The Hartree–Fock Method

The Hartree–Fock (HF) method is a mean-field approximation to the many-electron Schrödinger equation. Each electron moves in an average field created by all other electrons, leading to a set of one-electron equations.

This chapter provides the theoretical foundation for the HF implementation in OpenQuantum, including the electronic Hamiltonian, the Roothaan–Hall equations, the Fock matrix construction, and total energy evaluation.

Overview of Chapters

• Electronic Hamiltonian — The Hamiltonian in atomic units
• Roothaan–Hall Equations — LCAO expansion and matrix equations
• Fock Matrix — Core Hamiltonian and two-electron contributions
• Total Energy — Energy expression and nuclear repulsion

UHF and ROHF

For open-shell systems, see Unrestricted & Restricted Open-Shell HF which covers separate α/β orbitals, spin contamination, and the ROHF formalism.

Key Equations

The central equations are summarized here for quick reference.

Electronic Hamiltonian (atomic units)

$$\hat{H}=\sum _i\hat{h}(i)+\sum _{i<j}\frac{1}{r_{ij}}$$

where:

• $\hat{h}(i)=-\frac{1}{2}{\nabla }_i^2-{\sum }_A\frac{Z_A}{r_{iA}}$ is the one-electron operator (kinetic + nuclear attraction)
• $1/r_{ij}$ is the electron–electron repulsion

Roothaan–Hall Equations

In the LCAO approximation, molecular orbitals are expanded in atomic orbitals:

$${\varphi }_i=\sum _{\mu }{C}_{\mu i}{\chi }_{\mu }$$

This leads to the matrix equations:

$$\mathbf{FC}=\mathbf{SC}\mathbf{\epsilon }$$

where:

• $\mathbf{F}$ is the Fock matrix
• $\mathbf{C}$ is the MO coefficient matrix
• $\mathbf{S}$ is the overlap matrix
• $\mathbf{\epsilon }$ is the diagonal matrix of orbital energies

Fock Matrix Elements

$$F_{\mu \nu }=H_{\mu \nu }+G_{\mu \nu }$$

where:

• $H_{\mu \nu }=T_{\mu \nu }+V_{\mu \nu }$ is the core Hamiltonian (kinetic + nuclear attraction)
• $G_{\mu \nu }={\sum }_{\lambda \sigma }{P}_{\lambda \sigma }\left[(\mu \nu |\lambda \sigma )-\frac{1}{2}(\mu \lambda |\nu \sigma )\right]$ is the two-electron contribution

The density matrix is:

$$P_{\mu \nu }=2\sum _{i(\text{occ})}{C}_{\mu i}{C}_{\nu i}$$

Total Energy

$$E=\frac{1}{2}\mathrm{Tr}[\mathbf{P}(\mathbf{H}+\mathbf{F})]+E_{\text{nuc}}$$

where $E_{\text{nuc}}={\sum }_{A<B}\frac{Z_A{Z}_B}{R_{AB}}$ is the nuclear repulsion energy.