# how are the frequencies at a local maximum of PES like?

On the potential energy surface, if you find a local maximum and calculate its frequencies in Gaussian or something like that, will you get all negative frequencies or all positive frequencies? I know that at a local minimum the frequencies are all positive and that at a saddle point there are both negative and positive frequencies, but what about a maximum point?

The potential energy surface (PES) is a very complex thing and it is hard to be understood (if we can understand it at all). With quantum chemical approaches we can map out this surface. We differentiate between positive, real and imaginary (usually shown as negative) frequencies. There are no negative frequencies. What you may encounter analyzing the PSE, are one of the following extreme points.

• Local/Global Minimum
All derivatives have to be positive, the number of imaginary frequencies (NImag) is zero $i=0$. This corresponds to an observable state of the system/ molecule, i.e. it can be trapped and analysed with spectroscopy. Exciting any of these frequencies, the energy of the molecule will increase. But you already knew that.
• Transition State
A transition state is a first order saddle point on the PES. The Nimag is exactly one, $i=1$. It connects two local minima with each other. Try to imagine a molecule, like ammonia. This can undergo inversion, i.e. nitrogen inversion. The two pyramidal forms are local minima, which are connected via a planar form (nitrogen is $\ce{sp^2}$ hybridised), which is the transition state. At that point one of the frequencies is imaginary, which means, that it corresponds to lowering the energy while exciting this frequency. All other vibrations would still increase the energy of a molecule.
You cannot observe a state like this via common spectroscopic methods, but it still has a physical meaning. There are various theories in use to describe reactions of molecules. The most popular is the Transition State Theory, which was developed by H. Eyring and M. Polanyi.
It formally corresponds to a transition between transition states. Its NImag is $i=2$, so there are two vibrations that lower the energy. It usually has no physical meaning, as the molecule would probably decompose before reaching this state.
If one of these vibrations correspond to a rotation of a moiety, e.g. a methyl group, and is therefore close to zero, it corresponds to the so called rotational space and might be within range of the transition state.
• $n$-th Order Saddle point
It is like the above a transition between two $(n-1)$-th order saddle points and has even less physical meaning. You sometimes encounter these saddle points as artefacts of the used calculation model. It is strongly advised to not use these states for any explanation.
• Maximum
This is one cannot be found. The PES is usually like a well. The borders of this would be total fragmentation of the molecule. Imagine the homolytic dissociation of dihydrogen (a 2D PES). The maximum value of the PES corresponds to superimposed atoms and its energy value is at infinity.

An additional note. A transition state always connects two local minima. And vice versa, two local minima will always be connected via a series of transition states and local minima. It does not make much sense, looking at any other points than these, as they can most likely not interpreted in a physical meaningful way.

Comment by Ath

What I mean is not global maximum, but local maximum, which doesn't necessarily represent bond breaking...The local maximum I'm interested in is actually the rotation of part of a molecule. I wonder whether the frequencies are all positive or all negative at that point.

There are no maxima on the PES, since the maximum energy corresponds to superimposed atoms. A rotation of moieties is a transition state, or a first order saddle point, and has exactly one imaginary mode.

• Thank you for answering my question,Martin. What I mean is not global maximum, but local maximum, which doesn't necessarily represent bond breaking...The local maximum I'm interested in is actually the rotation of part of a molecule. I wonder whether the frequencies are all positive or all negative at that point. – OhLook Jul 18 '14 at 3:18
• See my edit. It is very important that you do not call those frequencies negative, they are imaginary. There are certain limitations to our programs, so that they usually are printed as a negative frequency to an easier distinguishing from the real ones. – Martin - マーチン Jul 18 '14 at 3:30
• No maxima on the PES? I know transition states are in the valleys, and there are definitely peaks around valleys. The peaks are the local maxima, right? – OhLook Jul 18 '14 at 3:40
• Those peaks lead to second order saddle points from first order saddle points, or to infinitely high potential energy walls, or to complete dissociation of the system (free atoms), which is a plateau. No maxima are possible. – Martin - マーチン Jul 18 '14 at 7:44