By 1964, the methods inherited from Herring, Slater, and their successors had made it possible to calculate the energy bands of crystals with a single electron moving in an effective potential, but the underlying problem — precisely solving a system of many mutually repelling electrons — still seemed, in principle, unsolvable without handling the full wave function of all electrons at once, a mathematical object whose complexity grows explosively with the number of particles. Pierre Hohenberg, then a visiting researcher at the École Normale Supérieure in Paris, and Walter Kohn, a regular Bell Labs consultant where he had worked closely with Conyers Herring and Joaquin Luttinger, proved in 1964 a result that radically changed the terms of the problem: they showed that the ground-state energy of a system of interacting electrons is uniquely and exactly determined by the system's electron density — a much simpler function, depending only on three spatial coordinates, in contrast to the full wave function, which depends on the coordinates of all electrons simultaneously. The Hohenberg-Kohn theorem proved that this universal density functional exists, but did not say how to calculate it in practice. That piece arrived the following year, when Kohn, back at the University of California, San Diego, formulated together with his new postdoctoral researcher Lu Jeu Sham a practical procedure: reformulating the problem as a set of self-consistent single-electron equations — formally analogous to the Hartree-Fock equations, but exact in principle — in which all the complexity of electron-electron interactions is encapsulated in a single exchange-correlation term. Although that exact term remains unknown and must be approximated in practice, the Kohn-Sham equations turned density functional theory (DFT) into the most widely used electronic-structure calculation method in contemporary physics, chemistry, and materials science, used today together with the pseudopotential and augmented-wave methods inherited from Herring and Slater to design semiconductors, catalysts, drugs, and batteries via computational simulation before synthesizing them in a laboratory.