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.

[mohanchen]

该测试算例破坏了 /08_EXX/08_KP_HSE_symm 的结果

信息如下：
结论
会调用 ELPA_Solver::generalized_eigenvector 的复数重载。

判定依据
输入：basis_type=lcao、ks_solver=genelpa、gamma_only=0（多 k）、nspin=1。genelpa 只在定义 __ELPA 时可用。
k 点类型：KPT 是 2 2 2 多 k，波函数类型为 std::complex，因此模板实参 T = std::complex。
代码路径（kpar=1，走非 pool 路径）：
hsolver_lcao.cpp:148-152：method == "genelpa" → DiagoElpa::diag。
diago_elpa.cpp:66-95：DiagoElpa<std::complex>::diag 在第 89 行调用 es.generalized_eigenvector(h_mat.p, s_mat.p, this->DecomposedState, eigen.data(), psi.get_pointer())，正是上一轮定位到的复数重载调用点 2。
运行日志印证：running_scf.log 中 HSolverLCAO hamiltSolvePsiK 被调用 63 次，与 21 SCF 步 × 3 个不可约 k 点吻合（每次 k 点求解都会进入一次该函数）。
不会命中的路径
DiagoElpa 实数重载：本算例为多 k 复数，不会走。
DiagoElpa<std::complex>::diag_pool：需要 kpar>1，INPUT 未设置，默认 kpar=1，不走 parakSolve。
hsolver::elpa_diag（diag_hs_para.cpp）：仅 dav_subspace 子空间对角化用，本算例不是 dav。
补充：该调用成立的前提是编译时启用了 __ELPA（ENABLE_ELPA），日志能正常出结果说明该二进制已启用 ELPA。

[mohanchen]

Please confirm the modification is correct. @Tao-Xia @dyzheng @maki49

[mohanchen]

The error is:

[mohanchen]

See comments.

[Growl1234]

• Looks like fallback path (DecomposedState == 3) is forgotten.
• Has all related comments (inline documentation) been updated?

Wrt EXX issues, I believe this was a bug and #7923 gave the right fix.