Multidimensional tunnelling of molecules aligned by strong electric fields

TL;DR

This work extends semiclassical instanton theory to full dimensionality for molecules in electric fields, enabling the calculation of tunnelling splittings between two degenerate orientations. It introduces the treatment of two zero modes—permutational and rotational—within a ring-polymer instanton framework, yielding a practical final expression for the splitting that incorporates the instanton action and fluctuation determinants. The method is validated against a perturbative model for H$_2$ and then applied to a non-perturbative, ab initio treatment, showing the critical role of field-dependent geometry and polarizability in shaping the barrier and the resulting splitting. The approach is further demonstrated on formaldehyde in a high-frequency laser field, revealing potentially observable heavy-atom tunnelling under suitably tuned fields, and offering a pathway to test instanton theory against experiments in multidimensional, field-driven tunnelling. Overall, the paper provides a robust, on-the-fly ab initio framework for predicting and interpreting field-induced tunnelling phenomena in both diatomic and polyatomic molecules.

Abstract

Strong electric fields can be used to align molecules. However, a non-polar molecule such as H$_2$ has no preference for its orientation. There are thus two equivalent configurations with equal energy separated by a potential-energy barrier. Quantum mechanically, the molecule can tunnel between these configurations resulting in a tunnelling splitting, which in the case of H$_2$, is the same as the ortho--para splitting. In this work, we generalize semiclassical instanton theory to calculate the energy splitting of molecules in electric fields in full dimensionality. This goes beyond a perturbative treatment of the field and takes into account changes in molecular geometry during the tunnelling process which influence its electrical properties and can have a significant impact on the result. We first study the case of H$_2$ in a static electric field and then show how it can be applied to larger polar molecules subjected to oscillating electric fields, where we find that even large-amplitude heavy-atom tunnelling can lead to observable splittings.

Figures (7)

• Figure 1: Illustration of the tunnelling splitting for a hydrogen molecule aligned in a strong electric field. The indistinguishable atoms are coloured to clearly illustrate the two degenerate states. Quantum mechanically, the molecule can tunnel between these, resulting in two delocalized eigenstates marked by the dashed lines. The quantity we seek is the splitting of these two levels, $\Delta$. Note that the angle is only defined on the interval $\theta=[0,\pi]$, although for illustration purposes we show mirror images of the potential and wavefunctions outside this range.
• Figure 2: Illustration of one of the infinitely many instanton paths. The blue beads symbolize the positions of the hydrogen atoms along the tunnelling pathway (N.B. many more beads are used in practice to reach convergence). The higher density of beads at the top and bottom shows that the imaginary-time velocity of the atoms decreases as the molecule approaches its aligned configurations. Moreover, from this figure, it should be clear how the action is invariant to rotations of the instanton around the $z$-axis.
• Figure 3: Illustration of the permutational zero mode. In the limit of an infinite number of beads at low temperature, there are infinitely many beads in each well. Therefore, shifting each bead to the position of the next bead leaves the polymer effectively unchanged and hence the action invariant. This symmetry results in a zero-frequency mode of the polymer Hessian.
• Figure 4: Plot of the model potential energy surface defined in Eq. \ref{['eq:pot']}. There are two minima corresponding to the aligned orientations of the hydrogen molecule. The optimized instanton pathway is shown in red. One can see that the bond length decreases slightly during the tunnelling process---similar to an ice skater pulling in their arms while performing a spin.
• Figure 5: Illustration of the two minimum-energy orientations of the formaldehyde molecule in an oscillating laser field. The purple line represents the axis of the electric field with the arrows in both directions symbolizing that the molecule feels no directionality of the field.