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Scientific validation

PyIntPak treats agreement with published operators and independent mathematics as the numerical standard. The historical Fortran program is not required to run the validation suite and is not the oracle for three- or four-electron integrals.

Comparison rule

Scientific comparisons use both an absolute and a relative tolerance:

abs(calculated - reference) <= atol + rtol * abs(reference)

The absolute term controls values close to zero. The relative term scales the allowed error for larger integrals. The authoritative constants are defined in tests/tolerances.py:

Check rtol atol Purpose
Independent reference 2e-10 2e-12 Dense Gaussian moments and direct Coulomb quadrature
Exact symmetry 2e-12 2e-13 Algebraically identical operator permutations
Molecular dataset 5e-11 5e-12 Frozen contracted He and H2 records
Vanishing-geminal limit 2e-9 2e-11 Continuity between zero and a 1e-11 exponent

These tolerances include headroom for differences among supported Python, NumPy, SciPy, compiler, and CPU combinations. A tolerance must not be loosened only to make a failing test pass: the numerical cause and the largest observed error must first be documented in the change.

Independent oracle

tests/independent_reference.py does not import PyIntPak. It constructs the full Gaussian exponent matrix, obtains correlated Cartesian moments from the Gaussian covariance, and evaluates a Coulomb edge through direct Laplace quadrature. This differs structurally from the production HRR/VRR dependency graphs and generalized Boys implementation.

The oracle validates:

  • all-s fundamental two-, three-, and four-electron integrals;
  • Cartesian F, G, H, and I functions on both bra and ket paths;
  • the Coulomb-geminal and pure-geminal three-electron operators;
  • all three published four-electron operators.

Identities and limits

The suite enumerates all independent bra/ket exchanges: 2^3 for each three-electron operator and 2^4 for each four-electron operator. It also tests particle permutations that preserve each labelled operator graph, including the required exchange of unequal geminal exponents.

Zero-geminal reductions are compared directly with the independent oracle. Continuity is checked at a geminal exponent of 1e-11, and positive all-s integrals are checked for decay as the geminal exponent becomes tight.

Molecular references

tests/reference_data/small_molecules.json contains selected normalized 6-31G contracted records for He and H2. It records all geometry, basis, Slater expansion, operator, and value data required by the test. See tests/reference_data/README.md for provenance and maintenance rules.

Running validation

Run the complete suite:

python3 -m unittest discover -s tests -v

Run the independent mathematical oracle tests only:

python3 -m unittest discover -s tests -p 'test_independent_validation.py' -v