This is my note from Prof. Ron Elber’s lecture on “varational transition state theory”.

In the system, there are more than one minimum. If we can find a surface on which the probability of state going to one minimum is equal to going to another. Finding such a surface can be formulate as

The surface S is usually in high dimension and complicated. A simplification is that the surface is a plane, that is, .

Therefore the problem for finding the S is equivalent to .

We reformulate the integral,

,

because it is a plane, no Jacob transformation, so .

So the minimization is to make sure, .

The index i here is due to the fact that in general one plane is not a good separator, so multiple planes (piece-wise) are needed, i is just the i-th plane. So reorganize the definition of I, we have

For simplicity, I consider as I.

The function can be rewritten as a limitation, so we have .

Some tricks here:

define some new symbols

rewrite the above with the new symbols:

Apply Talyor expansion to U, we have

Now we have

There exists such a simplification:

So we have

The above is simplified because

The must here means, in previous introduction of limit function for the function, the coefficient is not right. Anyway, the trick is that in the end,all k terms are reduced. I am too lazy to revisit each term and change them.

Because we expect . So just make sure the total force along the surface, here under the approximation the plane the total force is equal to zero.

I don’t know what it will turn out after we do though.