Practical Calculation of Geometric Phase

Question

I'm a graduate student working in the field of quantum chemistry, specifically in the field of non-adiabatic dynamics of molecular systems. I've run into a slight problem in a project that I've encountered, in that I need to calculate the geometric (Berry's) phase around a closed loop that contains a conical intersection of two adiabatic electronic states (within the Born-Oppenheimer approx). My current problem manifests in the choice of path for the integration. I must compute the path integral

$\tau_{ij}(\Gamma) = \oint_\Gamma d s_\mu \langle\Psi_i \vert \nabla^\mu \Psi_j\rangle = n\pi$

where $n$ is the number of conical intersections (or points where the Yang-Mills field tensor is non-zero) enclosed in the closed loop $\Gamma$. Here $\mu$ runs over all of the $3N$ nuclear cartesian components of the $N$ atoms in the molecular system.

From what I've read, this integration should be relatively trivial, given a conical intersection: simply map out a circle in parameter space around a conical intersection and perform the path integration. This seems strange to me as how can we say that any arbitrary path "surrounding" the conical intersection in (essentially) $\mathbb{R}^2$ will "contain" the conical intersection once embedded into the full dimensional manifold of the adiabatic electronic states?

I see how this would make sense if the exact conical intersection is know (i.e. in the analytic solution section), but the conical intersections that I obtain are approximate (the states are degenerate up to about a nano-Hartree), which leaves me with doubts as to whether or not the same simple contour would still be sufficient.

I guess the general question is, how do I guarantee that the loop that I generate actually "contains" the conical intersection (practically)?

Answer 1

A conical intersection in an multidimensional parameter space of dimension $M$ is an $M-2$ dimensional hypersurface. (In your example $M$ should be $3N-6$ after the removal of the center of mass translations and rigid rotations).

This is because every energy constraint is an $M-1$ dimensional hypersurface and the conical intersection is $M-2$ dimensional being the intersection of two constraints.

Assuming that the conical surface equation in the parameter space is:

$$V^2(x) = U^2(x) + W^2(x)$$

Then the conical intersection is given by

$$U(x_s) = W(x_s)=0$$

together with:

$$\nabla U(x_s) \ne 0, \nabla W(x_s) \ne 0$$

since the intersection is conical, ($x_s$ are the coordinates of the points on the conical intersection).

Consequently it is possible to parametrize every point in the vicinity of the conical intesection by $x_s$ and two additional coordinates $u,w$ along the gradients at $x_s$.

Thus basically, to determine the winding number of a loop in the neighborhood of a conical intersection, one needs to project it to the $u-w$ plane and find the two dimensonal winding number around $u=w=0$.