Hartree-Fock and Self-Consistent Field Theory

Hartree-Fock theory approximates a many-electron wavefunction as a single antisymmetric product of orbitals and solves for those orbitals self-consistently in the average field of all the others.

Definition

An approximation in which each electron is treated as moving in the mean field produced by the nuclei and all other electrons, giving the best single-determinant wavefunction in the variational sense.

Scope

Covers the variational mean-field treatment of electronic structure: the Slater determinant ansatz, the Fock operator, the Roothaan-Hall matrix equations for closed-shell systems, restricted and unrestricted formulations, and the iterative self-consistent-field procedure. Provides the reference wavefunction for most correlated methods.

Core questions

• How does the single-determinant ansatz incorporate the Pauli antisymmetry requirement?
• Why must the Hartree-Fock equations be solved iteratively to self-consistency?
• What is the difference between restricted and unrestricted Hartree-Fock?
• What physics is omitted by the mean-field approximation?

Key theories

Variational principle

The energy of any trial wavefunction is an upper bound to the true ground-state energy, justifying optimization of the orbitals to minimize the Hartree-Fock energy.

Roothaan-Hall equations

Recasting the integro-differential Hartree-Fock equations into a matrix eigenvalue problem over a finite basis enables practical molecular computation.

Mechanisms

The self-consistent-field cycle begins with a guess density, builds the Fock matrix, diagonalizes it to obtain new orbitals, forms a new density, and repeats until the density and energy converge.

Clinical relevance

Hartree-Fock supplies the qualitative molecular-orbital picture used throughout chemistry and serves as the starting point and reference for nearly all higher-accuracy correlated calculations.

History

Hartree introduced the self-consistent field for atoms in 1928; Fock and Slater added the antisymmetric determinantal form; Roothaan's 1951 matrix formulation made molecular Hartree-Fock calculations tractable on computers.

Key figures

• Douglas Hartree
• Vladimir Fock
• Clemens Roothaan
• John Slater

Seminal works

• roothaan1951
• szabo1996

Frequently asked questions

What is the correlation energy?

It is defined as the difference between the exact non-relativistic energy and the Hartree-Fock energy in the same basis, representing the electron correlation the mean field cannot capture.

When is unrestricted Hartree-Fock used?

For open-shell systems such as radicals, where allowing alpha and beta electrons to occupy spatially different orbitals gives a better energy, at the cost of spin contamination.