Equation of motion and the constraining field in ab initio spin dynamics

In $a\phantom{\rule{0}{0ex}}b$ $i\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}o$ spin dynamics, the effective field acting on a magnetic moment can either be calculated from the energy gradient or from the negative of the constraining field that stabilizes an out-of-equilibrium noncollinear magnetic configuration. Here, the authors show that the constraining field in DFT calculations is not exactly the negative of the effective magnetic field obtained from the energy gradient. They provide a theoretical explanation of this surprising result and discuss the implications for $a\phantom{\rule{0}{0ex}}b$ $i\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}o$ spin dynamics.