qiskit-community / qiskit-nature / 7083968559

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Fix the `AngularMomentum` (again) (#1292)

* fix: use the symmetric formula for S^2

Prior to this fix, the `AngularMomentum` operator was working on the
assumption that the number of alpha-spin particles would always exceed
the number of beta-spin particles (for non-singlet systems). However,
this is not guaranteed to be the case. This commit fixes this problem by
using a symmetric formula for S^2 which is the average of the two cases
(alpha- vs. beta-spin particles exceeding the other).

* feat: log a warning when the alpha-beta overlap is non-unitary

When dealing with active spaces obtained from unrestricted-spin
orbitals, the alpha-beta overlap matrix which is used to construct the
S^2 operator can become non-unitary (for example, when the active set of
alpha- and beta-spin orbitals do not span the same space). This can
result in an <S^2> value measured on the active space that is largely
different from (e.g.) the UHF <S^2> value. More importantly, when this
is the case, the difference between these two <S^2> values is "hidden"
in the inactive+inactive and inactive+active interactions. Especially
when spin contamination is present, this can lead to vastly unexpected
results and may indicate a poor choice of active space.

* Add reno

* test: add a regression test

* fix: update warning tolerance

* Linting and formatting

* docs: fix verbatim in RST

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