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Fix missing last term in B-spline factor for structure factor - #8210
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… by (1-1/N!)^-3
The denominator loop in Structure_Factor::bsplinecoef missed the last integer-point sample M_{N+1}(N) = 1/N! of the Cardinal B-spline, so the G=0 deconvolution factor was 1/(1-1/N!) per direction instead of 1, and S(G=0) -- the integrated electron number -- was scaled by (1-1/N!)^-3 (8x at nbspline=2). The method is exact at G=0 at any order by partition of unity, so this was an off-by-one implementation error, not low-order interpolation error.
Add a unit test asserting b(0)=1 for norder = 2,4,6,10,20 and a low-order OFDFT integration case 35_OF_LPS_bspline; regenerate references of the six existing nbspline=10 cases (energies shift by ~1e-4 eV, the intended behavior change).
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Fix missing last term in B-spline deconvolution factor for structure factor (nbspline).
Summary
Fix an error in the B-spline correction coefficients used by
Structure_Factor::bsplinecoef.The previous denominator sum omitted the last nonzero integer sample of the Cardinal B-spline.
This gives incorrect structure factors when
nbsplineis enabled, with a particularly large error at low orders.For
nbspline = 2, the zero-frequency structure factor is multiplied by 8, which also scales the integratedatomic-superposition initial density before any subsequent normalization.
Linked Issue
Fix #8092
Unit Tests and/or Case Tests for my changes
What's changed?
Governance Notes