TT-SCHA for Thermodynamics of Binary Alloys

The free energy of a crystal is an integral over the positions of all its atoms, in 3N dimensions. We evaluate it without random sampling for the ordered alloys B2 NiAl and gamma-TiAl with embedded-atom potentials, using a self-consistent harmonic approximation (SCHA) corrected to third order in temperature, calculated via thermal averages of tensor-train (TT) integrals built from one atom and its neighbors, which we call TT-SCHA. The result is for an infinite crystal without sampling error; its numerical error is set by the starting pivots of the interpolation (up to 9 MPa in the pressure, reduced by averaging three runs; SM Table S26). Against the molecular dynamics of 65,536 atoms (Frenkel-Ladd free energy at 600 K, integrated energy above), TT-SCHA free energy agrees to 0.13 meV per atom at 600-1000 K, using 420-500 times fewer core-seconds for one run. At zero pressure, results agree with published calculations that use the same embedded-atom potential: the published elastic constants of gamma-TiAl (to 1 GPa, by a static lattice calculation) and its thermal expansion from Monte Carlo sampling (to 0.012 percentage points).