How many transition states are there on a potential energy surface?

Much of chemistry can be described in terms of potential energy surfaces. They describe the energy of an electronic state of a set of molecules as a function of the positions of the atoms in the molecules. Local minima on the surface describe stable molecules (reactants and products of chemical reactions). Chemical reactions proceed by thermal activation over saddle points (transition states). Hence, an interesting and important question concerns how many possible transition states there might be on a surface? How are the number of transition states related to the number of local minima?

In the process of writing a paper on double proton transfer I have stumbled across a very general result that I have never seen stated before. For me there is some curious personal history because the result uses a theorem in the first paper I ever published, thirty years ago, resulting from my undergraduate honours [final year] thesis on general relativity! More on that below.

Here is the result. Consider a smooth surface, i.e. one with no conical intersections, and with isolated extremal points.

I illustrate this below with two model surfaces for double proton transfer.

For example, in the bottom figure, 4+1-4=1.

Hence, if varying the system parameters introduces an extra maxima or minima then one additional saddle point must also appear. One can intuitively see how this works in two dimensions but it turns out it is true in any dimension.

This relation is a consequence of differential topology [essentially the Poincare-Hopf index theorem]. The minima and maxima are associated with an index +1 and saddle points with -1.

The general theorem I proved 30 years ago states that if a smooth function f(r) (where r is a vector) tends to infinity as the magnitude of r tends to infinity or if the gradient of f points outward

over a closed surface (curve in two dimensions), then the extrema of f inside that closed surface, must satisfy the above relation.

How might a potential energy surface satisfy this general requirement on f(r)=Energy(bond lengths)? Essentially it is because as one greatly stretches or compresses chemical bonds the energy of the system will become large.

Aside: it was really strange for me looking at my old paper, published in the Journal of the Australian Mathematical Society. I actually can't believe I wrote it! It is so formal and mathematical. There are parts of it I now struggle to understand. The theorem was not motivated by chemistry but rather proving a general theorem in general relativity that a gravitational lens must produce an odd number of images.

So, has anyone seen this result for potential energy surfaces stated before? I could not find it in David Wales' nice book Energy Landscapes.