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

• From: Tobias Kraemer <Tobias.Kraemer * mu.ie>
• Subject: CCL:G: How to calculate S^2 (S squared) value of a broken-symmetry state?
• Date: Wed, 19 Jun 2019 18:14:54 +0000

 Dear Ankur,   I found a mistake/typo in my previous post to you, and I hope I haven’t confused you with this. What I meant is that in the case of the open-shell diradical, the pure S = 0 state (not the the broken-symmetry determinant) is a mixture of 50% |a(1)b(2)> and 50% |b(1)a(2)> (a and b symbolising alpha- and beta-spin for the first and second particle in different spatial orbitals, respectively). The open-shell diradical is therefore a linear combination of two determinants. The broken-symmetry determinant, which only uses say the|a(1)b(2)> part, is likewise a linear combination of the singlet and triplet states with equal weights. Hence |a(1)b(2)> = SQRT(1/2)|0,0>  +  SQRT(1/2)|1,0>. The coefficients SQRT(1/2) in this case (and therefore weights) can be obtained from tables (Wigner coefficients, Clebsch-Gordon coefficients). Similar arguments then apply to spin states other than the one discussed above, ie. the Ms = ½ (not Ms = 3/2 as I wrote) broken-symmetry doublet. This is what I meant to say, hope this makes more sense now.   Best,   Tobias   Dr. Tobias Krämer Lecturer in Inorganic Chemistry Department of Chemistry Maynooth University, Maynooth, Co. Kildare, Ireland. E: tobias.kraemer #at# mu.ie   T: +353 (0)1 474 7517     From: owner-chemistry+tobias.kraemer==mu.ie #at# ccl.net On Behalf Of Tobias Kraemer Tobias.Kraemer],[mu.ie Sent: 19 June 2019 11:13 To: Tobias Kraemer Subject: CCL:G: How to calculate S^2 (S squared) value of a broken-symmetry state?   Dear Ankur,   I guess the formulation in the book is a bit misleading, and I would rather call this the weighted average. You can derive the eigenvalue of S^2 from the individual contributions of the uncoupled electronic states to the broken-symmetry wavefunction. The contribution of each of these states is given by the Wigner coefficients (or Clebsch-Gordon coefficients), which tell you how these spin vectors are coupled together. In the case of the open-shell diradical, the Ms = ½ state (the broken-symmetry determinant) is a mixture of 50% |a(1)b(2)> and %50 |b(1)a(2)> (a and b symbolising alpha- and beta-spin for the first and second particle, respectively). This stems from the fact that the alpha-beta combination is: |a(1)b(2)> = SQRT(1/2)|0,0>  +  SQRT(1/2)|1,0>   (see Tables for vector coupling) This indicates that the BS solution is a 50:50 mixture of the Ms=0 components of the singlet and triplet states. So the expectation value of could be obtained from 0.5x0.0 + 0.5x2.0 = 0.5(0.0+2.0) = 1.0, which is the average of both. When more than 2 spin centres are involved, things become somewhat more complex, but follow the same procedure. For the Ms=3/2 case (the broken-symmetry doublet) the expansion (using the vector coupling tables) involves 66% |1/2,1/2>  (for which S^2 of the pure doublet state is 0.75), mixed with 33% |3/2,1/2> (with 3.75 for the ‘pure’ quartet state). Hence, this is not simply a 50% mix any longer, but you should be able to work it out from here. At least I think this is the way the author in the book chapter you have cited looks at this.   Hope this clarifies it.   Tobias     Dr. Tobias Krämer Lecturer in Inorganic Chemistry Department of Chemistry Maynooth University, Maynooth, Co. Kildare, Ireland. E: tobias.kraemer #at# mu.ie   T: +353 (0)1 474 7517     From: owner-chemistry+tobias.kraemer==mu.ie #at# ccl.net On Behalf Of Ankur Gupta ankkgupt,,iu.edu Sent: 18 June 2019 21:35 To: Tobias Kraemer Subject: CCL: How to calculate S^2 (S squared) value of a broken-symmetry state?   Dear Dr. Krämer   Thank you for your reply. I was aware of the equation for S^2 (UHF) in Szabo. However, I was a bit confused by the language used in a few books and papers that I came across. Specifically, the paragraph in the following book (Page 238 of Molecular Water Oxidation Catalysis: A Key Topic for New Sustainable Energy Conversion Schemes, Editor Antoni Llobet),   explicitly states the following,   "For a broken-symmetry singlet contaminated by a triplet state, we would expect HS< S2 > to be 2 (the correct eigenvalue for a triplet state) and BS< S2 > to be about 1 (the average of a singlet and a triplet). In the same way, for a broken-symmetry doublet contaminated by a quartet state, we would expect HS< S2 > to be 3.75 (the correct eigenvalue for a quartet) and BS< S2 > to be 1.75 (the average of a doublet and a quartet)."   Best, Ankur   On Tue, Jun 18, 2019 at 3:26 PM Tobias Kraemer Tobias.Kraemer]![mu.ie wrote: Sent to CCL by: Tobias Kraemer [Tobias.Kraemer(!)mu.ie] 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. Tobias Dr. Tobias Krämer Lecturer in Inorganic Chemistry Department of Chemistry Maynooth University, Maynooth, Co. Kildare, Ireland. E: tobias.kraemer-.-mu.ie   T: +353 (0)1 474 7517 -----Original Message----- > From: owner-chemistry+tobias.kraemer==mu.ie-.-ccl.net On Behalf Of Ankur Kumar Gupta ankkgupt*iu.edu Sent: 18 June 2019 16:53 To: 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[-]iu.edu] > 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.http://www.ccl.net/cgi-bin/ccl/send_ccl_messagehttp://www.ccl.net/chemistry/sub_unsub.shtmlhttp://www.ccl.net/spammers.txt -= This is automatically added to each message by the mailing script =-