Large-scale first-principles electronic structure methods

First-principles electronic structure methods provide a quantum-mechanically grounded route to describing microscopic matter, but their application to structurally complex and large-scale systems is constrained by global operations whose cost usually scales cubically or worse. Linear-scaling electronic-structure methods seek to overcome this bottleneck by exploiting locality, or nearsightedness, in electronic matter. Their practical performance, however, is not determined by formal scaling alone. It depends on how local representations are reconciled with global fermionic constraints, including orthogonality, idempotency, and electron-number conservation, as well as with numerical prefactors, accuracy control, and implementation details. This review uses this locality-constraint balance as a guiding principle to revisit recent advances in large-scale first-principles methods. Fragment-based schemes, direct expansion and purification approaches, local subspace projection, stochastic representations, and variational optimization methods are discussed in terms of mathematical foundations, favorable regimes, limiting cases, and open challenges. Special attention is given to stochastic methods, which replace part of the deterministic enforcement of global constraints with controlled statistical averaging and thereby provide a complementary route to ultra-large systems. Linear-scaling treatments of long-range Coulomb interactions and exact exchange are also reviewed, together with their implications for hybrid-functional and many-body calculations. The review further examines how exascale computing, artificial intelligence, and quantum computing are broadening the routes to large-scale first-principles simulations. Exascale computing provides a hardware-driven path to larger systems and reshapes the practical implementation of linear-scaling algorithms, whereas artificial intelligence and quantum computing introduce learned or hybrid paradigms that may compete with, complement, or interface with traditional linear-scaling theory. By connecting algorithmic principles, practical applicability, and emerging interfaces, this article provides a framework for assessing the current status and future development of large-scale first-principles simulations.