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Physical Review Letters

Nuclear Fluxes in Diatomic Molecules Deduced from Pump-Probe Spectra with Spatiotemporal Resolutions down to 5 pm and 200 asec

Jörn Manz, Jhon Fredy Pérez-Torres, Yonggang Yang

Published Oct 10, 2013• 24 citations
Publisher

Abstract

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."

DOI
10.1103/physrevlett.111.153004

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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."
A full quantum treatment shows that coupled electronic and nuclear fluxes exhibit a strong sensitivity to a small mass change in a vibrating molecule. This has been exemplified with the existing isotopes of H2{\mathrm{H}}_{2}H2​+{}^{+}+ as well as few fictitious ones. We find that the fluxes undergo a significant change as one goes from one isotope of reduced mass $\ensuremath{\mu}$ to another. Other well-defined observables are likewise affected. It turns out that as a general rule, the heavier the isotope, the larger the flux, the smaller the dispersion, and the longer the revival period. While we were able to confirm analytically that the time at the first turning point scales as $\sqrt{\ensuremath{\mu}}$ and that the revival period changes linearly with $\ensuremath{\mu}$, the mechanism of other observables remains subtle as the result of quantum interference highlighted by the pronounced difference observed on the dispersion pattern.
Abstract Nonadiabatic effects in the nuclear dynamics of the H2+ molecular ion, initiated by ionization of the H2 molecule, is studied by means of the probability and flux distribution functions arising from the space fractional Schrödinger equation. In order to solve the fractional Schrödinger eigenvalue equation, it is shown that the quantum Riesz fractional derivative operator fulfills the usual properties of the quantum momentum operator acting on the bra and ket vectors of the abstract Hilbert space. Then, the fractional Fourier grid Hamiltonian method is implemented and applied to molecular vibrations. The eigenenergies and eigenfunctions of the fractional Schrödinger equation describing the vibrational motion of the H2+ and D2+ molecules are analyzed. In particular, it is shown that the position‐momentum Heisenberg's uncertainty relationship holds independently of the fractional Schrödinger equation. Finally, the probability and flux distributions are presented, demonstrating the applicability of the fractional Schrödinger equation for taking into account nonadiabatic effects.
Chemical dynamics, in principle, should be understood by solving the time-dependent Schrödinger equation for a molecular system, describing motion of the nuclei and electrons. However, the computational efforts to solve this partial second-order differential equation scales exponentially with the system size, which prevents us from getting exact numerical solutions for systems larger than 4-5 atoms. Thus, approximations simplifying the picture are necessary. The so-called Born-Oppenheimer approximation, separating motion of the electrons and nuclei is the central one: solution to the electronic Schrödinger equation defines the potential energy surface on which the nuclear motion unfolds, and there are standard quantum chemistry software packages for solving the electronic Schrödinger equation. For the nuclear Schrödinger equation, however, there are no widely applicable quantum-mechanical approaches, and most simulations are performed using classical Newtonian mechanics which is often adequate due to large nuclear masses. However, the nuclear quantum effects are significant for chemical processes involving light nuclei at low energies, and including these effects into simulation, even approximately, is highly desirable. In this dissertation, an approximate methodology of including quantum-mechanical effects within the quantum trajectory or the de Broglie-Bohm formulation of the Schrödinger equations is developed. Use of the trajectory framework makes the approach scalable to hundreds of degrees of freedom. The methodology is applied to study high-dimensional systems (solid He4 and others) relevant to chemistry.
A method of molecular dynamics with quantum corrections, practical for studies of large molecular systems, is reviewed. The approach is based on the Bohmian formulation of the time-dependent Schrödinger equation in which a wavefunction is represented by an ensemble of interdependent trajectories. The quantum effects come from the quantum potential acting on trajectories on par with the usual classical potential. The quantum potential is determined from the evolving nuclear wavefunction, i.e. from the quantum trajectory (QT) ensemble itself. For practical and conceptual reasons the quantum potential and corresponding quantum nuclear effect are computed only for the selected light nuclei. For studies of reactive chemical processes, the classical potential is computed on-the-fly using the density functional tight binding method of electronic structure. A massively parallel implementation, based on the message passing interface allows for efficient simulations of ensembles of thousands of trajectories describing systems of up to 200 atoms. As a biochemical application, the approximate QT approach is used to model the tunnelling-dominated proton transfer in soybean-lipoxygenase-1. A materials science application is represented by a study of the nuclear quantum effect on adsorption of hydrogen and deuterium on a C37H15 molecule, which is a model ‘flake’ of graphene.
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.