Two-Electron Integrals

Electron repulsion integrals (ERIs) are the computational bottleneck of Hartree–Fock and post-Hartree–Fock calculations:

$$(\mu \nu |\lambda \sigma )=\iint {\chi }_{\mu }({\mathbf{r}}_1){\chi }_{\nu }({\mathbf{r}}_1)\frac{1}{r_{12}}{\chi }_{\lambda }({\mathbf{r}}_2){\chi }_{\sigma }({\mathbf{r}}_2)d{\mathbf{r}}_{1}d{\mathbf{r}}_2$$

Permutational Symmetry

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 computed. This reduces storage and computation by a factor of 8.

ERI Engine Selection

OpenQuantum provides multiple ERI algorithms, selectable via the INT section:

INT
  Eri  auto  # auto, md, sp, rys
END

The selected engine applies to all ERI computation paths:

• In-core integral build
• Direct SCF
• Analytical nuclear gradients
• Fully analytical Hessians
• Semi-analytical (FD) Hessians

McMurchie–Davidson Algorithm

The MD algorithm uses Hermite Gaussian intermediates and is universal for all angular momenta. It builds ERIs by:

1. Computing Hermite integrals over Gaussian primitives
2. Contracting to shell-pair level
3. Applying permutational symmetry

Complexity scales as $O(L^6)$ for maximum angular momentum $L$.

SP Fast Paths

For S and P shells (angular momentum $L\le 1$), analytical formulas give exact ERIs with far fewer operations. These are ~10× faster than MD for organic molecules. For higher angular momentum shells, MD is used as fallback.

Rys Quadrature

Rys quadrature is a numerical integration method that becomes competitive for D/F+ shells. OpenQuantum implements the Phenix-style Rys algorithm with precomputed roots and weights. MD is used as fallback when the Rys path is unavailable.

Schwarz Screening

For direct SCF and large systems, ERIs are pre-screened using the Cauchy–Schwarz inequality:

$$(\mu \nu |\lambda \sigma )\le \sqrt{(\mu \nu |\mu \nu )}\sqrt{(\lambda \sigma |\lambda \sigma )}$$

Only ERIs above the threshold Acc2E (default 10⁻¹²) are computed. This dramatically reduces the number of ERIs for large systems.