Subject: SUMMARY: Vibrational entropy of a transition state

From: Herve Toulhoat <[email-masked]>
Date: Tue, 29 Jun 1999 13:03:15 +0200

Dear CCL'ers,

I summarize below the replies I obtained very quickly to my question. It
helped a lot.
CCL proved once again very effective in filling the gaps of my porous
personal knowledge.

I thank very much all those who supplied information. I hope this
summary will be useful to others.

I found thereafter that in MOPAC 6.0 the keyword TRANS=n allows to
delete the first n vibrations (i.e. the negative(s) frequencies) for a
thermochemical calculation. I am not sure this very useful feature is
present in other "off the self" codes available (could vendors and
authors manifest themselves on that point?).

With my best regards,

Dr Herve TOULHOAT

Directeur de Recherche Associe
Group Leader, Molecular Modeling and Computational Chemistry
Div. Computer Science and Applied Mathematics, IFP
Director, Groupement de Recherches CNRS G1209:
Dynamique Moleculaire Quantique Appliquee a la Catalyse
Research Group CNRS G1209
Ab Initio Molecular Dynamics Applied to Catalysis
(Scientific Partners: CNRS, IFP, Total, University of vienna, TU
Eindhoven)

INSTITUT FRANCAIS DU PETROLE                    Tel: +33-01-47-52-73-50
1 & 4 Avenue de Bois-Preau                      Fax: +33-01-47-52-70-22
92852 RUEIL-MALMAISON Cedex FRANCE       E-mail: [email-masked]

Visit IFP Website at http://www.ifp.fr

My original request was:
=========================================================================

The transition state theory (TST) allows to relate in principle
the rate constant of a reaction to the variation in free energy
between reactant(s) and the transition state. Of specific interest
is the variation of entropy between reactant(s) and TS, which relates to
the so-called pre-exponential
factor of the rate equation.

The entropy change for a gas phase reaction will be dominated by the
vibrational component.
The vibrational entropy of a molecular species can be evaluated from the
set of vibrational
normal modes. A TS however corresponds to a saddle point of the PES, and
exhibits therefore
one imaginary frequency (one negative eigenvalue in the hessian matrix).
This imaginary
frequency cannot be included in the statistical mechanical calculation
of the vibrational entropy.
Is it legitimate to simply forget about it?

More generally, is there a rigourous procedure for evaluating the
vibrational entropy of a TS?
============================================================================

I obtained very rapidly a clearcut answer through numerous replies:
===========================================================================

>From R.A.van Santen <[email-masked]>:

Dear Herve,

        In our book R.A. van Santen, H. Niemantsverdriet, Chemical
kinetics
and Catalysis we have explained why in the transition state entropy the
reaction coordinate is not included. The procedure is vigorous. I hope
to
see you Soon.

        Best regards,

                Rutger

Eindhoven University of Technology
Laboratory for Inorganic Chemistry and Catalysis
P.O. Box 513
5600 MB EINDHOVEN
The Netherlands

Tel.: +31 40 2472730
Fax:  +31 40 2455054

(Comment: I have read the book some times ago, certainly not carefully
enough!
 Indeed the answer is there page 149 for instance. Specialthanks to you
Rutger).
===============================================================================

>From Ned C.Haubein:

Dr. Toulhoat,

> This imaginary frequency cannot be included in the statistical
> mechanical calculation of the vibrational entropy. Is it legitimate to

> simply forget about it?

yes, you should omit the contribution of the motion along the reaction
coordinate.  For a rigorous derivation, see "Kinetics and Mechanism"  by

John W. Moore and Ralph G. Pearson, 3rd ed, Wiley, 1981 pp 165,178.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Ned C. Haubein
Graduate Student
Dept. of Chemical Engineering
Northwestern University
2145 Sheridan Rd / Rm E136
Evanston, IL 60208-3120
Phone: 847-467-1402
Fax:   847-491-3728
Email: [email-masked]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
===============================================================================

>From Wayne Steinmetz <[email-masked]>:

In Eyring's derivation of the TST, the contributions of the degree of
freedom
with an imaginary vibrational frequency is handled separately.  This
degree of freedom corresponds to translation in reaction space along the

reaction coordinate.  The particle-in-a-one-dimensional-box model is
used
to evaluate its partition function.  The result of this calculation and
the contribution of the velocity in reaction space at the transition
state
combine to yield the kT/h term.  Therefore, in the well known result
rate constant = (kT/h) x [exp(-delGact/kT)], translation in reaction
space, the degree of freedom associated with the imaginary frequency,
does NOT
contribute to the calculation of delGact, the standard Gibbs free energy
of
activation.  In this sense, the transition state is a peculiar species
with
only 3N-1 degree of freedom, not 3N as in the case of a normal molecule.

A useful reference for these calculations from first principles
is H. S. Johnston, Gas Phase Reaction Rate Theory, Ronald Press, New
York,
1966.  Johnston was the first to come up with a reaction mechanism for
the destruction of stratospheric ozone.  In his case, the villain was
NO.  His calculations were instrumental in ending the US participation
in
the development of the supersonic transport which would have dumped
large amounts of NO in the stratosphere.  Rowland later used his
approach
to establish that atomic chlorine would have the same effect.
In the development of his model, Johnston required rate  constants
that had not been measured at the time.  He therefore refined the
use of TST to obtain reasonable estimates of the missing data.

(Comment: I was delighted with this point of modern history. I suppose
Concorde is to be blamed for a part of the ozone hole. I am afraid
however of subsonic transport as well.)
=============================================================================

>From Karl Irikura <[email-masked]>:

Dear Dr. Toulhoat,

I've never seen any mathematical discussion of this (but I'm not a
kineticist).  The
quasi-equilibrium hypothesis of TST only makes sense physically if
motion along one degree of
freedom, the reaction coordinate, is prohibited, because any
displacement in that direction would
yield a molecular structure that is no considered a transition state.
If you agree with this, then
it's easy to accept that the (imaginary) "frequency" in the deleted
coordinate should also be
deleted.  This is what people usually do, so the vibrational partition
function includes only 3N-7
vibrational coordinates but is otherwise ordinary.

If you do not agree to delete that coordinate, then clearly the
"imaginary frequency" is
irrelevant, since it describes only the curvature of the surface and has
nothing to do with energy
eigenstates (which alone determine the partition function).  Instead,
the energy levels on the
unbound (one-dimensional) potential are a continuum of scattering
states.  You could work out
the corresponding partition function for a model potential, e.g., U =
-0.5*k*x*x (maybe this has
already been done in some textbook).  Perhaps the result would be
similar to that for a free
translation.  But I wonder if this approach would be consistent with the
usual simple treatment
(translational or vibrational or rotational) of the reactants along all
coordinates, including the
special one.  Please let me know if you learn whether this can be worked
out consistently.

Meilleurs voeux,

Karl Irikura

================================================================================

>From Carlos Gonzalez <[email-masked]>:

Dear Herve;

Once a reacting system goes from the reactant side (a stationary point
characterized by a all
positive frequencies) towards the TS, one of the modes becomes a
translation and this is
basically why it becomes imaginary (col 1, in mathematical language).
So, accordingly, this
mode can not and should not be included in the zero point energy
corrections for the calculation
of the entropy. A similar situation arises when you step down from a TS
following a reaction
path toward the minima. In these cases, of course overall translations,
and rotations should be
projected out of the Hessian, but you also need to project out the
translation along the reaction
coordinate (the one that correlates with the mode that gives an
imaginary frequency at the TS).
This is a general procedure that is used to compute the rate constants
as well as the DeltaG of
activatioan along the path needed in VTST calculations. You can read
Donald Truhlar's papers
on this subject. I hope this little note helps you.

Carlos Gonzalez.
==============================================================================

>From Frank Jensen <[email-masked]>:

        The missing frequency is the reaction coordinate, and is
part of the pre-exponential factor. So yes, it is legitimate
to not include it in the entropy. And note for bimolecular
reactions the translational entropy change by 30-40 eu., which
usually is much larger than the vibrational component.

> More generally, is there a rigourous procedure for evaluating the
> vibrational entropy of a TS?

        Beyond the rigid rotor-harmonic frequency approximation, there
are no 'standard' procedure. The next improvement could be anharmonic
corrections. Try looking up some of Truhlars work.

        Frank

================================================================================================

>From Jordi Villa (<[email-masked]>):

Dear Herve,

I recommend to you to read some work from Donald G. Truhlar
(http://comp.chem.umn.edu/~truhlar/)
specially the following references:

"The Current Status of Transition State Theory," D. G. Truhlar, W. L.
Hase, and J. T. Hynes, Journal of Physical Chemistry 87, 2664-2682
(1983). Erratum: 87, 5523 (1983).

"Generalized Transition State Theory," D. G. Truhlar, A. D. Isaacson,
and B. C. Garrett, in Theory of Chemical Reaction Dynamics,
     edited by M. Baer (CRC Press, Boca Raton, FL, 1985), Vol. 4, pp.
65-137.

"Transition State Theory," M. M. Kreevoy and D. G. Truhlar, in
Investigation of Rates and Mechanisms of Reactions, 4th edition,
     edited by C. F. Bernasconi (John Wiley and Sons, New York, 1986),
Part 1, pp. 13-95. [Tech. Chem. (N.Y.), 4th ed., 6/Pt. 1, 13-95
     (1986)]

"Dynamical Formulation of Transition State Theory: Variational
Transition States and Semiclassical Tunneling," S. C. Tucker and
     D. G. Truhlar, in New Theoretical Concepts for Understanding
Organic Reactions, edited by J. Bertrán and I. G. Csizmadia (Kluwer,
     Dordrecht, The Netherlands, 1989), pp. 291-346. [NATO ASI Ser. C
267, 291-346 (1989)]

"Current Status of TransitionState Theory," D. G. Truhlar, B. C.
Garrett, and S. J. Klippenstein, Journal of Physical Chemistry 100,
12771-12800
(1996). (Centennial Issue)

and to take a look at his computer progrtam POLYRATE
(http://comp.chem.umn.edu/polyrate/)

Best wishes

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