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Lu Jeu Sham and Walter Kohn developed self-consistent equations that make density functional theory computationally useful. The Kohn-Sham equations introduce an auxiliary system of noninteracting electrons with the same ground-state electron density as the interacting system. An effective potential includes the external potential, electrostatic repulsion and an exchange-correlation functional. Solving the equations and updating the electron density iteratively gives a self-consistent state. The difficult many-electron effects remain in the exchange-correlation functional, so practical calculations still need an approximation to it.^[ref]

A theoretical condensed matter physicist who, with Walter Kohn, established a foundation of density functional theory in 1964. Their variational principle expresses the ground-state energy of interacting electrons through the electron density, making a three-dimensional quantity central to a many-particle problem.^[ref]

His 1967 work also ruled out the relevant finite-temperature long-range order in one- and two-dimensional bosonic and fermionic systems under the theorem's assumptions. This helps explain why dimensionality matters when discussing Bose-Einstein condensation.^[ref]

Walter Kohn and Pierre Hohenberg established a foundation for density functional theory: under their ground-state assumptions, the electron density determines the external scalar potential up to an additive constant. They formulated the ground-state energy as a universal density functional plus the external-potential contribution, with a variational principle selecting the physical density. This replaces the full many-electron wave function as the basic variable by a function of only three spatial coordinates. The theorem establishes the framework, rather than providing an explicit exact functional for practical computation.^[ref]