Dynamical tunnelling in molecules: quantum routes to energy flow

Dynamical Tunneling in More than Two Degrees of Freedom

Recent progress towards understanding the mechanism of dynamical tunneling in Hamiltonian systems with three or more degrees of freedom (DoF) is reviewed. In contrast to systems with two degrees of freedom, the three or more degrees of freedom case presents several challenges. Specifically, in higher-dimensional phase spaces, multiple mechanisms for classical transport have significant implications for the evolution of initial quantum states. In this review, the importance of features on the Arnold web, a signature of systems with three or more DoF, to the mechanism of resonance-assisted tunneling is illustrated using select examples. These examples represent relevant models for phenomena such as intramolecular vibrational energy redistribution in isolated molecules and the dynamics of Bose-Einstein condensates trapped in optical lattices.

Nonstatistical Photoinduced Processes in Gaseous Organic Molecules

Processes that proceed in femtoseconds are usually referred to as being ultrafast, and they are investigated in experiments that involve laser pulses with femtosecond duration in so-called pump probe schemes, where a light pulse triggers a molecular process and a second light pulse interrogates the temporal evolution of the molecular population. The focus of this review is on the reactivity patterns that arise when energy is not equally distributed on all the available degrees of freedom as a consequence of the very short time scale in play and on how the localization of internal energy in a specific mode can be thought of as directing a process toward (or away from) a certain outcome. The nonstatistical aspects are illustrated with examples from photophysics and photochemistry for a range of organic molecules. The processes are initiated by a variety of nuclear motions that are all governed by the energy gradients in the Franck-Condon region. Essentially, the molecules will start to adapt to the new electronic environment on the excited state to eventually reach the equilibrium structure. It is this structural change that is enabling an ultrafast electronic transition in cases where the nuclear motion leads to a transition point with significant coupling between to electronic states and to ultrafast reaction if there is a coupling to a reactive mode at the transition point between the involved states. With the knowledge of the relation between electronic excitation and equilibrium structure, it is possible to predict how the nuclei move after excitation and often whether an ultrafast (and inherently nonstatistical) electronic transition or even a bond breakage will take place. In addition to the understanding of how nonstatistical photoinduced processes proceed from a given excited state, it has been found that randomization of the energy does not even always take place when the molecule takes part in processes that are normally considered statistical, such as for example nonradiative transitions between excited states. This means that energy can be localized in a specific degree of freedom on a state other than the one that is initially prepared. This is a finding that could kickoff the ultimate dream in applied photochemistry; namely light excitation that leads to the rupture of a specific bond.

Resonance-assisted tunneling in four-dimensional normal-form Hamiltonians

Nonlinear resonances in the classical phase space lead to a significant enhancement of tunneling. We demonstrate that the double resonance gives rise to a complicated tunneling peak structure. Such double resonances occur in Hamiltonian systems with an at least four-dimensional phase space. To explain the tunneling peak structure, we use the universal description of single and double resonances by the four-dimensional normal-form Hamiltonians. By applying perturbative methods, we reveal the underlying mechanism of enhancement and suppression of tunneling and obtain excellent quantitative agreement. Using a minimal matrix model, we obtain an intuitive understanding.

Unimolecular Rate Constants versus Energy and Pressure as a Convolution of Unimolecular Lifetime and Collisional Deactivation Probabilities. Analyses of Intrinsic Non-RRKM Dynamics

Following work by Slater and Bunker, the unimolecular rate constant versus collision frequency, k_uni(ω, E), is expressed as a convolution of unimolecular lifetime and collisional deactivation probabilities. This allows incorporation of nonexponential, intrinsically non-RRKM, populations of dissociating molecules versus time, N( t)/ N(0), in the expression for k_uni(ω, E). Previous work using this approach is reviewed. In the work presented here, the biexponential f₁ exp(- k₁ t) + f₂ exp(- k₂ t) is used to represent N( t)/ N(0), where f₁ k₁ + f₂ k₂ equals the RRKM rate constant k( E) and f₁ + f₂ = 1. With these two constraints, there are two adjustable parameters in the biexponential N( t)/ N(0) to represent intrinsic non-RRKM dynamics. The rate constant k₁ is larger than k( E) and k₂ is smaller. This biexponential gives k_uni(ω, E) rate constants that are lower than the RRKM prediction, except at the high and low pressure limits. The deviation from the RRKM prediction increases as f₁ is made smaller and k₁ made larger. Of considerable interest is the finding that, if the collision frequency ω for the RRKM plot of k_uni(ω, E) versus ω is multiplied by an energy transfer efficiency factor β_c, the RRKM k_uni(ω, E) versus ω plot may be scaled to match those for the intrinsic non-RRKM, biexponential N( t)/ N(0), plots. This analysis identifies the importance of determining accurate collisional intermolecular energy transfer (IET) efficiencies.

Relevance of the Resonance Junctions on the Arnold Web to Dynamical Tunneling and Eigenstate Delocalization

We study the competition and correspondence between the classical and quantum routes to intramolecular vibrational energy redistribution (IVR) in a three degrees of freedom model effective Hamiltonian. Specifically, we focus on the classical and the quantum dynamics near the resonance junctions on the Arnold web that are formed by an intersection of independent resonances. The regime of interest models the IVR dynamics from highly excited initial states near dissociation thresholds of molecular systems wherein both classical and purely quantum, involving dynamical tunneling, routes to IVR coexist. In the vicinity of a resonance junction, classical chaos is inevitably present, and hence one expects the quantum IVR pathways to have a strong classical component as well. We show that with increasing resonant coupling strengths the classical component of IVR leads to a transition from coherent dynamical tunneling to incoherent dynamical tunneling. Furthermore, we establish that the quantum IVR dynamics can be predicted based on the structures on the classical Arnold web. In addition, we investigate the nature of the highly excited eigenstates to identify the quantum signatures of the multiplicity-2 junctions. For the parameter regimes studies herein, by projecting the eigenstates onto the Arnold web, we find that eigenstates in the vicinity of the junctions are primarily delocalized due to dynamical tunneling.

Complex-path prediction of resonance-assisted tunneling in mixed systems

We present a semiclassical prediction of regular-to-chaotic tunneling in systems with a mixed phase space, including the effect of a nonlinear resonance chain. We identify complex paths for direct and resonance-assisted tunneling in the phase space of an integrable approximation with one nonlinear resonance chain. We evaluate the resonance-assisted contribution analytically and give a prediction based on just a few properties of the classical phase space. For the standard map excellent agreement with numerically determined tunneling rates is observed. The results should similarly apply to ionization rates and quality factors.

Perturbation-free prediction of resonance-assisted tunneling in mixed regular-chaotic systems

For generic Hamiltonian systems we derive predictions for dynamical tunneling from regular to chaotic phase-space regions. In contrast to previous approaches, we account for the resonance-assisted enhancement of regular-to-chaotic tunneling in a nonperturbative way. This provides the foundation for future semiclassical complex-path evaluations of resonance-assisted regular-to-chaotic tunneling. Our approach is based on a new class of integrable approximations which mimic the regular phase-space region and its dominant nonlinear resonance chain in a mixed regular-chaotic system. We illustrate the method for the standard map.

Bifurcations of families of 1D-tori in 4D symplectic maps

The regular structures of a generic 4d symplectic map with a mixed phase space are organized by one-parameter families of elliptic 1d-tori. Such families show prominent bends, gaps, and new branches. We explain these features in terms of bifurcations of the families when crossing a resonance. For these bifurcations, no external parameter has to be varied. Instead, the longitudinal frequency, which varies along the family, plays the role of the bifurcation parameter. As an example, we study two coupled standard maps by visualizing the elliptic and hyperbolic 1d-tori in a 3d phase-space slice, local 2d projections, and frequency space. The observed bifurcations are consistent with the analytical predictions previously obtained for quasi-periodically forced oscillators. Moreover, the new families emerging from such a bifurcation form the skeleton of the corresponding resonance channel.

Active Site Dynamical Effects in the Hydrogen Transfer Rate-limiting Step in the Catalysis of Linoleic Acid by Soybean Lipoxygenase-1 (SLO-1): Primary and Secondary Isotope Contributions

Using ab initio molecular dynamics (AIMD) simulations that facilitate the treatment of rare events, we probe the active site participation in the rate-determining hydrogen transfer step in the catalytic oxidation of linoleic acid by soybean lipoxygenase-1 (SLO-1). The role of two different active site components is probed. (a) On the hydrogen atom acceptor side of the active site, the hydrogen bonding propensity between the acceptor side hydroxyl group, which is bound to the iron cofactor, and the backbone carboxyl group of isoleucine (residue number 839) is studied toward its role in promoting the hydrogen transfer event. Primary and secondary (H/D) isotope effects are also probed and a definite correlation with subtle secondary H/D isotope effects is found. With increasing average nuclear kinetic energy, the increase in transfer probability is enhanced due to the presence of the hydrogen bond between the backbone carbonyl of I839 and the acceptor oxygen. Further increase in average nuclear kinetic energy reduces the strength of this secondary hydrogen bond which leads to a deterioration in hydrogen transfer rates and finally embrances an Arrhenius-like behavior. (b) On the hydrogen atom donor side, the coupling between vibrational modes predominantly localized on the donor-side linoleic acid group and the reactive mode is probed. There appears to be a qualitative difference in the coupling between modes that belong to linoleic acid and the hydrogen transfer mode, for hydrogen and deuterium transfer. For example, the donor side secondary hydrogen atom is much more labile (by nearly a factor of 5) during deuterium transfer as compared to the case for hydrogen transfer. This appears to indicate a greater coupling between the modes belonging to the linoleic acid scaffold and the deuterium transfer mode and also provides a new rationalization for the abnormal (nonclassical) secondary isotope effect results obtained by Knapp, Rickert, and Klinman in J. Am. Chem. Soc. , 2002 , 124 , 3865 . To substantiate our findings noted in point a above, we have suggested an I839 → A839 or I839 → V839 mutation. This will modify the bulkiness of hydrogen the bonding residue, allowing greater flexibility in the secondary hydrogen bond formation highlighted above and adversely affecting the reaction rate.

One Versus Two Photon Control of Dynamical Tunneling: Influence of the Irregular Floquet States

A useful approach to control quantum processes involves driving systems with two colored laser fields and varying the relative phase between the fields to control the quantum interferences. A particularly interesting class of bichromatic control schemes involves the so-called M versus N-photon control that results in laser-induced symmetry breaking and leads to directed transport; however, recent studies have shown that the mechanism of laser-induced symmetry breaking has a common classical and quantum origin. In this context, a relevant question is the extent to which such a detailed classical-quantum correspondence holds if the process to be controlled involves quantum tunneling. In this work, we address this issue in terms of controlling dynamical tunneling between field-induced islands of stability in the classical phase space of a model system, a periodically driven pendulum. This is also a paradigmatic model for Hamiltonian ratchets wherein the islands of stability, that is, nonlinear resonances, play a crucial role in the observed directed transport. We compute an appropriate control landscape for the process and show that despite breaking the relevant symmetries, there exist regions in the control landscape where the control fails. The lack of control can be understood in terms of the phase-space nature of the quantum Floquet states that participate in the dynamics of the initial wavepacket. We argue that robust regions of no control arise due to the phenomenon of chaos-assisted tunneling and comment on the possible influence of such regions on the directed transport in the model system.

Breaking of a bond: when is it statistical?

Unimolecular dissociation dynamics of a model three degree of freedom triatomic molecule is studied in order to understand the mechanisms for deviations from statisticality. Performing a wavelet based time–frequency analysis of the dynamics allows for the dynamics to be followed on the network of nonlinear resonances, also called as the Arnold web. The results indicate that the long lifetime trajectories spend a considerable amount of time trapped near junctions in the web. It is argued that characterizing the dynamics near such junctions might lead to deeper insights into the origins of nonstatistical dynamics.

Dynamical traps lead to the slowing down of intramolecular vibrational energy flow

The phenomenon of intramolecular vibrational energy redistribution (IVR) is at the heart of chemical reaction dynamics. Statistical rate theories, assuming instantaneous IVR, predict exponential decay of the population with the properties of the transition state essentially determining the mechanism. However, there is growing evidence that IVR competes with the reaction timescales, resulting in deviations from the exponential rate law. Dynamics cannot be ignored in such cases for understanding the reaction mechanisms. Significant insights in this context have come from the state space model of IVR, which predicts power law behavior for the rates with the power law exponent, an effective state space dimensionality, being a measure of the nature and extent of the IVR dynamics. However, whether the effective IVR dimensionality can vary with time and whether the mechanism for the variation is of purely quantum or classical origins are issues that remain unresolved. Such multiple power law scalings can lead to surprising mode specificity in the system, even above the threshold for facile IVR. In this work, choosing the well-studied thiophosgene molecule as an example, we establish the anisotropic and anomalous nature of the quantum IVR dynamics and show that multiple power law scalings do manifest in the system. More importantly, we show that the mechanism of the observed multiple power law scaling has classical origins due to a combination of trapping near resonance junctions in the network of classical nonlinear resonances at short to intermediate times and the influence of weak higher-order resonances at relatively longer times.

Global structure of regular tori in a generic 4D symplectic map

For the case of generic 4d symplectic maps with a mixed phase space, we investigate the global organization of regular tori. For this, we compute elliptic 1-tori of two coupled standard maps and display them in a 3d phase-space slice. This visualizes how all regular 2-tori are organized around a skeleton of elliptic 1-tori in the 4d phase space. The 1-tori occur in two types of one-parameter families: (α) Lyapunov families emanating from elliptic-elliptic periodic orbits, which are observed to exist even far away from them and beyond major resonance gaps, and (β) families originating from rank-1 resonances. At resonance gaps of both types of families either (i) periodic orbits exist, similar to the Poincaré-Birkhoff theorem for 2d maps, or (ii) the family may form large bends. In combination, these results allow for describing the hierarchical structure of regular tori in the 4d phase space analogously to the islands-around-islands hierarchy in 2d maps.

Visualization and comparison of classical structures and quantum states of four-dimensional maps

For generic 4D symplectic maps we propose the use of 3D phase-space slices, which allow for the global visualization of the geometrical organization and coexistence of regular and chaotic motion. As an example, we consider two coupled standard maps. The advantages of the 3D phase-space slices are presented in comparison to standard methods, such as 3D projections of orbits, the frequency analysis, and a chaos indicator. Quantum mechanically, the 3D phase-space slices allow for the comparison of Husimi functions of eigenstates of 4D maps with classical phase-space structures. This confirms the semiclassical eigenfunction hypothesis for 4D maps.

Instanton and noninstanton tunneling in periodically perturbed barriers: semiclassical and quantum interpretations

In multidimensional barrier tunneling, there exist two different types of tunneling mechanisms, instanton-type tunneling and noninstanton tunneling. In this paper we investigate transitions between the two tunneling mechanisms from the semiclassical and quantum viewpoints taking two simple models: a periodically perturbed Eckart barrier for the semiclassical analysis and a periodically perturbed rectangular barrier for the quantum analysis. As a result, similar transitions are observed with change of the perturbation frequency ω for both systems, and we obtain a comprehensive scenario from both semiclassical and quantum viewpoints for them. In the middle range of ω, in which the plateau spectrum is observed, noninstanton tunneling dominates the tunneling process, and the tunneling amplitude takes the maximum value. Noninstanton tunneling explained by stable-unstable manifold guided tunneling (SUMGT) from the semiclassical viewpoint is interpreted as multiphoton-assisted tunneling from the quantum viewpoint. However, in the limit ω→0, instanton-type tunneling takes the place of noninstanton tunneling, and the tunneling amplitude converges on a constant value depending on the perturbation strength. The spectrum localized around the input energy is observed, and there is a scaling law with respect to the width of the spectrum envelope, i.e., the width ∝ℏω. In the limit ω→∞, the tunneling amplitude converges on that of the unperturbed system, i.e., the instanton of the unperturbed system.

"Pump-probe" atom-centered density matrix propagation studies to gauge anharmonicity and energy repartitioning in atmospheric reactive adducts: case study of the OH + isoprene and OH + butadiene reaction intermediates

Time-resolved "pump-probe" ab initio molecular dynamics studies are constructed to probe the stability of reaction intermediates, the mechanism of energy transfer, and energy repartitioning, for moieties involved during the interaction of volatile organic compunds with hydroxyl radical. These systems are of prime importance in the atmosphere. Specifically, the stability of reaction intermediates of hydroxyl radical adducts to isoprene and butadiene molecules is used as a case study to develop novel computational techniques involving "pump-probe" ab initio molecular dynamics. Starting with the various possible hydroxyl radical adducts to isoprene and butadiene, select vibrational modes of each of the adducts are populated with excess energy to mimic the initial conditions of an experiment. The flow of energy into the remaining modes is then probed by subjecting the excited adducts to ab initio molecular dynamics simulations. It is found that the stability of the adducts arises directly due to the anhormonically driven coupling of the modes to facilitate repartitioning of the excess vibrational energy. This kind of vibrational repartitioning has a critical influence on the energy density.

Non-instanton tunneling: semiclassical and quantum interpretations

Tunneling essentially different from instanton-type tunneling, say noninstanton tunneling, is studied from both semiclassical and quantum viewpoints. Taking a periodically perturbed rounded-off step potential for which the instanton-type tunneling is substantially prohibited, we analyze change of the tunneling probability with change of the perturbation frequency based on the stable-unstable manifold-guided tunneling (SUMGT) theory, which we have recently introduced. In the large and small limits of the frequency, the tunneling rate rapidly decays, but it is markedly enhanced in an intermediate range. We will also make a quantum interpretation of the noninstanton tunneling by using an exactly solvable model--a periodically perturbed right-angled step potential. Analysis with this model shows that SUMGT is considered as a sort of photoassisted tunneling through a large energy gap induced with absorbing a huge number quanta, which is completely different from the instanton-type tunneling. Both approaches from the semiclassical and quantum viewpoints complement each other to cause a better understanding of noninstanton tunneling.

Poincaré-Birkhoff theorem in quantum mechanics

Quantum manifestations of the dynamics around resonant tori in perturbed hamiltonian systems, dictated by the Poincaré-Birkhoff theorem, are shown to exist. They are embedded in the interactions involving states which differ in a number of quanta equal to the order of the classical resonance. Moreover, the associated classical phase space structures are mimicked in the quasiprobability density functions and their zeros.