The Hartree-Fock self-consistent-field procedure is a fixed-point iteration on the one-particle density.

The fixed-point map

Given a density $\rho$, build the Fock operator $F[\rho]$, diagonalize to obtain orbitals ${\phi_i}$, then construct the new density

$$\rho’(\mathbf{r}) = \sum_{i \in \text{occ}} |\phi_i(\mathbf{r})|^2.$$

The map $T: \rho \mapsto \rho’$ is the SCF operator. A converged solution is a $\rho^$ with $T(\rho^) = \rho^*$.

Why this matters

Recognizing the SCF as a fixed-point iteration imports the entire toolkit of fixed-point acceleration methods:

• DIIS (Pulay) — extrapolate from the residual history.
• Anderson acceleration — multi-step Krylov-style averaging.
• Damping — under-relaxation to control oscillation.
• Level shifting — change the spectrum of $T’$ near the fixed point.

Each is a generic technique that does not need to know it’s solving an electronic-structure problem; it just needs to be told “here is a fixed-point map, here is the residual, accelerate.”