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of the N´ eel state. If Ψ is an eigenstate of the original model Hamiltonian (1) with energy E , the transformed state Ψ ′ = P$_{A}$$_{(}$$_{B}$$_{)}$ Ψ is clearly then an eigenstate of the transformed Hamiltonian H ′ = P$_{A}$$_{(}$$_{B}$$_{)}$HP − 1 A ( B ) with energy E . Importantly, the Heisenberg exchange coupling constants transform under (2) as 

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when orbital α lies on the A ( B ) sublattice of the N´ eel state. Both J α,β 2 on the B ( A ) sublattice and the Hund’s rule coupling J$_{0}$ remain fixed. Yet how does the spectrum of states transform under (2)? Let T$_{x}$ and T$_{y}$ denote translations along the x axis and along the y axis by one lattice constant a . Because T$_{x}$T$_{y}$ moves each sublattice to itself, it commutes with P$_{A}$ and with P$_{B}$ . If Ψ has a crystal momentum ¯ h k , we therefore have that T$_{x}$T$_{y}$ Ψ ′ = e i ( k$_{x}$ + k$_{y}$ ) $^{a}$Ψ $^{′}$. The crystal momentum then transforms under (2) as 

Also, the spin of the new state Ψ ′ remains unchanged because P$_{A}$ and P$_{B}$ both commute with S$_{i}$ = ∑ $_{α}$S$_{i}$ ( α ). It is also useful to consider the special case where all nearest-neighbor and next-nearest- 

neighbor exchange coupling constants are equal, respectively: J α,β 1 = J$_{1}$ and J α,β 2 = J$_{2}$ . The model Hamiltonian (1) then reduces to H = 1 $_{2}$J$_{0}$ ∑ $_{i}$S$_{i}$ · S$_{i}$ + $^{J$_{1}$}$∑ 〈 i,j $_{〉}$S$_{i}$ · S$_{j}$ + $^{J$_{2}$}$∑ 〈〈 i,j $_{〉〉}$S$_{i}$ · S$_{j}$ . Because S$_{i}$ + S$_{j}$ commutes with S$_{i}$ · S$_{i}$ , the latter then commutes with the Hamiltonian. The total spin at a given site i is hence a good quantum number. This means that the groundstate obeys Hund’s rule in the classical limit because states with maximum total spin at a given site minimize both the Hund’s-rule energy ( J$_{0}$ < 0) and the Heisenberg exchange energies in such a case. A violation of Hund’s rule will therefore require some variation in the Heisenberg exchange coupling constants among the different iron orbitals. 

## III. SPIN-1 IRON ATOMS 

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The iron 3d orbitals in ferro-pnictide materials experience a crystal field due to the neig- boring pnictide atoms that lie above/below the center of each unit square of iron atoms. Both DFT calculations and basic considerations indicate that the crystal field splits these levels so that the degenerate 3 d$_{xz}$ and 3 d$_{yz}$ orbitals lie just below the highest-energy 3 d orbital. 4 , 22 The 

