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We present a semi-empirical effective Hamiltonian to capture effects ofdisorder associated with Ba and Sr cations occupying $A$ sites in(Ba$_{x}$Sr$_{1-x}$)TiO$_3$ on its ferroelectric phase transition. Averagingbetween the parameters of first-principles effective Hamiltonians of endmembers BaTiO$_3$ and SrTiO$_3$, we include a term with an empirical parameterto capture the local polarization and strains arising from the differencebetween ionic radii of Ba and Sr. Using mixed-space molecular dynamics of theeffective Hamiltonian, we determine $T$-dependent ferroelectric phasetransitions in (Ba$_{x}$Sr$_{1-x}$)TiO$_3$ which are in good agreement withexperiment. Our scheme of determination of semi-empirical parameters ineffective Hamiltonian should be applicable to other perovskite-typeferroelectric solid solutions.

Molecular Dynamics Simulations of Chemically Disordered Ferroelectric (Ba,Sr)TiO$_3$ with a Semi-Empirical Effective Hamiltonian