on the order of 1% of the Brillouin zone. Also, the dependence on temperature shown by the electrical resistance of such parent compounds indicates proximity to a metal-insulator transition. 21 Undoped ferro-pnictide materials in the cSDW phase then may very well lie near a Mott transition, where conduction electrons experience moderately strong repulsive interactions. By continuity with the limit of strong inter-electron repulsion, we therefore believe that a local-moment description of magnetism in parent compounds to iron-based high- T$_{c}$ superconductors is a justifiable starting point at low temperature. The absence of a Stoner continuum of incoherent magnetic excitations reported in recent experimental studies of magnetic excitations in the cSDW phase of a ferro-pnictide material is consistent with this view. 14 Consider then the following multi-orbital spin-1/2 Hamiltonian that contains near- neighbor Heisenberg exchange among local iron moments within isolated layers in addition to Hund’s-rule coupling: 7 Above, S$_{i}$ ( α ) is the spin operator that acts on the spin-1/2 state of orbital α in the iron atom at site i . The latter runs over the square lattice of iron atoms that make up an isolated layer. The application of Hund’s rule is controlled by a local ferromagnetic Heisenberg exchange constant J$_{0}$ < 0, while nearest neighbor and next-nearest neighbor Heisenberg exchange across the links 〈 i, j 〉 and 〈〈 i, j 〉〉 is controlled by the tensor exchange constants J α,β 1 and J α,β 2 , respectively. The strength of the crystal field at each iron atom determines the number of orbitals per iron atom above. It can be as low as two for strong crystal fields compared to the Hund’s-rule coupling, in which case spin-1/2 moments exist on the iron 3 d$_{xz}$ and 3 d$_{yz}$ orbitals. 4 , 22 Four 3 d orbitals per iron atom carry spin-1/2 moments in the case of weak crystal fields compared to the Hund’s-rule coupling, on the other hand. 3 , 7 Last, the above model Hamiltonian reduces to the conventional J$_{1}$ - J$_{2}$ model over the square lattice when Hund’s rule is enforced, 8 , 9 , 13 − J$_{0}$ → ∞ , where the spin at each site is given by half the number of orbitals, S = 1 or S = 2. We now illustrate a useful transformation of the model Hamiltonians (1) over the square lattice that relates eigenstates and eigenvalues with different nearest-neighbor Heisenberg exchange coupling constants. Let p denote a permutation of the orbitals. Consider then the partial “reflection” P$_{A}$$_{(}$$_{B}$$_{)}$ : orbital α → p ( α ) on the A ( B ) sublattice (2) 4 H = 1 2 J$_{0}$ ∑ i [ ∑ α S$_{i}$ ( α ) ] 2 + ∑ 〈 i,j 〉 ∑ α,β J α,β 1 S$_{i}$ ( α ) · S$_{j}$ ( β ) + ∑ 〈〈 i,j 〉〉 ∑ α,β J α,β 2 S$_{i}$ ( α ) · S$_{j}$ ( β ) . (1)