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 <S^2> 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
<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 <Tobias.Kraemer- at -mu.ie>
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
<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 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 <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 <Tobias.Kraemer-.-mu.ie>
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
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