Subject: Summary:nth-order saddle point, n>1

From: Sofia Godinho <[email-masked]>
Date: Tue, 12 Jun 2001 12:19:57 +0100 (WEST)

Dear CCLers,
some days ago I posted a question regarding what would be a nth-order
saddle point, n>1. 
Here you have the summary of the answers. Thanks are due for their help, 
to - "in order of arrival":
Xavier Assfeld, James Robinson, Laurence Cuffe, David Shobe, Christoph 
van Wuellen, Andrus Metsala, Huub van Dam and Wai-To Chan.
________________
Original message:
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Dear CCLers,
I wonder what is the chemical/physical meaning of a nth-order
saddle point (n>1)?
That is...for instance, when I get four immaginary frequencies
does it mean that I have to "pass through" four vibrational modes
to arrive to a minimum? (Or to three to get to a transition state?)
Or it simply has nothing to do with it and it is only a mathematical
definition of a saddle point?
Or...?

May I thank you in advance for any help, 
        Sofia

_______
Summary:
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From: Xavier ASSFELD <[email-masked]>

A nth order saddle point correspond to the transition state
a n chemical reaction occuring simultaneously.
There was a discussion about it some months ago on CCL, and
as far as I remember, there was experimental evidences that
proved that 2nd order saddle points exist.
Nth order saddle point do not belong to the minimum energy
path, but are available configurations in highly energetic
(kinetic) chemical system.
Hope this helps.
--

                                      ...Xav
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From: "Robinson, James" <[email-masked]>

A transition state should have only ONE imaginary frequency.

It would be odd if a single TS could lead to different products. How would
the TS distinguish between paths to follow. I suspect that attempting to
rationalise some mathematical result will be academic. I doubt if it has
any
physio/chemical meaning and is probably just the result of computation.
Simple finding some computational result does not mean anything. Results
should be reconciled with real-world problems.

James
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From: Laurence Cuffe <[email-masked]>

For a bent triatomic molecule where the bent form is the most
stable then a linear arrangement may appear as an order two
saddle point, indicating that there is a two dimensional surface of
acceptable trajectories to the minimum rather than a single path.
(your on to of a hill and every way is down!)  more generally where I
have encountered higher order saddle points examination of the
molecular motions involved in the imaginary frequencies have
revealed that they corespond to rotations of separate groups in the
molecule each of which was at a local saddle point.
Hope this helps
Laurence Cuffe
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From: "Shobe, Dave" <[email-masked]>

I don't think it means anything other than that there are four
"independent"
ways of distorting the molecule that lower the energy.  (These can be
symmetry-equivalent pairs or triples of distortions--"independent" really
has to do with the orthogonality of vectors).  You can go directly to the
minimum from the fourth order saddle point without going through other
stationary points.  As an example, imagine you are at the top of a hill--a
maximum, but also a 2nd-order saddle point.  If you go in the right
direction you will reach a first-order saddle point, but in most cases
following the gradient downward takes you directly to a local minimum.

--David Shobe
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From: "Christoph van [iso-8859-1] Wuellen"
<[email-masked]>

>A transition state should have only ONE imaginary frequency.
>
the reason is that if it is a higher-order TS, there will be another one
with lower energy, and molecules are sharp enough to take that one.
---------------------------------------------------------------------------
From: Andrus Metsala <[email-masked]>

A transition state may have more than one imaginary frequency. For example
so-called monkey saddle has two imaginary frequencies. We may imagine
transition states even with more than two imaginary frequencies.

Andrus
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From: Huub van Dam <[email-masked]>

Lets restrict this problem to a 2 dimensional problem. Think of the
potential energy surface as hilly country side. Down in the valleys you
will find 2 positive frequencies. This means that moving either along the
north-south axes or the east-west axes or any linear combination of the
two, you will always have to go uphill. Passing from one valley to the
next between to hill tops, you will find a point with 1 imaginary
frequency. This means that there is one direction along which you can only
go down hill from that point. Along another direction you will go uphill
towards one of the hill tops.  Finally, when you are on a hill top you
will find 2 imaginary frequencies. This means that there are 2 linearly
independent directions both leading downhill from the point where you are.

In your example, where you have 4 imaginary frequencies this simply means
that there are 4 linearly independents directions along which you can only
go down in energy. That is all.

I hope this helps,
    Huub
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From: Wai-To Chan <[email-masked]>

You can easily pick up a higher-order saddle point structure.
Let me give you an illustration. Suppose you are trying to locate
the transition state (TS) structure of the hydrogen
transfer reaction of a large hydrocarbon molecule:

  RH + X ----> R---H--X ---->  R + HX

  You obtained a TS structure R---H---X with the 1st imaginary

vibration corresponding to the H-transfer motion. But your TS
optimization job could fail to minimize the energy of the
structure with respect to some of the degrees of freedom
other than the reaction coordiates. For instance, a methyl
group in R is locked in a higher energy conformation
and you obtain a second imaginary frequency corresponding to
the torsion of that methyl group.

If you obtained a TS structure with more than one imaginary frequencies
you should first examine their motions with a program like
MOLDEN or GAUSSVIEW. If you are lucky one of the vibrations
would be the one corresponding to the motion along the
reaction coordinate. You may then try a reoptimization of
this higher order saddle point to a 1st order structure.
For the RH+X illustration from above you may try partial
optimization of the 2nd order structure by relaxing any
constraints of the the motion of the problematic methyl group.
You then repeat your TS search job
using the reoptimized structure as the starting geometry.

Wai-To Chan
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Sofia Godinho
Ph-D student
ITQB-UNL-Portugal
email: [email-masked]