Tunneling splittings in vibrational spectra of non-rigid molecules

Theory of Nonadiabatic Tunneling Splitting

Estimating tunneling splittings is a long-standing quantum mechanical challenge for theoretical methods. Sometimes splittings are so small, i.e., within a fraction of a wavenumber, pushing the limits of experimental detection and computational precision. Currently, most computational methods are able, at best, to obtain only ground-state tunneling splittings, either for symmetric or asymmetric potentials. In this Letter, we introduce a unified theoretical approach, based on a two-state approximation that can be equally applied to symmetric and asymmetric diabatic potential crossing and for excited states, providing reliable estimates even for states near the energy crossing. The method opens the door to analytic approximations for the tunneling splitting of model potential systems. It provides a framework for the introduction of vibrational perturbation theory to the estimation of nonadiabatic tunneling splittings. It also provides new insight into the semiclassical theory, leading to an instanton based steepest descent expression applicable also to excited states. Numerical tests on model systems are promising, providing the groundwork for implementation to future multidimensional applications.

Competing tunneling trajectories in a two-dimensional potential with variable topology as a model for quantum bifurcations

We present a path-integral approach to treat a two-dimensional model of a quantum bifurcation. The model potential has two equivalent minima separated by one or two saddle points, depending on the value of a continuous parameter. Tunneling is, therefore, realized either along one trajectory or along two equivalent paths. The zero-point fluctuations smear out the sharp transition between these two regimes and lead to a certain crossover behavior. When the two saddle points are inequivalent one can also have a first order transition related to the fact that one of the two trajectories becomes unstable. We illustrate these results by numerical investigations. Even though a specific model is investigated here, the approach is quite general and has potential applicability for various systems in physics and chemistry exhibiting multistability and tunneling phenomena.

Coherent oscillations and incoherent tunneling in a one-dimensional asymmetric double-well potential

For a model one-dimensional asymmetric double-well potential we calculated the so-called survival probability (i.e., the probability for a particle initially localized in one well to remain there). We use a semiclassical (WKB) solution of the Schrödinger equation. It is shown that behavior essentially depends on transition probability, and on a dimensionless parameter Lambda that is a ratio of characteristic frequencies for low-energy nonlinear in-well oscillations and interwell tunneling. For the potential describing a finite motion (double-well) one has always a regular behavior. For Lambda<<1, there are well defined resonance pairs of levels and the survival probability has coherent oscillations related to resonance splitting. However, for Lambda>>1 there are no oscillations at all for the survival probability, and there is almost an exponential decay with the characteristic time determined by Fermi golden rule. In this case, one may not restrict himself to only resonance pair levels. The number of levels perturbed by tunneling grows proportionally to square root of [Lambda] (in other words, instead of isolated pairs there appear the resonance regions containing the sets of strongly coupled levels). In the region of intermediate values of Lambda one has a crossover between both limiting cases, namely, the exponential decay with subsequent long period recurrent behavior.