Roothaan-Hall: FC = SCε

The Fock operator lives in an infinite-dimensional function space. To compute with it, project it onto the finite basis: each molecular orbital becomes a vector of coefficients C, and the operator equation F̂φ = εφ becomes a matrix equation.

Because Gaussian basis functions overlap, you get a generalized eigenvalue problem FC = SCε, with S the overlap matrix. One diagonalization returns every orbital at once: the lowest are occupied, the rest are the virtual (empty) orbitals that correlated methods later excite into.