  We consider the mapping of tight-binding electronic structure theory to a
local spin Hamiltonian, based on the adiabatic approximation for spin degrees
of freedom in itinerant-electron systems. Local spin Hamiltonians are
introduced in order to describe the energy landscape of small magnetic
fluctuations, locally around a given spin configuration. They are designed for
linear response near a given magnetic state and in general insufficient to
capture arbitrarily strong deviations of spin configurations from the
equilibrium. In order to achieve this mapping, we include a linear term in the
local spin Hamiltonian that, together with the usual bilinear exchange tensor,
produces an improved accuracy of effective magnetic Weiss fields for
non-collinear states. We also provide examples from tight-binding electronic
structure theory, where our implementation of the calculation of exchange
constants is based on constraining fields that stabilize an out-of-equilibrium
spin configuration. We check our formalism by means of numerical calculations
for iron dimers and chains.
