Predicting complex ground state structures from first principles: genetic algorithm for finding accurate coarse-grained Hamiltonians

First-principles quantum-mechanical (QM) calculations allow to evaluate many interesting properties of a \emph{given} nano-scale configuration of atoms with high accuracy. However, \emph{predicting} stable structures or finite-$T$ thermodynamic configurational averages with QM accuracy remains a challenge: even for a binary solid with $N$ atoms per unit cell, $2^N$ distinct configurations must be evaluated. Such large numbers of calculations can be made affordable by mapping the QM Hamiltonian onto a computationally simpler ``coarse-grained'' Hamiltonian. The ability to predict ground state structures then depends on the shape of the coarse-grained Hamiltonian, but this shape is not a priori clear. For instance, a few simple assumed generic interactions will allow only a few simple ground states, but any more complex structures will be missed. For the generalized Ising model for binary alloys (cluster expansion), I show how a genetic algorithm$^{1,2}$ can identify the leading interactions which characterize a given system. I illustrate the method for the bcc binary alloys of Nb, Ta, Mo, W. A rich spectrum of ground state structures is found, including both well-known and unsuspected complex structures, far beyond what is envisioned from ``usual-suspect'' structure listings or from simple generic interactions. At the same time, order-disorder temperatures are significantly lower than those from simple intuition-based interactions, in agreement with experimental observations for these systems. This work was done at the National Renewable Energy Laboratory, supported by DOE-SC-BES, in collaboration with A. Zunger and G. Hart. \\ $^1$G. Hart, V. Blum, M. Walorski and A. Zunger, Nature Materials \textbf{4}, 391 (2005); $^2$V. Blum, G. Hart, M. Walorski and A. Zunger, Phys. Rev. B \textbf{72}, 165113 (2005).