A Real-Space Density-Functional Theory Program for the Ground-State Energy of the Helium Atom: Staged Construction, Validation, and the Self-Interaction Error

This paper documents a self-contained density-functional theory (DFT) code, built from scratch, that computes the ground-state energy of the helium atom taking nothing as input but the nuclear charge. The program is constructed in stages, each adding one piece of physics: a hydrogen-like radial Schrödinger solver (Numerov integration with a shooting/bisection eigenvalue search), the Hartree potential from the radial Poisson equation, the local-density-approximation (LDA) exchange of Slater, and the Perdew–Zunger parametrisation of the Ceperley–Alder correlation. Each stage is validated against a known analytic or reference limit and locked in by an automated test. The converged results are E = −2.8616 Ha for the self-interaction-corrected Hartree level (which, for a two-electron singlet, coincides with Hartree–Fock), E = −2.7236 Ha after adding Slater exchange, and E = −2.8342 Ha for the full LDA, compared with the Hartree–Fock limit −2.8617 Ha and the exact value −2.9037 Ha. The discussion shows that self-interaction-corrected Hartree is identical to restricted Hartree–Fock for a two-electron singlet, that the LDA total energy is less accurate than Hartree–Fock for so small a system because of its self-interaction error, and that the LDA effective potential has the wrong long-range asymptotics. Source code, test suite, and figure scripts are released for replication.