Pexsi stage4 pattern reuse opt

[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/updated unit tests for the PEXSI abstraction
  and density-matrix related code paths.
• Re-tested the official 10 LCAO examples with PEXSI,
  including Gamma-only and multi-k cases.
• Compared Gamma-only PEXSI results against an origin/
  develop based branch using identical input files.
• Tested selected systems from the 5.1 benchmark set
  with both conventional solver and PEXSI under multiple
  MPI sizes.
• Verified that the current Gamma PEXSI path does not
  regress against origin/develop on the same inputs for
  the tested Gamma cases.

What's changed?

• Added complex multi-k PEXSI support, including
  complex BCD-to-CCS / CCS-to-BCD matrix conversion,
  complex PEXSI driver calls, and density-matrix /
  energy-density-matrix writeback.
• Added a PEXSI solver interface layer to reduce
  direct coupling between DiagoPexsi and the concrete
  PEXSI backend, making the code easier to test and
  extend.
• Refactored DM/EDM buffer ownership to use safer
  container-managed storage while preserving the
  existing pointer-based call interface.
• Added k-point parallel PEXSI execution support and
  global chemical-potential handling for multi-k
  calculations.
• Aligned pexsi_temp with Fermi-Dirac smearing when
  appropriate.
• Optimized BCD-to-CCS H/S redistribution by packing H
  and S values in one redistribution path.
• Added timing labels for matrix conversion and PEXSI
  stages to help identify performance bottlenecks.
• Preserved Gamma real PEXSI behavior for
  options.spin, while applying the spin handling
  needed by the complex multi-k path separately.
• Documented performance, timeout causes, same-input
  Gamma comparison, and remaining bottlenecks in the
  LaTeX refactor report.

Any changes of core modules? (ignore if not

applicable)

• HSolver: extended the PEXSI diagonalization path
  to support complex multi-k calculations and added the
  PEXSI abstraction interface.
• ElecState: updated LCAO density-matrix handling so
  PEXSI-generated DM/EDM can be written back correctly.
• module_pexsi: added complex matrix conversion,
  complex PEXSI execution, timing instrumentation, and
  safer ownership of intermediate buffers.
• No intentional changes were made to unrelated
  ESolver, Hamilt, Operator, or Psi behavior.

[mohanchen]

We don't need your .tex file in this PR

[mohanchen]

What's the py code used for?

[mohanchen]

What's the py code used for?

[mohanchen]

We don't need tex file in the code