Fix: Remove nonzero tails beyond spherical Bessel projector cutoffs (Useful Information for updating DeePKS benchmark data)

[AsTonyshment]

Summary

This PR fixes generated DeePKS spherical-Bessel projector files that contained nonzero oscillatory samples beyond their declared cutoff radius. It preserves the requested radial spacing, keeps generated output meshes compatible with Simpson integration, repairs the affected versioned projector files, warns about inconsistent orbital metadata during reading, and refreshes all numerical references affected by the corrected projectors.

Background

Bessel_Basis::init_TableOne() extends its internal radial mesh with auxiliary points for Simpson integration. The same mesh was previously written directly to jle.orb, so a file declaring a 2.0 Bohr cutoff and a 0.01 Bohr spacing could contain 205 points and extend to 2.04 Bohr. Because the spherical Bessel function grows again after its zero at the cutoff, those extra points formed a nonzero tail and caused inconsistent results between grid and two-center projector integrations.

Changes

• Preserve the original dr when writing generated projectors.
• When the cutoff lies on the radial grid and the exact-cutoff mesh contains an odd number of points, set Mesh so that $(\mathrm{Mesh}-1)dr=r_{\mathrm{cut}}$ and omit auxiliary integration points from the file.
• When the exact-cutoff mesh would contain an even number of points, retain one additional zero-valued sample at $r_{\mathrm{cut}}+dr$ so that the output mesh remains compatible with Simpson integration.
• When the cutoff does not lie on the grid, retain the existing mesh but write zero for every sample beyond $r_{\mathrm{cut}}$.
• Add nonfatal warnings to the modern AtomicRadials reader and the legacy LCAO_Orbitals reader when the declared cutoff differs from $(\mathrm{Mesh}-1)dr$.
• Extend the Bessel-basis unit test to cover an exact-cutoff case that would otherwise produce an even mesh, verifying that the generated mesh is odd and the additional sample is zero.
• Repair all eight affected versioned jle.orb files in examples, fixtures, and DeePKS integration cases.
• Refresh 39 DeePKS unit-test references, 31 integration-test result.ref files, and 8 CSR references. Existing totaltimeref values are preserved.

Thread-sensitive integration references

The Gamma-point nspin=2 cases 29 and 31 follow different SCF trajectories with one and two OpenMP threads per MPI process. GitHub CI runs these cases with 4 MPI processes and OMP_NUM_THREADS=2, so their references were generated with the same configuration. No test threshold was relaxed.

Verification

• MODULE_IO_bessel_basis_test: 3/3 passed, including the even-mesh safeguard case.
• MODULE_NAO_atomic_radials: 5/5 passed.
• MODULE_AO_ORB_read_test --gtest_filter='LcaoOrbitalsTest.ReadOrbitals': passed.
• Existing DeePKS unit-test targets: 28/28 passed.
• Full 09_DeePKS integration suite with 4 MPI processes and OMP_NUM_THREADS=2: all 31 cases passed.
• Repository scan after repair: 0 inconsistent complete .orb files among the 81 files with complete cutoff and mesh metadata.
• git diff --check: passed.

Compatibility

Existing user projector files are not modified automatically. Readers only report inconsistent metadata and continue with the original samples. Newly generated files either end at the declared cutoff or retain only zero-valued samples beyond it, and their output meshes remain compatible with Simpson integration. No INPUT parameter or DeePKS model format is changed.

Documentation

No user-facing INPUT parameter or documented interface changes, so generated parameter documentation does not need an update. The compatibility behavior and the new reader warning are described in this PR body; intermediate diagnostic reports and figures are intentionally excluded from the PR.

Governance notes

The generator and reader changes do not introduce new cross-layer global dependencies or new header dependencies.