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Hartree–Fock Approximation

In the previous chapter, we introduced the Slater determinant, which provides a mathematically correct many-electron wavefunction that satisfies the Pauli Exclusion Principle.

However, one important question remains unanswered:

Which molecular orbitals should be used to construct the Slater determinant?

There are infinitely many possible choices of orbitals. The goal of the Hartree–Fock method is to determine the set of orbitals that produces the lowest possible electronic energy.

This is achieved using the Variational Principle, one of the most powerful concepts in quantum mechanics.

The Exact Problem

The electronic Schrödinger equation is

\[ \hat H_e \Psi = E\Psi \]

where

  • \(\hat H_e\) is the electronic Hamiltonian,
  • \(\Psi\) is the exact many-electron wavefunction,
  • \(E\) is the exact electronic energy.

For molecules containing many electrons,

an exact solution is generally impossible because of the electron–electron repulsion term.

Instead of solving the equation exactly, Hartree–Fock searches for the best approximate wavefunction.

The Variational Principle

The Hartree–Fock method is based on the Variational Principle, which states:

For any normalized trial wavefunction, the calculated energy is always greater than or equal to the true ground-state energy.

Mathematically,

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

where

  • \(E_0\) is the exact ground-state energy,
  • \(E_{\text{trial}}\) is the energy obtained from a trial wavefunction.

This principle guarantees that improving the trial wavefunction lowers the calculated energy.

Choosing the Trial Wavefunction

Hartree–Fock assumes that the trial wavefunction is a single Slater determinant,

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

The unknown quantities are therefore the spin orbitals

\[ \chi_1,\chi_2,\ldots,\chi_N \]

Instead of varying the entire wavefunction,

Hartree–Fock varies the orbitals inside the determinant until the total energy becomes as small as possible.

The Mean-Field Approximation

The greatest difficulty in solving the electronic Schrödinger equation is the electron–electron interaction.

Each electron is attracted to every nucleus,

while simultaneously being repelled by every other electron.

This means that the motion of one electron depends on the instantaneous positions of all the others.

Hartree–Fock simplifies this problem using the mean-field approximation.

Instead of treating every electron as interacting with the exact positions of all other electrons,

each electron is assumed to move in the average electrostatic field generated by the remaining electrons.

Note

In Hartree–Fock theory, electrons do not respond to the instantaneous motion of other electrons. Instead, they move in an average or self-consistent field.

Independent Particle Picture

Under the mean-field approximation,

the many-electron problem is transformed into a set of one-electron problems.

Each electron occupies its own molecular orbital,

but every orbital depends on the average distribution of all other electrons.

This approximation greatly simplifies the mathematics while retaining much of the essential physics.

Hartree–Fock Energy

Using a Slater determinant and applying the Variational Principle,

the total electronic energy becomes

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

This equation contains three distinct contributions.

One-Electron Energy

The first term,

\[ \sum_i \langle i|\hat h|i\rangle \]

represents the energy of each electron moving independently in the field of the nuclei.

It includes

  • electronic kinetic energy,
  • electron–nucleus attraction.

Coulomb Energy

The second contribution,

\[ J_{ij} \]

is called the Coulomb integral.

It represents the classical electrostatic repulsion between two electron clouds.

Mathematically,

\[ J_{ij} = \iint \frac {\phi_i^*(1)\phi_i(1)\phi_j^*(2)\phi_j(2)} {r_{12}} \,d\tau_1\,d\tau_2 \]

The Coulomb term is always positive because electrons repel one another.

Exchange Energy

The third contribution,

\[ K_{ij} \]

is called the exchange integral.

Unlike the Coulomb term,

it has no classical analogue.

The exchange interaction arises solely because the electronic wavefunction is antisymmetric.

Mathematically,

\[ K_{ij} = \iint \frac {\phi_i^*(1)\phi_j(1)\phi_j^*(2)\phi_i(2)} {r_{12}} \,d\tau_1\,d\tau_2 \]

The exchange interaction is therefore a purely quantum mechanical effect.

Info

The exchange interaction exists only between electrons having the same spin.

What Does Hartree–Fock Actually Do?

The Hartree–Fock procedure repeatedly performs three tasks:

  1. Guess a set of molecular orbitals.
  2. Compute the total electronic energy.
  3. Modify the orbitals until the energy can no longer be lowered.

When the energy reaches its minimum,

the orbitals are said to be self-consistent.

Advantages of the Hartree–Fock Approximation

The Hartree–Fock approximation offers several important advantages.

  • It satisfies the Pauli Exclusion Principle.
  • It provides physically meaningful molecular orbitals.
  • It is computationally efficient.
  • It forms the foundation of many post-Hartree–Fock methods.

Because of these advantages,

Hartree–Fock remains one of the most widely used starting points in quantum chemistry.

Limitations

Although Hartree–Fock is remarkably successful,

it neglects one important physical effect:

electron correlation.

In Hartree–Fock,

each electron experiences only the average electric field produced by all other electrons.

The instantaneous motion of electrons is ignored.

As a result,

Hartree–Fock energies are always higher than the exact ground-state energy.

Methods such as

  • MP2,
  • Configuration Interaction (CI),
  • Coupled Cluster (CC),
  • CASSCF,

include electron correlation and generally produce much more accurate results.

Preparing for the Next Step

The Hartree–Fock energy expression contains two unfamiliar quantities:

  • the Coulomb operator,
  • the Exchange operator.

Rather than evaluating these terms directly,

they are combined into a single operator known as the Fock operator.

The Fock operator transforms the Hartree–Fock equations into a compact form that can be solved iteratively.

This operator is the central mathematical object in Hartree–Fock theory and is derived in the next chapter.

Key Takeaways

  • Hartree–Fock searches for the lowest-energy Slater determinant.
  • The Variational Principle guarantees that minimizing the energy improves the wavefunction.
  • Electrons are treated using the mean-field approximation.
  • The Hartree–Fock energy consists of one-electron, Coulomb, and exchange contributions.
  • The next step is to combine these terms into the Fock operator.