[PR-6] feat: implement Schwarz screening for two-electron integral optimization

[San1357]

Summary

• Implemented SchwarzScreener class — uses Schwarz inequality to skip negligible two-electron integrals before computing them
• Implemented shell-pair prescreening — uses Gaussian decay factor exp(-a*b/(a+b) * |A-B|^2) to skip distant shell pairs
• Integrated Schwarz screening into electron_repulsion_integral_improved() via schwarz_threshold parameter (default 0.0 = disabled)
• Part of Issue WoC: 2-electron integrals: better performance and more flexibility #221 (Week 7),
• builds on PR [PR-5] feat: integrate OS+HGP algorithm into electron repulsion module #239 (OS+HGP integration)

What Changed

File	Lines	What
gbasis/integrals/_schwarz_screening.py	+218	New module: compute_schwarz_bound_shell_pair, compute_schwarz_bounds, shell_pair_significant, SchwarzScreener class with is_significant() and get_statistics()
gbasis/integrals/electron_repulsion.py	+67	Added _screener and _contraction_index_map class attributes to ElectronRepulsionIntegralImproved, Schwarz screening block in construct_array_contraction, schwarz_threshold parameter in electron_repulsion_integral_improved()
tests/test_schwarz_screening.py	+460	23 tests: TestShellPairPrescreening, TestSchwarzBounds, TestSchwarzScreener, TestScreeningCorrectness, TestBenchmarkImprovedAlgorithm
tests/test_two_elec_int_improved.py	+83	Added TestSchwarzScreening class: test_schwarz_no_effect, test_schwarz_screening_tight, test_schwarz_statistics

Mathematical Reference

The Schwarz inequality for electron repulsion integrals:

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

If $\sqrt{(\mu\nu|\mu\nu)} \cdot \sqrt{(\lambda\sigma|\lambda\sigma)} < \tau$, the quartet $(\mu\nu|\lambda\sigma)$ is skipped.

Shell-pair prescreening uses the Gaussian product decay factor:

$$\exp\left(-\frac{ab}{a+b}|\mathbf{A}-\mathbf{B}|^2\right)$$

References:

• Häser, M. & Ahlrichs, R. J. Comput. Chem. 1989, 10, 104.
• Gill, P. M. W.; Johnson, B. G.; Pople, J. A. Int. J. Quantum Chem. 1991, 40, 745.

How To Test

# Schwarz screening unit tests
python -m pytest tests/test_schwarz_screening.py -v

# Integration tests (includes TestSchwarzScreening)
python -m pytest tests/test_two_elec_int_improved.py -v

# All 44 tests together
python -m pytest tests/test_schwarz_screening.py tests/test_two_elec_int_improved.py -v

# Coverage
python -m pytest tests/test_schwarz_screening.py tests/test_two_elec_int_improved.py --cov=gbasis/integrals --cov-report=term-missing

Proof That It Works

Unit tests (SchwarzScreener)

Integration tests (screening inside ERI computation)

All 44 tests

Coverage

black & ruff

First Checklist

• ✅ Tests added for each new function
• ✅ All 44 tests pass
• ✅ Black formatted
• ✅ Ruff linting clean
• ✅ Existing tests not broken
• ✅ Docstrings in numpydoc format
• ✅ No new dependencies added (uses existing numpy)
• ✅ Pure Python only (no C/Fortran)

Scope Of this PR

• Implements Schwarz screening as optional optimization for electron_repulsion_integral_improved()
• Adds SchwarzScreener class with precomputed bounds matrix and statistics tracking
• Does not modify existing electron_repulsion_integral() — only improved version gets screening
• Screening is opt-in via schwarz_threshold parameter — default 0.0 disables it (backward compatible)
• Week 7 deliverable building on OS+HGP pipeline (PR [PR-5] feat: integrate OS+HGP algorithm into electron_repulsion module #239)

Second Checklist

• ✅ Write a good description of what the PR does.
• ✅ Add tests for each unit of code added (e.g. function, class)
• ☐ Update documentation
• ☐ Squash commits that can be grouped together
• ☐ Rebase onto master

Type of Changes

Type	Checked
🐛 Bug fix
✨ New feature	✓
🔨 Refactoring
📜 Docs