  A systematic method is presented for constructing effective Hamiltonians for
general phonon-related structural transitions. The key feature is the
application of group theoretical methods to identify the subspace in which the
effective Hamiltonian acts and construct for it localized basis vectors, which
are the analogue of electronic Wannier functions. The results of the symmetry
analysis for the perovskite, rocksalt, fluorite and A15 structures and the
forms of effective Hamiltonians for the ferroelectric transition in $PbTiO_3$
and $BaTiO_3$, the oxygen-octahedron rotation transition in $SrTiO_3$, the
Jahn-Teller instability in $La_{1-x}(Ca,Sr,Ba)_xMnO_3$ and the
antiferroelectric transition in $PbZrO_3$ are discussed. For the oxygen-
octahedron rotation transition in $SrTiO_3$, this method provides an
alternative to the rotational variable approach which is well behaved
throughout the Brillouin zone. The parameters appearing in the Wannier basis
vectors and in the effective Hamiltonian, given by the corresponding invariant
energy expansion, can be obtained for individual materials using first-
principles density-functional-theory total energy and linear response
techniques, or any technique that can reliably calculate force constants and
distortion energies. A practical approach to the determination of these
parameters is presented and the application to ferroelectric $PbTiO_3$
discussed.
