Hartree–Fock Approximation¶

The Slater Determinant introduced in the previous chapter provides a mathematically correct many-electron wavefunction that satisfies the Pauli Exclusion Principle and the antisymmetry requirement for identical electrons. However, the molecular orbitals contained within the determinant are still unknown.

The purpose of Hartree–Fock (HF) theory is to determine the set of molecular orbitals that produces the lowest possible electronic energy for a molecule.

Rather than attempting to solve the many-electron Schrödinger equation exactly, Hartree–Fock replaces the complex many-electron problem with a set of coupled one-electron equations that are solved iteratively until self-consistency is achieved.

Hartree–Fock therefore represents the first practical electronic structure method capable of treating molecules containing many electrons.

The Variational Principle¶

Hartree–Fock theory is based on one of the most fundamental principles of quantum mechanics: the Variational Principle.

It states that

The expectation value of the Hamiltonian calculated using any approximate wavefunction is always greater than or equal to the exact ground-state energy.

Mathematically,

\[ \boxed{ E[\Psi] = \frac{\langle \Psi|\hat H|\Psi\rangle} {\langle\Psi|\Psi\rangle} \ge E_0 } \]

where

\(E[\Psi]\) is the energy of the trial wavefunction,
\(E_0\) is the exact ground-state energy.

The goal is therefore to adjust the molecular orbitals until the calculated energy becomes as small as possible.

The Hartree–Fock Wavefunction¶

Hartree–Fock assumes that the electronic wavefunction can be represented by a single Slater Determinant,

\[ \Psi_{HF} = \frac1{\sqrt{N!}} \begin{vmatrix} \chi_1(1)&\chi_2(1)&\cdots&\chi_N(1)\\ \chi_1(2)&\chi_2(2)&\cdots&\chi_N(2)\\ \vdots&\vdots&\ddots&\vdots\\ \chi_1(N)&\chi_2(N)&\cdots&\chi_N(N) \end{vmatrix} \]

where

\(\chi_i\) are spin orbitals,
each electron occupies one spin orbital,
the determinant automatically satisfies antisymmetry.

The unknown quantities are the orbitals themselves.

The Hartree–Fock Approximation¶

Instead of allowing every electron to interact instantaneously with every other electron, Hartree–Fock assumes that

Each electron moves in the average electrostatic field generated by all other electrons.

This approximation transforms the many-electron problem into a collection of coupled one-electron equations.

Graphically,

Exact Problem

Electron 1 ↔ Electron 2 ↔ Electron 3 ↔ ...

        ↓

Hartree–Fock

Electron 1
      │
Average Field
      │
Electron 2
      │
Average Field
      │
Electron 3

Although simplified, this approximation captures much of the essential physics while remaining computationally tractable.

The Fock Operator¶

Applying the variational principle to the Slater Determinant leads to the Hartree–Fock equations,

\[ \boxed{ \hat F \chi_i = \varepsilon_i \chi_i } \]

where

\(\hat F\) is the Fock operator,
\(\chi_i\) is a molecular spin orbital,
\(\varepsilon_i\) is the corresponding orbital energy.

The Fock operator plays the same role in Hartree–Fock theory that the Hamiltonian plays in the exact Schrödinger equation.

Components of the Fock Operator¶

The Fock operator is written as

\[ \boxed{ \hat F = \hat h + \sum_j (\hat J_j-\hat K_j) } \]

where

Symbol	Meaning
\(\hat h\)	Core Hamiltonian
\(\hat J\)	Coulomb operator
\(\hat K\)	Exchange operator

Each term has a clear physical interpretation.

Core Hamiltonian¶

The core Hamiltonian contains the one-electron contributions,

\[ \hat h = \hat T_e + \hat V_{en} \]

which include

electron kinetic energy,
electron–nucleus attraction.

These terms would exist even if only one electron were present.

Coulomb Operator¶

The Coulomb operator,

\[ \hat J, \]

represents the average electrostatic repulsion between electrons.

It describes the classical interaction of one electron with the average charge distribution of all the others.

Exchange Operator¶

The exchange operator,

\[ \hat K, \]

has no classical analogue.

It arises solely because electrons are described by a Slater Determinant and therefore obey the antisymmetry requirement.

The exchange operator

lowers the total electronic energy,
enforces the Pauli Exclusion Principle,
correctly treats electrons having the same spin.

Hartree–Fock Equations¶

The Hartree–Fock equations are

\[ \boxed{ \hat F \chi_i = \varepsilon_i \chi_i } \]

Notice that these equations resemble the Schrödinger equation,

\[ \hat H\Psi = E\Psi. \]

However,

the Hamiltonian has been replaced by the Fock operator,
the many-electron wavefunction has been replaced by one-electron spin orbitals,
the orbital energies are solved instead of the total molecular energy directly.

Why the Equations Must Be Solved Iteratively¶

A complication immediately appears.

The Fock operator depends on the electron density,

but the electron density depends on the molecular orbitals,

and the molecular orbitals are obtained by solving the Fock equations.

This creates a circular dependency,

Orbitals
    │
    ▼
Electron Density
    │
    ▼
Fock Operator
    │
    ▼
New Orbitals

The equations therefore cannot be solved in a single step.

Instead, they are solved iteratively using the Self-Consistent Field (SCF) procedure.

Hartree–Fock Energy¶

After convergence, the Hartree–Fock electronic energy is

\[ E_{HF} = \sum_i \langle i|\hat h|i\rangle + \frac12 \sum_{ij} (J_{ij}-K_{ij}) + V_{nn} \]

where

the first term is the one-electron energy,
the second term accounts for electron–electron interactions,
the final term is the constant nucleus–nucleus repulsion.

Limitations of Hartree–Fock¶

Although Hartree–Fock represents a major improvement over the Hartree Product, it still has important limitations.

Most notably, it neglects dynamic electron correlation.

As a result,

reaction energies may be inaccurate,
weak intermolecular interactions are poorly described,
transition-state energies often contain significant errors.

More sophisticated methods such as MP2, Coupled Cluster, CASSCF, and Density Functional Theory recover much of this missing correlation energy.

Relationship to Density Functional Theory¶

Although this documentation focuses primarily on B3LYP Density Functional Theory, Hartree–Fock remains essential because the mathematical framework is nearly identical.

Both methods

use molecular orbitals,
expand orbitals in basis functions,
solve matrix eigenvalue equations,
employ SCF iterations,
produce converged electronic densities.

The principal difference is that DFT replaces the Hartree–Fock exchange term with an exchange–correlation functional.

Summary¶

Hartree–Fock theory determines the molecular orbitals by minimizing the total electronic energy using the Variational Principle. The resulting Hartree–Fock equations replace the many-electron Schrödinger equation with a set of coupled one-electron equations involving the Fock operator. Because the Fock operator depends on the electron density, the equations are solved iteratively through the Self-Consistent Field procedure. Although Hartree–Fock neglects dynamic electron correlation, it provides the mathematical foundation upon which modern Density Functional Theory is built.

Next Section¶

The next chapter introduces the Basis Set Expansion (LCAO Approximation), where molecular orbitals are expressed as linear combinations of atomic basis functions. This converts the Hartree–Fock equations from differential equations into matrix equations that can be solved efficiently on a computer.