Asymptotic analysis of the Berry curvature in the $E\otimes e$ Jahn-Teller model

The effective Hamiltonian for the linear $E\otimes e$ Jahn-Teller model describes the coupling between two electronic states and two vibrational modes in molecules or bulk crystal impurities. While in the Born-Oppenheimer approximation the Berry curvature has a delta function singularity at the conical intersection of the potential energy surfaces, the exact Berry curvature is a smooth peaked function. Numerical calculations revealed that the characteristic width of the peak is $\hslash {\mathcal{K}}^{1/2}/g{\mathcal{M}}^{1/2}$, where $\mathcal{M}$ is the mass associated with the relevant nuclear coordinates, $\mathcal{K}$ is the effective internuclear spring constant, and $g$ is the electronic-vibrational coupling. This result is confirmed here by an asymptotic analysis of the $\mathcal{M}\to \infty$ limit, an interesting outcome of which is the emergence of a separation of length scales. Being based on the exact electron-nuclear factorization, our analysis does not make any reference to adiabatic potential energy surfaces or nonadiabatic couplings. It is also shown that the Ham reduction factors for the model can be derived from the exact geometric phase.

Figure 1

Adiabatic potential energy surfaces for the linear $E\otimes e$ Jahn-Teller model with respect to vibrational normal mode coordinates $Q_2$ and $Q_3$.

Figure 2

Nuclear wave function $\chi \left(Q\right)$ and the electronic variable $\theta \left(Q\right)$ for increasing values of $\mathcal{M}$ (light red to dark red). The minimum of the adiabatic potential energy surface occurs at $Q/Q_0=1$.

Figure 3

The function $X_1\left(q\right)=X\left(q\right)-X_0\left(q\right)$ for ${\varepsilon }^2=0.0025$.

Figure 4

The exact function ${\chi }_{\mathrm{exact}}\left(q\right)$ (black) and the approximation ${\chi }_{\mathrm{out},0}\left(q\right)$ (red dashed).

Figure 5

The function $dln{\chi }^2/dq$ is plotted for the series of values $\varepsilon =(\frac{1}{120},\frac{1}{100},\frac{1}{80},\frac{1}{60},\frac{1}{40},\frac{1}{20})$ (dark red to light red).

Figure 6

The error ${\chi }_{\mathrm{uniform}}\left(q\right)-{\chi }_{\mathrm{exact}}\left(q\right)$.

Figure 7

The effective potential energy surface ${\mathcal{E}}_{\mathrm{eff},\mathrm{in}}\left(s\right)$ (black) and BO potential energy surface with (without) the centrifugal potential [dashed red (dotted blue)] are plotted for $\epsilon =1/20$.

Figure 8

The geometric contributions ${\mathcal{E}}_{\mathrm{geo},1}$ (top panel) and ${\mathcal{E}}_{\mathrm{geo},2}$ (bottom panel) to the effective potential energy surface are plotted for the same series of $\varepsilon$ values as in Fig. 5.