Draft: Enable complex multi-k PEXSI path for LCAO

[JohnHe233]

Reminder

• Have you linked an issue with this pull request?
• Have you added adequate unit tests and/or case tests for your pull request?
• Have you noticed possible changes of behavior below or in the linked issue?
• Have you explained the changes of codes in core modules of ESolver, HSolver,
  ElecState, Hamilt, Operator or Psi? (ignore if not applicable)

Linked Issue

Fix #...

Unit Tests and/or Case Tests for my changes

• Added unit coverage for the refactored PEXSI solver interface:
  • DiagoPexsiRefactorTest.UsesInjectedSolverAndOwnedDensityBuffers
  • Uses a fake IPexsiSolver to test solver injection and owned DM/EDM buffers
    without invoking the real PEXSI/MPI sparse solve path.
• Added tools/pexsi/check_lcao_10case_regression.py for LCAO PEXSI regression
  checks.
• Retested the official LCAO 10-case set with ks_solver pexsi.
  • 5/10 cases completed SCF: 001, 002, 003, 005, 008.
  • 002 passes the strict 5.0e-7 eV total-energy tolerance.
  • 004, 006, 007, 009 and 010 timed out under the local 4 MPI ranks / 1800 s test
    limit.

What's changed?

• Added a complex multi-k PEXSI path for LCAO.
  • Converts complex H/S matrices from ABACUS block-cyclic distribution to PEXSI CCS
    format.
  • Calls the complex expert-level PEXSI Fermi-operator workflow.
  • Retrieves complex DM/EDM matrices and maps them back to ABACUS data structures.
• Added a global chemical-potential search for multi-k PEXSI.
  • Uses one chemical potential per spin channel.
  • Accumulates electron number, derivative and energies over weighted k-points.
  • Adds Newton / bracket / bisection fallback handling.
• Fixed k-point and spin weighting for nspin=1 to avoid double-counting spin
  degeneracy.
• Implemented complex PEXSI DM/EDM handling in ElecStateLCAO::dm2rho.
• Refactored the PEXSI solver dependency.
  • Added pexsi::IPexsiSolver.
  • Changed DiagoPexsi to depend on the interface instead of directly depending on
    PEXSI_Solver.
• Reworked DM/EDM buffer ownership in DiagoPexsi.
  • Uses std::vector<std::vector<T>> for storage.
  • Preserves the existing std::vector<T*> DM/EDM interface for current callers.
• Added a deterministic guard for the Gamma-only real PEXSI path if it is incorrectly
  entered by a multi-k case.
• Optimized BCD-to-CCS H/S redistribution by packing H and S values into one
  MPI_Alltoallv.
• Fixed the countMatrixDistribution zero-value check from fabs(A[i] < 1e-31) to
  fabs(A[i]) < 1e-31.
• Added TransMAT2CCS timer coverage to separate ABACUS-side matrix conversion cost
  from PEXSI solve cost.

Possible behavior changes

• Multi-k LCAO inputs with ks_solver pexsi no longer fail immediately at the
  previous unsupported complex-path boundary.
• Gamma-only real PEXSI now fails earlier with a clearer error if a multi-k index
  enters the real path.
• Default PEXSI settings for fixed / low-temperature occupation are more conservative
  in this branch, which improves energy consistency for some cases but can
  significantly increase runtime.
• Large or metallic cases may still be too expensive with the current serial-over-k
  implementation and repeated PEXSI setup/factorization.

Any changes of core modules? (ignore if not applicable)

• HSolver:
  • Updated DiagoPexsi to support solver-interface injection, owned density
    buffers, complex multi-k PEXSI calls and global chemical-potential accumulation.
  • Updated HSolverLCAO integration to pass k-point weights and handle PEXSI DM/EDM
    output.
• ElecState:
  • Implemented complex PEXSI DM/EDM to density handling in ElecStateLCAO.
• PEXSI module:
  • Added IPexsiSolver.
  • Extended BCD/CCS transformations for complex matrices.
  • Added complex PEXSI expert-routine workflow and conversion timers.

[Flying-dragon-boxing]

Nice try! I will test the correctness of total energy on my machine later. A few small questions:

• You mentioned that you've optimized BCD-to-CCS H/S redistribution by packing H and S values into one
  MPI_Alltoallv, have you tested the speed up?
• For the cases that don't pass in 1800s, you may try reducing the pexsi_npole.
• You may test the correctness of force calculation after the total energy is properly dealt with.