On the equation of motion and the constraining field in ab-initio spin dynamics.

It is generally accepted that the effective magnetic field acting on a magnetic moment is given by the gradient of the energy with respect to the magnetization. However, in ab-initio spin dynamics within the adiabatic approximation the effective field is also known to be exactly the negative of the constraining field, which acts as a Lagrange multiplier to stabilize an out-of-equilibrium, non-collinear magnetic configuration. We show that for a physical Hamiltonian both these fields are exactly equivalent, while there can be a finite difference for mean-field Hamiltonians and also for density-functional theory (DFT) calculations. This inequality of the constraining field and the energy gradient is highly relevant for both ab-initio spin dynamics and the ab-initio calculation of exchange constants and effective magnetic Hamiltonians. We argue that the effective magnetic field and exchange constants have highest accuracy in DFT when calculated from the energy gradient and not from the constraining field.