Deep Tunneling Regime Research Articles

Deep nuclear resonant tunneling thermal rate constant calculations

AbstractA fast and robust time‐independent method to calculate thermal rate constants in the deep resonant tunneling regime for scattering reactions is presented. The method is based on the calculation of the cumulative reaction probability which, once integrated, gives the thermal rate constant. We tested our method with both continuous (single and double Eckart barriers) and discontinuous first derivative potentials (single and double rectangular barriers). Our results show that the presented method is robust enough to deal with extreme resonating conditions such as multiple barrier potentials. Finally, the calculation of the thermal rate constant for double Eckart potentials with several quasi‐bound states and the comparison with the time‐independent log‐derivative method are reported. An implementation of the method using the Mathematica Suite is included in the Supporting Information. © 2013 Wiley Periodicals, Inc.

A ring polymer molecular dynamics study of the isotopologues of the H + H2 reaction

The inclusion of Quantum Mechanical (QM) effects such as zero point energy (ZPE) and tunneling in simulations of chemical reactions, especially in the case of light atom transfer, is an important problem in computational chemistry. In this respect, the hydrogen exchange reaction and its isotopic variants constitute an excellent benchmark for the assessment of approximate QM methods. In particular, the recently developed ring polymer molecular dynamics (RPMD) technique has been demonstrated to give very good results for bimolecular chemical reactions in the gas phase. In this work, we have performed a detailed RPMD study of the H + H(2) reaction and its isotopologues Mu + H(2), D + H(2) and Heμ + H(2), at temperatures ranging from 200 to 1000 K. Thermal rate coefficients and kinetic isotope effects have been computed and compared with exact QM calculations as well as with quasiclassical trajectories and experiment. The agreement with the QM results is good for the heaviest isotopologues, with errors ranging from 15% to 45%, and excellent for Mu + H(2), with errors below 15%. We have seen that RPMD is able to capture the ZPE effect very accurately, a desirable feature of any method based on molecular dynamics. We have also verified Richardson and Althorpe's prediction [J. O. Richardson and S. C. Althorpe, J. Chem. Phys., 2009, 131, 214106] that RPMD will overestimate thermal rates for asymmetric reactions and underestimate them for symmetric reactions in the deep tunneling regime. The ZPE effect along the reaction coordinate must be taken into account when assigning the reaction symmetry in the multidimensional case.

Classical-like trajectory simulations for accurate computation of quantum reactive scattering probabilities

A trajectory ensemble method is introduced that enables accurate computation of microcanonical quantum reactive scattering quantities, using a classical-like simulation scheme. Individual quantum trajectories are propagated independently, using a Newton-like ODE which treats quantum dynamical effects along the reaction coordinate exactly, and preserves the phase space volume element. The sampling of initial conditions resembles a classical microcanonical simulation, but modified so as to incorporate quantization in the perpendicular mode coordinates. The method is exact for one-dimensional or separable systems, and achieves ∼1% accuracy for the coupled multidimensional benchmark applications considered here, even in the deep tunneling regime.

Quantum rate dynamics for proton transfer reaction in a model system: Effect of the rate promoting vibrational mode

We extended our previous calculation of the quantum rate dynamics for a model system of proton transfer (PT) reaction using the Liouville space hierarchical equations of motion method in this study. A rate promoting vibrational (RPV) mode that symmetrically coupled to the proton coordinate was included in the quantum dynamics calculations, in order to study the effect of enhanced tunneling by the proton donor-acceptor motion. Adding the RPV mode is observed to increase the PT rate and reduce the kinetic isotope effects. We also found that the PT dynamics is influenced by the dissipation of the RPV mode. Besides this extension, in the case without the RPV, we investigated whether the PT rate dynamics in the deep tunneling regime can reduce to an effective two-state spin-boson type of model and found that this is only possible at low reorganization energies.

Bimolecular reaction rates from ring polymer molecular dynamics: Application to H + CH4→ H2 + CH3

In a recent paper, we have developed an efficient implementation of the ring polymer molecular dynamics (RPMD) method for calculating bimolecular chemical reaction rates in the gas phase, and illustrated it with applications to some benchmark atom-diatom reactions. In this paper, we show that the same methodology can readily be used to treat more complex polyatomic reactions in their full dimensionality, such as the hydrogen abstraction reaction from methane, H + CH(4) → H(2) + CH(3). The present calculations were carried out using a modified and recalibrated version of the Jordan-Gilbert potential energy surface. The thermal rate coefficients obtained between 200 and 2000 K are presented and compared with previous results for the same potential energy surface. Throughout the temperature range that is available for comparison, the RPMD approximation gives better agreement with accurate quantum mechanical (multiconfigurational time-dependent Hartree) calculations than do either the centroid density version of quantum transition state theory (QTST) or the quantum instanton (QI) model. The RPMD rate coefficients are within a factor of 2 of the exact quantum mechanical rate coefficients at temperatures in the deep tunneling regime. These results indicate that our previous assessment of the accuracy of the RPMD approximation for atom-diatom reactions remains valid for more complex polyatomic reactions. They also suggest that the sensitivity of the QTST and QI rate coefficients to the choice of the transition state dividing surface becomes more of an issue as the dimensionality of the reaction increases.

Ring-polymer molecular dynamics rate-theory in the deep-tunneling regime: Connection with semiclassical instanton theory

We demonstrate that the ring-polymer molecular dynamics (RPMD) method is equivalent to an automated and approximate implementation of the "Im F" version of semiclassical instanton theory when used to calculate reaction rates in the deep-tunneling regime. This explains why the RPMD method is often reliable in this regime and also shows how it can be systematically improved. The geometry of the beads at the transition state on the ring-polymer potential surface describes a finite-difference approximation to the "instanton" trajectory (a periodic orbit in imaginary time beta variant Planck's over 2pi on the inverted potential surface). The deep-tunneling RPMD rate is an approximation to the rate obtained by applying classical transition-state theory (TST) in ring-polymer phase-space using the optimal dividing surface; this TST rate is in turn an approximation to a free-energy version of the Im F instanton rate. The optimal dividing surface is in general a function of several modes of the ring polymer, which explains why centroid-based quantum-TSTs break down at low temperatures for asymmetric reaction barriers. Numerical tests on one-dimensional models show that the RPMD rate tends to overestimate deep-tunneling rates for asymmetric barriers and underestimate them for symmetric barriers, and we explain that this is likely to be a general trend. The ability of the RPMD method to give a dividing-surface-independent rate in the deep-tunneling regime is shown to be a consequence of setting the bead-masses equal to the physical mass.

Bimolecular reaction rates from ring polymer molecular dynamics

We describe an efficient procedure for calculating the rates of bimolecular chemical reactions in the gas phase within the ring polymer molecular dynamics approximation. A key feature of the procedure is that it does not require that one calculate the absolute quantum mechanical partition function of the reactants or the transition state: The rate coefficient only depends on the ratio of these two partition functions which can be obtained from a thermodynamic integration along a suitable reaction coordinate. The procedure is illustrated with applications to the three-dimensional H + H(2), Cl + HCl, and F + H(2) reactions, for which well-converged quantum reactive scattering results are computed for comparison. The ring polymer rate coefficients agree with these exact results at high temperatures and are within a factor of 3 of the exact results at temperatures in the deep quantum tunneling regime, where the classical rate coefficients are too small by several orders of magnitude. This is probably already good enough to encourage future applications of the ring polymer theory to more complex chemical reactions, which it is capable of treating in their full dimensionality. However, there is clearly some scope for improving on the ring polymer approximation at low temperatures, and we end by suggesting a way in which this might be accomplished.

On the effect of nuclear bridge modes on donor–acceptor electronic coupling in donor–bridge–acceptor molecules

We report a theoretical study of intra-molecular electronic coupling in a symmetric DBA (donor–bridge–acceptor) complex, in which a donor electronic site is coupled to an acceptor site by way of intervening orbitals of a molecular bridge unit. In the off-resonant (deep tunneling) regime of electronic transport, the lowest unoccupied molecular orbitals (MO’s) of the DBA system are split into distinguishable donor/acceptor and bridge orbitals. The effect of geometrical changes at the bridge on the donor/acceptor electronic energy manifold is studied for local stretching and bending modes. It is demonstrated that the energy splitting in the manifold of donor/acceptor unoccupied MOs changes in response to such changes, as assumed in simple McConnell-type models. Limitations of the simple models are revealed where the electronic charging of the bridge orbitals correlates with increasing donor/acceptor orbital energy splitting only for stretching but not for bending bridge modes.

Computation of the tunneling H-transfer reaction kinetics in the fluorene molecular crystal

The semi-empirical computation of the rate constant is fully performed for the photochemical bimolecular hydrogen atom transfer reaction in condensed phase, proceeding in the deep tunneling regime. For the system considered (fluorene + acridine in its first triplet state, inside the fluorene molecular crystal) the kinetics have been thoroughly investigated experimentally over wide temperature range (http://bernd-prass.de). In the present theoretical work the potential energy surface is constructed by means of the reported earlier QM/MM approach, covering the interaction of the reaction pair with the crystalline environment. Direct evaluation of two-dimensional tunneling amplitudes on this surface is made by means of the accurate quantum-mechanical numerical method. The kinetic part of the rate calculation invokes Golden Rule methodology in its recently elaborated modification while consistently accounting for the dynamical energy exchange between the tunneling pattern and the medium. The careful calibration of transition amplitudes for H- and D-transfer reactions provided the parameters controlling their exponential dependence on the promotion mode (i.e. the distance between the terminal heavy atoms of the reaction centre), which establishes the effective tunneling length of the transfer process. The experimental temperature dependence of the reaction rates for both cases is well reproduced by tuning the parameters of the phonon frequency spectrum projected on the promotion mode. The fitted values of these spectral density parameters are contained within the range located in the preliminary independent computation of the lattice dynamics for the fluorene crystal. Most difficult is the part of the computation concerned with the absolute values of rate constants for both H- and D-reactions. The magnitude of the isotope effect can be reasonably interpreted at a qualitative level by complementing the amplitude quantum-mechanical data with empirical Frank–Condon (FC) factors inherent to the reorganization of several intra-molecular modes, which is known to accompany the background transfer reaction. However, this treatment, based on the oversimplified one-dimensional model of such reorganization, is not fully consistent, since the pertaining FC parameters were found to differ significantly for the cases of H- and D-transfer.

Site-directed electronic tunneling through a vibrating molecular network

The effect of electronic-nuclear coupling on electronic transport through a complex molecular network is studied. Electronic tunneling dynamics in a network of N donor/acceptor sites, connected by molecular bridges, is shown to be controlled by electronic-nuclear coupling at the bridges. Particularly, electronic coupling to an accepting nuclear mode at the contact site between the donor and the rest of the network is shown to affect the tunneling path selection to specific acceptors. In the "deep" tunneling regime, the network is mapped onto an N-level system using a recursive perturbation expansion, enabling analytical treatment of the electronic dynamics. The analytic formulation is applied for two model systems, demonstrating site-directed tunneling by electronic-nuclear coupling. Numerical simulations suggest that this phenomenon is not limited to the deep tunneling regime.

Chemical reaction rates from ring polymer molecular dynamics

We show how the ring-polymer molecular dynamics method can be adapted to calculate approximate Kubo-transformed flux-side correlation functions, and hence rate coefficients for condensed phase reactions. An application of the method to the standard model for a chemical reaction in solution--a quartic double-well potential linearly coupled to a bath of harmonic oscillators--is found to give results of comparable accuracy to those of the classical Wigner model and the centroid molecular dynamics method. However, since the present method does not require that one evaluate the Wigner transform of a thermal flux operator or that one perform a separate path integral calculation for each molecular dynamics time step, we believe it will prove easier to apply to more general problems than either of these alternative techniques. We also present a (logarithmic) discretization scheme for the Ohmic bath in the system-bath model that gives converged results with just nine bath modes--a surprisingly small number for a model of a condensed phase reaction. Finally, we present some calculations of the transmission through an Eckart barrier which show that the present method provides a satisfactory (although not perfect) description of the deep quantum tunneling regime. Part of the reason for the success of the method is that it gives the exact quantum-mechanical rate constant for the transmission through a parabolic barrier, as we demonstrate analytically in the Appendix.

Numerical Simulations of Electron Tunneling Currents in Water

This paper presents results of numerical simulations of electron tunneling through water that extend our previous calculations on such systems in several ways. First, a tip-substrate configuration is used; second, calculations are carried in the presence of an external potential bias; third, the image potential that reflects the interaction of the electron with the mobile metal electrons is taken into account in the static image approximation. Finally, all-to-all transmission probability calculations are performed in order to get an order-of-magnitude estimate of the current-voltage characteristics of this junction model. The computed currents are within the range of the few available experimental observations on scanning tunneling microscope (STM) currents in water, indicating that our calculation may have taken into account all the important physical attributes of such systems. In addition we examine the effect of the water medium on the spatial distribution of the tunneling flux. We find that while different water configurations scatter the tunneling electron in different ways, on the average the water affected loss of resolution is rather small in the deep tunneling regime but can be substantial in energy regimes where the tunneling is strongly affected by water-supported resonance structures.

The effect of quantum noise on the energy degeneration in a dissipative quantum two-level system

The influence of noise on the energy degeneration in a periodically driven two-state system is discussed. It is shown that the degenerate energy level of the system, split by an external monochromatic field, can be restored to the degenerate state by appropriate noise. In addition, the maximum oscillation amplitude at a series of values of the temperature in the deep tunneling regime exhibit the phenomenon of quantum multiple stochastic resonance.

Quantum Rate Constants from Short-Time Dynamics: An Analytic Continuation Approach

A method for calculating the quantum canonical rate constant of chemical reactions in a many body system by means of a short-time flux autocorrelation function combined with a maximum entropy numerical analytic continuation scheme is presented. The rate constant is expressed as the time integral of the real-time flux autocorrelation function. The real-time flux autocorrelation function is evaluated for short times fully quantum mechanically by path integral Monte Carlo simulations. The maximum entropy approach is then used to extract the rate from the short real-time flux autocorrelation data. We present two numerical tests, one for proton transfer in harmonic dissipative environments in the deep tunneling regime and the other for the two-level model of primary charge separation in the photosynthetic reaction center. The results obtained using the flux autocorrelation data up to the time of no more than βℏ are in excellent agreement with the exact quantum calculation over a wide range of parameters includ...

Dynamical effects in the calculation of quantum rates for electron transfer reactions

This paper presents results of path-integral quantum dynamics simulations of electron transfer rates in a simple class of model Hamiltonians for symmetric and asymmetric electron transfer systems. Our study is aimed at testing the practical usefulness of a centroid factorization of the electron transfer rate constant in the deep tunneling regime. To circumvent the sign problem in quantum Monte Carlo simulations of the electron flux, local filtering techniques have been employed. The simulations show that due to dynamical effects, the reactive flux far outside the transition state region also plays an important role in determining the rate. These results suggest that while the centroid formulation of the equilibrium (imaginary-time) quantum transition state theory (QTST) applied to the electron path is accurate for nonadiabatic electron transfer reactions, dynamical (real-time) effects can produce significant corrections to the QTST estimate for the rate outside this region. For model parameters characteristic for a chemical electron transfer system, we found that the rate is enhanced by a factor Ze≊4. The origin of these dynamical corrections can be understood within a perturbation theory in the number of kinks on the imaginary-time electron quantum path.