  We present a first-principles approach for the computation of the magnetic
Gibbs free energy of materials using magnetically constrained supercell
calculations. Our approach is based on an adiabatic approximation of slowly
varying local moment orientations, the so-called finite-temperature disordered
local moment picture. It describes magnetic phase transitions and how
electronic and/or magnetostructural mechanisms generate a discontinuous
(first-order) character. We demonstrate that the statistical mechanics of the
local moment orientations can be described by an affordable number of supercell
calculations containing noncollinear magnetic configurations. The applicability
of our approach is illustrated by firstly studying the ferromagnetic state in
bcc Fe. We then investigate the temperature-dependent properties of a
triangular antiferromagnetic state stabilizing in two antiperovskite systems
Mn$_3$AN (A = Ga, Ni). Our calculations provide the negative volume expansion
of these materials as well as the ab initio origin of the discontinuous
character of the phase transitions, electronic and/or magnetostructural, in
good agreement with experiment.
