In alloys cluster expansions (CE) are increasingly used to combine first-principles electronic-structure calculations and Monte Carlo methods to predict thermodynamic properties. As a basis-set expansion in terms of lattice geometrical clusters and effective cluster interactions, the CE is exact if infinite, but is tractable only if truncated. Yet until now a truncation procedure was not well defined and did not guarantee a reliable truncated CE. We present an optimal truncation procedure for CE basis sets that provides reliable thermodynamics. We then exemplify its importance in ${\mathrm{Ni}}_3\mathrm{V}$, where the CE has failed unpredictably, and now show agreement to a range of measured values, predict new low-energy structures, and explain the cause of previous failures.

The nudged-elastic band (NEB) method is modified with concomitant two climbing images (C2-NEB) to find a transition state (TS) in complex energy landscapes, such as those with a serpentine minimal energy path (MEP). If a single climbing image (C1-NEB) successfully finds the TS, then C2-NEB finds it too. However, improved stability of C2-NEB makes it suitable for more complex cases, where C1-NEB misses the TS because the MEP and NEB directions near the saddle point are different. Generally, C2-NEB not only finds the TS, but also guarantees, by construction, that the climbing images approach it from the opposite sides along the MEP. In addition, C2-NEB provides an accuracy estimate from the three images: the highest-energy one and its climbing neighbors. C2-NEB is suitable for fixed-cell NEB and the generalized solid-state NEB.

We calculate the transition temperature versus concentration ($T_c$ vs $c$) phase diagrams of several phase-segregating alloys [fcc Ca-Sr, Au-Pt, and Rh-(Pd,Cu,Ag,Au)] using a multiscale method combining first-principles calculations and Monte Carlo via the cluster expansion (CE). We study Pd-Rh, with its well-known high-$T$ miscibility gap, to verify the method’s reliability. We predict that Ca-Sr segregates at low temperatures. We then show that a rapid estimate of $T_c$ is obtained from enthalpies analytically derived from a CE, and, using thermodynamic integration, we determine under what circumstances this mean-field estimate is accurate compared to Monte Carlo results. Also, we discuss how an electronegativity difference of the alloying elements quickly assess when vibrational entropy effects should be included in the estimate of $T_c$.

This review of the current state of magnetocalorics is focused on materials exhibiting a giant magnetocaloric response near room temperature. To be economically viable for industrial applications and mass production, materials should have desired useful properties at a reasonable cost and should be safe for humans and the environment during manufacturing, handling, operational use, and after disposal. The discovery of novel materials is followed by a gradual improvement of properties by compositional adjustment and thermal or mechanical treatment. Consequently, with time, good materials become inferior to the best. There are several known classes of inexpensive materials with a giant magnetocaloric effect, and the search continues.