Spectrum stability with Feynman Sigma

[benroberts999]

Spectrum energies do not perfectly match with valence energies when using Feynman Sigma.
When using Goldstone sigma, however, they do.
This implies it's probably a numerical error stemming from Feynman Sigma - but is not obvious.
Typically difference is small, and doesn't seem to negatively affect much. However, when scaling sigma to exactly reproduce energy intervals, the scaling is done for valence states, so the spectrum states will not match exactly!
Possibly related to #22
Possibly related to #26

[benroberts999]

Agreement is to parts in $10^6$ comparing Finite-Difference Hartree-Fock energies to basis/spectrum energies.
Roughly the same level of agreement between Brueckner orbitals and Spectrum orbitals using Goldstone technique for $\Sigma$.
However, agreement drops to parts in $\sim 10^4$ when using Feynman technique.

Could be related to B-spline code, but more likely is due to calculation of $\Sigma \psi$.

HF	HF/Bruckner	Spectrum
$6s_{1/2}$	-0.1273682	-0.1273681	1E-06
$7s_{1/2}$	-0.0551874	-0.0551874	9E-07
$8s_{1/2}$	-0.0309525	-0.0309525	6E-07
$9s_{1/2}$	-0.0198146	-0.0198146	-1E-06
Goldstone(2)
$6s_{1/2}$	-0.1475863	-0.1475867	-3E-06
$7s_{1/2}$	-0.0593160	-0.0593161	-2E-06
$8s_{1/2}$	-0.0325347	-0.0325347	-6E-07
$9s_{1/2}$	-0.0205929	-0.0205930	-2E-06
Feynman(2)
$6s_{1/2}$	-0.1478421	-0.1478491	-5E-05
$7s_{1/2}$	-0.0593860	-0.0593874	-2E-05
$8s_{1/2}$	-0.0325636	-0.0325641	-2E-05
$9s_{1/2}$	-0.0206073	-0.0206076	-1E-05
Feynman($\infty$)
$6s_{1/2}$	-0.1432236	-0.1431959	2E-04
$7s_{1/2}$	-0.0584265	-0.0584218	8E-05
$8s_{1/2}$	-0.0321932	-0.0321915	5E-05
$9s_{1/2}$	-0.0204248	-0.0204240	4E-05

Full code input/output:
SpectrumStability.txt