The Hartree-Fock Method

Chem 3240 · Lecture 7.3

Davit Potoyan

The Many-Electron Problem

  • The Schrodinger equation is unsolvable for \(N > 1\) electrons.
  • The culprit: electron, electron repulsion couples every coordinate.
  • Idea: let each electron feel only the average field of all the others.
  • A many-body problem becomes many one-electron problems.

The Slater Determinant

  • The total wavefunction must be antisymmetric (Pauli).

\[\Psi(1, 2, \ldots, N) = \frac{1}{\sqrt{N!}} \begin{vmatrix} \psi_1(1) & \psi_2(1) & \cdots & \psi_N(1) \\ \psi_1(2) & \psi_2(2) & \cdots & \psi_N(2) \\ \vdots & \vdots & \ddots & \vdots \\ \psi_1(N) & \psi_2(N) & \cdots & \psi_N(N) \end{vmatrix}\]

  • Swapping two electrons swaps rows, flipping the sign automatically.

The Hartree-Fock Equation

  • Each orbital is an eigenfunction of one effective operator.

\[\hat{F}\,\psi_i = \varepsilon_i\,\psi_i\]

  • \(\hat{F}\) is the Fock operator.
  • \(\varepsilon_i\) are the orbital energies.

Inside the Fock Operator

\[\hat{F} = \hat{h} + \sum_{j}\left(\hat{J}_j - \hat{K}_j\right)\]

  • \(\hat{h}\): kinetic energy plus nuclear attraction.
  • \(\hat{J}_j\): Coulomb operator, classical electron, electron repulsion.
  • \(\hat{K}_j\): exchange operator, a purely quantum effect with no classical analogue.

The Self-Consistent Field

\(\hat{F}\) depends on the very orbitals it determines, so we iterate:

  1. Guess the orbitals \(\psi_i\).
  2. Build the Fock operator \(\hat{F}\).
  3. Solve \(\hat{F}\psi_i = \varepsilon_i\psi_i\).
  4. Repeat until the orbitals stop changing.

The Mean Field in Action

  • Helium: the simplest closed-shell test.
  • Each \(1s\) electron moves in the average field of the other.
  • The two-electron problem splits into two one-electron problems.

The Effective Potential

  • One electron feels the bare nuclear attraction…
  • …screened by the averaged charge cloud of the other.
  • The net pull is weaker than the bare nucleus: shielding.

The Total Energy

\[E_{HF} = \sum_{i}\langle \psi_i | \hat{h} | \psi_i \rangle + \frac{1}{2}\sum_{i,j}\left(J_{ij} - K_{ij}\right)\]

  • \(J_{ij}\): Coulomb integral, classical repulsion.
  • \(K_{ij}\): exchange integral, lowers the energy.
  • The factor \(\tfrac{1}{2}\) removes double counting of each pair.

Strengths and Limits

Strengths

  • Efficient: intractable many-body problem becomes coupled one-electron equations.
  • Exchange is captured exactly through the determinant.

Limits

  • No dynamic correlation: each electron sees only the average, not instantaneous, positions.
  • Post-HF methods (MP2, CCSD) add correlation on top.

Takeaway

Hartree-Fock replaces the impossible many-body problem with a self-consistent set of one-electron equations, where each electron moves in the averaged field of all the others. The Slater determinant enforces antisymmetry and captures exchange exactly, but correlation only on average.