Mag4: An Automated First-Principles Workflow to Extract Magnetic Interactions, from Pair Couplings to Four-Spin Ring Exchange

Abstract Chemists rationalize and design magnetic materials through structure-property relationships built on pairwise exchange couplings J, which energy-mapping methods now extract routinely from first principles. Yet whenever four magnetic centers close a loop, as in the CuO2 planes of the cuprates or the square nets of infinite-layer oxides, a four-spin ring (cyclic) exchange Jring can arise, reshaping magnetic ground states and corrupting the very J values used for design. What has been missing is a practical way to obtain it: a direct, local extraction of the kind the four-state method provides for J. We supply it by generalizing that method to a 16-state (24) scheme, and we automate the complete workflow in the openly available Mag4 package, a fully automatic implementation of the four-state family of methods: from a single cif file, Mag4 proposes the supercell that isolates the couplings, generates the density functional theory (DFT) inputs in the exchange-correlation flavour of the user’s choice, extracts J and Jring with the band-gap and local-moment diagnostics that certify them, and predicts the magnetic order, its propagation vector, and the critical temperature. Symmetry reduces the 16 configurations to six or eight inequivalent energies, so the cost is modest. The derivation also shows that the conventional four-state magnetic coupling, J, is itself ring-renormalized, by ∓2JringS2 with the sign set by the reference state. T-La2CuO4 confirms this quantitatively: three concordant routes agree on Jring to 0.2%, giving Jring/J1 = 0.25, and a four-state J1 quoted without naming its reference is wrong by 12% in this material. The direct 16-state extraction itself proves reference-dependent, the Néel and ferromagnetic baths bracketing the mapping value: a fourth-order fingerprint of interactions beyond the pair-plus-ring model, which additional reference baths resolve into a bare Jring and a converging tower of six- and eight-spin loop couplings. SrFeO2 (S = 2), with the same plaquette yet Jring/J = 0.006, provides the negative control: a plaquette is necessary for ring exchange, far from sufficient: a structure-property criterion that chemists can screen against when searching for materials hosting strong multi-spin magnetism.