Test: Add <phi|r_a|p_{l,m}> snap_psibeta unit test

[AsTonyshment]

Linked Issue

No linked issue; this PR is a focused RT-TDDFT unit-test and numerical-convention cleanup for the velocity-gauge beta-projector snapshot path.

Unit Tests and/or Case Tests for my changes

This PR extends the existing MODULE_LCAO_tddft_snap_psibeta_half_test GoogleTest target to cover real Ti orbital and UPF beta-projector data. The test compares snap_psibeta_half_tddft at zero vector potential against the existing cal_r_overlap_R::get_psi_r_beta two-center reference path for both projector overlaps and Cartesian position moments.

Focused check:

ctest --test-dir build -R '^MODULE_LCAO_tddft_snap_psibeta_half_test$' --output-on-failure

Exact Verification Performed

• Commands run: not run while drafting this PR text.
• Result summary: verification is pending on the focused GoogleTest target above.
• Checks not run, with reason: local PR metadata update only; no fresh build/test output was collected during this text update.

What's changed?

• Replaces the synthetic beta-projector coverage in snap_psibeta_half_tddft_test with real Ti orbital data and Ti UPF beta projectors.
• Compares zero-vector-potential results for nlm[0] = <phi|p_{l,m}> and nlm[1..3] = <phi|r_a|p_{l,m}>.
• Exercises the production quadrature settings, a denser radial grid, and a higher-order Lebedev angular grid.
• Makes the shared projector snapshot interface explicit about the UPF convention that beta projectors are stored as r * p_l(r).
• Updates the radial quadrature factor so the shared integrator evaluates the intended p_l(r) * r^2 dr volume-integral contribution from stored r * p_l(r) data.
• Registers the extra source dependencies required by the focused test target.
• Makes cal_r_overlap_R orbital-index maps robust when a test-built UnitCell has more orbitals than PARAM.globalv.nlocal.

Governance Checklist

• Global dependencies: the PR-level budget is non-increasing (GlobalV/GlobalC/PARAM added=10, removed=18, net_delta=-8 in the Agent Governance report). The flagged cal_r_overlap_R.cpp lines are migration-neutral formatting/partial-cleanup lines in a legacy writer/reference path that already depends on PARAM.globalv for matrix dimensions, output directories, spin polarization, and rank-gated sparse output.
• Default parameters: no new default arguments are added to existing interfaces.
• Headers: the reported cal_r_overlap_R.h include is not a new logical dependency; the header already needed source_lcao/module_ri/abfs-vector3_order.h for the public std::set<Abfs::Vector3_Order<int>> signature, and the diff only moves it within the include block.
• Line endings: new and changed text uses LF.
• Build linkage: the additional sources needed by MODULE_LCAO_tddft_snap_psibeta_half_test are listed in source/source_lcao/module_rt/test/CMakeLists.txt.
• Documentation: no documentation update required because this only changes internal RT-TDDFT projector integration behavior/tests and does not change user-facing INPUT syntax, runtime options, file formats, or documented workflows.
• CodeRabbit: request @coderabbitai review if automatic review does not start.

INPUT Parameter Changes

• Parameters added, removed, or changed: none.
• docs/parameters.yaml updated: no.
• docs/advanced/input_files/input-main.md updated: no.
• No input documentation update required because no INPUT parameter metadata, defaults, parsing behavior, availability, or user-facing INPUT documentation changed.

Core Module Impact

• Affected core modules: RT-TDDFT LCAO projector snapshot helpers in source_lcao/module_rt; test-only reference coverage through source_io/module_hs/cal_r_overlap_R.
• Risk summary: the production default quadrature remains the same, while the radial-projector convention is clarified and corrected for UPF beta projectors stored as r * p_l(r). The main behavioral risk is numerical drift in beta-projector overlaps, which is covered by comparison against the existing two-center reference path.
• Compatibility or performance impact: no INPUT or file-format compatibility impact. Production runtime cost is expected to remain unchanged for default quadrature settings; the denser radial and higher-order Lebedev grids are selected only by the test overload.

Governance Exception

• Rule: Global dependency budget warning and header dependency review warning.
• Reason: cal_r_overlap_R is a legacy sparse-output/reference helper that still reads PARAM.globalv, PARAM.inp, and GlobalV::DRANK directly for matrix dimensions, output paths, spin layout, and root-rank IO. This PR reduces the total global-reference budget and does not introduce a new ownership pattern; it only keeps the existing path usable as a two-center reference for the focused RT-TDDFT test.
• Scope: source/source_io/module_hs/cal_r_overlap_R.cpp and source/source_io/module_hs/cal_r_overlap_R.h.
• User or maintenance risk: low for users because there is no INPUT or file-format change. Maintenance risk is limited to retaining the existing global-dependency shape in this helper until the surrounding sparse-output API receives an explicit context object.
• Why the normal rule cannot be followed now: removing the remaining globals would require a broader API migration across cal_r_overlap_R, sparse matrix output, and call sites that is outside this focused test/numerical-fix PR.
• Follow-up cleanup plan: migrate cal_r_overlap_R to explicit context inputs for dimensions, spin polarization, output directories, and rank-gated IO in a dedicated cleanup PR.
• Requested approver: ABACUS maintainer/reviewer for the Agent Governance warning acknowledgement.