[BUG] nbspline gives wrong structure factors at low orders: #8092

Describe the bug

Summary

Enabling nbspline (Cardinal B-spline structure factor) with a low order produces wrong structure factors S(G) in both OFDFT and KSDFT.

Details

Example: tests/01_PW/032_PW_15_CF_CS_bspline (GaAs, 28 electrons)

1. Total energies are catastrophically wrong at n=2–6, from the first SCF step.

n=2
!FINAL_ETOT_IS 141273.0584391949523706 eV

n=4
!FINAL_ETOT_IS -3859.3156704751031612 eV

n=6
!FINAL_ETOT_IS -4193.0982667816042522 eV

n=8
!FINAL_ETOT_IS -4200.1922370850106745 eV

n=20
!FINAL_ETOT_IS -4200.2532789746392154 eV

no nbspline specified
!FINAL_ETOT_IS -4200.2531231037410180 eV

2. The initial electron count is multiplied by exactly (1−1/n!)⁻³ (n = nbspline): 8× for n=2, 1.136× for n=4, 1.004× for n=6.Taken from warning.log:

n=2
Electron number from rho = 224

n=4
Electron number from rho = 31.8132653899894

n=6
Electron number from rho = 28.1169914924782

n=8
Electron number from rho = 28.0020834366776

n=20
Electron number from rho = 28.0000000000001

no nbspline specified
Electron number from rho = 28.0000000000001

Here,
224/28=8; (1-1/2!)⁻³=8
31.8132653899894/28=1.136188; (1-1/4!)⁻³=1.136188
28.1169914924782/28=1.004178; (1-1/6!)⁻³=1.004178
...

This implies a missing term leading to the wrong Structural Factor, while the orders $\ge$ 8–10 are numerically unaffected, so the existing integration tests (which use nbspline 10 and 20) pass with correct references.

Expected behavior

No response

To Reproduce

Change nbspline to a rather low order for case tests/01_PW/032_PW_15_CF_CS_bspline.

Environment

No response

Additional Context

No response

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