The Hartree-Fock (HF) method is the simplest ab initio approximation that uses a single Slater determinant as the trial wave function. It is derived variationally from the many-electron Schrödinger equation.
Step 1: Many-electron Hamiltonian
\[
\hat{H} = \sum_{i=1}^{N} \hat{h}(i) + \sum_{i < j}^{N} \frac{1}{r_{ij}}
\]
where the one-electron operator is
\[
\hat{h}(i) = -\frac{1}{2}\nabla_i^2 - \sum_{A} \frac{Z_A}{r_{iA}}
\]
(kinetic energy + nuclear attraction).
Step 2: Variational energy for a Slater determinant
The HF energy is the expectation value:
The HF energy is the expectation value:
\[
E_{\text{HF}} = \langle \Psi | \hat{H} | \Psi \rangle
\]
Using the Slater-Condon rules (which follow directly from expanding the determinant), this simplifies to:
\[
E_{\text{HF}} = \sum_{i=1}^{N} h_{ii} + \frac{1}{2} \sum_{i,j=1}^{N} \left( J_{ij} - K_{ij} \right)
\]
where
- \(h_{ii} = \langle \chi_i | \hat{h} | \chi_i \rangle\) (one-electron integrals)
- \(J_{ij} = \langle \chi_i \chi_j | \frac{1}{r_{12}} | \chi_i \chi_j \rangle\) (Coulomb integrals)
- \(K_{ij} = \langle \chi_i \chi_j | \frac{1}{r_{12}} | \chi_j \chi_i \rangle\) (Exchange integrals)
Step 3: Variational minimization under orthonormality constraint
We minimize \(E_{\text{HF}}\) with respect to variations \(\delta\chi_k\) while keeping the spin-orbitals orthonormal: \[ \langle \chi_i | \chi_j \rangle = \delta_{ij} \] Introducing Lagrange multipliers \(\epsilon_{ij}\), the stationarity condition \(\delta E_{\text{HF}} = 0\) leads to the Fock equations:
We minimize \(E_{\text{HF}}\) with respect to variations \(\delta\chi_k\) while keeping the spin-orbitals orthonormal: \[ \langle \chi_i | \chi_j \rangle = \delta_{ij} \] Introducing Lagrange multipliers \(\epsilon_{ij}\), the stationarity condition \(\delta E_{\text{HF}} = 0\) leads to the Fock equations:
\[
\hat{F} \, \chi_i = \epsilon_i \, \chi_i
\]
where the Fock operator is
\[
\hat{F} = \hat{h} + \sum_{j=1}^{N} \left( \hat{J}_j - \hat{K}_j \right)
\]
with the Coulomb operator
\[
\hat{J}_j(1)\phi(1) = \left[ \int \frac{|\chi_j(2)|^2}{r_{12}} \, d\mathbf{r}_2 \right] \phi(1)
\]
and the exchange operator
\[
\hat{K}_j(1)\phi(1) = \left[ \int \frac{\chi_j^*(2) \phi(2)}{r_{12}} \, d\mathbf{r}_2 \right] \chi_j(1)
\]
Step 4: Self-consistent field (SCF) solution
The Fock operator depends on the orbitals themselves, so the equations are solved iteratively:
The Fock operator depends on the orbitals themselves, so the equations are solved iteratively:
- Guess initial orbitals → build \(\hat{F}\)
- Diagonalize \(\hat{F}\) → new orbitals
- Repeat until convergence (energy and orbitals stable)
The resulting orbitals are called canonical Hartree-Fock orbitals, and the Slater determinant built from them gives the best single-determinant approximation to the true ground-state wave function.