SCF: chasing your own tail

There is a circularity at the heart of Hartree-Fock: the Fock matrix is built from the electron density, but the density comes from diagonalizing the Fock matrix. F depends on its own solution.

So we iterate. Guess a density, build F, diagonalize for orbitals, rebuild the density, repeat until it stops changing — a fixed-point iteration P = F(P). This loop is the Self-Consistent Field (SCF) method; from a core-Hamiltonian guess it usually converges in a handful to a few dozen cycles.

Now run the real thing: an actual SCF, compiled from the Knot DSL to a 47 KB WebAssembly module — it converges live in your browser to the textbook −1.11671 hartree.