Abstract Previous results for nuclear fluxes during coherent tunnelling of molecules with symmetric double well potentials are extended to fluxes in asymmetric double well potentials. The theory is derived using the two-state approximation (TSA). The symmetric system serves as a reference. As an example, we consider the one-dimensional model of the tunnelling inversion of oriented ammonia, with semiclassical dipole coupling to an electric field. The tunnelling splitting increases with the dipole coupling by a factor The tunnelling time decreases by The nuclear density appears as the sum of two parts: The tunnelling part decreases as times the density of the symmetric reference, whereas the non-tunnelling part is the initial density times Likewise, the nuclear flux decreases by with essentially the same shape as for the symmetric reference, with maximum value at the potential barrier. Coherent nuclear tunnellings starting from the upper or lower wells of the asymmetric potential are equivalent. The results are universal, in the frame of the TSA, hence they allow straightforward extrapolations from one system to others. This is demonstrated by the prediction of isotope effects for five isotopomers of ammonia.
Abstract Previous results for nuclear fluxes during coherent tunnelling of molecules with symmetric double well potentials are extended to fluxes in asymmetric double well potentials. The theory is derived using the two-state approximation (TSA). The symmetric system serves as a reference. As an example, we consider the one-dimensional model of the tunnelling inversion of oriented ammonia, with semiclassical dipole coupling to an electric field. The tunnelling splitting increases with the dipole coupling by a factor The tunnelling time decreases by The nuclear density appears as the sum of two parts: The tunnelling part decreases as times the density of the symmetric reference, whereas the non-tunnelling part is the initial density times Likewise, the nuclear flux decreases by with essentially the same shape as for the symmetric reference, with maximum value at the potential barrier. Coherent nuclear tunnellings starting from the upper or lower wells of the asymmetric potential are equivalent. The results are universal, in the frame of the TSA, hence they allow straightforward extrapolations from one system to others. This is demonstrated by the prediction of isotope effects for five isotopomers of ammonia.
The quantum theory of concerted electronic and nuclear fluxes (CENFs) during coherent periodic tunnelling from reactants (R) to products (P) and back to R in molecules with asymmetric double-well potentials is developed. The results are deduced from the solution of the time-dependent Schrödinger equation as a coherent superposition of two eigenstates; here, these are the two states of the lowest tunnelling doublet. This allows the periodic time evolutions of the resulting electronic and nuclear probability densities (EPDs and NPDs) as well as the CENFs to be expressed in terms of simple sinusodial functions. These analytical results reveal various phenomena during coherent tunnelling in asymmetric double-well potentials, e.g., all EPDs and NPDs as well as all CENFs are synchronous. Distortion of the symmetric reference to a system with an asymmetric double-well potential breaks the spatial symmetry of the EPDs and NPDs, but, surprisingly, the symmetry of the CENFs is conserved. Exemplary application to the Cope rearrangement of semibullvalene shows that tunnelling of the ideal symmetric system can be suppressed by asymmetries induced by rather small external electric fields. The amplitude for the half tunnelling, half nontunnelling border is as low as 0.218 × 10(-8) V/cm. At the same time, the delocalized eigenstates of the symmetric reference, which can be regarded as Schrödinger's cat-type states representing R and P with equal probabilities, get localized at one or the other minima of the asymmetric double-well potential, representing either R or P.
Coherent tunnelling in molecular systems with cyclic and non-cyclic symmetric double well potentials may proceed with similar nuclear densities, but with entirely different flux densities. For sufficiently high potential barriers, the nuclear densities may even become indistinguishable, whereas the patterns of the flux densities at a given time remain pincer-motion type for the cyclic systems, but unidirectional for the non-cyclic one. This effect is traced back to symmetry breaking of the cyclic to the non-cyclic model. Accordingly, nuclear flux densities are much more sensitive to symmetry breaking than nuclear densities. For a proof of principle, the phenomenon is demonstrated by means of three one-dimensional models. The cyclic model I represents torsion in oriented B2Cl2F2, the non-cyclic model II is constructed from I by symmetry breaking and the non-cyclic model III represents tunnelling by inversion of oriented NH3.
We discuss the effect of molecular symmetry on coherent tunneling in symmetric double-well potentials whose two molecular equilibrium configurations are interconverted by nuclear permutations. This is illustrated with vibrational tunneling in ammonia molecules, electronic tunneling in the dihydrogen cation, and laser-induced rotational tunneling of homonuclear diatomics. In this contribution, we reexamine the textbook picture of coherent tunneling in such potentials, which is depicted with a wavepacket shuttling back and forth between the two potential-wells. We show that the common application of this picture to the aforementioned molecules contravenes the principle of the indistinguishability of identical particles. This conflict originates from the sole consideration of the dynamics of the tunneling-mode, connecting the double-well energy minima, and complete omission of all the remaining molecular degrees of freedom. This gives rise to double-well wavepackets that are nonsymmetric under nuclear permutations. To obey quantum statistics, we show that the double-well eigenstates composing these wavepackets must be entangled with the wavefunctions that describe all the omitted molecular modes. These wavefunctions have compensating and opposite nuclear permutation symmetry. This in turn leads to complete quenching of interference effects behind localization in one potential-well or another. Indeed, we demonstrate that the reduced density of probability of the symmetrized molecular wavefunction, where all the molecular coordinates but the tunneling-mode are integrated out, is symmetrically distributed over the two potential-wells, at all times. This applies to any multilevel wavepacket of isotropic or fully aligned symmetric double-well achiral molecules. However, in the case of coherent electronic or vibrational tunneling, fully aligned molecules may exhibit dynamical localization in the space-fixed frame, where the tunneling-mode density shuttles between the opposite directions of the alignment axis. This dynamical spatial-localization results from linear combinations of molecular states that have opposite parity. In summary, this study shows that dynamical localization of the tunneling-mode density on either of the two indistinguishable molecular equilibrium configurations of symmetric double-well achiral molecules is forbidden by quantum statistics, whereas its dynamical localization in the space-fixed frame is allowed by parity. The subtle distinction between these two types of localization has far-reaching implications in the interpretation of many ultrafast molecular dynamics experiments.
Nelson's stochastic mechanics formulates quantum dynamics as a real-time conservative diffusion process in which a particle undergoes Brownian-like motion with a fluctuation amplitude fixed by Planck's constant. While being mathematically equivalent to the Schrödinger formulation, this approach provides an alternative dynamical framework that enables the study of time-resolved quantities that are not straightforwardly defined within the standard operator-based approach. In the present work, Nelson's stochastic mechanics is employed to investigate tunneling-time statistics for bound states in double-well potentials. Using first-passage time theory within this framework, both the mean tunneling time, τˉ, and the full probability distribution, p(τ), are computed. The theoretical predictions are validated through extensive numerical simulations of stochastic trajectories for two representative potentials. For the square double-well potential, analytical expressions for τˉ are derived and are shown to be in excellent agreement with simulations. In the high-barrier limit, the results reveal a direct relation between the stochastic-mechanical and quantum-mechanical tunneling times, expressed as τQM=(π/2)τˉ, where τQM corresponds to half the oscillation period of the probability of finding the particle in either well. This relation is further confirmed for generic double-well systems through a WKB analysis. As a concrete application, the inversion dynamics of the ammonia molecule is analyzed, yielding an inversion frequency of approximately 24 GHz, in close agreement with experimental observations. These results highlight the potential of stochastic mechanics as a conceptually coherent and quantitatively consistent framework for analyzing tunneling phenomena in quantum systems.
An asymmetric double-well potential is considered, assuming that the minima of the wells are quadratic with a frequency ω and the difference of the minima is close to a multiple of �ω. A WKB wave function is constructed on both sides of the local maximum between the wells, by matching the WKB function to the exact wave functions near the classical turning points. The continuities of the wave function and its first derivative at the local maximum then give the energy-level splitting formula, which not only reproduces the instanton result for a symmetric potential, but also elucidates the appearance of resonances of tunneling in the asymmetric potential. Key words: quantum tunneling, energy splitting, double-well potential PACS: 03.65.Sq, 03.65.Ge, 82.20.Xr 1
Double-well potentials are widely used to model phenomena in physics and chemistry. The system is assumed to be formed in a metastable state, its dynamical evolution providing the clues for the interpretation of the experimental data. Quantum mechanics predicts coherent oscillations of probability between wells if the double-well potential is nearly symmetric and irreversible exponential decay if the final well has an infinite width. For very asymmetric double-well potentials, these two extreme behaviors are expected to coexist. The purpose of the present paper is to investigate this coexistence and its evolution as a function of the width of (or density of states in) the second well. In this sense, increasing the density of states can be regarded as a mechanism for coherence breakdown. The dynamical evolution of the metastable state can be simulated by solving numerically the time-dependent Schroedinger equation (TDSE). This approach is general, intuitive, and gives access to time scales and dynamical effects. It also allows the inclusion of phenomenological dissipation.