selfconsistentfield

Self-consistent field (SCF) refers to a class of computational methods used to determine the electronic structure of many-electron systems by solving a one-electron problem in a potential that depends on the electrons themselves. In an SCF approach, the full many-body problem is reduced to a set of equations in which the effective Hamiltonian (such as the Fock operator in Hartree-Fock theory or the Kohn-Sham operator in density functional theory) is a functional of the electron density. The solution is sought iteratively: an initial guess for the density or orbitals yields a new one-electron Hamiltonian, which in turn produces new orbitals and density, and the cycle repeats until convergence is achieved.

SCF methods are foundational in quantum chemistry and solid-state physics. In Hartree-Fock theory, the Fock operator depends on the occupied orbitals, and the SCF cycle solves for a self-consistent set of molecular orbitals. In Kohn-Sham density functional theory, the effective potential includes an exchange-correlation term, and the self-consistent solution yields the ground-state electron density and total energy. Post-Hartree-Fock methods may use SCF loops as part of their iterative frameworks, though they incorporate additional correlation treatments beyond mean field.

A typical SCF procedure includes: selecting an initial density or orbital guess; constructing the effective one-electron Hamiltonian; solving the eigenproblem to obtain new orbitals; updating the density; checking convergence against predefined criteria; and applying mixing or extrapolation schemes (such as DIIS) to stabilize and accelerate convergence. Convergence can be challenging for systems with near-degenerate states or metals, requiring techniques like damping, level shifting, orbital or density mixing, and orbital preconditioning.

SCF results depend on the chosen theoretical framework and approximations and are most reliable when the mean-field description appropriately captures the dominant physics of the system.