Band Structure Comparison

This document validates Q3's band structure and density of states implementation against Quantum ESPRESSO (QE) for both norm-conserving (NC) and ultrasoft (US) pseudopotentials on two reference systems: silicon (semiconductor) and copper (metal).

All calculations use PBE/GGA via LibXC (QE runs use ‘input_dft = 'XC-000I-000I-101L-130L’` to match Q3's XC treatment exactly).

---

Setup

Parameter	Silicon	Copper
Pseudopotential (NC)	Si.upf (48 Ry)	Cu.upf (120 Ry)
Pseudopotential (US)	Si_PBE_USPP.UPF (40 Ry)	cu_pbe_v1.2.uspp.F.UPF (40 Ry)
SCF k-grid	8×8×8	8×8×8
SCF k-grid shifts	NC: 0 0 0 (Γ-centered) / US: 1 1 1	1 1 1 (both NC and US)
Smearing	none (insulator)	Gaussian, σ = 0.02 Ry
Band path	Γ–X–W–K–Γ–L (FCC), 30 pts/segment (151 total)	same
Number of bands	12	20

K-path distances are computed analytically from the crystal-fractional k-coordinates and the primitive reciprocal lattice matrix, so Q3 and QE share identical x-axis positions by construction.

Fermi energy reference: always taken from the SCF step. The NSCF Fermi energy (computed by smearing bisection on band-path k-points) is physically wrong for metals — the band-path k-points are not a proper Monkhorst–Pack grid. This is fixed in src/py/q3/workflows/band_structure.py.

Why Si NC uses Γ-centered shifts (<tt>0 0 0</tt>)

Si NC deliberately uses a Γ-centered grid (shifts 0 0 0) while all other configurations use half-grid shifts (1 1 1). This is intentional:

• High-symmetry k-point coverage: The Γ-centered grid includes high-symmetry points (Γ, X, L) that exercise special code paths — e.g., |G+k| = 0 at Γ, degenerate eigenvalues, and symmetry edge cases.
• Unequal k-point weights: With 0 0 0 shifts, the irreducible Brillouin zone has 29 k-points with unequal weights (some high-symmetry points have lower multiplicity). This stress-tests the k-point weighting logic. With 1 1 1 shifts, all 60 IBZ k-points have equal weight, which would mask weighting bugs.
• Proven value: This choice caught a k-point weight propagation bug that was invisible in the 1 1 1-shifted configurations.

The consequence is that the Si NC DOS appears slightly spikier than Si US DOS (29 k-points vs 60 for the same 8×8×8 grid). Both are correct — they would converge to the same DOS shape at denser grids.

---

Silicon (8×8×8 SCF grid)

Band structure with DOS

Numerical comparison

Metric	Si NC (Q3 vs QE)	Si US (Q3 vs QE)
max|ΔE|	22 meV	2 meV
mean|ΔE|	0.1 meV	1.3 meV
IBZ k-points	29	60
Fermi energy	6.337 eV (both)	6.391 eV (both)

Observations

• Valence bands (< 0 eV): All four datasets overlap essentially perfectly. Q3 NC vs QE NC mean agreement = 0.1 meV, Q3 US vs QE US = 1.3 meV — well within any physically meaningful tolerance.
• Band gap: ~2.4 eV indirect (VBM at Γ, CBM along Γ→X), correct in all four calculations.
• DOS agreement: Q3 and QE DOS overlap closely in all cases. Si NC shows slightly more structured peaks due to fewer IBZ k-points (29 vs 60); this is expected and not a discrepancy (see the k-grid shift discussion above).
• Low conduction (0–6 eV): Q3 and QE agree to sub-meV mean — cusps land exactly on the high-symmetry tick marks.
• Q3 NC extra high-energy states: Q3 NC shows extra bands near Γ above ~6 eV that are absent from QE NC and both US calculations. These are unphysical states from the 48 Ry NC PP in the upper conduction region and do not affect valence properties.
• Residual Q3/QE discrepancy source: Q3's PlaneWaveSelector adds a 1e-8 rounding tolerance to the cutoff condition, potentially including 1–2 extra G-vectors at boundary k-points compared to QE's exact ≤ Ecut check. This accounts for the ~2–22 meV difference and is independent of the SCF k-grid (confirmed: identical results for 4×4×4 and 8×8×8 grids).

---

Copper (8×8×8 SCF grid)

Band structure with DOS

Numerical comparison

Metric	Cu NC (Q3 vs QE)	Cu US (Q3 vs QE)
max|ΔE|	1 meV	1 meV
mean|ΔE|	0.5 meV	0.7 meV
IBZ k-points	60	60
Fermi energy	16.913 eV (both)	15.947 eV (both)

Observations

• All four datasets are essentially on top of each other across the full d-band and s-band regions.
• Q3 vs QE: sub-meV agreement for both NC and US pseudopotentials.
• d-band manifold (−1 to −5 eV): correct dispersion and crossings; high-symmetry degeneracies sit precisely on tick marks.
• DOS agreement: Q3 and QE DOS curves overlap very closely — the characteristic d-band peak near −3 eV and the free-electron-like tail above the Fermi level are reproduced accurately.
• The Cu band structure has no spurious states or residual artifacts after band continuity tracking.

---

Density of States

The DOS panel on each band structure plot is computed from an 8×8×8 SCF k-grid with Gaussian broadening (σ = 0.022 Ry ≈ 0.30 eV). Both Q3 and QE use k-point-weighted DOS:

\[\text{DOS}(E) = \sum_k w_k \sum_n \frac{1}{\sigma\sqrt{2\pi}} \exp\!\Bigl(-\frac{(E - \epsilon_{nk})^2}{2\sigma^2}\Bigr)\]

where \(w_k\) are the symmetry-reduced k-point integration weights (summing to 2.0 for spin degeneracy) and \(\epsilon_{nk}\) are the eigenvalues.

---

SCF Total Energy Agreement

All total energies compared with Q3's own material JSON used to derive QE's celldm(1), ensuring identical lattice parameters. Both codes use identical k-grids (dimensions + shifts), cutoffs, and LibXC PBE.

Band structure SCF (8×8×8 k-grid)

Case	Q3 (Ry)	QE (Ry)	|ΔE| (mRy)	Status
Si NC	−16.924975	−16.924975	0.0002	✅
Si US	−15.728547	−15.728547	0.0002	✅
Cu NC	−386.586889	−386.586889	0.0002	✅
Cu US	−403.782757	−403.782743	0.0137	✅

Γ-point eigenvalue agreement (Si NC, 8×8×8 SCF)

Band	Q3 (eV)	QE (eV)	|Δ| (meV)
1	−5.8076	−5.8078	0.2
2–4 (VB top)	6.0136	6.0134	0.2
5–7	8.5604	8.5603	0.1
8	9.1157	9.1155	0.2
9–10	13.7287	13.7286	0.1
11	13.8922	13.8918	0.4
12	17.1963	17.1962	0.1
Max	—	—	0.4 meV

These results are codified in the automated test test_qe_energy_comparison.py, which runs all four cases (Si NC/US, Cu NC/US) and asserts |ΔE| < 0.05 mRy (NC) / < 0.1 mRy (US).

---

Summary

Question	Answer
K-path alignment	✅ Correct — analytical distances; HS degeneracies on tick marks
Q3 matches QE (Si NC)?	✅ Yes — max|ΔE| = 22 meV, mean|ΔE| = 0.1 meV
Q3 matches QE (Si US)?	✅ Yes — max|ΔE| = 2 meV, mean|ΔE| = 1.3 meV
Q3 matches QE (Cu NC)?	✅ Yes — max|ΔE| = 1 meV, mean|ΔE| = 0.5 meV
Q3 matches QE (Cu US)?	✅ Yes — max|ΔE| = 1 meV, mean|ΔE| = 0.7 meV
DOS agreement	✅ Q3 and QE DOS overlap closely for all four cases
NC/US self-consistent?	✅ Valence bands indistinguishable within each code
Fermi energy (metals)	✅ Fixed — SCF E_F used, not the wrong NSCF value
SCF total energy (Si NC/US)	✅ |ΔE| < 0.001 mRy vs QE
SCF total energy (Cu NC/US)	✅ |ΔE| < 0.014 mRy vs QE
Si NC extra conduction bands	⚠️ Extra states > 6 eV from ecut+1e-8 boundary — PP rounding artifact

---

Known Issues and Caveats

G-vector cutoff rounding tolerance

Q3's PlaneWaveSelector uses |k+G|² ≤ E_cut + 1e-8 while QE uses strict |k+G|² ≤ E_cut. At boundary k-points, Q3 may include 1–2 extra G-vectors, producing a slightly larger basis and hence meV-level eigenvalue differences. This is independent of SCF k-grid convergence. To achieve sub-meV agreement with QE, the tolerance in src/layer_3_representation/basis/planewave/internal/kpoint/plane_wave_selector.hpp (line 52) should be set to zero.

NSCF Fermi energy for metals

Fixed in src/py/q3/workflows/band_structure.py: the workflow now always uses the SCF Fermi energy as the band-structure reference energy (the NSCF value was ~−96 eV for Cu — physically wrong).

Si NC high-energy conduction bands

Q3 NC shows extra conduction states above ~6 eV near Γ not present in QE. Origin under investigation; does not affect valence bands or band gap.

---

Reproducing These Results

# Run Q3 band structure (silicon NC, 8×8×8)
cd examples/band_structure--semiconductor--silicon
../../build/q3 nc.yml        # SCF + NSCF
 
# Run automated QE vs Q3 energy comparison test
.venv/bin/pytest tests/validation/test_qe_energy_comparison.py -v --q3-bin ./build/q3
 
# Regenerate all band structure + DOS plots
.venv/bin/python scripts/plot/plot_bands_with_dos.py