gus_massa took the time to actually read this on the last thread and asked
three things. That thread locked before I answered, which was my fault for
being slow, so here they are.
You suggested a short version, two or three pages, that goes straight to the
Table 1 results. I wrote one. It's about three pages and it's here:
https://github.com/PureStateLabs/sqd-spin-referee/blob/main/...
You asked how many states sit near that energy, and whether there's a graphic.
Yes. The four lowest roots of the flagship subspace span 12.5 mHa, with the
ground gap at 0.285 mHa, and their spin values are S^2 = 1.371, 2.363, 4.738
and 11.590. Climbing the energy ladder climbs the spin ladder. The figure is
fig_flagship_s4.png in the archive: panel A is that ladder, B is the residual
against S^2, C is the per-root variance.
You said you couldn't picture how S greater than 1 arises, and wondered if it
needed extra freedom in the d orbitals. It doesn't. [2Fe-2S] is two high-spin
Fe(III) centres, each d5 with S = 5/2. Coupling 5/2 with 5/2 gives total
S = 0,1,2,3,4,5, so S^2 runs 0, 2, 6, 12, 20, 30. That's the whole Heisenberg
ladder and it's ordinary Clebsch-Gordan, no d-orbital subtlety needed. It also
puts your objection on solid ground: 4.7 isn't an exotic value, it sits between
the S=1 and S=2 rungs, which is exactly why it reads as a mixture rather than a
contaminated singlet.
You wrote that some contamination is expected but 4.7 was too much in your
opinion. I've since measured the second moment on those states, and the
variance is the part that settles it. A spin eigenstate has Var(S^2) = 0. The
low-lying roots here run 3.5 to 7.4. They aren't slightly perturbed
eigenstates, they're broad mixtures, and a scalar mean of 4.7 understates how
broad.
You suggested a short version, two or three pages, that goes straight to the Table 1 results. I wrote one. It's about three pages and it's here: https://github.com/PureStateLabs/sqd-spin-referee/blob/main/...
You asked how many states sit near that energy, and whether there's a graphic. Yes. The four lowest roots of the flagship subspace span 12.5 mHa, with the ground gap at 0.285 mHa, and their spin values are S^2 = 1.371, 2.363, 4.738 and 11.590. Climbing the energy ladder climbs the spin ladder. The figure is fig_flagship_s4.png in the archive: panel A is that ladder, B is the residual against S^2, C is the per-root variance.
You said you couldn't picture how S greater than 1 arises, and wondered if it needed extra freedom in the d orbitals. It doesn't. [2Fe-2S] is two high-spin Fe(III) centres, each d5 with S = 5/2. Coupling 5/2 with 5/2 gives total S = 0,1,2,3,4,5, so S^2 runs 0, 2, 6, 12, 20, 30. That's the whole Heisenberg ladder and it's ordinary Clebsch-Gordan, no d-orbital subtlety needed. It also puts your objection on solid ground: 4.7 isn't an exotic value, it sits between the S=1 and S=2 rungs, which is exactly why it reads as a mixture rather than a contaminated singlet.
You wrote that some contamination is expected but 4.7 was too much in your opinion. I've since measured the second moment on those states, and the variance is the part that settles it. A spin eigenstate has Var(S^2) = 0. The low-lying roots here run 3.5 to 7.4. They aren't slightly perturbed eigenstates, they're broad mixtures, and a scalar mean of 4.7 understates how broad.