Subject: SUMMARY: Second order TS's crossroads

From: Valentine Ananikov <[email-masked]>
Date: Tue, 10 Oct 2000 14:59:36 +0400

Hi!

In summary, the following references were suggested: 

1) F. Jensen, "Transition Structure Optimization Techniques",
in The Encyclopedia of Computational Chemistry,
(1998) 3114-3123.
2) J. Mol. Struct. 1995, 338, 117-130
3) J. Chem. Phys. 104, 1774-8 (1996)
4) J. Chem. Phys. 104, 8507-15 (1996)

Below are my original request and full replies I have got.

I thank to Frank Jensen, Per-Ola Norrby, Trevor Power,
Stefan Fau, David Woon and George Vacek for the replies !

best regards, Valentine. 

----------------------------------------------------------

  Dear CCL'ers,

There are two questions concerning second order TS's, which I
found being rather contradictory in the literature. So:

1) in some articles one may read something like that: "... we
found that reaction proceeds through the second order transition
state ... and gives desired products ..."

while the other authors state: "... second order saddle points
are of no interest for chemical reactions ..."

Where is the truth? May a chemical transformation proceed through
a second order transition state?

2) Next,

If there is a second order TS, does it mean that there are
also two independent first order TS's representing each of the 
imaginary vectors of the former? Is the statement always correct?

Any suggestions and references are very welcome.

best regards,
Valentine.

              -------------------------------

---------------------------------------------reply1------------
from: Frank Jensen, <[email-masked]>

        Depends on whether you have a 'static' or 'dynamic' picture.
In the static picture nuclei move infinitely slowly, i.e. along the IRC.
Since there is no unique IRC from a second order saddle point, it
may be unclear what reactant/product it connects. Furthermore, the
energy of a second order saddle point can always be lowered along one of
the imaginary modes, leading to a first order saddle point, a real TS.
Again, unless there is symmetry in the system, there is no unique
way of getting from a second order saddle point to a TS, i.e. there may be
one, two or more real TSs, connecting the same reactant/product or
different reactants/products.
In this picture, second order saddle points have no chemical significance,
as they always are higher in energy than the real TSs.

In the dynamic picture nuclei are moving on the PES, with a kinetic
energy depending on the temperature. In this case the reaction samples
a larger fraction of the surface, and a representation of the reaction
requires running many trajectories, and suitable averaging. In this case
one can imagine a second order saddle point being sufficiently low
in energy that it makes a significant contribution to the reaction, i.e. 
is traversed by a good fraction of the trajectories. In this case it is
of chemical significance.

In principle one could imagine two minima being connected dynamically
by a second order saddle point, but without having a real TS connecting
the two minima (there may be another path with real TSs going through
one or more intermediates). In such a case the second order saddle point
may be very important chemically, but I have not yet seen an explicit 
example of this.

        Frank
---------------------------------------------------------------

---------------------------------------------reply2------------
from: Per-Ola Norrby, [email-masked]

>1) in some articles one may read something like that: "... we
>found that reaction proceeds through the second order transition
>state ... and gives desired products ..."
>
>while the other authors state: "... second order saddle points
>are of no interest for chemical reactions ..."
>
>Where is the truth? May a chemical transformation proceed through
>a second order transition state?

        The second is more true than the former.  All low-energy 
points on the potential energy surface will be visited, but the lower 
the energy is, the higher the population will be.  If there are any 
second order TS's, there MUST also be some first order TS's, and they 
must have a lower energy, therefore be more important.  You get very 
close to the truth if you simply ignore higher-order maxima.

>2) Next,
>
>If there is a second order TS, does it mean that there are
>also two independent first order TS's representing each of the
>imaginary vectors of the former? Is the statement always correct?

        No.  If you verify that the second order maximum leads to the 
correct reactant and product, there must also be at least two 
flanking TSs (there could also be intermediates etc, you never know), 
but there is no good way to tell what the vector in the two flanking 
TSs will be.  If you have a very simple potential energy surface, you 
might try projecting the vector from reactant to product in the 
2D-space described by the two eigenvectors of the 2nd-order max.  The 
resulting vector could well be similar to BOTH the vectors of the 
TSs, whereas the orthogonal direction might connect the 2nd order max 
to the TSs, but you have no guarantee.  The high order max is not a 
good starting point for finding the 1st order TSs if you're using a 
Newton-Raphson-based method, but there are several other methods that 
might do it.  Frank Jensen wrote a review that includes some suitable 
methods, if I remember correctly it was in "Encyclopedia of 
Computational Chemistry".

        Per-Ola
---------------------------------------------------------------

---------------------------------------------reply3------------
from: Trevor D. Power, [email-masked]

Valentine,
        I actually ran into this head first when I was doing modeling of
organolithium reactions about a year ago. Many theoreticians will tell you
that a second-order transition state is a theoretical entity and cannot
be called a transition state nor a molecule anyway.  If fact the only
correct designation would probably be a second-order saddle point.  When
one is looking for a transition state that describes a particular reaction
and they instead find such a maximum, this usually means that the reaction
they are trying to model is actually not concerted and may go through two
transition states and 1 intermediate.  I would think that one needs to go
to extreme measures to find a second-order saddle point; such as symmetry
or geometry restrictions (a partial optimization --- which does not make 
a lot of sense).  
        I would expect that those papers describing a reaction to go
through second-order saddle points are in journals not set up to
rigorously review computational manuscripts (like J. Org. Chem. for 
example).  You would do well to find papers on this subject in J. Phys.
Chem or J. Chem. Phys.  Alan Isaacson, Miami University, wrote a few
papers (none are reviews though) that began my literature search on this
topic.
        I would agree with the "other authors" as stated below.  The
principle of least motion is violated in a second-order saddle point.  

Hope my posting does not spark too many fires out there.

Regards,
Trevor D. Power
---------------------------------------------------------------

---------------------------------------------reply4------------
from: Stefan Fau, [email-masked]

Hi Valentine,

to 1)

I think that 2nd order TSs are of little importance, chemically.
If you have a 2nd order TS with one imaginary mode connecting
your reactants and products, you can optimize along the other
imaginary mode and get a 1st order TS that should still connect
your reactants and products. (I don't know of any exceptions to
that, but maybe there are a few.) Due to the optimization, the energy
of this 1st order TS is lower, and therefore more reacting molecules
should pass along here (assuming that the effects of nuclear motion
are small enough).

Classically, every point (geometry) on the potential energy
hypersurface which is below the molecules energy is accessible
by that molecule. So if you supply your molecule with lots of energy
lots of geometries are possible. If you give just enough energy to
keep
a very slow reaction going virtually all product molecules will have
passed the 1st order TS.

to 2)

A 2nd order TS is not generally connected to four 1st order TSs (two
for each imaginary mode). I once had a 2nd order TS connected to
two mirror-symmetric 1st order TSs and two minima. The first order
TSs connected the same minima. I think that this scheme might be
common with 2nd order TSs of higher symmetry.

The example is published in
S. Fau, G. Frenking, Theochem - J. Mol. Struct. 1995, 338, 117-130.
I think the 2nd order TS was a C2v structure of B2CH4, the 1st order
TS was non-planar Cs and the minimum was planar Cs.

Stefan
---------------------------------------------------------------

---------------------------------------------reply5------------
from: David E. Woon, [email-masked]

I agree with the other respondents about the importance (or lack
thereof) of second-order saddle points, with one small exception
that was not mentioned. If an nth order saddle point is defined
as a stationary structure (all first derivatives zero) with n
negative eigenvalues of the second derivative matrix, then a
second order saddle point is still a true TS if the eigenvalues
are degenerate. This occasionally occurs due to symmetry.

Dave Woon
---------------------------------------------------------------

---------------------------------------------reply6------------
from: George Vacek, [email-masked]

>         Many theoreticians will tell you
> that a second-order transition state is a theoretical entity and cannot
> be called a transition state nor a molecule anyway.  If fact the only
> correct designation would probably be a second-order saddle point.  When
> one is looking for a transition state that describes a particular reaction
> and they instead find such a maximum, this usually means that the reaction
> they are trying to model is actually not concerted and may go through two
> transition states and 1 intermediate.  I would think that one needs to go
> to extreme measures to find a second-order saddle point;

A well stated response Trevor.  A couple papers where theoreticians concede that
experimentalists may have seen evidence of a second order saddle point are:

``The Anomalous Behavior of the Zeeman Anticrossing Spectra of \~A
$^1A_u$ Acetylene: Theoretical Considerations,'' G. Vacek,
C. D. Sherrill, Y. Yamaguchi and H. F. Schaefer, {\it J. Chem. Phys.}
{\bf 104}, 1774-8 (1996).

``The \~A $^1A_u$ State and the $T_2$ Potential Surface of Acetylene:
Implications for Triplet Perturbations in the Flourescence Spectra of
the \~A State,'' C. D. Sherrill, G. Vacek, Y. Yamaguchi,
H. F. Schaefer, J. F. Stanton and J. Gauss, {\it J. Chem. Phys.} {\bf
104}, 8507-15 (1996).

Regards,
George Vacek
---------------------------------------------------------------

.... that's all.

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