  Calculations of stable and metastable magnetic states as well as minimum
energy paths for transitions between states are carried out using a
noncollinear extension of the multiple-impurity Alexander-Anderson model and a
magnetic force theorem which is derived and used to evaluate the total energy
gradient with respect to orientation of magnetic moments -- an important tool
for efficient navigation on the energy surface. By using this force theorem,
the search for stable and metastable magnetic states as well as minimum energy
paths revealing the mechanism and activation energy of transitions can be
carried out efficiently. For Fe monolayer on W(110) surface, the model gives
magnetic moment as well as exchange coupling between nearest and next-nearest
neighbors that are in good agreement with previous density functional theory
calculations. When applied to nanoscale Fe islands on this surface, the
magnetic moment is predicted to be 10\% larger for atoms at the island rim,
explaining in part an experimentally observed trend in the energy barrier for
magnetization reversal in small islands. Surprisingly, the magnetic moment of
the atoms does not change much along the minimum energy path for the
transitions, which for islands containing more than 15 atom rows along either
$[001]$ or $[1\bar{1}0]$ directions involves the formation of a thin, temporary
domain wall. A noncollinear magnetic state is identified in a $7\times 7$
atomic row Fe island where the magnetic moments are arranged in an antivortex
configuration with the central ones pointing out of the $(110)$ plane. This
illustrates how the model can describe complicated exchange interactions even
though it contains only a few parameters. The minimum energy path between this
antivortex state and the collinear ground state is also calculated and the
thermal stability of the antivortex state estimated.
