Efficient cluster expansion for substitutional systems

Abstract

We demonstrate a cluster expansion technique that is capable of accurately predicting formation energies in binary substitutional systems "ven for those with large atomic relaxations. Conventional cluster expansions converge rapidly only in the absence of atomic relaxations, and they fail for 1ongperiod lattice-mismatched superlattices. When combined with first-principles total-energy methods, our method allows for very fast calculations for structures containing hundreds or thousands of atoms. The convergence and effectiveness of the cluster expansion are enhanced in two ways. First, the expansion is recast into reciprocal space, which allows for the inclusion of all important pair interactions. Second, a reciprocal-space formulation for elastic strain energy is introduced, allowing accurate predictions for both long-and short-period superlattices. We illustrate the power of the method by performing a cluster expansion that requires total-energy calculations for only 12 simple input structures, with at most eight atoms per unit cell. We then correctly predict the formation energies of relaxed long-period superlattices, low-symmetry intermixed superlattices, structures with varied compositions, substitutional impurities, and a~1000 atom/cell simulation of the random alloy. For a lattice with N sites, the problem of finding the energies of the 2+ possible configurations can be exactly mapped into a Ising Hamiltonian: E(o) = Jp+ ) J;S,(o) + ) J;,S;(o)S~(cr) j&i + ) J, , I,S,(rr)S, (o)SI,(o) + for configuration o, where the J's are "interaction energies, " and the first summation is over all sites in the lattice, the second over all pairs of sites, the third over all triplets, and so on. The primary advantage of thecluster expansion is that the interaction energies J are the same for all configurations o. Thus, once the J's are known, the energy E(o) of any configuration can be calculated almost immediately by simply calculating the spin products and summing Eq. (1). Because the Ising representation of the energy can be calculated rapidly, and is also a linear function of the spin products, one can readily (i) apply linear programming techniques to find ground-state structures P, (ii) use statistical-mechanics techniques (Monte Carlo and cluster variation methods) to calculate phase diagrams, ii i~a nd (iii) calculate the energy of an arbitrarily complex configuration.

Vibrational thermodynamics: coupling of chemical order and size effects

“…When relaxations are included the ECI become much longer range (the largest ECI is in fact in the fifth-neighbour shell). The introduction of longer-range interactions when relaxations are included has also been found in cluster expansions of the total energy [4,[48][49][50][51]. In addition, in the relaxed case the RMS error is almost constant after the fifthneighbour shell, even though it is about five times as large as the corresponding error in the unrelaxed case.…”

Stability of Sb-Te layered structures: First-principles study

“…with the case of SnO 2−x (Ref. 31)] or using more elaborate methods to account for the structural relaxations, 32,33 we show that fast convergence can be achieved by considering a ternary CE. Here we account for the energetic differences between structures with an odd or even sequence of Sb layers (due to Peierls relaxation) by switching the occupation variable for Sb from 0 to −1 if the Sb layer is a part of a sequence of an even number Sb layers, while the occupation variable for Te remains 1.…”

Cation and vacancy ordering inLixCoO2

“…29) allows one to retain more figures in the expansion than total energies, and also requires the pair interactions to be as convergent as possible in real space. Although more sophisticated versions of the cluster expansion approach 29 are available when one requires extreme accuracy and has access to a large database of structural energies, we use the simple real-space expansion of Eq.…”