Subject: Summary: Imaginary freq. and rate constant

From: "Ivar Martin" <[email-masked]>
Date: Thu, 25 May 95 15:42:40 EST

Dear CCL people,

to those of you who responded to my question about rate constant
calculation based on transition state vibrational properties, I am 
very grateful for sharing your knowledge!

I would like to send my thanks to the following people for 
their valuable coments:
                            Ryan Bettens
                            Joe Durant
                            Frank Jensen
                            Istvan Mayer
                            Arvi Rauk
                            John Rupley
                            Robert Q. Topper
                            Dom Zichi
                            
My original question was: 
     
     As it is expected, MOPAC calculates one imaginary vibrational
frequency at TS. Is it correct to use this frequency as pre-exponent
factor (frequency factor) in Arrhenius rate constant equation?

The unanimous answer was: NO, IT IS NOT CORRECT!

RATIONALE: 
-------------------------------------
From: Arvi Rauk <[email-masked]>

The imaginary frequency only measures the curvature 
of the reaction coordinate at the transition state. It may 
be used in the Bell formula for a quantum mechanical tunnelling
correction to the classical rate which would only be a funcion of the
barrier height.  
-------------------------------------
From: Frank Jensen <[email-masked]>

      The A*exp(-E/RT) expression is equivalent to the TS expression
kT/h * exp(deltaS/R) * exp(-deltaH/RT). Delta S and H may be calculated
from S and H of the reactant and TS. S and H for a given structure may
be calculated by means of stat. mech. from properties of the individual
molecules, i.e. moments of inertia and vibrational frequencies. The
important feature here is that there are 3N-6 vibrational contributions
to the reactant, but only 3N-7 for the TS, i.e. the imaginary frequency
is NOT involved. In TS theory it is assumed that the motion along
the reaction coordinate (the imaginary frequency at the TS) is treated
classically. Only if e.g. tunneling is considered does the imaginary
frequency enter, for example by means of a Bell correction etc. 
      So, to answer your question, no the imag. freq. has nothing
to do with A in the A*exp(-E/RT) expression. A is essentially related to
deltaS for the reaction.
---------------------------------------
From: Joe Durant <[email-masked]> 

No, the negative frequency associated with the reaction coordinate is
not used in the calculation of the sum of states for the transition
state!  I suggest a careful reading of Robinson and Holbrook
"Unimolecular Reactions", or the kinetics texts by Johnston or Benson.
I quote from Robinson and Holbrook, pg 12 "Q(double dagger) is the
partition function for all the degrees of freedom of the activated
complex except the reaction coordinate."  The translational degree of
freedom gives rise to the kT/h term in the rate constant expression.
The negative frequency is related to the barrier shape, and can be
used in tunneling calculations.
     A closely related point is that the negative frequency does not
contribute to the zero point energy of the transition state.
------------------------------------------
From: John Rupley <[email-masked]> 
  
Transition State Theory assumes in its model free flight at 
the barrier top, i.e., a flat barrier top (corresponding to a zero 
frequency!).  The frequency factor is the frequency of attempt at 
barrier crossing, and can be taken as the frequency defining the shape 
of the reactant _well_ in the direction of the reaction coordinate. 
This frequency is built into the deltaS# part of the deltaG# of the 
Arrhenius expression. 
     The imaginary frequency and the curvature of the potential
surface at the barrier top also enter into the Kramers and related
classical models, for reaction in a dense medium.
-----------------------------------------------
From: Dom Zichi <[email-masked]> 

      The correct frequency to be used in the transition state theory 
      rate expression, kTST, is that of the reactant well, not the 
      frequency describing the curvature at the barrier top.  This 
      is easily seen from the following.  A simple TST rate assumes 
      that every trajectory at the transition barrier q# which has a 
      forward flux toward products will always end up as product.
      This leads to the following expression for kTST,

            /    /              .       .
      kTST= | dq | dp e(-H/kBT) q theta(q) delta(q#) / Qrct
            /    /

      where theta is a step function which equals one for positive
      flux toward products and 0 for negative flux, delta(q#) is a 
      delta function of position along the reaction coordinate q at
      the barrier top and Qrct is the partition function for the 
      reactant well,

            /    /             
      Qrct= | dq | dp e(-Hr/kBT) 
            /    /

                  2 2      2
      For Hr = m w q /2 + p /2 valid to q#, i.e. 

      H(q)=  Hr(q)  q<q#
             Hp(q)  q>q#

      one can easily obtain kTST = (w/2*pi) exp(-H(q#)/kBT). The
frequency term counts the number of times a forward reaction is
attempted while the exponential term counts the number of
trajectories that make it to the barrier top relative to the reactant
well.
-----------------------------------------------
From: Istvan Mayer <[email-masked]>

The existence of an "imaginary frequency" at the TS is a reformulation
of the fact, that the enrgy has a maximum (and not a minimum) along a
given (normal) coordinate. Although it is expressed in frequency units,
it is really a measure of the curvature of that point only, and has
nothing to do directly with any reaction rate. The "quasi-thermodynamic"
formulation of Eyring's "absolute reaction rate" or "transitiuon state"
theory was a great-scale misunderstanding, which caused very much harm
to science. 20 years ago I wrote: "The activated complexes are not true
chemical species, do not fulfil the conditions defining a canonic
ensemble, and therefore cannot be considered as independent subjects of
the canonic distribution." (J. Chem. Phys. 60, 2564, 1974). This also
means that there is no meaning to attribute any thermodynamic quantities
to the TS.
     This does not mean that there can be no modern variants of
transition state theory, which are free of the deffects of the old one.
-----------------------------------------------
From: Ryan Bettens <[email-masked]> 

Assuming that the rate constant, k, is given by, k = A exp{-E_a/(k_b T)}
for all T. E_a is the barrier height and k_b is Boltzmann's constant. From 
RRKM theory A is given by, 

A = (alpha/h) f_{inf} G^*(E - E_a)/N(E - E_a).

Where,
alpha        is the reaction path degeneracy,
h            is planks constant,
f_{inf}      high-pressure centrifugal correction factor,
E            the internal excitation energy,
G^*(E - E_a) total number of states between 0 and (E - E_a) for the 
             transition state (hereafter TS),
N(E - E_a)   the number of states per unit energy range at (E - E_a) for 
             the reactant.
             
To get a nice interpretation for what A actually is we (a) ignore alpha, 
(b) ignore rotation (this cannot be done if the reactant is linear and the 
TS is non-linear and visa versa), (c) assume an harmonic oscillator and the 
approximation of Whitteen and Rabinovitch (1). A then becomes, 

A = {(E - E_a + a^* (E_z)^*)/(E - E_a + a E_z)}^(s - 1)
    x {Prod(i = 1 to s) nu_i}/{Prod(i = 1 to s - 1) (nu_i)^*}.
    
Where,
s   is the number of degrees of vibrational freedom of the reactant,
a   is an empirical correction factor from ref. (1),
E_z is the zero-point energy
^*  refer to the TS with one less degree of vibrational freedom.

Approximating the first term, which is taken to the power of (s - 1), as 
unity, A finally becomes the ratio of the product of harmonic frequencies 
of the reactant to the product of the harmonic frequencies of the TS. Note 
that there is (s - 1) frequencies in the TS (the "missing" degree of 
freedom is the mode responsible for the chemical change, i.e., the mode 
with the negative force constant) compared with the reactant's s 
frequencies. Furthermore, if we make the assumption that the TS's 
frequencies are not much different from the reactant's frequencies, except 
for the "missing" frequency, we find, 

A = nu_i

where nu_i is the mode in the reactant which when energized is responsible 
for the chemical change. Thus for a C-C single bond dissociating 
A ~ 900 cm^{-1} = 2.7 x 10^{13} s^{-1}. All of the above expressions come 
from Forst (2), which I suggest you take a look at if you really want a 
good understanding of the approximations made etc. 
     Ref.'s
(1)  G. Z. Whitten and B. S. Rabinovitch, J. Chem. Phys., V38 (1963) 2466.
(2)  W. Forst, Theory of Unimolecular Reactions, 1973, Academic Press, 
     New York.
---------------------------------------
From: Robert Topper <[email-masked]>

May I suggest that you have a look at some of the recent papers by
Donald Truhlar (Minnesota) and his group! They have developed two
codes for the calculation of reaction rate constants. One of these,
called MORATE, interfaces with MOPAC to carry out transition-state
theory calculations for polyatomic systems. At its highest level of
theory, MORATE includes corrections for multidimensional tunneling
and zero-point motion. However, this would entail getting information
about the potential energy all along the reaction coordinate and not
just at the transition state.  However, MORATE can also carry out
calculations using only the transition-state properties.

With the experience you have gained using MOPAC you could probably
use MORATE with ease. It is extremely well-documented and is
available from QCPE and the CPC library...  and the methods are also
described in a number of articles. They have been extensively
benchmarked (where possible) against "exact" quantum scattering
theory treatments and against experiment.  Overall, the methods work
quite well for absolute rate constants and do a terrific job of
getting at kinetic isotope effects.  Truhlar's group is constantly
developing the technology and improving the user-friendliness of the
code... and they have been doing so for a number of years now. It is
written almost entirely in machine-portable FORTRAN and is available
for a number of platforms.

That said, you must realize that the absolute, accurate calculation
of reaction rate constants can occasionally be frought with
difficulties.  For example; transition-state theory makes some
assumptions about the nature of reactive motion. In particular, TST
is only "rigorously" correct when no back-reaction is possible (i.e.,
when a molecule goes from reactants to products it never re-forms
into reactants.  See "Chemical Kinetics and Dynamics" by Steinfeld,
Francisco and Hase for a nice introductory discussion of this).  This
is sort of a tough situation to achieve experimentally. However, in
my own weird, brief experience in studying these effects I have never
seen a case where the correction was larger than an order of
magnitude...which is considered to be pretty good for rate constants.
And that order of magnitude was in pretty extreme cases. There are
other possibilities... the presence of "dynamical bottlenecks" to
reaction can slow things down (this is a nonlinear effect).  Overall
though, TST can be a very accurate and useful approximation.  One
case in which you are pretty much guaranteed that TST will work is
when the time scale of reaction is much much faster than the time
scale of back-reaction... this can happen when the products' phase
space is very much larger than that of the transition state.  Also,
any reaction that involves dissociation is a good candidate.
-----------------------------------
THANK YOU ALL AGAIN!
_______________________________
Dr. Ivar Martin
Department of Bioorganic Chemistry           tel: +372 2 526510
Institute of Chemistry                       fax: +372 2 536371
Akadeemia tee 15 EE0026                   e-mail: [email-masked]
Tallinn, ESTONIA