# Convergence of the Roothaan-Hall equations

Suppose that we are given a time-independent quantum mechanical system whose wavefunction depends on three space coordinates. Let $$F$$ be the Fock operator of the system. Suppose also that we have a set of functions $$f_k : \mathbb{R}^3 \to \mathbb{C}$$, $$k \in \mathbb{N}$$, so that $$\mathrm{clos} \;\mathrm{span} \{f_k : k \in \mathbb{N}\} = L^2(\mathbb{R}^3) .$$ Let $$n \in \mathbb{Z}$$ and $$B_n = \{ f_k : k = 0,\ldots,n\}$$. Let $$F_n C_n = E_n S_n C_n$$ be the Roothaan-Hall equations of the system in basis set $$B_n$$. What are the necessary and sufficient conditions for the solutions of the Roothaan-Hall equations to converge to the exact solutions of the Hartree-Fock equations of the system when $$n \to \infty$$? Does $$\{ f_k : k \in \mathbb{N}\}$$ have to be a Riesz basis of $$L^2(\mathbb{R}^3)$$? We may consider only the ground state of the system.