Refactor and Fix Parallel Handling of Plane Wave Counts in Eigenvalue Output

[A-006]

Description

This PR addresses the logic for handling and reporting k-point specific plane wave counts (ngk) in the ModuleIO::write_eig_iter and ModuleIO::write_eig_file functions. It ensures that plane wave basis sizes are correctly aggregated across MPI ranks and accurately reported in the output logs.

Technical Breakdown & Conclusions

• $nks$ vs $ngk$:
  • $nks$: Represents the number of symmetry-reduced k-points assigned to the local pool (current group of processors).
  • $ngk$: A std::vector<int> registry. Each element ngk[ik] stores the number of plane waves (G-vectors) used for the $ik$-th k-point (i.e., the basis set dimension).
• Parallel Logic:
  • During parallel execution, kv.ngk is resized to match the local pool size (nks). This explains why the size might appear small (e.g., size 2 for 2 k-points).
  • Since G-vectors for a single k-point are distributed across ranks within a pool, an MPI_Allreduce with SUM over POOL_WORLD is required to obtain the total physical basis size for each k-point.

Changes

• Corrected Basis Summation: Implemented MPI_Allreduce to aggregate ngk counts across ranks in the same pool, ensuring the output reports the true total number of plane waves per k-point.
• Refined Multi-Pool Collection: Updated the logic where Global Rank 0 gathers eigenvalues (ekb) and occupations (wg) from other pools to prevent output race conditions.
• Cleanup: Removed redundant debug std::cout statements regarding ngk sizes while maintaining improved logging formats.

Verification (Commit: 80281a5)

• Test Case: 009_Li27Ni9O54Mn9Co9 (PW basis).
• Environment: 28-process MPI run with KPAR enabled.
• Result: Verified that even when nks is small, the plane wave counts are correctly calculated and stored, with the final output reflecting the full physical basis set assigned to those k-points.