Relationship between constraining fields and effective fields in constrained density functional theory calculations for spin dynamics

For density functional theory (DFT) based adiabatic spin dynamics, we investigate the relationship between the effective fields driving the dynamics of the atomic magnetic moments and the constraining fields used to enforce a specific noncollinear magnetic configuration. We revisit a derivation and a numerical comparison by Streib suggesting that the effective field from the energy gradient with respect to the magnetic moment direction and the constraining fields differ for systems with mean-field terms in the Hamiltonian, which they extended to include DFT. We find that a derivation focused on the DFT case clarifies what quantities to extract from the computational software to best represent the two fields. In particular, the definition of the magnetic moment magnitude must be consistent when making the comparison. This reestablishes the relationship of the two fields as having equal amplitude but opposite directions, in agreement with several earlier works. We also show that the constraining fields can be rescaled to reflect the correct field acting on the full, physically meaningful magnetic moment, rather than on the reduced moment used for numerical stability. Our numerical investigation is extended beyond the original case of an iron dimer to supercells of body-centered cubic iron in both ferromagnetic and disordered magnetic backgrounds, which further supports our conclusions.