Theoretical study of geometric phase effects in the hydrogen-exchange reaction

Emergence of the Molecular Geometric Phase from Exact Electron-Nuclear Dynamics

Geometric phases play a crucial role in diverse fields. In molecules, they appear when a reaction path encircles an intersection between adiabatic potential energy surfaces and the molecular wave function experiences quantum-mechanical interference effects. This intriguing effect, closely resembling the magnetic Aharonov-Bohm effect, crucially relies on the adiabatic description of the dynamics, and it is an open issue whether and how it persists in an exact quantum dynamical framework. Recent works suggest that the molecular geometric phase is an artifact of the adiabatic approximation, thereby challenging the entire concept. Here, building upon a recent investigation (Martinazzo, R.; Burghardt, I. Phys. Rev. Lett. 2024, 132, 243002), we address this issue using the exact factorization of the total wave function. We introduce instantaneous gauge-invariant phases separately for the electrons and nuclei and use them to monitor the phase difference between the trailing edges of a wavepacket encircling a conical intersection between adiabatic surfaces. The transition from the time-dependent open-path phase differences to the closed-path limit is examined, revealing how the phase differences in the electronic and nuclear subspaces compensate for each other upon path closure. In this way, we unambiguously demonstrate the role of the geometric phase in the interference process and shed light on its persistence beyond the adiabatic approximation.

An improved method for reactive scatterings in ultra-cold conditions using the time-dependent approach

A new efficient method for considering the long-range effect of reactive scattering processes in ultra-cold conditions has been developed using the time-dependent quantum wave packet theory, where the initial wave packet could be placed at a position near the interaction region. This is in contrast to previous methods, where the initial wave packet has to be placed far from the interaction region. The new method reduces the numerical effort significantly. Typical reactions, such as S(1D) + H2, D+ + H2, and 7Li + 7Li2 (v0 = 1, j0 = 0), under cold or ultra-cold conditions, are used to demonstrate the numerical efficiency of the new method.

Spiers Memorial Lecture: New directions in molecular scattering

The field of molecular scattering is reviewed as it pertains to gas-gas as well as gas-surface chemical reaction dynamics. We emphasize the importance of collaboration of experiment and theory, from which new directions of research are being pursued on increasingly complex problems. We review both experimental and theoretical advances that provide the modern toolbox available to molecular-scattering studies. We distinguish between two classes of work. The first involves simple systems and uses experiment to validate theory so that from the validated theory, one may learn far more than could ever be measured in the laboratory. The second class involves problems of great complexity that would be difficult or impossible to understand without a partnership of experiment and theory. Key topics covered in this review include crossed-beams reactive scattering and scattering at extremely low energies, where quantum effects dominate. They also include scattering from surfaces, reactive scattering and kinetics at surfaces, and scattering work done at liquid surfaces. The review closes with thoughts on future promising directions of research.

Dynamics of the Molecular Geometric Phase

The fate of the molecular geometric phase in an exact dynamical framework is investigated with the help of the exact factorization of the wave function and a recently proposed quantum hydrodynamical description of its dynamics. An instantaneous, gauge-invariant phase is introduced for arbitrary paths in nuclear configuration space in terms of hydrodynamical variables, and shown to reduce to the adiabatic geometric phase when the state is adiabatic and the path is closed. The evolution of the closed-path phase over time is shown to adhere to a Maxwell-Faraday induction law, with nonconservative forces arising from the electron dynamics that play the role of electromotive forces. We identify the pivotal forces that are able to change the value of the phase, thereby challenging any topological argument. Nonetheless, negligible changes in the phase occur when the local dynamics along the probe loop is approximately adiabatic. That is, the geometric phase effects that arise in an adiabatic limiting situation remain suitable to effectively describe certain dynamic observables.

Strong Angular Oscillation of Rotationally Resolved Differential Cross Section in the H + HD → H2 + D Reaction at the Collision Energy of 2.07 eV: Evidence of Geometric Phase Effects

Geometric phase (GP) effects in chemical reactions are subtle quantum phenomena that are challenging to identify. In this work, we report a joint experimental and theoretical study of the H + HD → H2 + D reaction at a collision energy of 2.07 eV, which is far below the energy of the conical intersection of 2.53 eV. The rotationally state-resolved differential cross sections were measured by a crossed-beam experiment with the scheme of D-atom Rydberg tagging time-of-flight detection. Experimental angular distributions of three rotational states of H2 products exhibit notable variation near the backward scattering direction. Time-dependent quantum mechanics calculations (TDQMs) were carried out at the same collision energy, with and without the inclusion of GP. The experimental angular distributions are in good agreement with TDQM results with the inclusion of GP but do not agree with TDQM results without the inclusion of GP. This work demonstrates the existence of GP effects at energy far below the conical intersection.

Reactant-Product Decoupling Technique Using the Intermediate Coordinate Method

Although the reactant-product decoupling (RPD) technique was proposed over two decades ago, it remains an efficient approach for calculating product state-resolved information on some simple direct reactions using the quantum wave packet method. In the past, usually the RPD technique employed the collocation method to transform the wave function between reactant and product arrangements, which requires quite large computational efforts. In this work, the intermediate coordinate (IC) method is employed to realize the RPD technique. Numerical examples demonstrate that this new IC RPD (IRPD) technique has superior computational efficiency compared with the original method employing the collocation method. Especially, the new IRPD technique significantly saves disk space and computer memory. To illustrate the features of our new method, the total reaction probabilities of the H + H2, H + Br2, and F + H2 reactions with J = 0 and the differential cross sections of the H + H2 and F + H2 reactions at a series of collision energy are calculated and presented. With this efficient and effective new RPD technique, the Li + HF reaction, which involves sharp resonances with long-range wave functions in the van der Waals wells in both the reactant and product arrangements, is also calculated with several J at the product state-resolved level to reveal the ability of the RPD technique for describing resonance wave functions. With these numerical examples, it is found that, for the reaction with resonances, the RPD approach should be applied carefully. Otherwise, it is very possible that the resonances could disappear with the application of the RPD technique.

Impact of Geometric Phase on Dynamics of Complex-Forming Reactions: H + O2 → OH + O

Reaction dynamics on the ground electronic state might be significantly influenced by conical intersections (CIs) via the geometric phase (GP), as demonstrated for activated reactions (i.e., the H + H2 exchange reaction). However, there have been few investigations of GP effects in complex-forming reactions. Here, we report a full quantum dynamical study of an important reaction in combustion (H + O2 → OH + O), which serves as a proving ground for studying GP effects therein. The results reveal significant differences in reaction probabilities and differential cross sections (DCSs) obtained with and without GP, underscoring its strong impact. However, the GP effects are less pronounced for the reaction integral cross sections, apparently due to the integral of the DCS over the scattering angle. Further analysis indicated that the cross section has roughly the same contributions from the two topologically distinct paths around the CI, namely, the direct and looping paths.

Exploring Exact-Factorization-Based Trajectories for Low-Energy Dynamics near a Conical Intersection

We study low-energy dynamics generated by a two-dimensional two-state Jahn-Teller Hamiltonian in the vicinity of a conical intersection using quantum wave packet and trajectory dynamics. Recently, these dynamics were studied by comparing the adiabatic representation and the exact factorization, with the purpose to highlight the different nature of topological-phase and geometric-phase effects arising in the two theoretical representations of the same problem. Here, we employ the exact factorization to understand how to accurately model low-energy dynamics in the vicinity of a conical intersection using an approximate description of the nuclear motion that uses trajectories. We find that since nonadiabatic effects are weak but non-negligible, the trajectory-based description that invokes the classical approximation struggles to capture the correct behavior.

Diagonalizing the Born-Oppenheimer Hamiltonian via Moyal perturbation theory, nonadiabatic corrections, and translational degrees of freedom

This article describes a method for calculating higher order or nonadiabatic corrections in Born-Oppenheimer theory and its interaction with the translational degrees of freedom. The method uses the Wigner-Weyl correspondence to map nuclear operators into functions on the classical phase space and the Moyal star product to represent operator multiplication on those functions. These are explained in the body of the paper. The result is a power series in κ2, where κ = (m/M)1/4 is the usual Born-Oppenheimer parameter. The lowest order term is the usual Born-Oppenheimer approximation, while higher order terms are nonadiabatic corrections. These are needed in calculations of electronic currents, momenta, and densities. The separation of nuclear and electronic degrees of freedom takes place in the context of the exact symmetries (for an isolated molecule) of translations and rotations, and these, especially translations, are explicitly incorporated into our discussion. This article presents an independent derivation of the Moyal expansion in molecular Born-Oppenheimer theory. We show how electronic currents and momenta can be calculated within the framework of Moyal perturbation theory; we derive the transformation laws of the electronic Hamiltonian, the electronic eigenstates, and the derivative couplings under translations; we discuss in detail the rectilinear motion of the molecular center of mass in the Born-Oppenheimer representation; and we show how the elimination of the translational components of the derivative couplings leads to a unitary transformation that has the effect of exactly separating the translational degrees of freedom.

Observation of geometric phase effect through backward angular oscillations in the H + HD → H2 + D reaction

Quantum interference between reaction pathways around a conical intersection (CI) is an ultrasensitive probe of detailed chemical reaction dynamics. Yet, for the hydrogen exchange reaction, the difference between contributions of the two reaction pathways increases substantially as the energy decreases, making the experimental observation of interference features at low energy exceedingly challenging. We report in this paper a combined experimental and theoretical study on the H + HD → H2 + D reaction at the collision energy of 1.72 eV. Although the roaming insertion pathway constitutes only a small fraction (0.088%) of the overall contribution, angular oscillatory patterns arising from the interference of reaction pathways were clearly observed in the backward scattering direction, providing direct evidence of the geometric phase effect at an energy of 0.81 eV below the CI. Furthermore, theoretical analysis reveals that the backward interference patterns are mainly contributed by two distinct groups of partial waves (J ~ 10 and J ~ 19). The well-separated partial waves and the geometric phase collectively influence the quantum reaction dynamics.

On the Nature of Geometric and Topological Phases in the Presence of Conical Intersections

The observable nature of topological phases related to conical intersections in molecules is studied. Topological phases should be ubiquitous in molecular processes, but their elusive character has often made them a topic of discussion. To shed some light on this issue, we simulate the dynamics governed by a Jahn-Teller Hamiltonian and analyze it employing two theoretical representations of the molecular wave function: the adiabatic and the exact factorization. We find fundamental differences between effects related to topological phases arising exclusively in the adiabatic representation, and thus not related to any physical observable, and geometric phases within the exact factorization that can be connected to an observable quantity. We stress that while the topological phase of the adiabatic representation is an intrinsic property of the Hamiltonian, the geometric phase of the exact factorization depends on the dynamics that the system undergoes and is connected to the circulation of the nuclear momentum field.

Geometric Phase Effects in the H + H2+ Reaction on the Lowest Triplet Ground State

To our knowledge, this is the first time geometric phase (GP) effects in the H + H2+ reaction on its lowest triplet ground state with collision energies lower than 1.85 eV have been studied using the quantum wave packet method and vector potential approach. We obtained the total reaction probabilities including and not including GP (NGP) effects for J ≤ 4. Visible GP effects could be seen at the lower energy regime but are tiny at the higher one. Moreover, they are more obvious in the product rovibrational state-resolved reaction probabilities, and the relative resonance magnitudes between GP and NGP results change with product rotational state values alternatively. The main reasons are the interferences between the one- and two-transition-state (1-TS and 2-TS) reaction paths, in that at the lower energy regime, the reaction probabilities from the 2-TS pathway show peaks of comparable probabilities compared with that of the 1-TS pathway. In addition, the "out-of-phase" trend observed in the H + H2 reaction does not exist rigorously in this system. Importantly, the visible GP effects exist in this H3+ system, which makes it a very useful candidate reaction for nonadiabatic investigations in both theory and experiment.

Electronic nonadiabatic effects in the state-to-state dynamics of the H + H2 → H2 + H exchange reaction with a vibrationally excited reagent

Out of the many major breakthroughs that the hydrogen-exchange reaction has led to, electronic nonadiabatic effects that are mainly due to the geometric phase has intrigued many. In this work we investigate such effects in the state-to-state dynamics of the H + H2 (v = 3, 4, j = 0) → H2 (v', j') + H reaction with a vibrationally excited reagent at energies corresponding to thermal conditions. The dynamical calculations are performed by a time-dependent quantum mechanical method both on the lower adiabatic potential energy surface (PES) and also using a two-states coupled diabatic theoretical model to explicitly include all the nonadiabatic couplings present in the 1E' ground electronic manifold of the H3 system. The nonadiabatic couplings are considered here up to the quadratic term; however, the effect of the latter on the reaction dynamics is found to be very small. Adiabatic population analysis showed a minimal participation of the upper adiabatic surface even for the vibrationally excited reagent. A strong nonadiabatic effect appears in the state-to-state reaction probabilities and differential cross sections (DCSs). This effect is manifested as "out-of-phase" oscillations in the DCSs between the results of the uncoupled and coupled surface situations. The oscillations persist as a function of both scattering angle and collision energy in both the backward and forward scattering regions. The origins of these oscillations are examined in detail. The oscillations that appear in the forward direction are found to be different from those due to glory scattering, where the latter showed a negligibly small nonadiabatic effect. The nonadiabatic effects are reduced to a large extent when summed over all product quantum states, in addition to the cancellation due to integration over the scattering angle and partial wave summation.

Competing quantum effects in heavy-atom tunnelling through conical intersections

Thermally activated chemical reactions are typically understood in terms of overcoming potential-energy barriers. However, standard rate theories break down in the presence of a conical intersection (CI) because these processes are inherently nonadiabatic, invalidating the Born-Oppenheimer approximation. Moreover, CIs give rise to intricate nuclear quantum effects such as tunnelling and the geometric phase, which are neglected by standard trajectory-based simulations and remain largely unexplored in complex molecular systems. We present new semiclassical transition-state theories based on an extension of golden-rule instanton theory to describe nonadiabatic tunnelling through CIs and thus provide an intuitive picture for the reaction mechanism. We apply the method in conjunction with first-principles electronic-structure calculations to the electron transfer in the bis(methylene)-adamantyl cation. Our study reveals a strong competition between heavy-atom tunnelling and geometric-phase effects.

Topological Metamaterials

The topological properties of an object, associated with an integer called the topological invariant, are global features that cannot change continuously but only through abrupt variations, hence granting them intrinsic robustness. Engineered metamaterials (MMs) can be tailored to support highly nontrivial topological properties of their band structure, relative to their electronic, electromagnetic, acoustic and mechanical response, representing one of the major breakthroughs in physics over the past decade. Here, we review the foundations and the latest advances of topological photonic and phononic MMs, whose nontrivial wave interactions have become of great interest to a broad range of science disciplines, such as classical and quantum chemistry. We first introduce the basic concepts, including the notion of topological charge and geometric phase. We then discuss the topology of natural electronic materials, before reviewing their photonic/phononic topological MM analogues, including 2D topological MMs with and without time-reversal symmetry, Floquet topological insulators, 3D, higher-order, non-Hermitian and nonlinear topological MMs. We also discuss the topological aspects of scattering anomalies, chemical reactions and polaritons. This work aims at connecting the recent advances of topological concepts throughout a broad range of scientific areas and it highlights opportunities offered by topological MMs for the chemistry community and beyond.

Geometric Phase Effect in Attosecond Stimulated X-ray Raman Spectroscopy

Conical intersections (CIs) are diabolical points in the potential energy surfaces generally caused by point-wise degeneracy of different electronic states, and give rise to the geometric phases (GPs) of molecular wave functions. Here we theoretically propose and demonstrate that the transient redistribution of ultrafast electronic coherence in attosecond Raman signal (TRUECARS) spectroscopy is capable of detecting the GP effect in excited state molecules by applying two probe pulses including an attosecond and a femtosecond X-ray pulse. The mechanism is based on a set of symmetry selection rules in the presence of nontrivial GPs. The model of this work can be realized for probing the geometric phase effect in the excited state dynamics of complex molecules with appropriate symmetries, using attosecond light sources such as free-electron X-ray lasers.

Representation and conservation of angular momentum in the Born-Oppenheimer theory of polyatomic molecules

This paper concerns the representation of angular momentum operators in the Born-Oppenheimer theory of polyatomic molecules and the various forms of the associated conservation laws. Topics addressed include the question of whether these conservation laws are exactly equivalent or only to some order of the Born-Oppenheimer parameter κ = (m/M)^1/4 and what the correlation is between angular momentum quantum numbers in the various representations. These questions are addressed in both problems involving a single potential energy surface and those with multiple, strongly coupled surfaces and in both the electrostatic model and those for which fine structure and electron spin are important. The analysis leads to an examination of the transformation laws under rotations of the electronic Hamiltonian; of the basis states, both adiabatic and diabatic, along with their phase conventions; of the potential energy matrix; and of the derivative couplings. These transformation laws are placed in the geometrical context of the structures in the nuclear configuration space that are induced by rotations, which include the rotational orbits or fibers, the surfaces upon which the orientation of the molecule changes but not its shape, and the section, an initial value surface that cuts transversally through the fibers. Finally, it is suggested that the usual Born-Oppenheimer approximation can be replaced by a dressing transformation, that is, a sequence of unitary transformations that block-diagonalize the Hamiltonian. When the dressing transformation is carried out, we find that the angular momentum operator does not change. This is a part of a system of exact equivalences among various representations of angular momentum operators in Born-Oppenheimer theory. Our analysis accommodates large-amplitude motions and is not dependent on small-amplitude expansions about an equilibrium position. Our analysis applies to noncollinear configurations of a polyatomic molecule; this covers all but a subset of measure zero (the collinear configurations) in the nuclear configuration space.

The states that hide in the shadows: the potential role of conical intersections in the ground state unimolecular decay of a Criegee intermediate

Criegee intermediates are of great significance to Earth's troposphere - implicated in altering the tropospheric oxidation cycle and in forming low volatility products that typically condense to form secondary organic aerosols (SOAs). As such, their chemistry has attracted vast attention in recent years. In particular, the unimolecular decay of thermal and vibrationally-excited Criegee intermediates has been the focus of several experimental and computational studies, and it is now recognized that Criegee intermediates undergo unimolecular decay to form OH radicals. In this contribution we reveal insight into the chemistry of Criegee intermediates by highlighting the hitherto neglected multi-state contribution to the ground state unimolecular decay dynamics of the Criegee intermediate products. The two key intermediates of present focus are dioxirane and vinylhydroperoxide - known to be active intermediates that mediate the unimolecular decay of CH₂OO and CH₃CHOO, respectively. In both cases the unimolecular decay path encounters conical intersections, which may play a pivotal role in the ensuing dynamics. This hitherto unrecognized phenomenon may be vital in the way in which the reactivity of Criegee intermediates are modelled and is likely to affect the ensuing dynamics associated with the unimolecular decay of a given Criegee intermediate.

Quantum Wave Packet Treatment of Cold Nonadiabatic Reactive Scattering at the State-To-State Level

Cold and ultracold collisions are dominated by quantum effects, such as resonances, tunneling, and nonadiabatic transitions between different electronic states. Due to the extremely long de Broglie wavelength in such processes, quantum reactive scattering is most conveniently characterized using the time-independent close-coupling (TICC) methods. However, the TICC approach is difficult for systems with a large number of channels because of its steep numerical scaling laws. Here, a recently proposed quantum wave packet (WP) approach for solving adiabatic reactive scattering problems at low collision energies is extended to include nonadiabatic transitions. To impose the outgoing boundary conditions, the total scattering wavefunction is split into three parts, the interaction, the asymptotic, and the long-range regions. Each region is associated with a different set of basis functions, which could be optimized separately. In this way, an extremely long grid can be used to accommodate the characteristic long de Broglie wavelengths in the scattering coordinate. The better numerical scaling laws of the WP approach have the potential for handling larger nonadiabatic reactive systems at low temperatures in the future.

Full-dimensional quantum stereodynamics of the non-adiabatic quenching of OH(A2Σ+) by H2

The Born–Oppenheimer approximation, assuming separable nuclear and electronic motion, is widely adopted for characterizing chemical reactions in a single electronic state. However, the breakdown of the Born–Oppenheimer approximation is omnipresent in chemistry, and a detailed understanding of the non-adiabatic dynamics is still incomplete. Here we investigate the non-adiabatic quenching of electronically excited OH(A²Σ⁺) molecules by H₂ molecules using full-dimensional quantum dynamics calculations for zero total nuclear angular momentum using a high-quality diabatic-potential-energy matrix. Good agreement with experimental observations is found for the OH(X²Π) ro-vibrational distribution, and the non-adiabatic dynamics are shown to be controlled by stereodynamics, namely the relative orientation of the two reactants. The uncovering of a major (in)elastic channel, neglected in a previous analysis but confirmed by a recent experiment, resolves a long-standing experiment–theory disagreement concerning the branching ratio of the two electronic quenching channels.

The breakdown of the Born–Oppenheimer approximation is omnipresent in chemistry and detailed understanding of non-adiabatic dynamics is still incomplete. Now, the non-adiabatic quenching of electronically excited OH(A²Σ⁺) molecules by H₂ has been investigated using full-dimensional quantum dynamics calculations and a high-quality diabatic-potential-energy matrix, providing insight into the branching ratio of the two electronic quenching channels.

Dynamics and kinetics of the Si(1D) + H2/D2 reactions on a new global ab initio potential energy surface

Recent studies on the exothermic complex-forming reactions have improved our understanding of complex-forming reactions greatly, however, so far a similar level of study on endothermic ones has been rather limited. In this work, the endothermic complex-forming reaction Si(1D) + H2 → SiH + H and its deuterated isotopic variant are investigated by quantum dynamics (QD) and ring polymer molecular dynamics (RPMD) calculations on a new global ab initio potential energy surface (PES) for the ground electronic state, which is constructed based on 8996 symmetry unique points computed at the icMRCI+Q/aug-cc-pVQZ level. The PES reproduces our ab initio data very well in the dynamically important regions, on which the ro-vibrational energy levels of SiH2 are calculated and general good agreement with experiment is obtained. The integral cross sections and product angular and state distributions are computed in a wide range of collision energies, and interesting dynamics behaviors are revealed. The rate coefficients are also investigated, which display an exponential rise from 2.09 × 10-20 to 6.00 × 10-12 cm3 s-1 for the Si(1D) + H2 reaction as the temperature increases from 300 to 1500 K, in contrast to the nearly temperature-independent behavior of exothermic complex-forming reactions. In addition, the applicability of the RPMD approach is demonstrated, and the kinetic isotope effect is investigated, the ratio of which decreases from 7.89 (300 K) to 1.70 (1500 K). The effects of tunneling and initial rotational excitation are also discussed.

Non-adiabatic quantum interference in the ultracold Li + LiNa → Li2 + Na reaction

Electronically non-adiabatic effects play an important role in many chemical reactions. However, how these effects manifest in cold and ultracold chemistry remains largely unexplored. Here for the first time we present from first principles the non-adiabatic quantum dynamics of the reactive scattering between ultracold alkali-metal LiNa molecules and Li atoms. We show that non-adiabatic dynamics induces quantum interference effects that dramatically alter the ultracold rotationally resolved reaction rate coefficients. The interference effect arises from the conical intersection between the ground and an excited electronic state that is energetically accessible even for ultracold collisions. These unique interference effects might be exploited for quantum control applications such as a quantum molecular switch. The non-adiabatic dynamics are based on full-dimensional ab initio potential energy surfaces for the two electronic states that includes the non-adiabatic couplings and an accurate treatment of the long-range interactions. A statistical analysis of rotational populations of the Li2 product reveals a Poisson distribution implying the underlying classical dynamics are chaotic. The Poisson distribution is robust and amenable to experimental verification and appears to be a universal property of ultracold reactions involving alkali metal dimers.

Mechanistic Insights into Ultracold Chemical Reactions under the Control of the Geometric Phase

Ultracold chemical reactions involve collision temperatures approaching absolute zero, and for molecular systems that exhibit a barrierless and exoergic reaction path significant reactivity can occur. In addition, many molecules contain a conical intersection, and the associated geometric phase has been shown to significantly alter the outcome of ultracold reactions. Here we report a quantum dynamics study for the ultracold O + OH → H + O₂ reaction. An analysis of the scattering wave functions reveals explicitly the nature of the quantum interference between the direct and looping reaction pathways around the conical intersection and thus illustrates how the reaction proceeds under the control of the geometric phase for the first time. The wave function analysis should generalize to other ultracold reactions that contain a conical intersection. Our findings indicate that quantum control techniques such as an optical lattice trap or the initial state orientation may be effective in controlling the reactivity.

Electronic spin separation induced by nuclear motion near conical intersections

Though the concept of Berry force was proposed thirty years ago, little is known about the practical consequences of this force as far as chemical dynamics are concerned. Here, we report that when molecular dynamics pass near a conical intersection, a massive Berry force can appear as a result of even a small amount of spin-orbit coupling (<10⁻³ eV), and this Berry force can in turn dramatically change pathway selection. In particular, for a simple radical reaction with two outgoing reaction channels, an exact quantum scattering solution in two dimensions shows that the presence of a significant Berry force can sometimes lead to spin selectivity as large as 100%. Thus, this article opens the door for organic chemists to start designing spintronic devices that use nuclear motion and conical intersections (combined with standard spin-orbit coupling) in order to achieve spin selection. Vice versa, for physical chemists, this article also emphasizes that future semiclassical simulations of intersystem crossing (which have heretofore ignored Berry force) should be corrected to account for the spin polarization that inevitably arises when dynamics pass near conical intersections.

Spin polarization is at the basis of quantum information and underlies some natural processes, but many aspects still need to be explored. Here, the authors, by quantum mechanical computations, show that even a weak spin-orbit coupling near a conical intersection can induce large spin selection, with consequences for spin manipulation in photochemical or electrochemical reactions.

Effect of Reagent Vibration and Rotation on the State-to-State Dynamics of the Hydrogen Exchange Reaction, H + H2 → H2 + H

State-to-state dynamics of the benchmark hydrogen exchange reaction H + H₂ (v = 0-4, j = 0-3) → H₂ (v', j') + H is investigated with the aid of the real wave packet approach of Gray and Balint-Kurti (J. Chem. Phys. 1998, 108, 950-962) and electronic ground BKMP2 potential energy surface of Boothroyd et al. (J. Chem. Phys. 1996, 104, 7139-7152). Initial state-selected and product state-resolved reaction probabilities, integral cross section, and product diatom vibrational and rotational level populations at a few collision energies are reported to elucidate the energy disposal mechanism. State-specific thermal rate constants are also calculated and compared with the available literature results. Coriolis coupling terms of the nuclear Hamiltonian are included, and calculations are parallelized over the helicity quantum number, Ω'. Attempts are made, in particular, to study the effect of reagent vibrational and rotational excitations on the dynamical attributes. It is found that the calculations become computationally expensive with reagent vibrational and rotational excitation. Reagent vibrational excitation is found to enhance the reactivity and has significant impact on the energy disposal to the vibrational and rotational degrees of freedom of the product. The interplay of reagent translational and vibrational energy on the product vibrational distribution unfolds an important aspect of the energy disposal mechanism. The effect of reagent rotation on the state-to-state dynamics is found not to be very significant, and the weak effect turns out to be specific to v'.

Advances and New Challenges to Bimolecular Reaction Dynamics Theory

Dynamics of bimolecular reactions in the gas phase are of foundational importance in combustion, atmospheric chemistry, interstellar chemistry, and plasma chemistry. These collision-induced chemical transformations are a sensitive probe of the underlying potential energy surface(s). Despite tremendous progress in past decades, our understanding is still not complete. In this Perspective, we survey the recent advances in theoretical characterization of bimolecular reaction dynamics, stimulated by new experimental observations, and identify key new challenges.

Theoretical investigation of the H + HD → D + H2 chemical reaction for astrophysical applications: A state-to-state quasi-classical study

We report a large set of state-to-state rate constants for the H + HD reactive collision, using Quasi-Classical Trajectory (QCT) simulations on the accurate H₃ global potential energy surface of Mielke et al. [J. Chem. Phys. 116, 4142 (2002)]. High relative collision energies (up to ≈56 000 K) and high rovibrational levels of HD (up to ≈50 000 K), relevant to various non thermal equilibrium astrophysical media, are considered. We have validated the accuracy of our QCT calculations with a new efficient adaptation of the Multi Configuration Time Dependent Hartree (MCTDH) method to compute the reaction probability of a specific reactive channel. Our study has revealed that the high temperature regime favors the production of H₂ in its highly rovibrationnally excited states, which can de-excite radiatively (cooling the gas) or collisionally (heating the gas). Those new state-to-state QCT reaction rate constants represent a significant improvement in our understanding of the possible mechanisms leading to the destruction of HD by its collision with a H atom.

Observation of the geometric phase effect in the H+HD→H2+D reaction below the conical intersection

It has long been known that there is a conical intersection (CI) between the ground and first excited electronic state in the H₃ system. Its associated geometric phase (GP) effect has been theoretically predicted to exist below the CI since a long time. However, the experimental evidence has not been established yet and its dynamical origin is waiting to be elucidated. Here we report a combined crossed molecular beam and quantum reactive scattering dynamics study of the H+HD → H₂+D reaction at 2.28 eV, which is well below the CI. The GP effect is clearly identified by the observation of distinct oscillations in the differential cross section around the forward direction. Quantum dynamics theory reveals that the GP effect arises from the phase alteration of a small part of the wave function, which corresponds to an unusual roaming-like abstraction pathway, as revealed by quasi-classical trajectory calculations.

The geometric phase effect associated with a conical intersection between the ground and first excited electronic state has been predicted in the H₃ system below the conical intersection energy. The authors, by a crossed molecular beam technique and quantum dynamic calculations, provide experimental evidence and insight into its origin.

How interference reveals geometric phase

Quantum phase effects are probed at energies below the H + HD reaction conical intersection

Quantum interference in H + HD → H2 + D between direct abstraction and roaming insertion pathways

Understanding quantum interferences is essential to the study of chemical reaction dynamics. Here, we provide an interesting case of quantum interference between two topologically distinct pathways in the H + HD → H2 + D reaction in the collision energy range between 1.94 and 2.21 eV, manifested as oscillations in the energy dependence of the differential cross section for the H2 (v′ = 2, j′ = 3) product (where v′ is the vibrational quantum number and j′ is the rotational quantum number) in the backward scattering direction. The notable oscillation patterns observed are attributed to the strong quantum interference between the direct abstraction pathway and an unusual roaming insertion pathway. More interestingly, the observed interference pattern also provides a sensitive probe of the geometric phase effect at an energy far below the conical intersection in this reaction, which resembles the Aharonov–Bohm effect in physics, clearly demonstrating the quantum nature of chemical reactivity.

Insights into the Mechanism of Nonadiabatic Photodissociation from Product Vibrational Distributions. The Remarkable Case of Phenol

The fate of a photoexcited molecule is often strongly influenced by electronic degeneracies, such as conical intersections, which break the Born-Oppenheimer separation of electronic and nuclear motion. Detailed information concerning internal energy redistribution in a nonadiabatic process can be extracted from the product state distribution of a photofragment in photodissociation. Here, we focus on the nonadiabatic photodissociation of phenol and discuss the internal excitation of the phenoxyl fragment using both symmetry analysis and wave packet dynamics. It is shown that unique and general selection rules exist, which can be attributed to the geometric phase in the adiabatic representation. Further, our results provide a reinterpretation of the experimental data, shedding light on the impact of conical intersections on the product state distribution.

Nonadiabatic Ultracold Quantum Reactive Scattering of Hydrogen with Vibrationally Excited HD(v = 5-9)

The results from electronically non-adiabatic and adiabatic quantum reactive scattering calculations are presented for the H + HD(v = 5-9) → H + HD(v', j') reaction at ultracold collision energies from 10 nK to 60 K. Several experimentally verifiable signatures of the geometric phase are reported in the total and vibrationally and rotationally resolved rate coefficients. Most notable is the predicted 2 orders of magnitude enhancement of the rotationally resolved ultracold rates of odd symmetry relative to those of even symmetry. Prominent shape resonances appear at higher collision energies (100 mK to 20 K), which could be measured experimentally. Significant geometric phase effects are also reported on the resonance energies and lifetimes. In particular, an enhancement (suppression) of the l = 1 (l = 2) shape resonances for HD(v = 5, 6) is predicted for even symmetry relative to those of odd symmetry.

Geometric Phase Effects in Ultracold Chemical Reactions

The role of the geometric phase effect in chemical reaction dynamics has long been a topic of active experimental and theoretical investigations. The topic has received renewed interest in recent years in cold and ultracold chemistry where it was shown to play a decisive role in state-to-state chemical dynamics. We provide a brief review of these developments focusing on recent studies of O + OH and hydrogen exchange in the H + H 2 and D + HD reactions at cold and ultracold temperatures. Non-adiabatic effects in ultracold chemical dynamics arising from the conical intersection between two electronic potential energy surfaces are also briefly discussed. By taking the hydrogen exchange reaction as an illustrative example it is shown that the inclusion of the geometric phase effect captures the essential features of non-adiabatic dynamics at collision energies below the conical intersection.

Up to a Sign. The Insidious Effects of Energetically Inaccessible Conical Intersections on Unimolecular Reactions

It is now well established that conical intersections play an essential role in nonadiabatic radiationless decay where their double-cone topography causes them to act as efficient funnels channeling wave packets from the upper to the lower adiabatic state. Until recently, little attention was paid to the effect of conical intersections on dynamics on the lower state, particularly when the total energy involved is significantly below that of the conical intersection seam. This energetic deficiency is routinely used as a sufficient condition to exclude consideration of excited states in ground state dynamics. In this account, we show that, this energy criterion notwithstanding, energy inaccessible conical intersections can and do exert significant influence on lower state dynamics. The origin of this influence is the geometric phase, a signature property of conical intersections, which is the fact that the real-valued electronic wave function changes sign when transported along a loop containing a conical intersection, making the wave function double-valued. This geometric phase is permitted by an often neglected property of the real-valued adiabatic electronic wave function; namely, it is determined only up to an overall sign. Noting that in order to change sign a normalized, continuous function must go through zero, for loops of ever decreasing radii, demonstrating the need for an electronic degeneracy (intersection) to accompany the geometric phase. Since the total wave function must be single-valued a compensating geometry dependent phase needs to be included in the total electronic-nuclear wave function. This Account focuses on how this consequence of the geometric phase can modify nuclear dynamics energetically restricted to the lower state, including tunneling dynamics, in directly measurable ways, including significantly altering tunneling lifetimes, thus confounding the relation between measured lifetimes and barrier heights and widths, and/or completely changing product rotational distributions. Some progress has been made in understanding the origin of this effect. It has emerged that for a system where the lower adiabatic potential energy surface exhibits a topography comprised of two saddle points separated by a high energy conical intersection, the effect of the geometric phase can be quite significant. In this case topologically distinct paths through the two adiabatic saddle points may lead to interference. This was pointed out by Mead and Truhlar almost 50 years ago and denoted the Molecular Aharonov-Bohm effect. Still, the difficulty in anticipating a significant geometric phase effect in tunneling dynamics due to energetically inaccessible conical intersections leads to the attribute insidious that appears in the title of this Account. Since any theory is only as relevant as the prevalence of the systems it describes, we include in this Account examples of real systems where these effects can be observed. The accuracy of the reviewed calculations is high since we use fully quantum mechanical dynamics and construct the geometric phase using an accurate diabatic state fit of high quality ab initio data, energies, energy gradients, and interstate couplings. It remains for future work to establish the prevalence of this phenomenon and its deleterious effects on the conventional wisdom discussed in this work.

Geometric phase effects on photodissociation dynamics of diatomics

We investigated the effect of the geometric phase (GP) on photodissociation dynamics at a light-induced conical intersection (LICI) through exact quantum dynamical calculations. By taking the one-photon photodissociation of H 2 + ionic molecules as an example, we explored the conditions wherein the LICI associated GP affects dissociation dynamics. We found that GP leads to a phase shift between the angular distributions of GP included and GP excluded photofragments. This effect is more pronounced when the energy of the initial vibrational level is above the energy of the LICI point.

Observation of the geometric phase effect in the H + HD → H2+ D reaction

Theory has established the importance of geometric phase (GP) effects in the adiabatic dynamics of molecular systems with a conical intersection connecting the ground- and excited-state potential energy surfaces, but direct observation of their manifestation in chemical reactions remains a major challenge. Here, we report a high-resolution crossed molecular beams study of the H + HD → H2+ D reaction at a collision energy slightly above the conical intersection. Velocity map ion imaging revealed fast angular oscillations in product quantum state–resolved differential cross sections in the forward scattering direction for H2products at specific rovibrational levels. The experimental results agree with adiabatic quantum dynamical calculations only when the GP effect is included.

Several levels of theory for description of isotope effects in ozone: Effect of resonance lifetimes and channel couplings

In this paper, two levels of theory are developed to determine the role of scattering resonances in the process of ozone formation. At the lower theory level, we compute resonance lifetimes in the simplest possible way, by neglecting all couplings between the diabatic vibrational channels in the problem. This permits to determine the effect of "shape" resonances, trapped behind the centrifugal barrier and populated by quantum tunneling. At the next level of theory, we include couplings between the vibrational channels, which permits to determine the role of Feshbach resonances and interaction of different reaction pathways on the global PES of ozone. Pure shape resonances are found to contribute little to the overall recombination process since they occur rather infrequently in the spectrum, in the vicinity of the top of the centrifugal barrier only. Moreover, the associated isotope effects are found to disagree with experimental data. By contrast, Feshbach-type resonances are found to make dominant contribution to the process. They occur in a broader range of spectrum, and their density of states is much higher. The properties of Feshbach resonances are studied in detail. They explain the isotopic ζ -effect, giving theoretical prediction in good agreement with experiments for both singly and doubly substituted ozone molecules. Importantly, Feshbach resonances also contribute to the isotopic η -effect, moving theoretical predictions in the right direction. Some differences with experimental data remain, which indicates that there may be another additional source of the η -effect.

Unambiguous Signature of the Berry Phase in Intense Laser Dissociation of Diatomic Molecules

We report strong evidence of Berry phase effects in intense laser dissociation of D₂⁺ molecules, manifested as Aharonov-Bohm-like oscillations in the photofragment angular distribution (PAD). Our calculations show that this interference pattern strongly depends on the parity of the diatom initial rotational state, (-1) ^j. Indeed, the PAD local maxima (minima) observed in one case ( j odd) correspond to local minima (maxima) in the other case ( j even). Using simple topological arguments, we clearly show that such interference conversion is a direct signature of the Berry phase. The sole effect of the latter on the rovibrational wave function is a sign change of the relative phase between two interfering components, which wind in opposite senses around a light-induced conical intersection (LICI). Therefore, encirclement of the LICI leads to constructive ( j odd) or destructive ( j even) self-interference of the initial nuclear wavepacket in the dissociative limit. To corroborate our theoretical findings, we suggest an experiment of strong-field indirect dissociation of D₂⁺ molecules, comparing the PAD of the ortho and para molecular species in directions nearly perpendicular to the laser polarization axis.

Signatures of a Conical Intersection in Adiabatic Dissociation on the Ground Electronic State

Conical intersections are known to cause nonadiabatic transitions, but their effects on adiabatic dynamics are often ignored. Using the overtone-induced dissociation of the hydroxymethyl radical as an example, we demonstrate that ground-state O-H bond rupture is significantly affected by a conical intersection with an electronically excited state along the dissociation path, despite the much lower energy of the dissociating state than that of the conical intersection. In addition to lifetime differences, the geometric phase leads to a different H₂CO rotational state distribution compared with that obtained using the standard single-state adiabatic model, which constitutes a signature of the conical intersection.

Symmetry and the geometric phase in ultracold hydrogen-exchange reactions

Quantum reactive scattering calculations are reported for the ultracold hydrogen-exchange reaction and its non-reactive atom-exchange isotopic counterparts, proceeding from excited rotational states. It is shown that while the geometric phase (GP) does not necessarily control the reaction to all final states, one can always find final states where it does. For the isotopic counterpart reactions, these states can be used to make a measurement of the GP effect by separately measuring the even and odd symmetry contributions, which experimentally requires nuclear-spin final-state resolution. This follows from symmetry considerations that make the even and odd identical-particle exchange symmetry wavefunctions which include the GP locally equivalent to the opposite symmetry wavefunctions which do not. It is shown how this equivalence can be used to define a constant which quantifies the GP effect and can be obtained solely from experimentally observable rates. This equivalence reflects the important role that discrete symmetries play in ultracold chemistry and highlights the key role that ultracold reactions can play in understanding fundamental aspects of chemical reactivity more generally.