One of the outstanding advancements in electronic-structure density-functional methods is the Sankey-Niklewski (SN) approach [Sankey and Niklewski, Phys. Rev. B 40, 3979 (1989)]; a method for computing total energies and forces, within an ab initio tight-binding formalism. Over the past two decades, several improvements to the method have been proposed and utilized to calculate materials ranging from biomolecules to semiconductors. In particular, the improved method (called FIREBALL) uses separable pseudopotentials and goes beyond the minimal sp 3 basis set of the SN method, allowing for double numerical (DN) basis sets with the addition of polarization orbitals and d-orbitals to the basis set. Herein, we report a review of the method, some improved theoretical developments, and some recent application to a variety of systems.ß 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction With the increase in computational power, greater efforts have been made by the electronicstructure community to optimize the performance of quantum mechanical methods. Quantum mechanical methods have become increasingly reliable as a complementary tool to experimental research. A variety of methods exist ranging in complexity from semi-empirical methods to density-functional theory (DFT) methods to highly-accurate methods going beyond the one-electron picture. Judicious approximations enable the computational materials science community to more efficiently examine a wider range of materials questions.Otto F. Sankey was one of the early visionaries by, firstly, demonstrating that molecular-dynamics (MD) simulations can be coupled efficiently with electronic-structure methods to optimize structures and evaluate energetics of materials [1]. Secondly, his judicious approximations in the

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Structurally disordered materials continue to pose fundamental questions [1][2][3][4] , including that of how different disordered phases ("polyamorphs") can coexist and transform from one to another 5-9 . As a widely studied case, amorphous silicon (a-Si) forms a fourfold-coordinated, covalent network at ambient conditions and much higher-coordinated, metalliclike phases under pressure 10-12 . However, a detailed mechanistic understanding of the structural transitions in disordered silicon has been lacking, due to intrinsic limitations of even the most advanced experimental and computational techniques. Here, we show how atomistic machine-learning (ML) models can break through this long-standing barrier, describing liquid-amorphous and amorphous-amorphous transitions with quantum-mechanical accuracy for a system of 100,000 atoms (ten-nanometre length scale). Our simulations reveal a three-step transformation sequence for a-Si under increasing external pressure. First, polyamorphic low-and high-density amorphous (LDA and HDA) regions are found to coexist, rather than appearing sequentially. Then, we observe a structural collapse into a distinct, very-high-density amorphous (VHDA) phase. Finally, our simulations indicate the transient nature of this VHDA phase: it rapidly nucleates crystallites, ultimately leading to the formation of a poly-crystalline structure, consistent with experiments [13][14][15] but not seen in earlier simulations 11,[16][17][18] . An ML model for electronic densities of states (DOS) confirms the onset of metallicity during VHDA formation and subsequent crystallisation. These results shed new light on liquid and amorphous states of silicon, and, in a wider context, they exemplify a holistic, ML-driven approach to predictive materials mod-

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