# What is the branching space of a conical section?

I am currently engaged in reading a research paper that focuses on the computation of the Potential Energy Surfaces (PES) of excited states. Within this paper, there is a discussion about a concept called the "branching space" associated with conical intersections. Unfortunately, I am not familiar with this concept and the information provided in the corresponding Wikipedia article (https://en.wikipedia.org/wiki/Conical_intersection) is quite brief.

I would greatly appreciate it if someone could kindly provide me with a more detailed explanation of the branching space and its significance in the context of conical intersections. Thank you in advance for your assistance.

A paper called "Potential energy surfaces near intersections" [1] provides the answer.

At the conical intersection, the energies of two potential energy surfaces parameterized by configuration coordinates $$\{q_i : i = 1, \dots, N\}$$ are equal: say, $$E_0(q_1, \dots, q_N) = E_1(q_1, \dots, q_N)$$.

This intersection is not completely isolated. In a small region of configuration space near the intersection called the intersection coordinate subspace,

$$E_0(q_1, \dots, q_i + \delta, \dots, q_N) = E_1(q_1, \dots, q_i + \delta, \dots, q_N),$$

for most, but not all, of the coordinates $$q_i$$. (Think of $$\delta$$ as a small perturbation.) One can argue by symmetry that this subspace has dimension $$N - 2$$, at least most of the time.

The remaining two dimensions form the branching space. If you like, the branching space is defined by the directions you can go to escape the energy degeneracy immediately. Visually, they should be the dimensions that look like a cone: both PES are degenerate and (locally) flat in all other directions!

[1] G. J. Atchity, S. S. Xantheas, and K. Ruedenberg. J. Chem. Phys. 95, 1861 (1991)

• Thanks for you explanation! Commented Nov 18, 2023 at 19:47

I'm a bit late to the party but if you're interested I can provide some further information. In general the PES(in the Born Oppenheimer approximation) has dimensionality 3N-6, where N is defined to be the number of atoms in your molecule/system. The 3N term comes from the fact that each atom has 3 cartesian coordinates (x,y,z), we then must discount the translation components (3) and rotational components (3), which is why the 6 is subtracted. If you had a linear molecule (straight line) then your actual dimensionality would be 3N-5, because you'd only discount 3 translational components and 2 rotational components. The reasoning for this requires some geometry which you can find online and I won't bore you with.

On the other hand the dimensionality of the CI space is 3N-8 (for a nonlinear molecule), this makes logical sense as it should be a subset of the PES, as it's the intersection of the states. Because the dimensionality of the CI space is 3N-8 we have a 2D space which we haven't accounted for, this is the branching space, this acts to remove the degeneracy(meaning make the energies not equal anymore) of the system, so in a way if you are travelling across any of those 2 directions, you will be ''leaving'' the Conical Intersection (which remember it's a space of intersection between 2 energy states).

The point of the branching space is that it's simply the space which acts to remove the degeneracy.

Hopefully this cleared up any misconceptions you had and let me know if you need any further information/resources to wrap your head around.

This website has some pictures (especially the double cone one) which give you a better understanding of what's going on :