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Physical Review A

Molecular isotopic effects on coupled electronic and nuclear fluxes

Anatole Kenfack, Ingo Barth, Falko Marquardt, Beate Paulus

Published Dec 3, 2010• 15 citations
Publisher

Abstract

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 ${\mathrm{H}}_{2}$${}^{+}$ 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.

DOI
10.1103/physreva.82.062502

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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 {\mathrm{H}}_{2}$${}^{+} as well as few fictitious ones. We find that the fluxes undergo a significant change as one goes from one isotope of reduced mass \ensuremathμ\ensuremath{\mu}\ensuremathμ 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 \ensuremathμ\sqrt{\ensuremath{\mu}}\ensuremathμ​ and that the revival period changes linearly with \ensuremathμ\ensuremath{\mu}\ensuremathμ, the mechanism of other observables remains subtle as the result of quantum interference highlighted by the pronounced difference observed on the dispersion pattern.
Over the last few decades, Born-Oppenheimer (BO) and Born-Oppenheimer-Huang (BOH) treatments for electron-nuclear equation of motion have been considered as the cornerstone for qualitative as well as quantitative exploration and understanding of chemical and physical processes. The BO approximation due to the large mass difference between electrons and nuclei as well as the energy separation among the electronic states could able to predict some of the processes (chemical and physical) satisfactorily at lower energy regime. Such approximation (BO) quite often fails for those cases with the involvement of excited electronic states, in particular, due to the coupling among the states. On the other hand, the BO treatment provides two important quantities, namely, the adiabatic potential energy surfaces (PESs) and the nonadiabatic coupling terms (NACTs) determining the forces, which control the motion of the nuclei in a molecule. The NACTs play a major role in photoexcitation spectra, charge transfer chemical reactions, ferroelectric property and other processes, where significant anomalies are observed between single and multiple PES(s) results. Since the NACTs often exhibit singularity at the close vicinity of degenerate states forming conical intersections (CIs), such coupling terms could break the symmetry of the molecular processes changing the observables qualitatively.
A comparative study of the vibrational deactivation of 14N2+^{14}\mathrm{N}_{2}^{+}14N2+​ ($\ensuremath{\nu}=1$, j=0j=0j=0) in collision with 3He^{3}\mathrm{He}3He and 4He^{4}\mathrm{He}4He atoms is investigated. Close-coupling calculations are carried out in the ${10}^{\ensuremath{-}6}--2000\text{\ensuremath{-}}{\mathrm{cm}}^{\ensuremath{-}1}$ kinetic energy range. At very low kinetic energy, it is found that the cross sections for the collision involving 3He^{3}\mathrm{He}3He is strongly enhanced by the presence of a virtual state associated with the van der Waals well. The change of mass from 3He^{3}\mathrm{He}3He to 4He^{4}\mathrm{He}4He also erases a strong shape resonance at $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$ supported by the van der Waals well. As a result, the law of variation of the associated vibrational quenching rate coefficient is found to be very different for these two systems. This behavior is further analyzed by drawing the real part of the scattering length as a function of the variable mass of a fictitious He atom. Two zero energy resonances are identified for a mass of, respectively, about 3.18 and 3.85 atomic mass units.
We analyze the entanglement between electronic and nuclear motions in molecular wave functions widely used by theoretical chemists, namely, (i) Born-Oppenheimer factorization in the adiabatic picture, (ii) the transformation into a diabatic picture, (iii) the use of a Born-Huang expansion, and (iv) the eigenfunction of the full molecular Hamiltonian. Our showcase is based on two one-electron one-dimensional molecular Hamiltonians (H2+ and the Shin-Metiu model). We find that within the Born-Oppenheimer approximation, any molecular state (although factorizable) is always entangled, and its entanglement content may be assessed by the variation of the electronic wave function along the different nuclear geometries, with the nuclear wave function indeed playing the role of a tester. The presence of avoided crossings among the adiabatic potential energy curves brings about dramatic changes in the entanglement content of the wave function: sharp avoided crossings favor a diabatic picture (real crossings between potential energy curves), while in broad avoided crossings, the adiabatic picture prevails. The total eigenfunction of the molecular Shin-Metiu Hamiltonian indicates that nuclear densities accommodate well within the diabatic curves for strong adiabatic couplings but within adiabatic curves for weak ones. Consequently, we find that the electron-nuclei entanglement content is a valid witness to unveil strong or weak nonadiabatic couplings in molecules. In terms of entanglement, we also find that the Born-Huang expansion, based on Born-Oppenheimer adiabatic electronic states, does not provide a correct trend of entanglement compared with that of the total molecular eigenfunction, thus indicating a very slow convergence of this expansion.
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."
Isotope shifts have been measured in the velocity-dependent total cross sections of the m1{m}_{1}m1​=1 magnetic substate of He(3 3^{3}3P) formed in low keV He+{\mathrm{He}}^{+}He++H2{\mathrm{H}}_{2}H2​ or D2{\mathrm{D}}_{2}D2​ collisions. Similar isotope shifts have been predicted on theoretical grounds and observed in ion-atom collision experiments, where united-atom rotational coupling is the primary excitation mechanism. By extending the basic theoretical ideas to ion-molecule collisions, we can interpret our measured shifts as resulting from rotational coupling within the triatomic HeH2+{\mathrm{HeH}}_{2}^{+}HeH2+​ complex.
We demonstrate selective control over rotational motion of small, linear molecules. By means of sequential excitation of the rotational motion by ultrashort pulses, we first prepare transiently aligned molecules with periodically revived angular distribution. Upon further, properly timed excitation, the rotational energy can be increased or decreased, depending on the exact timing of the second pulse. We show how this approach can be applied for selective rotational control of a single component in a molecular mixture. We discuss this selectivity in the context of molecular isotopes ( 14 N 2 , 15 N 2 ), where the difference in isotopic mass gives rise to different rotational revival times. We further apply the method to the selective addressing of molecular spin isomers (para, ortho 15 N 2 ) in a mixture, where wavefunction symmetry differences replace the mass differences as the origin of the selectivity. In both cases the method is demonstrated experimentally and the results are analysed theoretically.