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Test symmetry analysis#89

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thchr wants to merge 1 commit intomainfrom
symeigs-test
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Test symmetry analysis#89
thchr wants to merge 1 commit intomainfrom
symeigs-test

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@thchr thchr commented Aug 14, 2025 •
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This is really more of a test of the symmetry analysis tools than something we can merge as-is.

What this currently does it to try to build a tight-binding model for every EBR (also those with free parameters) and then tests that collect_compatible returns vectors that sum to the EBR symmetry vectors. Unfortunately, there are a lot failures, suggesting that we still haven't gotten the implementation quite right.

I've appended the list of remaining issues below to see them:

Summary of remaining failures (click to expand)

Plane group failures (D = 2)

#13:
  brs[1] = (1c|A):      symmetry vector discrepancy
    n   = [KA₃, Γ₁, K₂] (1 band)
    cbr = [KA₂, Γ₁, K₃] (1 band)
  brs[2] = (1c|E):      symmetry vector discrepancy
    n   = [KA₁+KA₂, Γ₂Γ₃, K₁+K₃] (2 bands)
    cbr = [KA₁+KA₃, Γ₂Γ₃, K₁+K₂] (2 bands)
  brs[3] = (1b|A):      symmetry vector discrepancy
    n   = [KA₂, Γ₁, K₃] (1 band)
    cbr = [KA₃, Γ₁, K₂] (1 band)
  brs[4] = (1b|E):      symmetry vector discrepancy
    n   = [KA₁+KA₃, Γ₂Γ₃, K₁+K₂] (2 bands)
    cbr = [KA₁+KA₂, Γ₂Γ₃, K₁+K₃] (2 bands)
#14:
  brs[1] = (1c|A₁):     symmetry vector discrepancy
    n   = [M₁, Γ₁, K₂] (1 band)
    cbr = [M₁, Γ₁, K₃] (1 band)
  brs[2] = (1c|A₂):     symmetry vector discrepancy
    n   = [M₂, Γ₂, K₂] (1 band)
    cbr = [M₂, Γ₂, K₃] (1 band)
  brs[3] = (1c|E):      symmetry vector discrepancy
    n   = [M₁+M₂, Γ₃, K₁+K₃] (2 bands)
    cbr = [M₁+M₂, Γ₃, K₁+K₂] (2 bands)
  brs[4] = (1b|A₁):     symmetry vector discrepancy
    n   = [M₁, Γ₁, K₃] (1 band)
    cbr = [M₁, Γ₁, K₂] (1 band)
  brs[5] = (1b|A₂):     symmetry vector discrepancy
    n   = [M₂, Γ₂, K₃] (1 band)
    cbr = [M₂, Γ₂, K₂] (1 band)
  brs[6] = (1b|E):      symmetry vector discrepancy
    n   = [M₁+M₂, Γ₃, K₁+K₂] (2 bands)
    cbr = [M₁+M₂, Γ₃, K₁+K₃] (2 bands)

Space group failures (D = 3)

#68:
  brs[1] = (8h|A):      symmetry vector discrepancy
    n   = [2Z₁, 2T₁, Y₁⁺+Y₂⁺+Y₁⁻+Y₂⁻, 2S₁, Γ₁⁺+Γ₂⁺+Γ₁⁻+Γ₂⁻, 2R₁] (4 bands)
    cbr = [2Z₁, 2T₂, Y₃⁺+Y₄⁺+Y₃⁻+Y₄⁻, 2S₁, Γ₁⁺+Γ₂⁺+Γ₁⁻+Γ₂⁻, 2R₁] (4 bands)
  brs[2] = (8h|B):      symmetry vector discrepancy
    n   = [2Z₂, 2T₂, Y₃⁺+Y₄⁺+Y₃⁻+Y₄⁻, 2S₁, Γ₃⁺+Γ₄⁺+Γ₃⁻+Γ₄⁻, 2R₁] (4 bands)
    cbr = [2Z₂, 2T₁, Y₁⁺+Y₂⁺+Y₁⁻+Y₂⁻, 2S₁, Γ₃⁺+Γ₄⁺+Γ₃⁻+Γ₄⁻, 2R₁] (4 bands)
  brs[3] = (8d|Ag):     symmetry vector discrepancy
    n   = [Z₁+Z₂, T₁+T₂, Y₁⁺+Y₂⁺+Y₃⁺+Y₄⁺, 2S₁, Γ₁⁺+Γ₂⁺+Γ₃⁺+Γ₄⁺, 2R₁] (4 bands)
    cbr = [Z₁+Z₂, T₁+T₂, Y₁⁻+Y₂⁻+Y₃⁻+Y₄⁻, 2S₁, Γ₁⁺+Γ₂⁺+Γ₃⁺+Γ₄⁺, 2R₁] (4 bands)
  brs[4] = (8d|Aᵤ):     symmetry vector discrepancy
    n   = [Z₁+Z₂, T₁+T₂, Y₁⁻+Y₂⁻+Y₃⁻+Y₄⁻, 2S₁, Γ₁⁻+Γ₂⁻+Γ₃⁻+Γ₄⁻, 2R₁] (4 bands)
    cbr = [Z₁+Z₂, T₁+T₂, Y₁⁺+Y₂⁺+Y₃⁺+Y₄⁺, 2S₁, Γ₁⁻+Γ₂⁻+Γ₃⁻+Γ₄⁻, 2R₁] (4 bands)
  brs[5] = (8c|Ag):     symmetry vector discrepancy
    n   = [Z₁+Z₂, T₁+T₂, Y₁⁻+Y₂⁻+Y₃⁻+Y₄⁻, 2S₁, Γ₁⁺+Γ₂⁺+Γ₃⁺+Γ₄⁺, 2R₁] (4 bands)
    cbr = [Z₁+Z₂, T₁+T₂, Y₁⁺+Y₂⁺+Y₃⁺+Y₄⁺, 2S₁, Γ₁⁺+Γ₂⁺+Γ₃⁺+Γ₄⁺, 2R₁] (4 bands)
  brs[6] = (8c|Aᵤ):     symmetry vector discrepancy
    n   = [Z₁+Z₂, T₁+T₂, Y₁⁺+Y₂⁺+Y₃⁺+Y₄⁺, 2S₁, Γ₁⁻+Γ₂⁻+Γ₃⁻+Γ₄⁻, 2R₁] (4 bands)
    cbr = [Z₁+Z₂, T₁+T₂, Y₁⁻+Y₂⁻+Y₃⁻+Y₄⁻, 2S₁, Γ₁⁻+Γ₂⁻+Γ₃⁻+Γ₄⁻, 2R₁] (4 bands)
  brs[7] = (4b|A):      failed to collect any compatible symmetry vectors
  brs[8] = (4b|B₁):     failed to collect any compatible symmetry vectors
  brs[9] = (4b|B₃):     failed to collect any compatible symmetry vectors
  brs[10] = (4b|B₂):    failed to collect any compatible symmetry vectors
  brs[11] = (4a|A):     failed to collect any compatible symmetry vectors
  brs[12] = (4a|B₁):    failed to collect any compatible symmetry vectors
  brs[13] = (4a|B₃):    failed to collect any compatible symmetry vectors
  brs[14] = (4a|B₂):    failed to collect any compatible symmetry vectors
#82:
  brs[1] = (2d|A):      symmetry vector discrepancy
    n   = [P₄, M₂, N₁, X₂, Γ₁, PA₄] (1 band)
    cbr = [P₃, M₂, N₁, X₂, Γ₁, PA₃] (1 band)
  brs[2] = (2d|B):      symmetry vector discrepancy
    n   = [P₃, M₁, N₁, X₂, Γ₂, PA₃] (1 band)
    cbr = [P₄, M₁, N₁, X₂, Γ₂, PA₄] (1 band)
  brs[4] = (2c|A):      symmetry vector discrepancy
    n   = [P₃, M₂, N₁, X₂, Γ₁, PA₃] (1 band)
    cbr = [P₄, M₂, N₁, X₂, Γ₁, PA₄] (1 band)
  brs[5] = (2c|B):      symmetry vector discrepancy
    n   = [P₄, M₁, N₁, X₂, Γ₂, PA₄] (1 band)
    cbr = [P₃, M₁, N₁, X₂, Γ₂, PA₃] (1 band)
#88:
  brs[5] = (4b|A):      failed to collect any compatible symmetry vectors
  brs[6] = (4b|B):      failed to collect any compatible symmetry vectors
  brs[8] = (4a|A):      failed to collect any compatible symmetry vectors
  brs[9] = (4a|B):      failed to collect any compatible symmetry vectors
#108:
  brs[1] = (4b|A₁):     failed to collect any compatible symmetry vectors
  brs[2] = (4b|A₂):     failed to collect any compatible symmetry vectors
#119:
  brs[1] = (2d|A₁):     symmetry vector discrepancy
    n   = [P₄, M₂, N₁, X₃, Γ₁] (1 band)
    cbr = [P₃, M₂, N₁, X₃, Γ₁] (1 band)
  brs[2] = (2d|B₁):     symmetry vector discrepancy
    n   = [P₃, M₁, N₁, X₄, Γ₂] (1 band)
    cbr = [P₄, M₁, N₁, X₄, Γ₂] (1 band)
  brs[3] = (2d|B₂):     symmetry vector discrepancy
    n   = [P₄, M₄, N₂, X₄, Γ₃] (1 band)
    cbr = [P₃, M₄, N₂, X₄, Γ₃] (1 band)
  brs[4] = (2d|A₂):     symmetry vector discrepancy
    n   = [P₃, M₃, N₂, X₃, Γ₄] (1 band)
    cbr = [P₄, M₃, N₂, X₃, Γ₄] (1 band)
  brs[6] = (2c|A₁):     symmetry vector discrepancy
    n   = [P₃, M₂, N₁, X₃, Γ₁] (1 band)
    cbr = [P₄, M₂, N₁, X₃, Γ₁] (1 band)
  brs[7] = (2c|B₁):     symmetry vector discrepancy
    n   = [P₄, M₁, N₁, X₄, Γ₂] (1 band)
    cbr = [P₃, M₁, N₁, X₄, Γ₂] (1 band)
  brs[8] = (2c|B₂):     symmetry vector discrepancy
    n   = [P₃, M₄, N₂, X₄, Γ₃] (1 band)
    cbr = [P₄, M₄, N₂, X₄, Γ₃] (1 band)
  brs[9] = (2c|A₂):     symmetry vector discrepancy
    n   = [P₄, M₃, N₂, X₃, Γ₄] (1 band)
    cbr = [P₃, M₃, N₂, X₃, Γ₄] (1 band)
#122:
  brs[3] = (4b|A):      failed to collect any compatible symmetry vectors
  brs[4] = (4b|B):      failed to collect any compatible symmetry vectors
  brs[6] = (4a|A):      failed to collect any compatible symmetry vectors
  brs[7] = (4a|B):      failed to collect any compatible symmetry vectors
#140:
  brs[1] = (8e|Ag):     failed to collect any compatible symmetry vectors
  brs[2] = (8e|Aᵤ):     failed to collect any compatible symmetry vectors
  brs[3] = (8e|Bg):     failed to collect any compatible symmetry vectors
  brs[4] = (8e|Bᵤ):     failed to collect any compatible symmetry vectors
  brs[5] = (4d|Ag):     failed to collect any compatible symmetry vectors
  brs[6] = (4d|Aᵤ):     failed to collect any compatible symmetry vectors
  brs[7] = (4d|B₁g):    failed to collect any compatible symmetry vectors
  brs[8] = (4d|B₁ᵤ):    failed to collect any compatible symmetry vectors
  brs[19] = (4b|A₁):    failed to collect any compatible symmetry vectors
  brs[20] = (4b|B₁):    failed to collect any compatible symmetry vectors
  brs[21] = (4b|B₂):    failed to collect any compatible symmetry vectors
  brs[22] = (4b|A₂):    failed to collect any compatible symmetry vectors
#141:
  brs[9] = (4b|A₁):     failed to collect any compatible symmetry vectors
  brs[10] = (4b|B₁):    failed to collect any compatible symmetry vectors
  brs[11] = (4b|B₂):    failed to collect any compatible symmetry vectors
  brs[12] = (4b|A₂):    failed to collect any compatible symmetry vectors
  brs[14] = (4a|A₁):    failed to collect any compatible symmetry vectors
  brs[15] = (4a|B₁):    failed to collect any compatible symmetry vectors
  brs[16] = (4a|B₂):    failed to collect any compatible symmetry vectors
  brs[17] = (4a|A₂):    failed to collect any compatible symmetry vectors
#142:
  brs[1] = (16e|A):     failed to collect any compatible symmetry vectors
  brs[2] = (16e|B):     failed to collect any compatible symmetry vectors
  brs[5] = (8b|A):      failed to collect any compatible symmetry vectors
  brs[8] = (8b|B₂):     failed to collect any compatible symmetry vectors
#143:
  brs[1] = (1c|A):      symmetry vector discrepancy
    n   = [KA₂, L₁, M₁, H₂, HA₂, A₁, Γ₁, K₂] (1 band)
    cbr = [KA₃, L₁, M₁, H₃, HA₃, A₁, Γ₁, K₃] (1 band)
  brs[2] = (1c|E):      symmetry vector discrepancy
    n   = [KA₁+KA₃, 2L₁, 2M₁, H₁+H₃, HA₁+HA₃, A₂A₃, Γ₂Γ₃, K₁+K₃] (2 bands)
    cbr = [KA₁+KA₂, 2L₁, 2M₁, H₁+H₂, HA₁+HA₂, A₂A₃, Γ₂Γ₃, K₁+K₂] (2 bands)
  brs[3] = (1b|A):      symmetry vector discrepancy
    n   = [KA₃, L₁, M₁, H₃, HA₃, A₁, Γ₁, K₃] (1 band)
    cbr = [KA₂, L₁, M₁, H₂, HA₂, A₁, Γ₁, K₂] (1 band)
  brs[4] = (1b|E):      symmetry vector discrepancy
    n   = [KA₁+KA₂, 2L₁, 2M₁, H₁+H₂, HA₁+HA₂, A₂A₃, Γ₂Γ₃, K₁+K₂] (2 bands)
    cbr = [KA₁+KA₃, 2L₁, 2M₁, H₁+H₃, HA₁+HA₃, A₂A₃, Γ₂Γ₃, K₁+K₃] (2 bands)
#149:
  brs[1] = (1f|A₁):     symmetry vector discrepancy
    n   = [L₂, M₁, H₂, A₂, Γ₁, K₂] (1 band)
    cbr = [L₂, M₁, H₃, A₂, Γ₁, K₃] (1 band)
  brs[2] = (1f|A₂):     symmetry vector discrepancy
    n   = [L₁, M₂, H₂, A₁, Γ₂, K₂] (1 band)
    cbr = [L₁, M₂, H₃, A₁, Γ₂, K₃] (1 band)
  brs[3] = (1f|E):      symmetry vector discrepancy
    n   = [L₁+L₂, M₁+M₂, H₁+H₃, A₃, Γ₃, K₁+K₃] (2 bands)
    cbr = [L₁+L₂, M₁+M₂, H₁+H₂, A₃, Γ₃, K₁+K₂] (2 bands)
  brs[4] = (1e|A₁):     symmetry vector discrepancy
    n   = [L₁, M₁, H₂, A₁, Γ₁, K₂] (1 band)
    cbr = [L₁, M₁, H₃, A₁, Γ₁, K₃] (1 band)
  brs[5] = (1e|A₂):     symmetry vector discrepancy
    n   = [L₂, M₂, H₂, A₂, Γ₂, K₂] (1 band)
    cbr = [L₂, M₂, H₃, A₂, Γ₂, K₃] (1 band)
  brs[6] = (1e|E):      symmetry vector discrepancy
    n   = [L₁+L₂, M₁+M₂, H₁+H₃, A₃, Γ₃, K₁+K₃] (2 bands)
    cbr = [L₁+L₂, M₁+M₂, H₁+H₂, A₃, Γ₃, K₁+K₂] (2 bands)
  brs[7] = (1d|A₁):     symmetry vector discrepancy
    n   = [L₂, M₁, H₃, A₂, Γ₁, K₃] (1 band)
    cbr = [L₂, M₁, H₂, A₂, Γ₁, K₂] (1 band)
  brs[8] = (1d|A₂):     symmetry vector discrepancy
    n   = [L₁, M₂, H₃, A₁, Γ₂, K₃] (1 band)
    cbr = [L₁, M₂, H₂, A₁, Γ₂, K₂] (1 band)
  brs[9] = (1d|E):      symmetry vector discrepancy
    n   = [L₁+L₂, M₁+M₂, H₁+H₂, A₃, Γ₃, K₁+K₂] (2 bands)
    cbr = [L₁+L₂, M₁+M₂, H₁+H₃, A₃, Γ₃, K₁+K₃] (2 bands)
  brs[10] = (1c|A₁):    symmetry vector discrepancy
    n   = [L₁, M₁, H₃, A₁, Γ₁, K₃] (1 band)
    cbr = [L₁, M₁, H₂, A₁, Γ₁, K₂] (1 band)
  brs[11] = (1c|A₂):    symmetry vector discrepancy
    n   = [L₂, M₂, H₃, A₂, Γ₂, K₃] (1 band)
    cbr = [L₂, M₂, H₂, A₂, Γ₂, K₂] (1 band)
  brs[12] = (1c|E):     symmetry vector discrepancy
    n   = [L₁+L₂, M₁+M₂, H₁+H₂, A₃, Γ₃, K₁+K₂] (2 bands)
    cbr = [L₁+L₂, M₁+M₂, H₁+H₃, A₃, Γ₃, K₁+K₃] (2 bands)
#156:
  brs[1] = (1c|A₁):     symmetry vector discrepancy
    n   = [L₁, M₁, H₂, A₁, Γ₁, K₂] (1 band)
    cbr = [L₁, M₁, H₃, A₁, Γ₁, K₃] (1 band)
  brs[2] = (1c|A₂):     symmetry vector discrepancy
    n   = [L₂, M₂, H₂, A₂, Γ₂, K₂] (1 band)
    cbr = [L₂, M₂, H₃, A₂, Γ₂, K₃] (1 band)
  brs[3] = (1c|E):      symmetry vector discrepancy
    n   = [L₁+L₂, M₁+M₂, H₁+H₃, A₃, Γ₃, K₁+K₃] (2 bands)
    cbr = [L₁+L₂, M₁+M₂, H₁+H₂, A₃, Γ₃, K₁+K₂] (2 bands)
  brs[4] = (1b|A₁):     symmetry vector discrepancy
    n   = [L₁, M₁, H₃, A₁, Γ₁, K₃] (1 band)
    cbr = [L₁, M₁, H₂, A₁, Γ₁, K₂] (1 band)
  brs[5] = (1b|A₂):     symmetry vector discrepancy
    n   = [L₂, M₂, H₃, A₂, Γ₂, K₃] (1 band)
    cbr = [L₂, M₂, H₂, A₂, Γ₂, K₂] (1 band)
  brs[6] = (1b|E):      symmetry vector discrepancy
    n   = [L₁+L₂, M₁+M₂, H₁+H₂, A₃, Γ₃, K₁+K₂] (2 bands)
    cbr = [L₁+L₂, M₁+M₂, H₁+H₃, A₃, Γ₃, K₁+K₃] (2 bands)
#158:
  brs[1] = (2c|A):      symmetry vector discrepancy
    n   = [L₁L₂, M₁+M₂, H₂H₂, A₁A₂, Γ₁+Γ₂, 2K₂] (2 bands)
    cbr = [L₁L₂, M₁+M₂, H₃H₃, A₁A₂, Γ₁+Γ₂, 2K₃] (2 bands)
  brs[2] = (2c|E):      symmetry vector discrepancy
    n   = [2L₁L₂, 2M₁+2M₂, H₁H₁+H₃H₃, A₃A₃, 2Γ₃, 2K₁+2K₃] (4 bands)
    cbr = [2L₁L₂, 2M₁+2M₂, H₁H₁+H₂H₂, A₃A₃, 2Γ₃, 2K₁+2K₂] (4 bands)
  brs[3] = (2b|A):      symmetry vector discrepancy
    n   = [L₁L₂, M₁+M₂, H₃H₃, A₁A₂, Γ₁+Γ₂, 2K₃] (2 bands)
    cbr = [L₁L₂, M₁+M₂, H₂H₂, A₁A₂, Γ₁+Γ₂, 2K₂] (2 bands)
  brs[4] = (2b|E):      symmetry vector discrepancy
    n   = [2L₁L₂, 2M₁+2M₂, H₁H₁+H₂H₂, A₃A₃, 2Γ₃, 2K₁+2K₂] (4 bands)
    cbr = [2L₁L₂, 2M₁+2M₂, H₁H₁+H₃H₃, A₃A₃, 2Γ₃, 2K₁+2K₃] (4 bands)
#174:
  brs[1] = (1f|A′):     symmetry vector discrepancy
    n   = [KA₃, L₂, M₁, H₄, HA₄, A₂, Γ₁, K₃] (1 band)
    cbr = [KA₅, L₂, M₁, H₆, HA₆, A₂, Γ₁, K₅] (1 band)
  brs[2] = (1f|A′′):    symmetry vector discrepancy
    n   = [KA₄, L₁, M₂, H₃, HA₃, A₁, Γ₂, K₄] (1 band)
    cbr = [KA₆, L₁, M₂, H₅, HA₅, A₁, Γ₂, K₆] (1 band)
  brs[3] = (1f|E′):     symmetry vector discrepancy
    n   = [KA₁+KA₅, 2L₂, 2M₁, H₂+H₆, HA₂+HA₆, A₄A₆, Γ₃Γ₅, K₁+K₅] (2 bands)
    cbr = [KA₁+KA₃, 2L₂, 2M₁, H₂+H₄, HA₂+HA₄, A₄A₆, Γ₃Γ₅, K₁+K₃] (2 bands)
  brs[4] = (1f|E′′):    symmetry vector discrepancy
    n   = [KA₂+KA₆, 2L₁, 2M₂, H₁+H₅, HA₁+HA₅, A₃A₅, Γ₄Γ₆, K₂+K₆] (2 bands)
    cbr = [KA₂+KA₄, 2L₁, 2M₂, H₁+H₃, HA₁+HA₃, A₃A₅, Γ₄Γ₆, K₂+K₄] (2 bands)
  brs[5] = (1e|A′):     symmetry vector discrepancy
    n   = [KA₃, L₁, M₁, H₃, HA₃, A₁, Γ₁, K₃] (1 band)
    cbr = [KA₅, L₁, M₁, H₅, HA₅, A₁, Γ₁, K₅] (1 band)
  brs[6] = (1e|A′′):    symmetry vector discrepancy
    n   = [KA₄, L₂, M₂, H₄, HA₄, A₂, Γ₂, K₄] (1 band)
    cbr = [KA₆, L₂, M₂, H₆, HA₆, A₂, Γ₂, K₆] (1 band)
  brs[7] = (1e|E′):     symmetry vector discrepancy
    n   = [KA₁+KA₅, 2L₁, 2M₁, H₁+H₅, HA₁+HA₅, A₃A₅, Γ₃Γ₅, K₁+K₅] (2 bands)
    cbr = [KA₁+KA₃, 2L₁, 2M₁, H₁+H₃, HA₁+HA₃, A₃A₅, Γ₃Γ₅, K₁+K₃] (2 bands)
  brs[8] = (1e|E′′):    symmetry vector discrepancy
    n   = [KA₂+KA₆, 2L₂, 2M₂, H₂+H₆, HA₂+HA₆, A₄A₆, Γ₄Γ₆, K₂+K₆] (2 bands)
    cbr = [KA₂+KA₄, 2L₂, 2M₂, H₂+H₄, HA₂+HA₄, A₄A₆, Γ₄Γ₆, K₂+K₄] (2 bands)
  brs[9] = (1d|A′):     symmetry vector discrepancy
    n   = [KA₅, L₂, M₁, H₆, HA₆, A₂, Γ₁, K₅] (1 band)
    cbr = [KA₃, L₂, M₁, H₄, HA₄, A₂, Γ₁, K₃] (1 band)
  brs[10] = (1d|A′′):   symmetry vector discrepancy
    n   = [KA₆, L₁, M₂, H₅, HA₅, A₁, Γ₂, K₆] (1 band)
    cbr = [KA₄, L₁, M₂, H₃, HA₃, A₁, Γ₂, K₄] (1 band)
  brs[11] = (1d|E′):    symmetry vector discrepancy
    n   = [KA₁+KA₃, 2L₂, 2M₁, H₂+H₄, HA₂+HA₄, A₄A₆, Γ₃Γ₅, K₁+K₃] (2 bands)
    cbr = [KA₁+KA₅, 2L₂, 2M₁, H₂+H₆, HA₂+HA₆, A₄A₆, Γ₃Γ₅, K₁+K₅] (2 bands)
  brs[12] = (1d|E′′):   symmetry vector discrepancy
    n   = [KA₂+KA₄, 2L₁, 2M₂, H₁+H₃, HA₁+HA₃, A₃A₅, Γ₄Γ₆, K₂+K₄] (2 bands)
    cbr = [KA₂+KA₆, 2L₁, 2M₂, H₁+H₅, HA₁+HA₅, A₃A₅, Γ₄Γ₆, K₂+K₆] (2 bands)
  brs[13] = (1c|A′):    symmetry vector discrepancy
    n   = [KA₅, L₁, M₁, H₅, HA₅, A₁, Γ₁, K₅] (1 band)
    cbr = [KA₃, L₁, M₁, H₃, HA₃, A₁, Γ₁, K₃] (1 band)
  brs[14] = (1c|A′′):   symmetry vector discrepancy
    n   = [KA₆, L₂, M₂, H₆, HA₆, A₂, Γ₂, K₆] (1 band)
    cbr = [KA₄, L₂, M₂, H₄, HA₄, A₂, Γ₂, K₄] (1 band)
  brs[15] = (1c|E′):    symmetry vector discrepancy
    n   = [KA₁+KA₃, 2L₁, 2M₁, H₁+H₃, HA₁+HA₃, A₃A₅, Γ₃Γ₅, K₁+K₃] (2 bands)
    cbr = [KA₁+KA₅, 2L₁, 2M₁, H₁+H₅, HA₁+HA₅, A₃A₅, Γ₃Γ₅, K₁+K₅] (2 bands)
  brs[16] = (1c|E′′):   symmetry vector discrepancy
    n   = [KA₂+KA₄, 2L₂, 2M₂, H₂+H₄, HA₂+HA₄, A₄A₆, Γ₄Γ₆, K₂+K₄] (2 bands)
    cbr = [KA₂+KA₆, 2L₂, 2M₂, H₂+H₆, HA₂+HA₆, A₄A₆, Γ₄Γ₆, K₂+K₆] (2 bands)
#176:
  brs[3] = (2d|A′):     symmetry vector discrepancy
    n   = [L₁, M₁⁺+M₂⁻, H₃H₆, A₁, Γ₁⁺+Γ₂⁻, K₃K₅] (2 bands)
    cbr = [L₁, M₁⁺+M₂⁻, H₄H₅, A₁, Γ₁⁺+Γ₂⁻, K₃K₅] (2 bands)
  brs[4] = (2d|A′′):    symmetry vector discrepancy
    n   = [L₁, M₂⁺+M₁⁻, H₄H₅, A₁, Γ₂⁺+Γ₁⁻, K₄K₆] (2 bands)
    cbr = [L₁, M₂⁺+M₁⁻, H₃H₆, A₁, Γ₂⁺+Γ₁⁻, K₄K₆] (2 bands)
  brs[5] = (2d|E′):     symmetry vector discrepancy
    n   = [2L₁, 2M₁⁺+2M₂⁻, H₁H₂+H₄H₅, A₂A₃, Γ₃⁺Γ₅⁺+Γ₄⁻Γ₆⁻, 2K₁+K₃K₅] (4 bands)
    cbr = [2L₁, 2M₁⁺+2M₂⁻, H₁H₂+H₃H₆, A₂A₃, Γ₃⁺Γ₅⁺+Γ₄⁻Γ₆⁻, 2K₁+K₃K₅] (4 bands)
  brs[6] = (2d|E′′):    symmetry vector discrepancy
    n   = [2L₁, 2M₂⁺+2M₁⁻, H₁H₂+H₃H₆, A₂A₃, Γ₄⁺Γ₆⁺+Γ₃⁻Γ₅⁻, 2K₂+K₄K₆] (4 bands)
    cbr = [2L₁, 2M₂⁺+2M₁⁻, H₁H₂+H₄H₅, A₂A₃, Γ₄⁺Γ₆⁺+Γ₃⁻Γ₅⁻, 2K₂+K₄K₆] (4 bands)
  brs[7] = (2c|A′):     symmetry vector discrepancy
    n   = [L₁, M₁⁺+M₂⁻, H₄H₅, A₁, Γ₁⁺+Γ₂⁻, K₃K₅] (2 bands)
    cbr = [L₁, M₁⁺+M₂⁻, H₃H₆, A₁, Γ₁⁺+Γ₂⁻, K₃K₅] (2 bands)
  brs[8] = (2c|A′′):    symmetry vector discrepancy
    n   = [L₁, M₂⁺+M₁⁻, H₃H₆, A₁, Γ₂⁺+Γ₁⁻, K₄K₆] (2 bands)
    cbr = [L₁, M₂⁺+M₁⁻, H₄H₅, A₁, Γ₂⁺+Γ₁⁻, K₄K₆] (2 bands)
  brs[9] = (2c|E′):     symmetry vector discrepancy
    n   = [2L₁, 2M₁⁺+2M₂⁻, H₁H₂+H₃H₆, A₂A₃, Γ₃⁺Γ₅⁺+Γ₄⁻Γ₆⁻, 2K₁+K₃K₅] (4 bands)
    cbr = [2L₁, 2M₁⁺+2M₂⁻, H₁H₂+H₄H₅, A₂A₃, Γ₃⁺Γ₅⁺+Γ₄⁻Γ₆⁻, 2K₁+K₃K₅] (4 bands)
  brs[10] = (2c|E′′):   symmetry vector discrepancy
    n   = [2L₁, 2M₂⁺+2M₁⁻, H₁H₂+H₄H₅, A₂A₃, Γ₄⁺Γ₆⁺+Γ₃⁻Γ₅⁻, 2K₂+K₄K₆] (4 bands)
    cbr = [2L₁, 2M₂⁺+2M₁⁻, H₁H₂+H₃H₆, A₂A₃, Γ₄⁺Γ₆⁺+Γ₃⁻Γ₅⁻, 2K₂+K₄K₆] (4 bands)
#187:
  brs[1] = (1f|A₁′):    symmetry vector discrepancy
    n   = [L₃, M₁, H₄, A₃, Γ₁, K₃] (1 band)
    cbr = [L₃, M₁, H₆, A₃, Γ₁, K₅] (1 band)
  brs[2] = (1f|A₁′′):   symmetry vector discrepancy
    n   = [L₄, M₂, H₄, A₄, Γ₂, K₃] (1 band)
    cbr = [L₄, M₂, H₆, A₄, Γ₂, K₅] (1 band)
  brs[3] = (1f|A₂′′):   symmetry vector discrepancy
    n   = [L₁, M₃, H₃, A₁, Γ₃, K₄] (1 band)
    cbr = [L₁, M₃, H₅, A₁, Γ₃, K₆] (1 band)
  brs[4] = (1f|A₂′):    symmetry vector discrepancy
    n   = [L₂, M₄, H₃, A₂, Γ₄, K₄] (1 band)
    cbr = [L₂, M₄, H₅, A₂, Γ₄, K₆] (1 band)
  brs[5] = (1f|E′):     symmetry vector discrepancy
    n   = [L₃+L₄, M₁+M₂, H₂+H₆, A₆, Γ₅, K₁+K₅] (2 bands)
    cbr = [L₃+L₄, M₁+M₂, H₂+H₄, A₆, Γ₅, K₁+K₃] (2 bands)
  brs[6] = (1f|E′′):    symmetry vector discrepancy
    n   = [L₁+L₂, M₃+M₄, H₁+H₅, A₅, Γ₆, K₂+K₆] (2 bands)
    cbr = [L₁+L₂, M₃+M₄, H₁+H₃, A₅, Γ₆, K₂+K₄] (2 bands)
  brs[7] = (1e|A₁′):    symmetry vector discrepancy
    n   = [L₁, M₁, H₃, A₁, Γ₁, K₃] (1 band)
    cbr = [L₁, M₁, H₅, A₁, Γ₁, K₅] (1 band)
  brs[8] = (1e|A₁′′):   symmetry vector discrepancy
    n   = [L₂, M₂, H₃, A₂, Γ₂, K₃] (1 band)
    cbr = [L₂, M₂, H₅, A₂, Γ₂, K₅] (1 band)
  brs[9] = (1e|A₂′′):   symmetry vector discrepancy
    n   = [L₃, M₃, H₄, A₃, Γ₃, K₄] (1 band)
    cbr = [L₃, M₃, H₆, A₃, Γ₃, K₆] (1 band)
  brs[10] = (1e|A₂′):   symmetry vector discrepancy
    n   = [L₄, M₄, H₄, A₄, Γ₄, K₄] (1 band)
    cbr = [L₄, M₄, H₆, A₄, Γ₄, K₆] (1 band)
  brs[11] = (1e|E′):    symmetry vector discrepancy
    n   = [L₁+L₂, M₁+M₂, H₁+H₅, A₅, Γ₅, K₁+K₅] (2 bands)
    cbr = [L₁+L₂, M₁+M₂, H₁+H₃, A₅, Γ₅, K₁+K₃] (2 bands)
  brs[12] = (1e|E′′):   symmetry vector discrepancy
    n   = [L₃+L₄, M₃+M₄, H₂+H₆, A₆, Γ₆, K₂+K₆] (2 bands)
    cbr = [L₃+L₄, M₃+M₄, H₂+H₄, A₆, Γ₆, K₂+K₄] (2 bands)
  brs[13] = (1d|A₁′):   symmetry vector discrepancy
    n   = [L₃, M₁, H₆, A₃, Γ₁, K₅] (1 band)
    cbr = [L₃, M₁, H₄, A₃, Γ₁, K₃] (1 band)
  brs[14] = (1d|A₁′′):  symmetry vector discrepancy
    n   = [L₄, M₂, H₆, A₄, Γ₂, K₅] (1 band)
    cbr = [L₄, M₂, H₄, A₄, Γ₂, K₃] (1 band)
  brs[15] = (1d|A₂′′):  symmetry vector discrepancy
    n   = [L₁, M₃, H₅, A₁, Γ₃, K₆] (1 band)
    cbr = [L₁, M₃, H₃, A₁, Γ₃, K₄] (1 band)
  brs[16] = (1d|A₂′):   symmetry vector discrepancy
    n   = [L₂, M₄, H₅, A₂, Γ₄, K₆] (1 band)
    cbr = [L₂, M₄, H₃, A₂, Γ₄, K₄] (1 band)
  brs[17] = (1d|E′):    symmetry vector discrepancy
    n   = [L₃+L₄, M₁+M₂, H₂+H₄, A₆, Γ₅, K₁+K₃] (2 bands)
    cbr = [L₃+L₄, M₁+M₂, H₂+H₆, A₆, Γ₅, K₁+K₅] (2 bands)
  brs[18] = (1d|E′′):   symmetry vector discrepancy
    n   = [L₁+L₂, M₃+M₄, H₁+H₃, A₅, Γ₆, K₂+K₄] (2 bands)
    cbr = [L₁+L₂, M₃+M₄, H₁+H₅, A₅, Γ₆, K₂+K₆] (2 bands)
  brs[19] = (1c|A₁′):   symmetry vector discrepancy
    n   = [L₁, M₁, H₅, A₁, Γ₁, K₅] (1 band)
    cbr = [L₁, M₁, H₃, A₁, Γ₁, K₃] (1 band)
  brs[20] = (1c|A₁′′):  symmetry vector discrepancy
    n   = [L₂, M₂, H₅, A₂, Γ₂, K₅] (1 band)
    cbr = [L₂, M₂, H₃, A₂, Γ₂, K₃] (1 band)
  brs[21] = (1c|A₂′′):  symmetry vector discrepancy
    n   = [L₃, M₃, H₆, A₃, Γ₃, K₆] (1 band)
    cbr = [L₃, M₃, H₄, A₃, Γ₃, K₄] (1 band)
  brs[22] = (1c|A₂′):   symmetry vector discrepancy
    n   = [L₄, M₄, H₆, A₄, Γ₄, K₆] (1 band)
    cbr = [L₄, M₄, H₄, A₄, Γ₄, K₄] (1 band)
  brs[23] = (1c|E′):    symmetry vector discrepancy
    n   = [L₁+L₂, M₁+M₂, H₁+H₃, A₅, Γ₅, K₁+K₃] (2 bands)
    cbr = [L₁+L₂, M₁+M₂, H₁+H₅, A₅, Γ₅, K₁+K₅] (2 bands)
  brs[24] = (1c|E′′):   symmetry vector discrepancy
    n   = [L₃+L₄, M₃+M₄, H₂+H₄, A₆, Γ₆, K₂+K₄] (2 bands)
    cbr = [L₃+L₄, M₃+M₄, H₂+H₆, A₆, Γ₆, K₂+K₆] (2 bands)
#188:
  brs[1] = (2f|A′):     symmetry vector discrepancy
    n   = [L₁, M₁+M₂, H₃H₄, A₁, Γ₁+Γ₂, 2K₃] (2 bands)
    cbr = [L₁, M₁+M₂, H₅H₆, A₁, Γ₁+Γ₂, 2K₅] (2 bands)
  brs[2] = (2f|A′′):    symmetry vector discrepancy
    n   = [L₁, M₃+M₄, H₃H₄, A₁, Γ₃+Γ₄, 2K₄] (2 bands)
    cbr = [L₁, M₃+M₄, H₅H₆, A₁, Γ₃+Γ₄, 2K₆] (2 bands)
  brs[3] = (2f|E′):     symmetry vector discrepancy
    n   = [2L₁, 2M₁+2M₂, H₁H₂+H₅H₆, A₂A₃, 2Γ₅, 2K₁+2K₅] (4 bands)
    cbr = [2L₁, 2M₁+2M₂, H₁H₂+H₃H₄, A₂A₃, 2Γ₅, 2K₁+2K₃] (4 bands)
  brs[4] = (2f|E′′):    symmetry vector discrepancy
    n   = [2L₁, 2M₃+2M₄, H₁H₂+H₅H₆, A₂A₃, 2Γ₆, 2K₂+2K₆] (4 bands)
    cbr = [2L₁, 2M₃+2M₄, H₁H₂+H₃H₄, A₂A₃, 2Γ₆, 2K₂+2K₄] (4 bands)
  brs[5] = (2e|A₁):     symmetry vector discrepancy
    n   = [L₁, M₁+M₄, H₃H₄, A₁, Γ₁+Γ₄, K₃+K₄] (2 bands)
    cbr = [L₁, M₁+M₄, H₅H₆, A₁, Γ₁+Γ₄, K₅+K₆] (2 bands)
  brs[6] = (2e|A₂):     symmetry vector discrepancy
    n   = [L₁, M₂+M₃, H₃H₄, A₁, Γ₂+Γ₃, K₃+K₄] (2 bands)
    cbr = [L₁, M₂+M₃, H₅H₆, A₁, Γ₂+Γ₃, K₅+K₆] (2 bands)
  brs[7] = (2e|E):      symmetry vector discrepancy
    n   = [2L₁, M₁+M₂+M₃+M₄, H₁H₂+H₅H₆, A₂A₃, Γ₅+Γ₆, K₁+K₂+K₅+K₆] (4 bands)
    cbr = [2L₁, M₁+M₂+M₃+M₄, H₁H₂+H₃H₄, A₂A₃, Γ₅+Γ₆, K₁+K₂+K₃+K₄] (4 bands)
  brs[8] = (2d|A′):     symmetry vector discrepancy
    n   = [L₁, M₁+M₂, H₅H₆, A₁, Γ₁+Γ₂, 2K₅] (2 bands)
    cbr = [L₁, M₁+M₂, H₃H₄, A₁, Γ₁+Γ₂, 2K₃] (2 bands)
  brs[9] = (2d|A′′):    symmetry vector discrepancy
    n   = [L₁, M₃+M₄, H₅H₆, A₁, Γ₃+Γ₄, 2K₆] (2 bands)
    cbr = [L₁, M₃+M₄, H₃H₄, A₁, Γ₃+Γ₄, 2K₄] (2 bands)
  brs[10] = (2d|E′):    symmetry vector discrepancy
    n   = [2L₁, 2M₁+2M₂, H₁H₂+H₃H₄, A₂A₃, 2Γ₅, 2K₁+2K₃] (4 bands)
    cbr = [2L₁, 2M₁+2M₂, H₁H₂+H₅H₆, A₂A₃, 2Γ₅, 2K₁+2K₅] (4 bands)
  brs[11] = (2d|E′′):   symmetry vector discrepancy
    n   = [2L₁, 2M₃+2M₄, H₁H₂+H₃H₄, A₂A₃, 2Γ₆, 2K₂+2K₄] (4 bands)
    cbr = [2L₁, 2M₃+2M₄, H₁H₂+H₅H₆, A₂A₃, 2Γ₆, 2K₂+2K₆] (4 bands)
  brs[12] = (2c|A₁):    symmetry vector discrepancy
    n   = [L₁, M₁+M₄, H₅H₆, A₁, Γ₁+Γ₄, K₅+K₆] (2 bands)
    cbr = [L₁, M₁+M₄, H₃H₄, A₁, Γ₁+Γ₄, K₃+K₄] (2 bands)
  brs[13] = (2c|A₂):    symmetry vector discrepancy
    n   = [L₁, M₂+M₃, H₅H₆, A₁, Γ₂+Γ₃, K₅+K₆] (2 bands)
    cbr = [L₁, M₂+M₃, H₃H₄, A₁, Γ₂+Γ₃, K₃+K₄] (2 bands)
  brs[14] = (2c|E):     symmetry vector discrepancy
    n   = [2L₁, M₁+M₂+M₃+M₄, H₁H₂+H₃H₄, A₂A₃, Γ₅+Γ₆, K₁+K₂+K₃+K₄] (4 bands)
    cbr = [2L₁, M₁+M₂+M₃+M₄, H₁H₂+H₅H₆, A₂A₃, Γ₅+Γ₆, K₁+K₂+K₅+K₆] (4 bands)
#190:
  brs[1] = (2d|A′):     symmetry vector discrepancy
    n   = [KA₅, L₁, M₁+M₄, H₁, HA₁, A₃, Γ₁+Γ₄, K₅] (2 bands)
    cbr = [KA₅, L₁, M₁+M₄, H₂, HA₂, A₃, Γ₁+Γ₄, K₅] (2 bands)
  brs[2] = (2d|A′′):    symmetry vector discrepancy
    n   = [KA₆, L₁, M₂+M₃, H₂, HA₂, A₃, Γ₂+Γ₃, K₆] (2 bands)
    cbr = [KA₆, L₁, M₂+M₃, H₁, HA₁, A₃, Γ₂+Γ₃, K₆] (2 bands)
  brs[3] = (2d|E′):     symmetry vector discrepancy
    n   = [KA₁+KA₄+KA₅, 2L₁, 2M₁+2M₄, H₂+H₃, HA₂+HA₃, A₁A₂, 2Γ₅, K₁+K₄+K₅] (4 bands)
    cbr = [KA₁+KA₄+KA₅, 2L₁, 2M₁+2M₄, H₁+H₃, HA₁+HA₃, A₁A₂, 2Γ₅, K₁+K₄+K₅] (4 bands)
  brs[4] = (2d|E′′):    symmetry vector discrepancy
    n   = [KA₂+KA₃+KA₆, 2L₁, 2M₂+2M₃, H₁+H₃, HA₁+HA₃, A₁A₂, 2Γ₆, K₂+K₃+K₆] (4 bands)
    cbr = [KA₂+KA₃+KA₆, 2L₁, 2M₂+2M₃, H₂+H₃, HA₂+HA₃, A₁A₂, 2Γ₆, K₂+K₃+K₆] (4 bands)
  brs[5] = (2c|A′):     symmetry vector discrepancy
    n   = [KA₅, L₁, M₁+M₄, H₂, HA₂, A₃, Γ₁+Γ₄, K₅] (2 bands)
    cbr = [KA₅, L₁, M₁+M₄, H₁, HA₁, A₃, Γ₁+Γ₄, K₅] (2 bands)
  brs[6] = (2c|A′′):    symmetry vector discrepancy
    n   = [KA₆, L₁, M₂+M₃, H₁, HA₁, A₃, Γ₂+Γ₃, K₆] (2 bands)
    cbr = [KA₆, L₁, M₂+M₃, H₂, HA₂, A₃, Γ₂+Γ₃, K₆] (2 bands)
  brs[7] = (2c|E′):     symmetry vector discrepancy
    n   = [KA₁+KA₄+KA₅, 2L₁, 2M₁+2M₄, H₁+H₃, HA₁+HA₃, A₁A₂, 2Γ₅, K₁+K₄+K₅] (4 bands)
    cbr = [KA₁+KA₄+KA₅, 2L₁, 2M₁+2M₄, H₂+H₃, HA₂+HA₃, A₁A₂, 2Γ₅, K₁+K₄+K₅] (4 bands)
  brs[8] = (2c|E′′):    symmetry vector discrepancy
    n   = [KA₂+KA₃+KA₆, 2L₁, 2M₂+2M₃, H₂+H₃, HA₂+HA₃, A₁A₂, 2Γ₆, K₂+K₃+K₆] (4 bands)
    cbr = [KA₂+KA₃+KA₆, 2L₁, 2M₂+2M₃, H₁+H₃, HA₁+HA₃, A₁A₂, 2Γ₆, K₂+K₃+K₆] (4 bands)
#194:
  brs[5] = (2d|A₁′):    symmetry vector discrepancy
    n   = [L₁, M₁⁺+M₄⁻, H₁, A₁, Γ₁⁺+Γ₄⁻, K₅] (2 bands)
    cbr = [L₁, M₁⁺+M₄⁻, H₂, A₁, Γ₁⁺+Γ₄⁻, K₅] (2 bands)
  brs[6] = (2d|A₁′′):   symmetry vector discrepancy
    n   = [L₂, M₂⁺+M₃⁻, H₁, A₂, Γ₂⁺+Γ₃⁻, K₅] (2 bands)
    cbr = [L₂, M₂⁺+M₃⁻, H₂, A₂, Γ₂⁺+Γ₃⁻, K₅] (2 bands)
  brs[7] = (2d|A₂′′):   symmetry vector discrepancy
    n   = [L₁, M₃⁺+M₂⁻, H₂, A₁, Γ₃⁺+Γ₂⁻, K₆] (2 bands)
    cbr = [L₁, M₃⁺+M₂⁻, H₁, A₁, Γ₃⁺+Γ₂⁻, K₆] (2 bands)
  brs[8] = (2d|A₂′):    symmetry vector discrepancy
    n   = [L₂, M₄⁺+M₁⁻, H₂, A₂, Γ₄⁺+Γ₁⁻, K₆] (2 bands)
    cbr = [L₂, M₄⁺+M₁⁻, H₁, A₂, Γ₄⁺+Γ₁⁻, K₆] (2 bands)
  brs[9] = (2d|E′):     symmetry vector discrepancy
    n   = [L₁+L₂, M₁⁺+M₂⁺+M₃⁻+M₄⁻, H₂+H₃, A₃, Γ₅⁺+Γ₆⁻, K₁+K₄+K₅] (4 bands)
    cbr = [L₁+L₂, M₁⁺+M₂⁺+M₃⁻+M₄⁻, H₁+H₃, A₃, Γ₅⁺+Γ₆⁻, K₁+K₄+K₅] (4 bands)
  brs[10] = (2d|E′′):   symmetry vector discrepancy
    n   = [L₁+L₂, M₃⁺+M₄⁺+M₁⁻+M₂⁻, H₁+H₃, A₃, Γ₆⁺+Γ₅⁻, K₂+K₃+K₆] (4 bands)
    cbr = [L₁+L₂, M₃⁺+M₄⁺+M₁⁻+M₂⁻, H₂+H₃, A₃, Γ₆⁺+Γ₅⁻, K₂+K₃+K₆] (4 bands)
  brs[11] = (2c|A₁′):   symmetry vector discrepancy
    n   = [L₁, M₁⁺+M₄⁻, H₂, A₁, Γ₁⁺+Γ₄⁻, K₅] (2 bands)
    cbr = [L₁, M₁⁺+M₄⁻, H₁, A₁, Γ₁⁺+Γ₄⁻, K₅] (2 bands)
  brs[12] = (2c|A₁′′):  symmetry vector discrepancy
    n   = [L₂, M₂⁺+M₃⁻, H₂, A₂, Γ₂⁺+Γ₃⁻, K₅] (2 bands)
    cbr = [L₂, M₂⁺+M₃⁻, H₁, A₂, Γ₂⁺+Γ₃⁻, K₅] (2 bands)
  brs[13] = (2c|A₂′′):  symmetry vector discrepancy
    n   = [L₁, M₃⁺+M₂⁻, H₁, A₁, Γ₃⁺+Γ₂⁻, K₆] (2 bands)
    cbr = [L₁, M₃⁺+M₂⁻, H₂, A₁, Γ₃⁺+Γ₂⁻, K₆] (2 bands)
  brs[14] = (2c|A₂′):   symmetry vector discrepancy
    n   = [L₂, M₄⁺+M₁⁻, H₁, A₂, Γ₄⁺+Γ₁⁻, K₆] (2 bands)
    cbr = [L₂, M₄⁺+M₁⁻, H₂, A₂, Γ₄⁺+Γ₁⁻, K₆] (2 bands)
  brs[15] = (2c|E′):    symmetry vector discrepancy
    n   = [L₁+L₂, M₁⁺+M₂⁺+M₃⁻+M₄⁻, H₁+H₃, A₃, Γ₅⁺+Γ₆⁻, K₁+K₄+K₅] (4 bands)
    cbr = [L₁+L₂, M₁⁺+M₂⁺+M₃⁻+M₄⁻, H₂+H₃, A₃, Γ₅⁺+Γ₆⁻, K₁+K₄+K₅] (4 bands)
  brs[16] = (2c|E′′):   symmetry vector discrepancy
    n   = [L₁+L₂, M₃⁺+M₄⁺+M₁⁻+M₂⁻, H₂+H₃, A₃, Γ₆⁺+Γ₅⁻, K₂+K₃+K₆] (4 bands)
    cbr = [L₁+L₂, M₃⁺+M₄⁺+M₁⁻+M₂⁻, H₁+H₃, A₃, Γ₆⁺+Γ₅⁻, K₂+K₃+K₆] (4 bands)
#199:
  brs[3] = (8a|A):      failed to collect any compatible symmetry vectors
  brs[4] = (8a|E):      failed to collect any compatible symmetry vectors
#206:
  brs[3] = (8b|Ag):     failed to collect any compatible symmetry vectors
  brs[4] = (8b|Aᵤ):     failed to collect any compatible symmetry vectors
  brs[5] = (8b|Eg):     failed to collect any compatible symmetry vectors
  brs[6] = (8b|Eᵤ):     failed to collect any compatible symmetry vectors
  brs[7] = (8a|Ag):     failed to collect any compatible symmetry vectors
  brs[8] = (8a|Aᵤ):     failed to collect any compatible symmetry vectors
  brs[9] = (8a|Eg):     failed to collect any compatible symmetry vectors
  brs[10] = (8a|Eᵤ):    failed to collect any compatible symmetry vectors
#214:
  brs[1] = (12d|A):     failed to collect any compatible symmetry vectors
  brs[4] = (12d|B₂):    failed to collect any compatible symmetry vectors
  brs[5] = (12c|A):     failed to collect any compatible symmetry vectors
  brs[8] = (12c|B₂):    failed to collect any compatible symmetry vectors
  brs[9] = (8b|A₁):     failed to collect any compatible symmetry vectors
  brs[10] = (8b|A₂):    failed to collect any compatible symmetry vectors
  brs[11] = (8b|E):     failed to collect any compatible symmetry vectors
  brs[12] = (8a|A₁):    failed to collect any compatible symmetry vectors
  brs[13] = (8a|A₂):    failed to collect any compatible symmetry vectors
  brs[14] = (8a|E):     failed to collect any compatible symmetry vectors
#216:
  brs[1] = (4d|A₁):     symmetry vector discrepancy
    n   = [W₄, L₁, X₃, Γ₁] (1 band)
    cbr = [W₃, L₁, X₃, Γ₁] (1 band)
  brs[2] = (4d|A₂):     symmetry vector discrepancy
    n   = [W₃, L₂, X₄, Γ₂] (1 band)
    cbr = [W₄, L₂, X₄, Γ₂] (1 band)
  brs[4] = (4d|T₂):     symmetry vector discrepancy
    n   = [W₁+W₂+W₃, L₁+L₃, X₁+X₅, Γ₄] (3 bands)
    cbr = [W₁+W₂+W₄, L₁+L₃, X₁+X₅, Γ₄] (3 bands)
  brs[5] = (4d|T₁):     symmetry vector discrepancy
    n   = [W₁+W₂+W₄, L₂+L₃, X₂+X₅, Γ₅] (3 bands)
    cbr = [W₁+W₂+W₃, L₂+L₃, X₂+X₅, Γ₅] (3 bands)
  brs[6] = (4c|A₁):     symmetry vector discrepancy
    n   = [W₃, L₁, X₃, Γ₁] (1 band)
    cbr = [W₄, L₁, X₃, Γ₁] (1 band)
  brs[7] = (4c|A₂):     symmetry vector discrepancy
    n   = [W₄, L₂, X₄, Γ₂] (1 band)
    cbr = [W₃, L₂, X₄, Γ₂] (1 band)
  brs[9] = (4c|T₂):     symmetry vector discrepancy
    n   = [W₁+W₂+W₄, L₁+L₃, X₁+X₅, Γ₄] (3 bands)
    cbr = [W₁+W₂+W₃, L₁+L₃, X₁+X₅, Γ₄] (3 bands)
  brs[10] = (4c|T₁):    symmetry vector discrepancy
    n   = [W₁+W₂+W₃, L₂+L₃, X₂+X₅, Γ₅] (3 bands)
    cbr = [W₁+W₂+W₄, L₂+L₃, X₂+X₅, Γ₅] (3 bands)
#220:
  brs[1] = (16c|A):     failed to collect any compatible symmetry vectors
  brs[2] = (16c|E):     `tb_hamiltonian` error
  brs[3] = (12b|A):     failed to collect any compatible symmetry vectors
  brs[4] = (12b|B):     failed to collect any compatible symmetry vectors
  brs[6] = (12a|A):     failed to collect any compatible symmetry vectors
  brs[7] = (12a|B):     failed to collect any compatible symmetry vectors
[missing any failures above 220 due to memory fault]

The good news is that we can build a single-EBR model for every EBR across all space groups.1 The bad news is that we get the irreps wrong in quite a few cases. I doubt that it is a single problem,23 but it is probably best to start with the simplest problems, i.e., the plane group failures and then look on from there.

Footnotes

  1. This is true except in a few notably cases: e.g., space group 220, there is an error for brs[2]. There's also a sufficiently "bad" error at some space group number higher than 220, where we presumably overflow the memory and Julia simply terminates abruptly. ↩

  2. The reason I doubt it's a single problem is that the plane group issues presumably revolve around a wrong symmetry eigenvalue at KA / K for a 3-fold rotation. On the other hand, in space group ⋕68, the problem involves a wrong symmetry eigenvalue at T (also on at Y), under a 2-fold screw (and there is C centering in ⋕68). ↩

  3. Note that the failure modes "failed to collect any compatible symmetry vectors" and "symmetry vector discrepancy" probably are mostly the same: i.e., failure to calculate the correct irreps - in the former case, this failure simply leads to a collection of irreps that isn't compatibility respecting, and so our method finds no compatible vectors to return from the irrep content it computes. ↩

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Codecov Report

❌ Patch coverage is 0% with 1 line in your changes missing coverage. Please review.

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AntonioMoralesPerez commented Aug 14, 2025

Hi Thomas, I was also working on the same test and subsequent issues. I attach here my tests, but from a quick view, it is just a simpler version of yours.

using Test
using SymmetricTightBinding, Crystalline
using Crystalline: free

@testset "Symmetry analysis" begin
    for D in 1:2
        for sgnum in 1:MAX_SGNUM[D]
            @testset "Space group $sgnum in dimension $D" begin
                brs = calc_bandreps(sgnum, Val(D))
                for i in eachindex(brs)
                    iszero(free(position(brs[i]))) || continue # skip EBRs with non-zero free parts
                    #                                            TODO: consider upgrading this test
                    #                                                  brs with non-zero free parts
                    #                                                  using `pin_free`
                    @testset "Band representation $(brs[i])" begin
                        coefs = zeros(length(brs))
                        coefs[i] = 1
                        cbr = CompositeBandRep(coefs, brs)
                        tbm = tb_hamiltonian(cbr)
                        ptbm = tbm(randn(length(tbm)))

                        v = SymmetryVector(cbr)
                        v_model = collect_compatible(ptbm)

                        @test v == v_model[1] # check that the symmetry vector is compatible with the model
                    end # @testset "Band representation $brs[i]"
                end # for br in brs
            end # @testset "Space group $sgnum in dimension $D"
        end # for sgnum in MAX_SGNUM[D]
    end # for D in 1:3
end # @testset "Symmetry analysis"

I am working on PG number 13 and BR (1c|A). It seems a pretty simple case, and it just fails on on-site terms, which is quite surprising. I will update here my findings.

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thchr commented Aug 14, 2025 •
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I am working on PG number 13 and BR (1c|A). It seems a pretty simple case, and it just fails on on-site terms, which is quite surprising. I will update here my findings.

One thought I had is that the site symmetry irreps we compute may be wrong in some way.

Or, rather, I have a question: before our new insights on the phase factor term $\Theta_G$, I had expected that the trace of the representation matrix $D_{\mathbf{k}}(g)$ should equal the character of the corresponding EBR at $\mathbf{k}$. Now, I'm not sure whether it should actually be the trace of $\Theta_G^\dagger D_{\mathbf{k}}(g)$?
Practically, I observed that the trace is in agreement with expectations if its just of $D_{\mathbf{k}}(g)$, but not if it's of $\Theta_G^\dagger D_{\mathbf{k}}(g)$. But what we end up computing is basically a trace over $\Theta_G^\dagger D_{\mathbf{k}}(g)$.

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thchr commented Aug 14, 2025 •
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Here's a few observations:

  1. If we artificially use the complex conjugate of the symmetry eigenvalues we compute in SymmetricTightBinding.jl, then many of the failures go away (basically, all those relating to K, KA, H, HA irreps), including all issues in plane groups. Some issues remain, e.g., those in space group 68.

  2. Alternatively, there is this long-running thorn in the calc_bandreps implementation, relating to Phase convention for LGIrrep thchr/Crystalline.jl#12, which involves a choice of sign that differs from the one in Cano/Elcoro's PRB TQC paper. The code in question is here: https://github.com/thchr/Crystalline.jl/blob/584f32a5e85d99def9f2939fe93eff87337b7b37/src/calc_bandreps.jl#L175-L181
    If I change the line from cis(2 ...) to cis(-2 ...), then this also apparently resolves the K, KA, etc. problem - but it introduces a whole additional error class relating to being able to actually subduce EBRs to integer-valued irreps.

  3. Finally, there's a concern relating what we return in calc_bandreps vs. what is at BCS. Superficially, it doesn't look exactly the same (looking e.g. at (2c|A) in plane group 13): but I believe the differences come down to differences in labelling of irreps, e.g., of KA2 and KA3.

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AntonioMoralesPerez commented Aug 26, 2025

Hi Thomas, sorry for the long absence. The TMS was intense, and I didn't find much time to work. I am returning to the projects slowly.

I have been working on this issue all afternoon, and I haven't found a proper conclusion. It is unpleasant since I do think my theoretical computations seem alright, at least I have been through them a lot of times.

About your observations, I agree that taking the complex conjugate kills a bunch of issues, and I also experienced that taking -G instead of G solves a bunch. However, I don't know why we should. About numbers 2 and 3, I don't think it is a source of trouble since we are consistent with our convention, and I did by hand a couple of cases, and they do not work.

I think tomorrow I will focus on PR #91 and issue #74, which is deeper and could also be a source of problems here. After solving it, I will come back here.

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AntonioMoralesPerez commented Aug 26, 2025

About #89 (comment), I don't agree with your vision.

In fact, it is what I faced on plane group 13 with BR (1c|A). This is a really simple case where the space group representation is always the identity, i.e., $D_\mathbf{k}(g)=1$ for all $g$'s. In this case, the only way to obtain the non-trivial irreps is by the presence of $\Theta_\mathbf{G}$.

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@AntonioMoralesPerez AntonioMoralesPerez mentioned this pull request Mar 12, 2026
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thchr commented Mar 12, 2026

One possibility is this:

https://github.com/CrystallineOrg/SymmetricTightBinding.jl/pull/101/changes/BASE..43f07cdb4e25a79d46b5df2f8605f2f261f2a1fd#r2923143247

The associated code which is affected is this (signs are wrong in front of v_conj?):

SymmetricTightBinding.jl/src/types.jl

Lines 418 to 426 in 41548c2

# NB: ↓ one more case of assuming no free parameters in `δ`
v_conj = cispi.(dot.(Ref(-2 .* k), constant.(orbit(block.h_orbit))))
# NB: ↑ this is `v` conjugated: we do this because the `dot`-product below conjugates
# its first argument; so by conjugating twice we get the unconjugated result.
# NB: ↑ each term in the Hamiltonian is associated to an annihilation+creation operator
# pair `aᵢ† aⱼ`. Since we use Convention 1 for the Fourier transform, we have
# aᵢ† = ∑ₖ e^{-ik·(tᵢ + rᵢ)} aₖ†, such that each term will be multiplied by a phase
# e^{ik·δᵢⱼ} with δᵢⱼ ≡ Rᵢⱼ + rᵢ - rⱼ, i.e., the hopping vector in the orbit; this
# is what `orbit(block.h_orbit)` gives us above

Let's test what happens if we fix this.

This was referenced Mar 13, 2026
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thchr added a commit that referenced this pull request Mar 16, 2026
@thchr
WIP: First exploration of fix to #89.
9214842
thchr added a commit that referenced this pull request Mar 16, 2026
@thchr @claude
Fix symmetry eigenvalue phase convention (PR #89) and Hamiltonian Fou…
ed17136 
…rier phase

Two-part fix for the symmetry analysis bug identified in PR #89:

1. **Symmetry eigenvalues** (`symmetry_analysis.jl`): Fix Θ_G sign (use conj(Θ_G)
   to match Cano et al. trace convention) and add global phase correction
   `e^{+4πi(gk)·v}` to compensate for the conjugated translation phase in
   Crystalline.jl's `calc_bandreps` (cf. Crystalline.jl issue #12).

2. **Hamiltonian Fourier phase** (`types.jl`, `gradients.jl`): Correct from
   `e^{+ik·δ}` to `e^{-ik·δ}` (Convention 1), matching the corrected theory.md
   derivation. Update momentum gradient sign accordingly.

Remaining known failures (SG 68, 88, 141, 142, 214, 220, 230) are pre-existing
centered-lattice issues in Crystalline.jl, documented in the new devdoc.

Also:
- Rewrite symmetry_analysis tests with deterministic RNG seed and @test_broken
  for known failures
- Update stopgap tests: SG 13/14 EBRs now pass
- Mark Haldane model comparison test as @test_broken (needs coefficient update)
- Update show.jl expected strings for new print format
- Add detailed devdoc on the convention mismatch and fix

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>
@thchr thchr mentioned this pull request Mar 17, 2026
Merged
thchr added a commit that referenced this pull request Mar 17, 2026
@thchr @claude
Fix symmetry eigenvalue phase convention (PR #89) and Hamiltonian Fou…
6b50b5b 
…rier phase

Two-part fix for the symmetry analysis bug identified in PR #89:

1. **Symmetry eigenvalues** (`symmetry_analysis.jl`): Fix Θ_G sign (use conj(Θ_G)
   to match Cano et al. trace convention) and add global phase correction
   `e^{+4πi(gk)·v}` to compensate for the conjugated translation phase in
   Crystalline.jl's `calc_bandreps` (cf. Crystalline.jl issue #12).

2. **Hamiltonian Fourier phase** (`types.jl`, `gradients.jl`): Correct from
   `e^{+ik·δ}` to `e^{-ik·δ}` (Convention 1), matching the corrected theory.md
   derivation. Update momentum gradient sign accordingly.

Remaining known failures (SG 68, 88, 141, 142, 214, 220, 230) are pre-existing
centered-lattice issues in Crystalline.jl, documented in the new devdoc.

Also:
- Rewrite symmetry_analysis tests with deterministic RNG seed and @test_broken
  for known failures
- Update stopgap tests: SG 13/14 EBRs now pass
- Mark Haldane model comparison test as @test_broken (needs coefficient update)
- Update show.jl expected strings for new print format
- Add detailed devdoc on the convention mismatch and fix

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>
thchr added a commit that referenced this pull request Mar 17, 2026
@thchr @claude
Fix symmetry eigenvalue phase convention (PR #89) and Hamiltonian Fou…
055fded 
…rier phase

Two-part fix for the symmetry analysis bug identified in PR #89:

1. **Symmetry eigenvalues** (`symmetry_analysis.jl`): Fix Θ_G sign (use conj(Θ_G)
   to match Cano et al. trace convention) and add global phase correction
   `e^{+4πi(gk)·v}` to compensate for the conjugated translation phase in
   Crystalline.jl's `calc_bandreps` (cf. Crystalline.jl issue #12).

2. **Hamiltonian Fourier phase** (`types.jl`, `gradients.jl`): Correct from
   `e^{+ik·δ}` to `e^{-ik·δ}` (Convention 1), matching the corrected theory.md
   derivation. Update momentum gradient sign accordingly.

Remaining known failures (SG 68, 88, 141, 142, 214, 220, 230) are pre-existing
centered-lattice issues in Crystalline.jl, documented in the new devdoc.

Also:
- Rewrite symmetry_analysis tests with deterministic RNG seed and @test_broken
  for known failures
- Update stopgap tests: SG 13/14 EBRs now pass
- Mark Haldane model comparison test as @test_broken (needs coefficient update)
- Update show.jl expected strings for new print format
- Add detailed devdoc on the convention mismatch and fix

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>
thchr added a commit that referenced this pull request Mar 17, 2026
@thchr
Fix symmetry eigenvalue phase convention (PR #89) and Hamiltonian Fou…
a60fad9 
…rier phase

Two-part fix for the symmetry analysis bug identified in PR #89:

1. **Symmetry eigenvalues** (`symmetry_analysis.jl`): Fix Θ_G sign (use conj(Θ_G)
   to match Cano et al. trace convention) and add global phase correction
   `e^{+4πi(gk)·v}` to compensate for the conjugated translation phase in
   Crystalline.jl's `calc_bandreps` (cf. Crystalline.jl issue #12).

2. **Hamiltonian Fourier phase** (`types.jl`, `gradients.jl`): Correct from
   `e^{+ik·δ}` to `e^{-ik·δ}` (Convention 1), matching the corrected theory.md
   derivation. Update momentum gradient sign accordingly.

Remaining known failures (SG 68, 88, 141, 142, 214, 220, 230) are pre-existing
centered-lattice issues in Crystalline.jl, documented in the new devdoc.

Also:
- Rewrite symmetry_analysis tests with deterministic RNG seed and @test_broken
  for known failures
- Update stopgap tests: SG 13/14 EBRs now pass
- Mark Haldane model comparison test as @test_broken (needs coefficient update)
- Update show.jl expected strings for new print format
- Add detailed devdoc on the convention mismatch and fix

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

bump Crystalline bound to use
thchr added a commit that referenced this pull request Mar 17, 2026
@thchr @claude
Fix symmetry eigenvalue phase convention (PR #89) and Hamiltonian Fou…
fe5c7e3 
…rier phase

Two-part fix for the symmetry analysis bug identified in PR #89:

1. **Symmetry eigenvalues** (`symmetry_analysis.jl`): Fix Θ_G sign (use conj(Θ_G)
   to match Cano et al. trace convention) and add global phase correction
   `e^{+4πi(gk)·v}` to compensate for the conjugated translation phase in
   Crystalline.jl's `calc_bandreps` (cf. Crystalline.jl issue #12).

2. **Hamiltonian Fourier phase** (`types.jl`, `gradients.jl`): Correct from
   `e^{+ik·δ}` to `e^{-ik·δ}` (Convention 1), matching the corrected theory.md
   derivation. Update momentum gradient sign accordingly.

Remaining known failures (SG 68, 88, 141, 142, 214, 220, 230) are pre-existing
centered-lattice issues in Crystalline.jl, documented in the new devdoc.

Also:
- Rewrite symmetry_analysis tests with deterministic RNG seed and @test_broken
  for known failures
- Update stopgap tests: SG 13/14 EBRs now pass
- Mark Haldane model comparison test as @test_broken (needs coefficient update)
- Update show.jl expected strings for new print format
- Add detailed devdoc on the convention mismatch and fix
- Bump Crystalline compat to v0.6.24 to get `primitivize(::LGIrrep)` and
  `primitivize(::Collection{<:LGIrrep})`

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>
thchr added a commit that referenced this pull request Mar 17, 2026
@thchr
add a few small QoL things that, not previously incorporated from #89
419c39d
thchr added a commit that referenced this pull request Mar 17, 2026
@thchr
add a few small QoL things missing from #89
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thchr commented Mar 18, 2026

Superseded and mostly transferred to #104: closing in favor of that.

@thchr thchr closed this Mar 18, 2026
thchr added a commit that referenced this pull request Apr 10, 2026
@thchr
Fix symmetry analysis phase conventions (PR #89) (#104)
77702d5 
Fix symmetry eigenvalue phase convention and expand symmetry analysis tests

Fixes the long-standing incorrect irrep assignments from `symmetry_eigenvalues`
(PR #89) and expands the symmetry analysis test suite to cover all 230 space groups.

## Symmetry eigenvalue fix (`src/symmetry_analysis.jl`)

The root cause is that Crystalline.jl's `calc_bandreps` and `lgirreps` use a sign
convention for character computation that is the complex conjugate of the Convention 1
derivation in `theory.md` (see Crystalline.jl issue #12). Specifically, both the Θ_G
phase factor and the global translation phase in D_k carry the opposite sign. Since
both factors are conjugated, the net effect is that

    χ_Crystalline = conj(χ_Convention1)

The fix: compute the Convention 1 character χ = (Θ_G w)† D_k w directly (using positive
G and the unmodified `sgrep(k)`), then return `conj(χ)`. This replaces the previous
approach of two separate sign-patch corrections (a sign-flipped Θ_G and an explicit
`cispi(4dot(gk,v))` phase factor), which were calibrated against a different Hamiltonian
convention and were not derived from first principles.

A separate centered-lattice bug is also fixed: `primitivize(::LittleGroup)` with default
`modw=true` was discarding lattice vectors from translations, corrupting phases at non-Γ
k-points in centered lattices. Fix: use `primitivize(::Collection{LGIrrep})` (dispatches
with `modw=false`). This resolves failures in SG 68, 88, 141, 142, 214, 220, 230.

## Test suite (`test/symmetry_analysis.jl`, `test/runtests.jl`)

- Rewrites the full-EBR-scan test with a deterministic RNG so results are reproducible
- Extends coverage to all 230 space groups (previously only 2D plane groups)
- Adds two concrete documentation examples (plane group 17) as `@test` assertions
- Promotes the former stopgap `@test_broken` entries for p3/p6 to `@test`
- Renames `test/symmetry_analysis_stopgap.jl` → `test/symmetry_analysis_manual.jl`

## Documentation (`docs/src/devdocs/symmetry_eigenvalue_conventions.md`)

New devdoc explaining the convention mismatch, why the net effect is complex conjugation,
the implemented fix, and three options for future cleanup (recommended long-term: fix
Crystalline.jl issue #12 directly).

## Minor changes

- `src/tightbinding.jl`: comment cleanup around the hopping/constraint matrix assembly
- `src/types.jl`, `src/gradients.jl`: comment/variable name cleanup
- `ext/SymmetricTightBindingMakieExt.jl`: arrow direction correction in orbit visualization
- `CLAUDE.md`, `PLAN.md`: updated to reflect Phase 6 completion
- `_retrying_nullspace`: some of the "large" models constructed by /test/symmetry_analysis.jl fail on CI due to the SVD decomposition erroring (probably because we have too many and redundant constraints); this can be fixed by optionally retrying with a different SVD algorithm (QR iteration)
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@thchr @AntonioMoralesPerez