Fix bug (deltaspin): restore LCAO spin rotation invariance

[Tao-Xia]

Reminder

• I have read AGENTS.md and docs/developers_guide/agent_governance.md.
• I have linked an issue or explained why this PR does not need one.
• I have added adequate unit tests and/or case tests, or explained why not.
• I have listed the exact verification commands run and their results.
• I have described user-visible behavior changes, including INPUT parameter changes.
• I have explained core-module impact for ESolver, HSolver, ElecState, Hamilt, Operator, Psi, or other source/ changes.
• I have requested any needed governance exception below.

Linked Issue

Fixes #7913

Unit Tests and/or Case Tests for my changes

• Commands run:
  • cmake --build build-elpa-cpu --target deltaspin_core_test MODULE_HSOLVER_LCAO abacus_basic_para -j 12
  • build-elpa-cpu/source/source_lcao/module_deltaspin/test/deltaspin_core_test
  • From source/source_hsolver/test: build-elpa-cpu/source/source_hsolver/test/MODULE_HSOLVER_LCAO
  • From the same directory: mpirun -np 4 build-elpa-cpu/source/source_hsolver/test/MODULE_HSOLVER_LCAO --gtest_filter='DiagoElpaComplexTest.UsesAuthoritativeUpperTriangle:DiagoTest/DiagoKPointsTest.LCAO/0'
  • Two 12-rank bcc-Fe LCAO DeltaSpin integration runs with equivalent x and y target moments, no SOC, and ks_solver genelpa.
• Result summary:
  • DeltaSpin: 22/22 tests passed.
  • LCAO eigensolver: 7/7 single-rank tests passed; 2/2 selected 4-rank tests passed.
  • Both new regression tests were also checked against the old behavior and failed as intended.
  • bcc-Fe x/y total energies differ by only 1.82e-8 eV; both target moments converge with transverse residuals of order 1e-6 mu_B.
• Checks not run, with reason:
  • Full repository test suite was not run because the change is confined to the complex genelpa transform and DeltaSpin coefficient mapping; the focused module tests and end-to-end integration runs exercise both paths.

What's changed?

• Use PZHEMM with the authoritative upper triangle when the complex custom ELPA solver forms the generalized Hamiltonian. This prevents stale lower-triangle entries from entering the eigensolve without allocating an extra full matrix.
• Define the Cartesian-Pauli-to-spinor conversion once and use it in the LCAO DeltaSpin operator.
• Add an invariant test requiring <psi | lambda dot sigma | psi> = lambda dot M, including the sign-sensitive y channel.
• Add a complex genelpa regression that deliberately corrupts the non-authoritative lower triangle and compares with the upper-triangle LAPACK reference.

There are no INPUT parameter or serialized-data changes. The user-visible correction is restored non-SOC spin-rotation invariance for LCAO DeltaSpin with genelpa.

Governance Notes

• INPUT/docs changes: none; no user-facing option or default changes.
• Core module impact: HSolver changes one complex generalized-transform kernel from general to Hermitian distributed multiplication; OperatorLCAO uses a shared DeltaSpin Pauli conversion. ESolver, ElecState, and Psi interfaces are unchanged.
• Exceptions requested: none.

[dyzheng]

LGTM，This PR may resolve the numerical instability issue in non‑collinear spin calculations with the LCAO basis set. Thank you for your submission.