CCL: How to calculate S^2 (S squared) value of a broken-symmetry state?

 Sent to CCL by: Tobias Kraemer [Tobias.Kraemer(!)]
 Dear Ankur,
 You will find an exact definition of the expectation value for S^2 for UHF
 determinants on page 107 of Szabo's text "Modern Quantum Chemistry".
 Essentially, in addition to the exact value of S^2, you will also need to
 consider the number of unpaired beta-spin electrons N(beta), as well as the
 overlap integrals between (occupied) alpha/beta spin orbitals.
 Say in the case of an open-shell singlet diradical (the two atoms at large
 separation), you would get S(S+1) + 1 = 0.0 + 1.0 (0.0 for the singlet, and
 N(beta) = 1). I am not sure where you get the value for the
 Broken-symmetry doublet from, since the weighter average doesn't give you the
 correct value here. In this case you would apply S(S+1) + 1 = 0.75 + 1.0 = 1.75.
 I am assuming that the value is not decreased by
 the spatial overlap terms. Generally the observed values are far from the ideal,
 if the spatial overlap between the occupied alpha and beta manifold for real
 systems is taken into account.
 Hope this helps.
 Dr. Tobias Krämer
 Lecturer in Inorganic Chemistry
 Department of Chemistry
 Maynooth University, Maynooth, Co. Kildare, Ireland.
 E: tobias.kraemer]~[   T: +353 (0)1 474 7517
 -----Original Message-----
 > From:]~[
 <]~[> On Behalf Of Ankur Kumar
 Gupta ankkgupt*
 Sent: 18 June 2019 16:53
 To: Tobias Kraemer <Tobias.Kraemer]~[>
 Subject: CCL: How to calculate S^2 (S squared) value of a broken-symmetry state?
 Sent to CCL by: "Ankur Kumar Gupta" [ankkgupt[-]]
 > From what I have read, the S^2 value of a broken-symmetry singlet
 (contaminated by a triplet) is 1.0, which is calculated to be the average of the
 singlet and triplet S^2. Similarly, the S^2 of broken-symmetry doublet
 (contaminated by the quartet state) turns out to be 1.75 (average of a doublet
 and a quartet). I would like to know how these average values are being
 calculated. I understand that these are probably weighted averages (as
 1.750.5(0.75+3.75), where 0.75 and 3.75 are S^2 values of doublet and quartet,
 respectively), but I don't know how that weighting is being done. I would be
 grateful if someone could provide a detailed explanation (mathematical
 derivation) of this.