Is the Molecular Berry Phase an Artifact of the Born-Oppenheimer Approximation?

We demonstrate that the molecular Berry phase and the corresponding nonanalyticity in the electronic Born-Oppenheimer wave function is, in general, not a true topological feature of the exact solution of the full electron-nuclear Schrödinger equation. For a numerically exactly solvable model we show that a nonanalyticity, and the associated geometric phase, only appear in the limit of infinite nuclear mass, while a perfectly smooth behavior is found for any finite nuclear mass.

Figure 1

(a) The first (blueish) and second (reddish) excited BO potential energy surfaces, (b) the vector field representation for $p$-orbital-like wave functions at a certain $\mathbf{R}$, and the BO electronic wave functions in the vector field representation for the first excited BO state (c) and the second excited BO state (d). The phase changes discontinuously across the line of red vectors ($L_1$ and $L_2$ in text).

Figure 2

The factorized nuclear densities (left), the corresponding electronic wave functions represented by the vector fields $\int \mathbf{r}{\mathrm{\Phi }}_N(\mathbf{r};\underset{\_}{\mathbf{R}})d\mathbf{r}$ (middle) and the potential energy surfaces (right) for the selected full wave functions ${\mathrm{\Psi }}_{\text{mol}}^A$ and ${\mathrm{\Psi }}_{\text{mol}}^B$ (from top to bottom) with $M=10.0$. The blueish and reddish surfaces are the first and second excited BO potential energy surfaces, respectively, while the pink surfaces are the exact potential energy surfaces near ${\mathbf{R}}_{\text{eq}}^{+}$.

Figure 3

The vector $\int \mathbf{r}{\mathrm{\Phi }}_A(\mathbf{r};\mathbf{R})d\mathbf{r}$ is plotted for various ionic masses, $M$. The values of $M$ for the panels (a), (b), (c), and (d) are 1.0, 10.0, 20.0 and 50.0, respectively. The dashed red line indicates, as guide for the eye, the change of curvature as the ionic mass increases.

Figure 4

The factorized nuclear densities (first row), and the difference between the exact potential energy surface and the 1st excited BO potential energy surface (${\epsilon }_A^{\text{ex}}-{\epsilon }_1^{\mathrm{BO}}$) (second row) for various nuclear masses ($M=1.0$, 10.0, 20.0, and 50.0 from left to right).