From owner-chemistry@ccl.net Tue Jun 18 12:53:00 2019 From: "Ankur Kumar Gupta ankkgupt*iu.edu" To: CCL Subject: CCL: How to calculate S^2 (S squared) value of a broken-symmetry state? Message-Id: <-53764-190618115316-8588-xhZUSBzW49HRM9djqi/3lg%server.ccl.net> X-Original-From: "Ankur Kumar Gupta" Date: Tue, 18 Jun 2019 11:53:15 -0400 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. From owner-chemistry@ccl.net Tue Jun 18 15:06:00 2019 From: "Tobias Kraemer Tobias.Kraemer]![mu.ie" To: CCL Subject: CCL: How to calculate S^2 (S squared) value of a broken-symmetry state? Message-Id: <-53765-190618150505-20039-bFQrZCGGMdMn51hbM5GiaQ/./server.ccl.net> X-Original-From: Tobias Kraemer Content-Language: en-US Content-Transfer-Encoding: 8bit Content-Type: text/plain; charset="iso-8859-1" Date: Tue, 18 Jun 2019 19:04:57 +0000 MIME-Version: 1.0 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 From owner-chemistry@ccl.net Tue Jun 18 16:37:00 2019 From: "Ankur Gupta ankkgupt,,iu.edu" To: CCL Subject: CCL: How to calculate S^2 (S squared) value of a broken-symmetry state? Message-Id: <-53766-190618163512-21056-rEZtPXuP9XoREoLDZTTnSw~~server.ccl.net> X-Original-From: Ankur Gupta Content-Type: multipart/alternative; boundary="00000000000051a218058b9f0ced" Date: Tue, 18 Jun 2019 16:34:52 -0400 MIME-Version: 1.0 Sent to CCL by: Ankur Gupta [ankkgupt]=[iu.edu] --00000000000051a218058b9f0ced Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable Dear Dr. Kr=C3=A4mer 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), https://books.google.com/books?id=3D7WNiAwAAQBAJ&pg=3DPT391&lpg=3DPT391&dq= =3D%22doublet%22+%22quartet%22+%22average%22+%22spin%22+%22broken%22&source= =3Dbl&ots=3D_k8KqUVV8c&sig=3DACfU3U3DS2BdYNMfnlqWhcD2ThGkdvDwTA&hl=3Den&sa= =3DX&ved=3D2ahUKEwj1r8jX6vPiAhVEMawKHalBBxIQ6AEwBnoECAkQAQ#v=3Donepage&q=3D= %22doublet%22%20%22quartet%22%20%22average%22%20%22spin%22%20%22broken%22&f= =3Dfalse explicitly states the following, "For a broken-symmetry singlet contaminated by a triplet state, we would ex= pect 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 < owner-chemistry,+,ccl.net> 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 UH= F > 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 t= he > overlap integrals between (occupied) alpha/beta spin orbitals. > Say in the case of an open-shell singlet diradical (the two atoms at larg= e > separation), you would get S(S+1) + 1 =3D 0.0 + 1.0 (0.0 for the singlet,= and > N(beta) =3D 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 =3D 0.75 = + > 1.0 =3D 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 manifol= d > for real systems is taken into account. > > Hope this helps. > > Tobias > > Dr. Tobias Kr=C3=A4mer > 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=3D=3Dmu.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 value= s > 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 i= s > being done. I would be grateful if someone could provide a detailed > explanation (mathematical derivation) of this.http://www.ccl.net/chemist= ry/sub_unsub.shtmlhttp://www.ccl.net/spammers.txt > > > -=3D This is automatically added to each message by the mailing script = =3D-> > > --00000000000051a218058b9f0ced Content-Type: text/html; charset="UTF-8" Content-Transfer-Encoding: quoted-printable
Dear Dr.=C2=A0Kr=C3=A4mer

Thank y= ou for your reply. I was aware of the equation for S^2 (UHF) in Szabo. Howe= ver, I was a bit confused by the language used in a few books and papers th= at I came across. Specifically, the paragraph in the following book (Page 2= 38 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=C2= =A0expect HS< 2 > to be = 2 (the correct eigenvalue for a triplet state) and BS< S2 > to be= about=C2=A01 (the average of a sing= let and a triplet). In the same way, for a broken-symmetry doublet contamin= ated by a quartet state, we would expect HS< S2 >= ; to be 3.75 (the correct=C2=A0eigenvalue for a quartet) and BS< S<= /span>2 > to be 1.75 = (the average of a doublet and a quartet)."
Best,
Ankur

On Tue, Jun 18, 2019 at 3:26 PM Tobi= as Kraemer Tobias.Kraemer]![mu.i= e <owne= r-chemistry,+,ccl.net> 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 c= onsider the number of unpaired beta-spin electrons N(beta), as well as the = overlap integrals between (occupied) alpha/beta spin orbitals.=C2=A0
Say in the case of an open-shell singlet diradical (the two atoms at large = separation), you would get S(S+1) + 1 =3D 0.0 + 1.0 (0.0 for the singlet, a= nd N(beta) =3D 1). I am not sure where you get the value for the
Broken-symmetry doublet from, since the weighter average doesn't give y= ou the correct value here. In this case you would apply S(S+1) + 1 =3D 0.75= + 1.0 =3D 1.75. I am assuming that the value is not decreased by
the spatial overlap terms. Generally the observed values are far from the i= deal, if the spatial overlap between the occupied alpha and beta manifold f= or real systems is taken into account.

Hope this helps.

Tobias

Dr. Tobias Kr=C3=A4mer
Lecturer in Inorganic Chemistry
Department of Chemistry

Maynooth University, Maynooth, Co. Kildare, Ireland.
E: tobias.kraemer-.-mu.ie=C2=A0=C2=A0 T: +353 (0)1 474 7517


-----Original Message-----
> From: owner-chemistry+tobias.kraemer=3D=3Dmu.ie-.-ccl.net <owner-chemistry= +tobias.kraemer=3D=3Dmu.ie-.-ccl.net> On Behalf Of Ankur Kumar Gupta ankkgupt*<= a href=3D"http://iu.edu" rel=3D"noreferrer" target=3D"_blank">iu.edu Sent: 18 June 2019 16:53
To: Tobias Kraemer <Tobias.Kraemer-.-mu.ie>
Subject: CCL: How to calculate S^2 (S squared) value of a broken-symmetry s= tate?


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 o= f the singlet and triplet S^2. Similarly, the S^2 of broken-symmetry double= t (contaminated by the quartet state) turns out to be 1.75 (average of a do= ublet and a quartet). I would like to know how these average values are bei= ng 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 quarte= t, respectively), but I don't know how that weighting is being done. I = would be grateful if someone could provide a detailed explanation (mathemat= ical derivation) of this.http://www.ccl.net/cgi-bin/c= cl/send_ccl_messagehttp://www.ccl.net/chemistry/sub_unsub.shtmlhttp://www.c= cl.net/spammers.txt


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