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Molecular Physics

Effects of molecular symmetry on the directions of nuclear flux densities during tunnelling in double well potentials

Thomas Grohmann, Jörn Manz, Axel Schild

Published May 2, 2013• 15 citations
Publisher

Abstract

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.

DOI
10.1080/00268976.2013.800599

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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.
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.
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.
Quantum tunneling between mirror-related wells of a torsional potential sets the timescale for coherent interconversion of chiral conformers. We ask whether slower tunneling also weakens the transition polarizability in hydrogen peroxide, hydrogen disulfide, and hydrogen diselenide. We use relaxed CCSD(T)//MP2 potentials and a kinetic model that removes overall molecular rotation to obtain the torsional states, then use static polarizabilities to calculate transition matrix elements between them. In the molecular frame used here, symmetry makes the matrix element of the even dipole component vanish between opposite-parity torsional states, while the odd polarizability component has a nonzero matrix element. The two quantities probe different parts of the torsional landscape: tunneling depends on penetration through the barriers, while the polarizability matrix element forms mainly within the wells. Consequently, tunneling slows sharply across the molecular series even as the matrix element grows. Higher torsional doublets and deuterium substitution further show that large changes in tunneling time can occur with little change in the transition polarizability matrix element. Calculated torsional energies for H2_22​O2_22​ and D2_22​O2_22​ agree with experimental values. Thus, slowing coherent interconversion need not weaken the intrinsic transition polarizability.
When molecules move, their nuclei flow. The corresponding quantum observable, i.e., the nuclear flux density, was introduced by Schrödinger in 1926, but until now, it has not been measured. Here the first experimental results are deduced from high-resolution pump-probe measurements of the time-dependent nuclear densities in a vibrating diatomic molecule or molecular ion. The nuclear densities are converted to flux densities by means of the continuity equation. The flux densities are much more sensitive to time-dependent quantum effects than the densities. Applications to the sodium molecule and the deuterium molecular ion unravel four new effects; e.g., at the turns from bond stretch to compression, the flux of the nuclei exhibits multiple changes of directions, from small to large bond lengths, a phenomenon that we call the "quantum accordion."