It applies to closed-shell molecules or atoms where all molecular orbitals or atomic orbitals, respectively, are doubly occupied.
The method was developed independently by Clemens C. J. Roothaan and George G. Hall in 1951, and is thus sometimes called the Roothaan-Hall equations.
[1][2][3] The Roothaan equations can be written in a form resembling generalized eigenvalue problem, although they are not a standard eigenvalue problem because they are nonlinear: where F is the Fock matrix (which depends on the coefficients C due to electron-electron interactions), C is a matrix of coefficients, S is the overlap matrix of the basis functions, and
is the (diagonal, by convention) matrix of orbital energies.
These equations are essentially a special case of a Galerkin method applied to the Hartree–Fock equation using a particular basis set.