Deep Tunneling Regime Research Articles

Semiclassical reaction rate constant of non-adiabatic system by means of initial value representation

We applied the semiclassical initial value representation (SC-IVR) method with the classical electron analog to deal with electronically non-adiabatic reactive barriers and calculated the thermal reaction rate constant. The symmetric form of the flux–flux correlation function is used, and the matrix element of the Boltzmannized flux operator is derived using imaginary-time path integral techniques. This combined use of non-adiabatic SC-IVR and non-adiabatic path-integral sampling (NA-IVR&PI) is beyond the treatment of mean-field, and we analyze its performance in comparison with some existing mean-field methods for different scenarios of potential energy surface. We found a systematically better performance. One exception is the deep tunneling regime, where the mean-field ring-polymer instanton is superior. In the golden rule limit, where all these mean-field methods break down, the significant advantage of NA-IVR&PI is shown by using the electronic states for description of the dividing surface.

Nuclear Quantum Effects in Hydroxide Hydrate Along the H-Bond Bifurcation Pathway.

Path integral (PI) simulations are used to explore nuclear quantum effects (NQEs) in hydroxide hydrate and its perdeuterated isotopomer along the H-bond bifurcation pathway. Toward this, a new potential energy surface using the symmetric gradient domain machine learning method with ab initio data at the CCSD(T)/aug-cc-pVTZ level is built. From PI umbrella sampling (US) simulations, free energy profiles along the bifurcation coordinate are explored as a function of temperature. At ambient temperature, the bifurcation barrier is increased upon inclusion of NQEs. At low temperatures in the deep tunneling regime, the barrier is strongly decreased and flattened. These trends are examined, and the role of the O-O distance is also investigated through two-dimensional US simulations.

Coherent state representation of thermal correlation functions with applications to rate theory.

A coherent state phase space representation of operators, based on the Husimi distribution, is used to derive an exact expression for the symmetrized version of thermal correlation functions. In addition to the time and temperature independent phase space representation of the two operators whose correlation function is of interest, the integrand includes a non-negative distribution function where only one imaginary time and one real time propagation are needed to compute it. The methodology is exemplified for the flux side correlation function used in rate theory. The coherent state representation necessitates the use of a smeared Gaussian flux operator whose coherent state phase space representation is identical to the classical flux expression. The resulting coherent state expression for the flux side correlation function has a number of advantages as compared to previous formulations. Since only one time propagation is needed, it is much easier to converge it with a semiclassical initial value representation. There is no need for forward-backward approximations, and in principle, the computation may be implemented on the fly. It also provides a route for analytic semiclassical approximations for the thermal rate, as exemplified by a computation of the transmission factor through symmetric and asymmetric Eckart barriers using a thawed Gaussian approximation for both imaginary and real time propagations. As a by-product, this example shows that one may obtain "good" tunneling rates using only above barrier classical trajectories even in the deep tunneling regime.

Dissipative tunneling rates through the incorporation of first-principles electronic friction in instanton rate theory. II. Benchmarks and applications.

In Paper I [Litman et al., J. Chem. Phys. (in press) (2022)], we presented the ring-polymer instanton with explicit friction (RPI-EF) method and showed how it can be connected to the ab initio electronic friction formalism. This framework allows for the calculation of tunneling reaction rates that incorporate the quantum nature of the nuclei and certain types of non-adiabatic effects (NAEs) present in metals. In this paper, we analyze the performance of RPI-EF on model potentials and apply it to realistic systems. For a 1D double-well model, we benchmark the method against numerically exact results obtained from multi-layer multi-configuration time-dependent Hartree calculations. We demonstrate that RPI-EF is accurate for medium and high friction strengths and less accurate for extremely low friction values. We also show quantitatively how the inclusion of NAEs lowers the crossover temperature into the deep tunneling regime, reduces the tunneling rates, and, in certain regimes, steers the quantum dynamics by modifying the tunneling pathways. As a showcase of the efficiency of this method, we present a study of hydrogen and deuterium hopping between neighboring interstitial sites in selected bulk metals. The results show that multidimensional vibrational coupling and nuclear quantum effects have a larger impact than NAEs on the tunneling rates of diffusion in metals. Together with Paper I [Litman et al., J. Chem. Phys. (in press) (2022)], these results advance the calculations of dissipative tunneling rates from first principles.

Highly superlinear giant terahertz photoconductance in GaAs quantum point contacts in the deep tunneling regime

A highly superlinear in radiation intensity photoconductance induced by terahertz laser radiation with moderate intensities has been observed in quantum point contacts made of GaAs quantum wells operating in the deep tunneling regime. For very low values of the normalized dark conductance $G_{\rm dark}/ G_0 \approx 10^{-6}$, with the conductance quantum $G_0=2e^2/h$, the photoconductance scales exponentially with the radiation intensity, so that already at $ 100 \text{ mW}/\text{cm}^2$ it increases by almost four orders of magnitude. This effect is observed for a radiation electric field oriented along the source drain direction. We provide model considerations of the effect and attribute it to the variation of the tunneling barrier height by the radiation field made possible by local diffraction effects. We also demonstrate that cyclotron resonance due to an external magnetic field manifests itself in the photoconductance completely suppressing the photoresponse.

Exploration of interference structure of a hydrogen atom with different wavelengths in a linearly polarized laser field

We investigate the ionization process of an atom with different wavelengths in a linearly polarized laser field. With the change of wavelength, the interference structure in photoelectron momentum distributions (PMDs) changes obviously. The temporal double-slit fringes can be clearly observed in the relatively high-energy region below the energy of 2U p for the laser wavelength , but we only observe the temporal double-slit fringes in the relatively low-energy region below the energy of 2U p for the laser wavelength . By analyzing the number of relative contribution of directly ionized electrons, we find a sensitive dependence of the direct electron on the laser wavelength. In addition, the temporal double-slit fringes can be also observed in the photoelectron momentum distributions in the deep tunneling regime ().

Benchmarking the Surface Hopping Method to Include Nuclear Quantum Effects.

We have benchmarked the surface hopping method to capture nuclear quantum effects in the spin-Boson model in the deep tunneling regime. The thermal populations and the rate constants calculated using the surface hopping method are compared with those calculated using Boltzmann theory and Fermi's golden rule, respectively. Additionally, we have proposed a simple kinetic model that partially includes nuclear quantum effects within Marcus theory, and the results of the surface hopping method are analyzed under the framework of this simple kinetic model. A broad range of parameters are investigated to identify the regimes for the successes and failures of the surface hopping method. This work shows that with the accurate treatment of decoherence and velocity reversal, surface hopping can generally capture the nuclear quantum effects in the deep tunneling and weak diabatic coupling regime.

Multidimensional Hydrogen Tunneling in Supported Molecular Switches: The Role of Surface Interactions.

The nuclear tunneling crossover temperature (T_{c}) of hydrogen transfer reactions in supported molecular-switch architectures can lie close to room temperature. This calls for the inclusion of nuclear quantum effects (NQEs) in the calculation of reaction rates even at high temperatures. However, computations of NQEs relying on standard parametrized dimensionality-reduced models quickly become inadequate in these environments. In this Letter, we study the paradigmatic molecular switch based on porphycene molecules adsorbed on metallic surfaces with full-dimensional calculations that combine density-functional theory for the electrons with the semiclassical ring-polymer instanton approximation for the nuclei. We show that the double intramolecular hydrogen transfer (DHT) rate can be enhanced by orders of magnitude due to surface fluctuations in the deep-tunneling regime. We also explain the origin of an Arrhenius temperature dependence of the rate below T_{c} and why this dependence differs at different surfaces. We propose a simple model to rationalize the temperature dependence of DHT rates spanning diverse fcc [110] surfaces.

Quantum momentum distribution and quantum entanglement in the deep tunneling regime.

In this paper, we consider the momentum operator of a quantum particle directed along the displacement of two of its neighbors. A modified open-path path integral molecular dynamics is presented to sample the distribution of this directional momentum distribution, where we derive and use a new estimator for this distribution. Variationally enhanced sampling is used to obtain this distribution for an example molecule, malonaldehyde, in the very low temperature regime where deep tunneling happens. We find no secondary feature in the directional momentum distribution and that its absence is due to quantum entanglement through a further study of the reduced density matrix.

Unexpected dynamical effects change the lambda-doublet propensity in the tunneling region for the O(3P) + H2 reaction.

One of the most relevant features of the O(3P) + H2 reaction is that it occurs on two different potential energy surfaces (PESs) of symmetries A' and A'' that correlate reactants and products. The respective saddle points, which correspond to a collinear arrangement, are the same for both PESs, whilst the barrier height rises more abruptly on the 3A' PES than on the 3A'' PES. Accordingly, the reactivity on the 3A'' PES should be always higher than on the 3A' PES. In this work, we present accurate quantum-scattering calculations showing that this is not always the case for rotationless reactants, where dynamical factors near the reaction threshold cause the 3A' PES to dominate at energies around the barrier. Further calculation of cross sections and Λ-doublet populations has allowed us to establish how the reaction mechanism changes from the deep tunneling regime to hyperthermal energies.

Semiclassical transition state theory based on fourth order vibrational perturbation theory: Model system studies beyond symmetric Eckart barrier.

The following one-dimensional model potential barriers are studied to compute the tunneling probabilities using semiclassical transition state theory (SCTST) based on fourth order vibrational perturbation theory (VPT4): (i) Morse barrier, (ii) asymmetric Eckart barrier, (iii) adjacent well barrier, and (iv) first order approximation to the symmetric Eckart barrier. The current investigation extends a previous study performed for the symmetric Eckart barrier. The performance of VPT4-SCTST is investigated for a broad range of energies, including the very deep tunneling regime (approximately 1% of the barrier height), and the results are compared to the exact quantum mechanical treatment, as well as to the popular second order vibrational perturbation theory (VPT2)-SCTST approach. While the VPT4-SCTST results are in excellent agreement with their quantum mechanical counterparts for a wide range of energies above and below the barrier, it is found for systems (ii) and (iv) that higher order terms (in ) in energy expansion are important for similarly excellent agreement in the deep tunneling regime. For the asymmetric Eckart barrier, the convergence properties of SCTST are analyzed in terms of its analytically known Siegert eigenvalues. A simple empirical modification of VPT2-SCTST advocated in the aforementioned previous work for the symmetric Eckart barrier is also tested here for the new model systems, and it is shown not to provide a uniformly systematic improvement.

Path-integral isomorphic Hamiltonian for including nuclear quantum effects in non-adiabatic dynamics.

We describe a path-integral approach for including nuclear quantum effects in non-adiabatic chemical dynamics simulations. For a general physical system with multiple electronic energy levels, a corresponding isomorphic Hamiltonian is introduced such that Boltzmann sampling of the isomorphic Hamiltonian with classical nuclear degrees of freedom yields the exact quantum Boltzmann distribution for the original physical system. In the limit of a single electronic energy level, the isomorphic Hamiltonian reduces to the familiar cases of either ring polymer molecular dynamics (RPMD) or centroid molecular dynamics Hamiltonians, depending on the implementation. An advantage of the isomorphic Hamiltonian is that it can easily be combined with existing mixed quantum-classical dynamics methods, such as surface hopping or Ehrenfest dynamics, to enable the simulation of electronically non-adiabatic processes with nuclear quantum effects. We present numerical applications of the isomorphic Hamiltonian to model two- and three-level systems, with encouraging results that include improvement upon a previously reported combination of RPMD with surface hopping in the deep-tunneling regime.

Tunneling effects in the unimolecular decay of (CH3)2COO Criegee intermediates to OH radical products.

Unimolecular decay of the dimethyl substituted Criegee intermediate (CH3)2COO is observed at energies significantly below the transition state barrier associated with hydrogen atom transfer [Y. Fang et al., J. Chem. Phys. 144, 061102 (2016)] with time-resolved detection of the resultant OH radical products. (CH3)2COO is prepared at specific energies in the 3900-4600 cm-1 region through IR excitation of combination bands involving CH stretch and another lower frequency mode, and the OH products are detected by UV laser-induced fluorescence. OH appearance times on the order of microseconds are observed in this deep tunneling regime, which are about 100 times slower than that in the vicinity of the barrier. The experimental rates are in good accord with Rice-Ramsperger-Kassel-Marcus (RRKM) calculations of the microcanonical dissociation rates for (CH3)2COO that include tunneling. Master equation modeling based on these microcanonical rates is used to predict the thermal decay rate of (CH3)2COO to OH products under atmospheric conditions of 276 s-1 at 298 K (high pressure limit). Thermal unimolecular decay of (CH3)2COO to OH products is shown to have significant contributions from tunneling at energies much below the barrier to H-atom transfer.

Deep tunneling in the unimolecular decay of CH3CHOO Criegee intermediates to OH radical products.

Unimolecular decay of Criegee intermediates produced in alkene ozonolysis is known to be a significant source of OH radicals in the troposphere. In this work, unimolecular decay of the methyl-substituted Criegee intermediate, syn-CH3CHOO, to OH products is shown to occur at energies significantly below the transition state barrier for a 1,4 hydrogen transfer that leads to these products [Y. Fang et al., J. Chem. Phys. 144, 061102 (2016)]. The rate of appearance of OH products arising from tunneling through the barrier is obtained through direct time-domain measurements following the vibrational activation of syn-CH3CHOO. IR excitation of syn-CH3CHOO at energies nearly 2000 cm-1 below the barrier is achieved through combination bands involving CH stretch and another lower frequency mode, and the resultant OH products are detected by UV laser-induced fluorescence. The observed syn-CH3CHOO combination bands in the 4100-4350 cm-1 region are identified by comparison with the computed IR absorption spectrum. The experimental decay rates are found to be ca. 106 s-1 in this deep tunneling regime, which is approximately 100-times slower than that in the vicinity of the barrier.The experimental results are consistent with statistical Rice-Ramsperger-Kassel-Marcus (RRKM) calculations of the microcanonical decay rates with tunneling through the barrier, and notable deviations may originate from the sparsity in the density of states for syn-CH3CHOO at lower energies. Thermal unimolecular decay of syn-CH3CHOO is predicted to have significant contribution from microcanonical rates at energies that are much below the barrier.

Correspondence of below-threshold high-order-harmonic generation and frustrated tunneling ionization

Among many of the nonlinear phenomena induced by strong laser pulses, two of the important processes are the harmonic generation and the creation of neutral atoms in the Rydberg states. We carry out a joint study of the below-threshold high-order-harmonic (BTH) generation and the production of low-lying Rydberg atoms driven by an intense few-cycle midinfrared laser pulse. Our results are based on the numerical solution to the three-dimensional time-dependent Schr\odinger equation within the single active electron approximation and a semiclassical simulation. The yields of BTH and low-lying Rydberg atoms are found to have a similar carrier envelope phase dependence. We find that both processes can be faithfully described semiclassically in the deep tunneling regime. The trajectory analysis shows that these two processes share the same series of trajectories and can be simultaneously manipulated by the driving pulse shape. Our finding bridges the below-threshold high-order-harmonic generation and the frustrated tunneling ionization.

How to remove the spurious resonances from ring polymer molecular dynamics.

Two of the most successful methods that are presently available for simulating the quantum dynamics of condensed phase systems are centroid molecular dynamics (CMD) and ring polymer molecular dynamics (RPMD). Despite their conceptual differences, practical implementations of these methods differ in just two respects: the choice of the Parrinello-Rahman mass matrix and whether or not a thermostat is applied to the internal modes of the ring polymer during the dynamics. Here, we explore a method which is halfway between the two approximations: we keep the path integral bead masses equal to the physical particle masses but attach a Langevin thermostat to the internal modes of the ring polymer during the dynamics. We justify this by showing analytically that the inclusion of an internal mode thermostat does not affect any of the established features of RPMD: thermostatted RPMD is equally valid with respect to everything that has actually been proven about the method as RPMD itself. In particular, because of the choice of bead masses, the resulting method is still optimum in the short-time limit, and the transition state approximation to its reaction rate theory remains closely related to the semiclassical instanton approximation in the deep quantum tunneling regime. In effect, there is a continuous family of methods with these properties, parameterised by the strength of the Langevin friction. Here, we explore numerically how the approximation to quantum dynamics depends on this friction, with a particular emphasis on vibrational spectroscopy. We find that a broad range of frictions approaching optimal damping give similar results, and that these results are immune to both the resonance problem of RPMD and the curvature problem of CMD.

Mixed quantum classical calculation of proton transfer reaction rates: from deep tunneling to over the barrier regimes.

We present mixed quantum classical calculations of the proton transfer (PT) reaction rates represented by a double well system coupled to a dissipative bath. The rate constants are calculated within the so called nontraditional view of the PT reaction, where the proton motion is quantized and the solvent polarization is used as the reaction coordinate. Quantization of the proton degree of freedom results in a problem of non-adiabatic dynamics. By employing the reactive flux formulation of the rate constant, the initial sampling starts from the transition state defined using the collective reaction coordinate. Dynamics of the collective reaction coordinate is treated classically as over damped diffusive motion, for which the equation of motion can be derived using the path integral, or the mixed quantum classical Liouville equation methods. The calculated mixed quantum classical rate constants agree well with the results from the numerically exact hierarchical equation of motion approach for a broad range of model parameters. Moreover, we are able to obtain contributions from each vibrational state to the total reaction rate, which helps to understand the reaction mechanism from the deep tunneling to over the barrier regimes. The numerical results are also compared with those from existing approximate theories based on calculations of the non-adiabatic transmission coefficients. It is found that the two-surface Landau-Zener formula works well in calculating the transmission coefficients in the deep tunneling regime, where the crossing point between the two lowest vibrational states dominates the total reaction rate. When multiple vibrational levels are involved, including additional crossing points on the free energy surfaces is important to obtain the correct reaction rate using the Landau-Zener formula.

Maximum tunneling velocities in symmetric double well potentials

We consider coherent tunneling of one-dimensional model systems in non-cyclic or cyclic symmetric double well potentials. Generic potentials are constructed which allow for analytical estimates of the quantum dynamics in the non-relativistic deep tunneling regime, in terms of the tunneling distance, barrier height and mass (or moment of inertia). For cyclic systems, the results may be scaled to agree well with periodic potentials for which semi-analytical results in terms of Mathieu functions exist. Starting from a wavepacket which is initially localized in one of the potential wells, the subsequent periodic tunneling is associated with tunneling velocities. These velocities (or angular velocities) are evaluated as the ratio of the flux densities versus the probability densities. The maximum velocities are found under the top of the barrier where they scale as the square root of the ratio of barrier height and mass (or moment of inertia), independent of the tunneling distance. They are applied exemplarily to several prototypical molecular models of non-cyclic and cyclic tunneling, including ammonia inversion, Cope rearrangement of semibullvalene, torsions of molecular fragments, and rotational tunneling in strong laser fields. Typical maximum velocities and angular velocities are in the order of a few km/s and from 10 to 100THz for our non-cyclic and cyclic systems, respectively, much faster than time-averaged velocities. Even for the more extreme case of an electron tunneling through a barrier of height of one Hartree, the velocity is only about one percent of the speed of light. Estimates of the corresponding time scales for passing through the narrow domain just below the potential barrier are in the domain from 2 to 40fs, much shorter than the tunneling times.

Relativistic features and time delay of laser-induced tunnel ionization

The electron dynamics in the classically forbidden region during relativistic tunnel-ionization is investigated. The classical forbidden region in the relativistic regime is identified by defining a gauge invariant total energy operator. Introducing position dependent energy levels inside the tunneling barrier, we demonstrate that the relativistic tunnel-ionization can be well described by a one-dimensional intuitive picture. This picture predicts that, in contrast to the well-known nonrelativistic regime, the ionized electron wave packet arises with a momentum shift along the laser's propagation direction. This is compatible with results from a strong field approximation calculation where the binding potential is assumed to be zero-ranged. Further, the tunneling time delay, stemming from Wigner's definition, is investigated for model configurations of tunneling and compared with results obtained from the exact propagator. By adapting Wigner's time delay definition to the ionization process, the tunneling time is investigated in the deep-tunneling and in the near-threshold-tunneling regimes. It is shown that while in the deep-tunneling regime signatures of the tunneling time delay are not measurable at remote distance, it is detectable, however, in the latter regime.

Ionization with low-frequency fields in the tunneling regime

Strong-field ionisation surprises with richness beyond current understanding despite decade long investigations. Ionisation with mid-IR light has promptly revealed unexpected kinetic energy structures that seem related to unanticipated quantum trajectories of the electrons. We measure first 3D momentum distributions in the deep tunneling regime (γ = 0.3) and observe surprising new electron dynamics of near-zero momentum electrons and extremely low momentum structures, below the eV, despite very high quiver energies of 95 eV. Such level of high-precision measurements at only 1 meV above the threshold, despite 5 orders higher ponderomotive energies, has now become possible with a specifically developed ultrafast mid-IR light source in combination with a reaction microscope, thereby permitting a new level of investigations into mid-IR recollision physics.