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The analytic polarizability (exact) and td=sos should agree to within
a small tolerance. The default for td=sos is perform a sum over
states including all excited states which formally will converge to
the exact result. In practice, there may be some small noise in the
sos calculation so the agreement is not always numerically perfect.

The approximate polarizability is a very crude approximation based on
the sum over states formalism. It is not equivalent to td=sos. We do
not suggest using the approximate polarizability. This approximate
estimation was useful many years ago when polarizability calculations
were more cpu intensive and is of limited use now that analytic
polarizability calculations are highly efficient.
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