The Fock Matrix

The Fock matrix elements are:

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

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:

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

Core Hamiltonian

The core Hamiltonian matrix elements are computed directly from one-electron integrals:

$$H_{\mu \nu }=T_{\mu \nu }+V_{\mu \nu }$$

These integrals are evaluated using the Obara–Saika recurrence relations and are independent of the electron density.

Two-Electron Contribution

The two-electron part of the Fock matrix is built from the density matrix and electron repulsion integrals (ERIs):

$$G_{\mu \nu }=\sum _{\lambda \sigma }{P}_{\lambda \sigma }\left[(\mu \nu |\lambda \sigma )-\frac{1}{2}(\mu \lambda |\nu \sigma )\right]$$

The ERIs $(\mu \nu |\lambda \sigma )$ are computed using:

• McMurchie–Davidson algorithm (default, universal for all angular momenta)
• SP fast paths for S/P shells (up to 10× speedup)
• Rys quadrature for D/F+ shells
• Direct SCF mode for large systems (recomputes ERIs each iteration)

ERIs have 8-fold permutational symmetry:

$$(\mu \nu |\lambda \sigma )=(\nu \mu |\lambda \sigma )=(\mu \nu |\sigma \lambda )=(\nu \mu |\sigma \lambda )=(\lambda \sigma |\mu \nu )=(\sigma \lambda |\mu \nu )=(\lambda \sigma |\nu \mu )=(\sigma \lambda |\nu \mu )$$

Only unique ERIs are stored, and symmetry is exploited in all ERI-heavy computations (energy, gradients, Hessians).

Symmetry Acceleration

When point-group symmetry is enabled, only symmetry-unique shell quartets are computed. The detected point group is used to skip symmetry-equivalent ERI shell quartets during energy, gradient, and Hessian computations, and to assign irreducible representation labels (A₁, B₁, B₂, …) to molecular orbitals.