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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.

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