Integral Evaluation

OpenQuantum computes all required molecular integrals using modern recurrence relations and efficient algorithms.

Overview of Chapters

One-Electron Integrals — Overlap, kinetic, nuclear attraction
Two-Electron Integrals — ERIs with MD, SP fast paths, Rys quadrature

One-Electron Integrals

One-electron integrals are evaluated using the Obara–Saika recurrence relations which provide an efficient and numerically stable way to compute integrals for arbitrary angular momentum.

Integral	Formula	Method
Overlap	$S_{μν} = \int χ_{μ} (r) χ_{ν} (r) d r$	Obara–Saika
Kinetic	$T_{μν} = \int χ_{μ} (r) (- \frac{1}{2} \nabla^{2}) χ_{ν} (r) d r$	Obara–Saika
Nuclear Attraction	$V_{μν} = \int χ_{μ} (r) \sum_{A} (- \frac{Z _{A}}{∣ r - R _{A} ∣}) χ_{ν} (r) d r$	Obara–Saika

Two-Electron Integrals

Electron repulsion integrals (ERIs):

$(μν ∣ λσ) = \iint χ_{μ} (r_{1}) χ_{ν} (r_{1}) \frac{1}{r _{12}} χ_{λ} (r_{2}) χ_{σ} (r_{2}) d r_{1} d r_{2}$

ERIs have 8-fold permutational symmetry. OpenQuantum provides multiple algorithms:

Algorithm	Best For	Key Feature
McMurchie–Davidson	All systems (default)	Universal, correct for all angular momenta
SP Fast Paths	Organic molecules (H,C,N,O,F)	10× ERI speedup for S/P shells only
Rys Quadrature	Transition metals, D/F+ basis sets	Rys-first path with MD fallback

The selected ERI engine applies to all ERI computation paths: in-core integral build, direct SCF, analytical nuclear gradients, fully analytical Hessians, and semi-analytical (finite-difference of gradients) Hessians.

See Two-Electron Integrals for detailed algorithm descriptions.