 0 2 7 0 0 0 0 0 0 3 7 0 0 0 |3,1,1 2 0 0 0 0 2 7 0 0 0 0 0 0 5 7 0 0 |3,2, 1 2 0 0 0 0 5 7 0 0 0 0 0 0 2 7 0 0 |3,2,1 2 0 0 0 0 0 1 7 0 0 0 0 0 0 6 7 0 |3,3, 1 2 0 0 0 0 0 6 7 0 0 0 0 0 0 1 7 0 |3,3,1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1

Operator Ŝx in the basis of = 3 spin-orbitals |l,m,ms

|3, 3, 1 2 |3, 3,1 2 |3, 2, 1 2 |3, 2,1 2 |3, 1, 1 2 |3, 1,1 2 |3, 0, 1 2 |3, 0,1 2 |3,1, 1 2 |3,1,1 2 |3,2, 1 2 |3,2,1 2 |3,3, 1 2 |3,3,1 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |3, 3, 1 2 0 1 2 0 0 0 0 0 0 0 0 0 0 0 0 |3, 3,1 2 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 |3, 2, 1 2 0 0 0 1 2 0 0 0 0 0 0 0 0 0 0 |3, 2,1 2 0 0 1 2 0 0 0 0 0 0 0 0 0 0 0 |3, 1, 1 2 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 |3, 1,1 2 0 0 0 0 1 2 0 0 0 0 0 0 0 0 0 |3, 0, 1 2 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 |3, 0,1 2 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 |3,1, 1 2 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 |3,1,1 2 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 |3,2, 1 2 0 0 0 0 0 0 0 0 0 0 0 1 2 0 0 |3,2,1 2 0 0 0 0 0 0 0 0 0 0 1 2 0 0 0 |3,3, 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 |3,3,1 2 0 0 0 0 0 0 0 0 0 0 0 0 1 2 0

Operator Ŝy in the basis of = 3 spin-orbitals |l,m,ms

|3, 3, 1 2 |3, 3,1 2 |3, 2, 1 2 |3, 2,1 2 |3, 1, 1 2 |3, 1,1 2 |3, 0, 1 2 |3, 0,1 2 |3,1, 1 2 |3,1,1 2 |3,2, 1 2 |3,2,1 2 |3,3, 1 2 |3,3,1 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |3, 3, 1 2 0 i 2 0 0 0 0 0 0 0 0 0 0 0 0 |3, 3,1 2 i 2 0 0 0 0 0 0 0 0 0 0 0 0 0 |3, 2, 1 2 0 0 0 i 2 0 0 0 0 0 0 0 0 0 0 |3, 2,1 2 0 0 i 2 0 0 0 0 0 0 0 0 0 0 0 |3, 1, 1 2 0 0 0 0 0 i 2 0 0 0 0 0 0 0 0 |3, 1,1 2 0 0 0 0 i 2 0 0 0 0 0 0 0 0 0 |3, 0, 1 2 0 0 0 0 0 0 0 i 2 0 0 0 0 0 0 |3, 0,1 2 0 0 0 0 0 0 i 2 0 0 0 0 0 0 0 |3,1, 1 2 0 0 0 0 0 0 0 0 0 i 2 0 0 0 0 |3,1,1 2 0 0 0 0 0 0 0 0 i 2 0 0 0 0 0 |3,2, 1 2 0 0 0 0 0 0 0 0 0 0 0 i 2 0 0 |3,2,1 2 0 0 0 0 0 0 0 0 0 0 i 2 0 0 0 |3,3, 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 i 2 |3,3,1 2 0 0 0 0 0 0 0 0 0 0 0 0 i 2 0

Operator Ŝz in the basis of = 3 spin-orbitals |l,m,ms

|3, 3, 1 2 |3, 3,1 2 |3, 2, 1 2 |3, 2,1 2 |3, 1, 1 2 |3, 1,1 2 |3, 0, 1 2 |3, 0,1 2 |3,1, 1 2 |3,1,1 2 |3,2, 1 2 |3,2,1 2 |3,3, 1 2 |3,3,1 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |3, 3, 1 2 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 |3, 3,1 2 0 1 2 0 0 0 0 0 0 0 0 0 0 0 0 |3, 2, 1 2 0 0 1 2 0 0 0 0 0 0 0 0 0 0 0 |3, 2,1 2 0 0 0 1 2 0 0 0 0 0 0 0 0 0 0 |3, 1, 1 2 0 0 0 0 1 2 0 0 0 0 0 0 0 0 0 |3, 1,1 2 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 |3, 0, 1 2 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 |3, 0,1 2 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 |3,1, 1 2 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 |3,1,1 2 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 |3,2, 1 2 0 0 0 0 0 0 0 0 0 0 1 2 0 0 0 |3,2,1 2 0 0 0 0 0 0 0 0 0 0 0 1 2 0 0 |3,3, 1 2 0 0 0 0 0 0 0 0 0 0 0 0 1 2 0 |3,3,1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2

Operator Ŝx in the basis of = 3 SO eigenfunctions |j,mj

|5 2,5 2 |5 2,3 2 |5 2,1 2 |5 2, 1 2 |5 2, 3 2 |5 2, 5 2 |7 2,7 2 |7 2,5 2 |7 2,3 2 |7 2,1 2 |7 2, 1 2 |7 2, 3 2 |7 2, 5 2 |7 2, 7 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |5 2,5 2 0 5 14 0 0 0 0 3 14 0 1 72 0 0 0 0 0 |5 2,3 2 5 14 0 2 7 0 0 0 0 15 2 7 0 3 2 7 0 0 0 0 |5 2,1 2 0 2 7 0 3 14 0 0 0 0 5 7 0 3 7 0 0 0 |5 2, 1 2 0 0 3 14 0 2 7 0 0 0 0 3 7 0 5 7 0 0 |5 2, 3 2 0 0 0 2 7 0 5 14 0 0 0 0 3 2 7 0 15 2 7 0 |5 2, 5 2 0 0 0 0 5 14 0 0 0 0 0 0 1 72 0 3 14 |7 2,7 2 3 14 0 0 0 0 0 0 1 27 0 0 0 0 0 0 |7 2,5 2 0 15 2 7 0 0 0 0 1 27 0 3 7 0 0 0 0 0 |7 2,3 2 1 72 0 5 7 0 0 0 0 3 7 0 15 14 0 0 0 0 |7 2,1 2 0 3 2 7 0 3 7 0 0 0 0 15 14 0 2 7 0 0 0 |7 2, 1 2 0 0 3 7 0 3 2 7 0 0 0 0 2 7 0 15 14 0 0 |7 2, 3 2 0 0 0 5 7 0 1 72 0 0 0 0 15 14 0 3 7 0 |7 2, 5 2 0 0 0 0 15 2 7 0 0 0 0 0 0 3 7 0 1 27 |7 2, 7 2 0 0 0 0 0 3 14 0 0 0 0 0 0 1 27 0

Operator Ŝy in the basis of = 3 SO eigenfunctions |j,mj

|5 2,5 2 |5 2,3 2 |5 2,1 2 |5 2, 1 2 |5 2, 3 2 |5 2, 5 2 |7 2,7 2 |7 2,5 2 |7 2,3 2 |7 2,1 2 |7 2, 1 2 |7 2, 3 2 |7 2, 5 2 |7 2, 7 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |5 2,5 2 0 i5 14 0 0 0 0 i 3 14 0 i 72 0 0 0 0 0 |5 2,3 2 i5 14 0 i2 7 0 0 0 0 1 7i15 2 0 1 7i3 2 0 0 0 0 |5 2,1 2 0 i2 7 0 3i 14 0 0 0 0 i5 7 0 i3 7 0 0 0 |5 2, 1 2 0 0 3i 14 0 i2 7 0 0 0 0 i3 7 0 i5 7 0 0 |5 2, 3 2 0 0 0 i2 7 0 i5 14 0 0 0 0 1 7i3 2 0 1 7i15 2 0 |5 2, 5 2 0 0 0 0 i5 14 0 0 0 0 0 0 i 72 0 i 3 14 |7 2,7 2 i 3 14 0 0 0 0 0 0 i 27 0 0 0 0 0 0 |7 2,5 2 0 1 7i15 2 0 0 0 0 i 27 0 i3 7 0 0 0 0 0 |7 2,3 2 i 72 0 i5 7 0 0 0 0 i3 7 0 i15 14 0 0 0 0 |7 2,1 2 0 1 7i3 2 0 i3 7 0 0 0 0 i15 14 0 2i 7 0 0 0 |7 2, 1 2 0 0 i3 7 0 1 7i3 2 0 0 0 0 2i 7 0 i15 14 0 0 |7 2, 3 2 0 0 0 i5 7 0 i 72 0 0 0 0 i15 14 0 i3 7 0 |7 2, 5 2 0 0 0 0 1 7i15 2 0 0 0 0 0 0 i3 7 0 i 27 |7 2, 7 2 0 0 0 0 0 i 3 14 0 0 0 0 0 0 i 27 0

Operator Ŝz in the basis of = 3 SO eigenfunctions |j,mj

|5 2,5 2 |5 2,3 2 |5 2,1 2 |5 2, 1 2 |5 2, 3 2 |5 2, 5 2 |7 2,7 2 |7 2,5 2 |7 2,3 2 |7 2,1 2 |7 2, 1 2 |7 2, 3 2 |7 2, 5 2 |7 2, 7 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |5 2,5 2 5 14 0 0 0 0 0 0 6 7 0 0 0 0 0 0 |5 2,3 2 0 3 14 0 0 0 0 0 0 10 7 0 0 0 0 0 |5 2,1 2 0 0 1 14 0 0 0 0 0 0 23 7 0 0 0 0 |5 2, 1 2 0 0 0 1 14 0 0 0 0 0 0 23 7 0 0 0 |5 2, 3 2 0 0 0 0 3 14 0 0 0 0 0 0 10 7 0 0 |5 2, 5 2 0 0 0 0 0 5 14 0 0 0 0 0 0 6 7 0 |7 2,7 2 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 |7 2,5 2 6 7 0 0 0 0 0 0 5 14 0 0 0 0 0 0 |7 2,3 2 0 10 7 0 0 0 0 0 0 3 14 0 0 0 0 0 |7 2,1 2 0 0 23 7 0 0 0 0 0 0 1 14 0 0 0 0 |7 2, 1 2 0 0 0 23 7 0 0 0 0 0 0 1 14 0 0 0 |7 2, 3 2 0 0 0 0 10 7 0 0 0 0 0 0 3 14 0 0 |7 2, 5 2 0 0 0 0 0 6 7 0 0 0 0 0 0 5 14 0 |7 2, 7 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2

Operator L^x in the basis of = 3 spin-orbitals |l,m,ms

|3, 3, 1 2 |3, 3,1 2 |3, 2, 1 2 |3, 2,1 2 |3, 1, 1 2 |3, 1,1 2 |3, 0, 1 2 |3, 0,1 2 |3,1, 1 2 |3,1,1 2 |3,2, 1 2 |3,2,1 2 |3,3, 1 2 |3,3,1 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |3, 3, 1 2 0 0 3 2 0 0 0 0 0 0 0 0 0 0 0 |3, 3,1 2 0 0 0 3 2 0 0 0 0 0 0 0 0 0 0 |3, 2, 1 2 3 2 0 0 0 5 2 0 0 0 0 0 0 0 0 0 |3, 2,1 2 0 3 2 0 0 0 5 2 0 0 0 0 0 0 0 0 |3, 1, 1 2 0 0 5 2 0 0 0 3 0 0 0 0 0 0 0 |3, 1,1 2 0 0 0 5 2 0 0 0 3 0 0 0 0 0 0 |3, 0, 1 2 0 0 0 0 3 0 0 0 3 0 0 0 0 0 |3, 0,1 2 0 0 0 0 0 3 0 0 0 3 0 0 0 0 |3,1, 1 2 0 0 0 0 0 0 3 0 0 0 5 2 0 0 0 |3,1,1 2 0 0 0 0 0 0 0 3 0 0 0 5 2 0 0 |3,2, 1 2 0 0 0 0 0 0 0 0 5 2 0 0 0 3 2 0 |3,2,1 2 0 0 0 0 0 0 0 0 0 5 2 0 0 0 3 2 |3,3, 1 2 0 0 0 0 0 0 0 0 0 0 3 2 0 0 0 |3,3,1 2 0 0 0 0 0 0 0 0 0 0 0 3 2 0 0

Operator L^y in the basis of = 3 spin-orbitals |l,m,ms

|3, 3, 1 2 |3, 3,1 2 |3, 2, 1 2 |3, 2,1 2 |3, 1, 1 2 |3, 1,1 2 |3, 0, 1 2 |3, 0,1 2 |3,1, 1 2 |3,1,1 2 |3,2, 1 2 |3,2,1 2 |3,3, 1 2 |3,3,1 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |3, 3, 1 2 0 0 i3 2 0 0 0 0 0 0 0 0 0 0 0 |3, 3,1 2 0 0 0 i3 2 0 0 0 0 0 0 0 0 0 0 |3, 2, 1 2 i3 2 0 0 0 i5 2 0 0 0 0 0 0 0 0 0 |3, 2,1 2 0 i3 2 0 0 0 i5 2 0 0 0 0 0 0 0 0 |3, 1, 1 2 0 0 i5 2 0 0 0 i3 0 0 0 0 0 0 0 |3, 1,1 2 0 0 0 i5 2 0 0 0 i3 0 0 0 0 0 0 |3, 0, 1 2 0 0 0 0 i3 0 0 0 i3 0 0 0 0 0 |3, 0,1 2 0 0 0 0 0 i3 0 0 0 i3 0 0 0 0 |3,1, 1 2 0 0 0 0 0 0 i3 0 0 0 i5 2 0 0 0 |3,1,1 2 0 0 0 0 0 0 0 i3 0 0 0 i5 2 0 0 |3,2, 1 2 0 0 0 0 0 0 0 0 i5 2 0 0 0 i3 2 0 |3,2,1 2 0 0 0 0 0 0 0 0 0 i5 2 0 0 0 i3 2 |3,3, 1 2 0 0 0 0 0 0 0 0 0 0 i3 2 0 0 0 |3,3,1 2 0 0 0 0 0 0 0 0 0 0 0 i3 2 0 0

Operator L^z in the basis of = 3 spin-orbitals |l,m,ms

|3, 3, 1 2 |3, 3,1 2 |3, 2, 1 2 |3, 2,1 2 |3, 1, 1 2 |3, 1,1 2 |3, 0, 1 2 |3, 0,1 2 |3,1, 1 2 |3,1,1 2 |3,2, 1 2 |3,2,1 2 |3,3, 1 2 |3,3,1 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |3, 3, 1 2 3 0 0 0 0 0 0 0 0 0 0 0 0 0 |3, 3,1 2 0 3 0 0 0 0 0 0 0 0 0 0 0 0 |3, 2, 1 2 0 0 2 0 0 0 0 0 0 0 0 0 0 0 |3, 2,1 2 0 0 0 2 0 0 0 0 0 0 0 0 0 0 |3, 1, 1 2 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |3, 1,1 2 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |3, 0, 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |3, 0,1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |3,1, 1 2 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |3,1,1 2 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |3,2, 1 2 0 0 0 0 0 0 0 0 0 0 2 0 0 0 |3,2,1 2 0 0 0 0 0 0 0 0 0 0 0 2 0 0 |3,3, 1 2 0 0 0 0 0 0 0 0 0 0 0 0 3 0 |3,3,1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 3

Operator L^x in the basis of = 3 SO eigenfunctions |j,mj

|5 2,5 2 |5 2,3 2 |5 2,1 2 |5 2, 1 2 |5 2, 3 2 |5 2, 5 2 |7 2,7 2 |7 2,5 2 |7 2,3 2 |7 2,1 2 |7 2, 1 2 |7 2, 3 2 |7 2, 5 2 |7 2, 7 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |5 2,5 2 0 45 7 0 0 0 0 3 14 0 1 72 0 0 0 0 0 |5 2,3 2 45 7 0 82 7 0 0 0 0 15 2 7 0 3 2 7 0 0 0 0 |5 2,1 2 0 82 7 0 12 7 0 0 0 0 5 7 0 3 7 0 0 0 |5 2, 1 2 0 0 12 7 0 82 7 0 0 0 0 3 7 0 5 7 0 0 |5 2, 3 2 0 0 0 82 7 0 45 7 0 0 0 0 3 2 7 0 15 2 7 0 |5 2, 5 2 0 0 0 0 45 7 0 0 0 0 0 0 1 72 0 3 14 |7 2,7 2 3 14 0 0 0 0 0 0 3 7 0 0 0 0 0 0 |7 2,5 2 0 15 2 7 0 0 0 0 3 7 0 63 7 0 0 0 0 0 |7 2,3 2 1 72 0 5 7 0 0 0 0 63 7 0 315 7 0 0 0 0 |7 2,1 2 0 3 2 7 0 3 7 0 0 0 0 315 7 0 12 7 0 0 0 |7 2, 1 2 0 0 3 7 0 3 2 7 0 0 0 0 12 7 0 315 7 0 0 |7 2, 3 2 0 0 0 5 7 0 1 72 0 0 0 0 315 7 0 63 7 0 |7 2, 5 2 0 0 0 0 15 2 7 0 0 0 0 0 0 63 7 0 3 7 |7 2, 7 2 0 0 0 0 0 3 14 0 0 0 0 0 0 3 7 0

Operator L^y in the basis of = 3 SO eigenfunctions |j,mj

|5 2,5 2 |5 2,3 2 |5 2,1 2 |5 2, 1 2 |5 2, 3 2 |5 2, 5 2 |7 2,7 2 |7 2,5 2 |7 2,3 2 |7 2,1 2 |7 2, 1 2 |7 2, 3 2 |7 2, 5 2 |7 2, 7 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |5 2,5 2 0 4i5 7 0 0 0 0 i 3 14 0 i 72 0 0 0 0 0 |5 2,3 2 4i5 7 0 8i2 7 0 0 0 0 1 7i15 2 0 1 7i3 2 0 0 0 0 |5 2,1 2 0 8i2 7 0 12i 7 0 0 0 0 i5 7 0 i3 7 0 0 0 |5 2, 1 2 0 0 12i 7 0 8i2 7 0 0 0 0 i3 7 0 i5 7 0 0 |5 2, 3 2 0 0 0 8i2 7 0 4i5 7 0 0 0 0 1 7i3 2 0 1 7i15 2 0 |5 2, 5 2 0 0 0 0 4i5 7 0 0 0 0 0 0 i 72 0 i 3 14 |7 2,7 2 i 3 14 0 0 0 0 0 0 3i 7 0 0 0 0 0 0 |7 2,5 2 0 1 7i15 2 0 0 0 0 3i 7 0 6i3 7 0 0 0 0 0 |7 2,3 2 i 72 0 i5 7 0 0 0 0 6i3 7 0 3i15 7 0 0 0 0 |7 2,1 2 0 1 7i3 2 0 i3 7 0 0 0 0 3i15 7 0 12i 7 0 0 0 |7 2, 1 2 0 0 i3 7 0 1 7i3 2 0 0 0 0 12i 7 0 3i15 7 0 0 |7 2, 3 2 0 0 0 i5 7 0 i 72 0 0 0 0 3i15 7 0 6i3 7 0 |7 2, 5 2 0 0 0 0 1 7i15 2 0 0 0 0 0 0 6i3 7 0 3i 7 |7 2, 7 2 0 0 0 0 0 i 3 14 0 0 0 0 0 0 3i 7 0

Operator L^z in the basis of = 3 SO eigenfunctions |j,mj

|5 2,5 2 |5 2,3 2 |5 2,1 2 |5 2, 1 2 |5 2, 3 2 |5 2, 5 2 |7 2,7 2 |7 2,5 2 |7 2,3 2 |7 2,1 2 |7 2, 1 2 |7 2, 3 2 |7 2, 5 2 |7 2, 7 2 ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ |5 2,5 2 20 7 0 0 0 0 0 0 6 7 0 0 0 0 0 0 |5 2,3 2 0 12 7 0 0 0 0 0 0 10 7 0 0 0 0 0 |5 2,1 2 0 0