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NaFe$_{0.5}$Cu$_{0.5}$As represents a rare exception in the metallic ironpnictide family, in which a small insulating gap is opened. Based onfirst-principles study, we provide a comprehensive theoretical characterizationof this insulating compound. The Fe$^{3+} $spin degree of freedom is quantifiedas a quasi-1D $S=\frac{5}{2}$ Heisenberg model. The itinerant As hole state isdownfolded to a $p_{xy}$-orbital hopping model on a square lattice. A uniqueorbital-dependent Hund's coupling between the spin and the hole is revealed.Several important material properties are analyzed, including (a) factorsaffecting the small $p-d$ charge-transfer gap; (b) role of the extra interchainFe; and (c) the quasi-1D spin excitation in the Fe chains. The experimentalmanifestations of these properties are discussed.

Electronic and spin dynamics in the insulating iron pnictide NaFe$_{0.5}$Cu$_{0.5}$As