[PR-4] feat: implement HRR (Eq. 67) and complete OS+HGP two-electron integral pipeline

[San1357]

Summary

• Implemented _horizontal_recursion_relation() — HRR (Eq. 67), transfers angular momentum to center B and D
• Implemented compute_two_electron_integrals_os_hgp() — complete OS+HGP pipeline entry point
• Full pipeline: Boys → VRR → ETR → Contract → HRR
• Part of Issue WoC: 2-electron integrals: better performance and more flexibility #221 (Week 4),
• builds on PR [PR-2] feat: implement VRR for OS+HGP two-electron integral pipeline #234 (VRR) and,
• ETR PR [PR-3] feat: implement ETR (Eq. 66) and primitive contraction for OS+HGP pipeline #237

What Changed

File	Lines	What
gbasis/integrals/_two_elec_int_improved.py	+200	HRR + complete pipeline implementation
tests/test_two_elec_int_improved.py	+200	14 tests (VRR + ETR + HRR + pipeline)

Mathematical Reference

The Horizontal Recursion Relation (Eq. 67) is:

$$[a, b+1_i | c, d] = [a+1_i, b | c, d] + (A-B)_i [a, b | c, d]$$

where angular momentum is transferred from center A to center B (and similarly C to D).

How To Test

# Run tests
pytest tests/test_two_elec_int_improved.py -v

# Check coverage
python -m pytest tests/test_two_elec_int_improved.py --cov=gbasis.integrals._two_elec_int_improved --cov-report=term-missing

# Check formatting & linting
black --check gbasis/integrals/_two_elec_int_improved.py tests/test_two_elec_int_improved.py
ruff check gbasis/integrals/_two_elec_int_improved.py tests/test_two_elec_int_improved.py

# Verify existing tests not broken
pytest tests/ --timeout=120

Proof That It Works

Tests (16 passed)

Check coverage

Black: Clean & Ruff: Clean

Existing tests: Not broken

307 passed, 279 skipped, 1 xfailed, 4 warnings in 67.52s (0:01:07)

First Checklist

• Tests added for each new function
• All 9 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

• This PR completes the full OS+HGP pipeline for two-electron integrals (VRR → ETR → Contract → HRR)
• Adds _vertical_recursion_relation (VRR)
• Adds _electron_transfer_recursion (ETR)
• Adds _optimized_contraction with per-center vectorized norm computation
• Adds _horizontal_recursion_relation (HRR)
• Adds compute_two_electron_integrals_os_hgp (full pipeline)
• Does NOT modify any existing code
• Does NOT change any existing tests

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
	🐛 Bug fix
✓	✨ New feature
	🔨 Refactoring
	📜 Docs