  We present a semi-empirical effective Hamiltonian to capture effects of
disorder associated with Ba and Sr cations occupying $A$ sites in
(Ba$_{x}$Sr$_{1-x}$)TiO$_3$ on its ferroelectric phase transition. Averaging
between the parameters of first-principles effective Hamiltonians of end
members BaTiO$_3$ and SrTiO$_3$, we include a term with an empirical parameter
to capture the local polarization and strains arising from the difference
between ionic radii of Ba and Sr. Using mixed-space molecular dynamics of the
effective Hamiltonian, we determine $T$-dependent ferroelectric phase
transitions in (Ba$_{x}$Sr$_{1-x}$)TiO$_3$ which are in good agreement with
experiment. Our scheme of determination of semi-empirical parameters in
effective Hamiltonian should be applicable to other perovskite-type
ferroelectric solid solutions.
