Reliable First-Principles Alloy Thermodynamics via Truncated Cluster Expansions

Abstract

In alloys cluster expansions (CE) are increasingly used to combine first-principles electronicstructure 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 Ni3V, 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. [3,10] ordering. As a means for multiscaling based on density-functional theory (DFT) electronic-structure energetics, the CE is a basis-set expansion in n-body clusters (associated with n Bravais lattice points) and effective cluster interactions (ECI) that specify configurational energies. Except for implicit DFT errors in the energy database, the CE is exact for an infinite basis, but impractical if not vastly truncated [11,12]. Although there are many successes, a truncated CE can and has unpredictably failed.We present a new method for an optimal truncation of the basis set that gives reliable thermodynamics. We then detail its importance in face-centered-cubic (fcc) Ni 3 V, a system with order-disorder transition from disordered A1 phase to ordered DO 22 phase at T c of 1318 K [13]. Previous CE for Ni 3 V [14,15,16] had errors of 40 − 1000% for a range of thermodynamic properties, prompting a search for missing physics [16]. We show that our new method allows more reliable predictions, including that of key low-energy configurational excitations. As a synopsis, we compare in Table I our CE results, along with the previous ones, with experimental values of T c and ∆E L12−DO22 SRO , the energy difference between DO 22 and metastable L1 2 as assessed from the short-range order (SRO) measurements [17,18]. The new CE now agrees with a range of experimentally assessed values (more below). We find that prior failure in Ni 3 V is due to inappropriate truncation of the cluster basis set and overfitting to get the ECI -underscoring again the need for careful application of basis-set methods. We have tested this new CE method on a few cubic and noncubic binaries and ternaries and found it to be especially important when multibody ECI are significant.

Compressive sensing as a paradigm for building physics models

“…Furthermore, CE is a stringent test case for compressive sensing because a significant amount of effort has been expended developing advanced model building techniques, which have been implemented in sophisticated general-purpose computer codes. [15][16][17][21][22][23][24][25] …”

A machine learning approach to graph-theoretical cluster expansions of the energy of adsorbate layers

“…Cluster expansions are, however, known to converge fast, so that the expansion can be truncated at a finite distance and at low orders of many-body patterns. [50][51][52][53] Additionally, the system of equations is likely to contain linear dependencies: for instance, there could exist a configuration whose pattern count is a linear combination of the counts of two other configurations. This may, in principle, be tackled by avoiding superposition of configurations while sampling.…”

Bayesian approach to cluster expansions

“…30 It was found that this constraint significantly improves the quality of the fit. ͑b͒ Single width.…”