The unique and unanticipated properties of multiple principal component alloys have reinvigorated the field of alloy design and drawn strong interest across scientific disciplines. The vast compositional parameter space makes these alloys a unique area of exploration by means of computational design. However, as of now a method to compute efficiently, yet with high accuracy the thermodynamic properties of such alloys has been missing. One of the underlying reasons is the lack of accurate and efficient approaches to compute vibrational free energies—including anharmonicity—for these chemically complex multicomponent alloys. In this work, a density-functional-theory based approach to overcome this issue is developed based on a combination of thermodynamic integration and a machine-learning potential. We demonstrate the performance of the approach by computing the anharmonic free energy of the prototypical five-component VNbMoTaW refractory high entropy alloy.

Free energies of bulk materials are nowadays routinely computed by density functional theory. In particular for metals, electronic excitations can significantly contribute to the free energy. For an ideal static lattice, this contribution can be obtained at low computational cost, e.g., from the electronic density of states derived at T=0 K or by utilizing the Sommerfeld approximation. The error introduced by these approximations at elevated temperatures is rarely known. The error arising from the ideal lattice approximation is likewise unexplored but computationally much more challenging to overcome. In order to shed light on these issues we have computed the electronic free energies for all 3d,4d, and 5d transition elements on the ideal lattices of the bcc, fcc, and hcp structures using finite-temperature density-functional theory. For a subset of elements we have explored the impact of explicit thermal vibrations on the electronic free energies by using ab initio molecular dynamics simulations. We provide an analysis of the observed chemical trends in terms of the electronic density of states and the canonical d band model and quantify the errors in the approximate methods. The electronic contribution to the heat capacities and the corresponding errors due to the different approximations are studied as well.