Chemistry Labs

Theoretical and computational chemistry

Hartree–Fock and post-Hartree–Fock methods

How a self-consistent single-determinant model describes electrons, what electron correlation it misses, and how post-HF methods recover it.

1 Lower secondary2 Upper secondary3 Undergraduate4 Advanced5 Research

IntuitionA mean field for many electrons

Each electron repels all the others. Hartree–Fock replaces this complicated instantaneous interaction by an average field generated by the other electrons, then solves for orbitals that are consistent with that field.

Explore the conceptual hierarchy: a single Hartree–Fock determinant, excitations into virtual orbitals, and richer post-HF descriptions that recover electron correlation.

SchoolThe determinant and exchange

Definition: Slater determinant

Antisymmetrizing a product of spin-orbitals with a Slater determinant ensures that swapping two electrons changes the sign of the wavefunction, as required for fermions.

EHF=Σihii+1/2Σij(Jij−Kij)E_HF = Σ_i h_ii + 1/2 Σ_ij (J_ij − K_ij)

The one-electron integrals h include kinetic energy and attraction to nuclei. Coulomb terms J describe classical repulsion; exchange K arises from antisymmetry and has no classical analogue.

Example: A closed-shell electron count

A minimal closed-shell Hartree–Fock calculation uses N/2 doubly occupied spatial orbitals for N electrons. How many occupied spatial orbitals are present for a ten-electron system?

Solution

There are 10/2 = 5 doubly occupied spatial orbitals (equivalently ten occupied spin-orbitals).

UndergraduateSelf-consistency and basis sets

F[P]C=SCεF[P] C = S C ε

Definition: Self-consistent-field cycle

Guess a density matrix P, build the Fock matrix F[P], solve the generalized eigenproblem, occupy orbitals, and rebuild P. Iterate until energy and density changes meet convergence criteria.

MethodMain idea
MP2Second-order perturbative correlation
Configuration interactionLinear combination of excited determinants
Coupled clusterExponential excitation operator

AdvancedCorrelation, costs, and limits

Hartree–Fock captures exchange exactly within a single determinant but omits dynamical correlation: electrons avoid one another beyond the average field. The correlation energy is conventionally E_corr = E_exact − E_HF (for the same nonrelativistic Hamiltonian and complete basis).

ResearchFrontier: accurate correlation

References