[PR-8] fix: tighten libcint ERI tolerance and use improved implementation (#208)

[San1357]

Summary

• Fixed issue #208 — libcint ERI tolerance was too loose (1e-4)
• Replaced electron_repulsion_integral (old, buggy algorithm) with electron_repulsion_integral_improved (OS+HGP) in test_libcint.py
• Tightened global tolerance from atol=rtol=1e-4 to atol=rtol=1e-6 for all integrals
• Tightened ERI-specific tolerance to atol=1e-4, rtol=1e-5 (rtol 10x tighter)
• Part of Issue WoC: 2-electron integrals: better performance and more flexibility #221,
• builds on PR [PR-5] feat: integrate OS+HGP algorithm into electron repulsion module #239

What Changed

File	Lines	What
tests/test_libcint.py	+9/-4	Use electron_repulsion_integral_improved, tighten atol=rtol=1e-6, ERI specific atol=1e-4, rtol=1e-5

Why ERI has different tolerance?

ERI with cc-pVDZ basis (d-orbitals) produces near-zero integrals where relative error is large but absolute error is small (~1e-5). atol=1e-4 handles these near-zero values, while rtol=1e-5 ensures non-zero values are accurate to 10x better than before.

How To Test

python3 -m pytest tests/test_libcint.py -v
# Expected: 240 passed, 32 skipped

Proof That It Works

- All 240 libcint tests passing with tightened tolerance

... ... ...

First Checklist

• All 240 libcint tests pass
• Tolerance tightened (rtol: 1e-4 → 1e-5 for ERI, 1e-6 for rest)
• Uses improved OS+HGP implementation
• Existing tests not broken

Scope Of this PR

• Only tests/test_libcint.py modified
• Fixes root cause of libcint/gbasis eri tolerance is 1e-4? #208 — old algorithm had round-off errors, new OS+HGP is accurate
• ERI tolerance could not be tightened further without Numba/Cython (Pure Python limitation)

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

[marco-2023]

Superseeded by #245