  Quantum-chemical calculations aimed at deriving magnetic coupling constants
in exchange-coupled spin clusters commonly utilize a broken-symmetry (BS)
approach. This involves calculating several distinct collinear spin
configurations, predominantly by density-functional theory (DFT). The energies
of these configurations are interpreted in terms of the Heisenberg model to
determine coupling constants for spin pairs. However, this energy-based
procedure has inherent limitations, primarily in its inability to provide
information on isotropic spin interactions beyond those included in the
Heisenberg model. Biquadratic exchange or multi-center terms, for example, are
usually inaccessible and hence assumed to be negligible. The present work
introduces a novel approach employing BS mean-field solutions, specifically
Hartree-Fock wave functions, for the construction of effective spin
Hamiltonians. This expanded method facilitates the extraction of a broader
range of coupling parameters by considering not only the energies, but also
Hamiltonian and overlap elements between different BS states. We demonstrate
how comprehensive s = 1/2 Hamiltonians, including multi-center terms, can be
straightforwardly constructed from a complete set of BS solutions. The approach
is exemplified for small clusters within the context of the half-filled
single-band Hubbard model. This allows to contrast the current strategy against
exact results, thereby offering an enriched understanding of the
spin-Hamiltonian construction from BS solutions.
