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

TL;DR

This paper clarifies the relationship between the effective magnetic field and the constraining field in ab initio spin dynamics, highlighting differences in mean-field Hamiltonians and implications for accurate calculations in density-functional theory.

Contribution

It demonstrates the exact equivalence of these fields for non-mean-field Hamiltonians and discusses the implications for DFT-based magnetic property calculations.

Findings

Fields are equivalent for non-mean-field Hamiltonians

Finite difference exists for mean-field Hamiltonians

Energy gradient provides more accurate magnetic parameters in DFT

Abstract

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 Hamiltonians without mean-field parameters both of these fields are exactly equivalent, while there can be a finite difference for mean-field Hamiltonians. For density-functional theory (DFT) calculations the constraining field obtained from the auxiliary Kohn-Sham Hamiltonian is not exactly equivalent to the DFT energy gradient. This inequality 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 the highest accuracy in DFT when calculated from the energy gradient and not from the constraining field.