Improve CI vector sign convention consistency in csf_fci.

[MatthewRHermes]

For two microstates $m'$ and $m$ of a multiplet with spin $s$ ($m'\ge m$; $s\ge |m|$; $s\ge |m'|$; m=(nelec[0]-nelec[1])/2), the full CI vectors ought to be related to one another as

$$ \langle m'|\hat{S}_+^{(m'-m)}|m\rangle = \sqrt{\frac{(s+m')!}{(s-m')!}\frac{(s-m)!}{(s+m)!}} \ge 0 $$

where

$$ \hat{S}_+ \equiv \sum_p \hat{a}_p^\dagger \hat{b}_p $$

is the spin raising operator. Arranging for this relationship to be satisfied requires a consistent sign convention for FCI vectors in different Hilbert spaces and for CSFs in different electron configurations, which in turn requires accounting for the sign flips that occur corresponding to the anticommutation of electron creation operators when going from $m$ to $m'$ so that determinants consistently reflect the PySCF convention,

$$ \ldots \hat{a}_{i+1}^\dagger \hat{a}_{i}^\dagger \ldots \hat{b}_{j+1}^\dagger \hat{b}_{j}^\dagger \ldots |\mathrm{vac}\rangle $$

I did not originally account for this when writing the csf_fci module. The sign convention of csf_fci without this PR corresponds to

$$ \ldots \hat{a}_{u+1}^\dagger \hat{a}_{u}^\dagger \ldots \hat{b}_{v}^\dagger \hat{b}_{v+1}^\dagger \ldots \hat{a}_{i+1}^\dagger \hat{b}_{i+1}^\dagger \hat{a}_{i}^\dagger \hat{b}_{i}^\dagger \ldots |\mathrm{vac}\rangle $$

instead, where $u,v$ are singly-occupied and $i$ are doubly-occupied orbitals (note the unpaired $\beta$ electrons in the wrong order, among other differences). The CSFs in different electron configurations differ by an arbitrary sign. This doesn't prevent the module from working for its current purposes because all operator matrix elements are computed in the determinant basis and the sign is consistent, but it does frustrate any attempt to use CSF basis states directly as anything other than signless axes defining a vector space.

This PR modifies the csf_fci module to correctly reflect the PySCF sign/normal-ordering convention at all times, so that a single CI vector in the CSF basis can be transformed into the determinant basis for any $m$ with a consistent sign. These modifications make it possible to, in future, directly compute things like $\langle \vec{s}'|\hat{O}|\vec{s}\rangle$, where $\vec{s}$ is the unique CSF string, without returning to the determinant basis.

For validation purposes, this PR also adds the file spin_op.py to the csf_fci module, although theoretically the code in that file is really more at home in the fci/spin_op.py file of the core modules.

EDIT: clarified and expanded the explanation.

[sunqm]

Is this reset to random seed necessary?

[MatthewRHermes]

No it is not. I left that over from debugging; thanks for catching.