Transition state - multidimensional surfaces I've been reading an explanation about transition states in Smith's Organic Synthesis:


  A transition state is an energy maximum along the reaction coordinate. However, a chemical
  reaction where the reactant contains N atoms actually involves movement over an energy surface
  with 3N-6 dimensions, and a transition state is actually an energy minimum in each of the remaining
  (3N-7) dimensions. Liken a chemical reaction to a drive across a mountain range. The “goal” is to
  go from one valley to another, just as the goal of a chemical reaction is to go from one minimum
  on the energy surface (the reactants) to another minimum (the product). The road moves about a
  two-dimensional surface whereas the energy surface for a chemical reaction may involve dozens
  or even hundreds of dimensions. There is no need to climb to the top of a mountain (a maximum
  in both dimensions) to go from one valley to another. Rather, is sufficient to go through a pass
  between mountains (a maximum in one dimension but a minimum in the other). Similarly, it is
  only necessary to pass through an energy maximum in one coordinate (the reaction coordinate)
  for a chemical reaction.
  Because a transition state is an energy maximum (albeit only in one of many dimensions) it cannot
  function to “trap” a molecule, let alone collect a population of “trapped” molecules. This means
  that a transition state cannot even be observed experimentally let alone isolated or characterized.
  Quite simply, a transition state does not exist in the sense that a molecule exists.


This multidimensional explanation completely confounds me. Could someone explain this in a more understandable language? What is this surface with 3N-6 dimensions? How did the author use it to conclude that TS can't be observed experimentally?
The picture below shows a simple potential energy surface with a transition state at the maximum point on the reaction path, which is at the 'saddle point'. The reaction path is the solid black line. (Oscillations in this show that the reactant and product are vibrating). The contours are at different energy with large negative being lowest. This picture is based on a calculation using known experimental parameters. 

Along the reaction path the energy is as small as it can be, even though it increases as we go along, because by moving to the right or left (perpendicular to the general direction of the path)  the energy is even higher. So the saddle point is just like a pass between high mountain peaks. At the highest point on the reaction path the only way to lower the energy is to go down, either to return on the original path or to continue to products. The time it takes to cross the transition state is definitely sub picosecond, &lt; $10^{-12}$ s and possibly up to a 100 times faster.

It is not possible to visualise a multidimensional reaction path, so we have to make do with 2D plots, as below, of one reaction coordinate vs. the other with contours at different energy. In principle many similar plots could be needed for a complex reaction but very often most of the atoms in the molecule are bystanders and it possible then to only look at the separation and energy of the atoms in bonds being broken and made.



Trajectory of $\ce {Cl + H2}$ in a gas phase reaction. The H atom approaches the chlorine molecule at the right. The yellow circle shows the transition state.