Volume 227, Issue 11 - Milestones in Physical Chemistry: The Potential Energy Surface, from Eyring and Polanyi to the present day

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This article has been published in German language in Zeitschrift für Physikalische Chemie, Abteilung B, Band 12, S. 279–311, (1931).

Probing reaction potential energy surfaces is crucial for understanding chemical reaction dynamics and kinetics. The fundamental importance of the potential energy surface in understanding chemical reactivity was pointed out in the pioneering paper by Eyring and Polanyi in 1931 [1]. Accurate potential energy surfaces, however, are difficult to obtain. In the last few decades, more accurate quantum chemistry methods to calculate potential energy surfaces became available. In addition, quantum dynamics methods were also developed and allow us to study simple chemical reactions from first principles. Furthermore, modern crossed molecular beams studies provide an excellent testing ground for developing an accurate physical picture of elementary chemical reactions, through interplay between theory and experiment. In this paper, we review recent developments in the study of reaction dynamics of elementary chemical reactions through combined experimental and theoretical efforts. We will provide two examples to demonstrate the importance of accurate potential energy surfaces in understanding the essentials of reaction dynamics. One is the simplest chemical reaction: the H + H 2 reaction, the other is the benchmark system for chemical reaction resonance: the F + H 2 reaction. Through these studies, dynamics of chemical reactions can be understood at the most fundamental level.

Characteristic signatures of roaming are seen in the translational energy distribution and rotational energy distributions in the products CH 3 O + NO from the unimolecular dissociation of nitromethane in both theory and experiment. Calculations on a new global potential energy surface reveal the detailed roaming-isomerization dynamics in this process, and these are supported by the experimental observations. Calculations find a characteristic of roaming, namely production of vibrationally excited CH 3 O, which is consistent with experiment. This continues to challenge the conventional idea in transition state theory of a localized transition state bottleneck that is a cornerstone of chemical reaction theory.

The H + HD → HD( ν ′,  j ′) + H reaction has been studied experimentally and theoretically. Differential cross sections of HD( ν ′,  j ′) products have been measured by means of a Photoloc technique and calculated using a time-independent quantum mechanical theory. Three product states: HD( ν ′ = 1,  j ′ = 8) at a collision energy ( E coll ) of 1.97 eV; HD( ν ′ = 2,  j ′ = 3) at E coll  = 1.46 eV; and HD( ν ′ = 2,  j ′ = 5) at E coll  = 1.44 eV, show very good agreement between theory and experiment. The other two, highly rotationally excited states studied, HD( ν ′ = 1,  j ′ = 12) and HD( ν ′ = 1,  j ′ = 13) at E coll  = 1.97 eV, exhibit a noticeable disagreement between experiment and theory. This is consistent with our most recent findings on the H + D 2  → HD( ν ′,  j ′) + D reaction, wherein the differential cross sections of HD( ν ′ = 1, high  j ′) product states showed similar disagreement between the experiment and theory [J. Jankunas, M. Sneha, R. N. Zare, F. Bouakline, and S. C. Althorpe, J. Chem. Phys. 138 (2013) 094310]. In all five cases, however, we find overwhelming support that the experimental signal is a sum of reactive and inelastic scattering events. The interference term escapes detection, frustrating our attempt to observe geometric phase effects. Nevertheless, this work constitutes a first experimental example in which the indistinguishability of reactive and inelastic channels must be taken into account explicitly when constructing differential cross sections.

Hydrazine is an important rocket fuel, used as both a monopropellant and a bipropellant. This paper presents theoretical results to complement the extensive experimental studies of the gas phase and Ir catalyzed decompositions involved in the monopropellant applications of hydrazine. Gas phase electronic structure theory calculations that include electron correlation predict that numerous molecular and free radical reactions occur within the same energy range as the basic free radical pathways: NN bond breaking around 65 kcal/mol and NH bond breaking around 81 kcal/mol. The data suggest that a revision to existing kinetics modeling is desirable, based on the energetics and the new elementary steps reported herein. A supported Ir 6 octahedron model for the Shell 405 Iridium catalyst used in thrusters was developed. Self-Consistent Field and electron correlation calculations (with core potentials and associated basis sets) find a rich chemistry for hydrazine on this catalyst model. The model catalyst provides dramatically lower NN and NH bond cleavage energies and an even smaller barrier to breaking the NH bond by NH 2 abstractions. Thus, the low temperature decomposition over the catalyst is interpreted in terms of consecutive NH 2 abstractions to produce ammonia and nitrogen. The higher temperature channel, which has hydrogen and nitrogen products, may be due to a mixture of two mechanisms. These two mechanisms are successive NH cleavages with surface H + H recombinations, and the same type of assisted H 2 eliminations found to occur in the gas phase part of this study.

We report the calculation of the H-, Br-, and I-abstraction channels in the reaction of OH radicals with bromoiodomethane CH 2 IBr. The resulting energy profiles at 0 K were obtained by high-level all-electron ab initio methods including valence and core-valence electron correlation, scalar relativistic effects, spin-orbit coupling, spin-adaptation, vibration contributions, and tunneling corrections. In terms of activation enthalpy at 0 K, the energy profile for the Br-abstraction showed that this reaction pathway is not energetically favorable in contrast to the two other channels (H- and I-abstractions), which are competitive. The H-abstraction was strongly exothermic (−84.4 kJ mol – 1 ), while the I-abstraction was modestly endothermic (16.5 kJ mol – 1 ). On the basis of our calculations, we predicted the rate constants using canonical transition state theory over the temperature range 250–500K for each abstraction pathway. The overall rate constant at 298 K was estimated to be 3.40 × 10 −14 and 4.22 × 10 −14  cm 3  molecule – 1  s – 1 for complex and direct abstraction mechanisms, respectively. In addition, the overall rate constant computed at 277 K was used in the estimation of the atmospheric lifetime for CH 2 IBr. On the basis of our theoretical calculations, the atmospheric lifetime for the OH removal process is predicted to be close to 1 year. In terms of atmospheric lifetime, the OH reaction is not competitive with the Cl reaction and photolysis processes.

Chemical dynamics simulations, based on both an analytic potential energy surface (PES) and direct dynamics, were used to investigate the intrinsic non-RRKM dynamics of the Cl − –CH 3 Br ion-dipole complex, an important intermediate in the Cl −  + CH 3 Br S N 2 nucleophilic substitution reaction. This intermediate may dissociate to Cl −  + CH 3 Br or isomerize to the ClCH 3 –Br − ion-dipole complex. The decomposition of microcanonical ensembles of the Cl − –CH 3 Br intermediate were simulated, and the ensuing populations vs. time of the excited intermediate and Cl −  + CH 3 Br and ClCH 3 –Br − products were fit with multi-exponential functions. The intrinsic non-RRKM dynamics is more pronounced for the simulations with the analytic PES than by direct dynamics, with the populations for the former and latter primarily represented by tri- and bi-exponential functions, respectively. For the analytic PES and direct dynamics simulations, the intrinsic non-RRKM dynamics is more important for the isomerization pathway to form ClCH 3 –Br − than for dissociation to Cl −  + CH 3 Br. Since the decomposition probability of Cl − –CH 3 Br is non-exponential, the Cl − –CH 3 Br unimolecular rate constant depends on pressure, with both high and low pressure limits. The high pressure limit is the RRKM rate constant and for the simulations with the analytic PES the rate constant decreased by a factor of 3.0, 5.6, and 4.3 in going from the high to low pressure limit for total energies of 40, 60, and 80 kcal/mol. For the direct dynamics simulations these respective factors are 2.4, 1.4, and 1.2. A separable phase space model with intermolecular and intramolecular complexes describes some of the simulation results, but overall models advanced for intrinsic non-RRKM dynamics give incomplete representations of the intermediate and product populations vs. time determined from the simulations.

The creation of electronic excitations during the reaction of atomic hydrogen on and with coinage and noble metals has been studied using metal-insulator-metal heterostructures. A characteristic current trace is observed when the outer metal surface of the structure is exposed to a 20 s pulse of H atoms. Comparison to the chemical kinetics allows to disentangle the contributions from the different chemical processes to this current. In the case of the coinage metals studied the observation is interpreted to suggest that predominantly electrons excited in connection with the hydrogen recombination reaction are the source of the current. For Pt such phenomenon is not observed. This difference allows insights into the role of the transition state for the non-adiabaticity in this simple chemical reaction.

The correct prediction of elementary processes occurring when H 2 scatters from a metal surface is one of the main challenges of surface science. In the field, the scattering of H 2 from Cu(111) has been considered a prototype system for activated dissociative chemisorption. Experimental and theoretical work suggested that a proper description of some scattering experiments on this system might require going beyond the static surface approximation, to consider how the motion of the Cu atoms affects the scattering event. Previous work suggested that important effects of phonons on the dynamics can be incorporated in the Potential Energy Surface (PES) by including four degrees of freedom, that have approximately additive couplings with the hydrogen molecule: the 3 dimensional motion of the nearest 1 st layer copper atom and the displacement of the nearest 2 nd layer copper atom along the direction perpendicular to the surface [3]. In the present work, we extend the 6D dynamical model by including the perpendicular motion of the 2 nd layer surface atom and we study this novel dynamical model with two techniques: an approximate method based on the Phonon Sudden Approximation (PSA) and an exact description using 7D wavepacket quantum dynamics. We consider how the inclusion and the excitation of the lattice degree of freedom affect some relevant processes: dissociative chemisorption, vibrational excitation of H 2 and state-to-state scattering probabilities fully resolved with respect to the vibrational states of the surface. We show that the PSA works in an excellent way for the system, thereby suggesting that this might be a viable way to study higher dimensional quantum models, incorporating four surface degrees of freedom that appear to be most relevant for H 2 scattering.

The present study demonstrates a priori prediction of the mechanism of Paternò–Büchi (PB) reaction by automated search for reaction pathways using the artificial force induced reaction (AFIR) method, for the simplest set of reactants, H 2 CO + C 2 H 4 . The search found all previously suggested pathways automatically without using any guess about the reaction mechanism. Even the reaction product was not assumed. In addition, a comprehensive view of the reaction path network obtained in the present extensive search gave new insights into the mechanism. Barriers on excited states explained the preference of the initial bonding sites. A seam of crossing between the lowest triplet and first singlet excited state is suggested to make a significant contribution to an intersystem crossing and the reaction mechanism of the present system.

Adsorbed molecules are involved in many reactions on solid surfaces that are of great technological importance. As such, there has been tremendous effort worldwide to learn how to theoretically predict rates for reactions involving adsorbed molecules. Theoretical calculations of rate constants require knowing both their activation energy and prefactor. Recent advances in ab initio computational methods ( e.g. , density functional theory with periodic boundary conditions and van der Waals corrections) promise to soon provide activation energies for surface reactions with sufficient accuracy to have real predictive ability. However, to predict reaction rates, we also need accurate predictions of prefactors. We recently discovered that the standard entropies of adsorbed molecules ( S ad 0 ) linearly track the entropy of the gas-phase molecule at the same temperature ( T ), such that S ad 0 ( T ) = 0.70  S gas 0 ( T ) − 3.3 R (R = the gas constant), with a standard deviation of only 2 R over a range of 50 R. This correlation, which applies only to conditions where their surface residence times are shorter than ∼ 1000 s, provides a powerful new method for estimating the partition functions for adsorbates and the kinetic prefactors for their reactions. For desorption, we show that the prefactors obtained with DFT using transition state theory (TST) and the harmonic oscillator approximation to get the partition function predicts prefactors for desorption that are of order 10 3 times larger than experimental values while our approach gives much better estimates. We also explore the applications of this approach to estimate prefactors within TST for the main classes of adsorbate reactions: desorption, diffusion, dissociation and association, and discuss its limitations. We discuss general issues associated with applying TST to rate laws and multi-step mechanisms in surface chemistry, and argue that rates of adsorbate reactions which are often taken to be proportional to coverage ( θ ) might better be taken as proportional to θ /(1 −  θ ) (unless the adsorbate forms islands), to account for the configurational entropy or excluded volume effects on the adsorbate's chemical potential.

Noble metal surfaces play a central role in heterogeneous catalysis. Lasers of the appropriate resonance frequency efficiently generate surface plasmons. These, in turn, may generate hot electrons, which can drive catalytic reactions at low temperatures. In this work, we demonstrate how embedding methods allow for the use of accurate ab-initio correlated wavefunction methods to describe excited-state potential energy surfaces of molecule–surface interactions. As model system, we consider the hot-electron-induced dissociation of hydrogen on Au(111), which has recently been demonstrated experimentally. We discuss merits and limitations of several different correlated wavefunction schemes. Our results show that dissociation barriers may be substantially reduced upon electron excitation and suggest a method to calculate the hot electron energies required for catalytic reactions.

We report an analytic potential energy surface (PES) based on several hundred DFT energies for H interacting with a Au(111) surface. Effective medium theory is used to fit the DFT data, which were obtained for the Au atoms held at their equilibrium positions. This procedure also provides an adequate treatment of the PES for displacements of Au atoms that occur during scattering of H atoms. The fitted PES is compared to DFT energies obtained from ab initio molecular dynamics trajectories. We present molecular dynamics simulations of energy and angle resolved scattering probabilities at five incidence angles at an incidence energy, E i  = 5 eV, and at a surface temperature, T S  = 10 K. Simple single bounce trajectories are important at all incidence conditions explored here. Double bounce events also make up a significant fraction of the scattering. A qualitative analysis of the double-bounce events reveals that most occur as collisions of an H-atom with two neighboring surface gold atoms. The energy losses observed are consistent with a simple binary collision model, transferring typically less than 150 meV to the solid per bounce.

We present an experimental study of the initial dissociative sticking probability of H 2 on the Pd–Cu(111) surface alloy at Pd coverages between 1–10%. The measurements have been performed using a supersonic molecular beam with an incident energy range E i  = 75–163 meV. In agreement with a recent STM study, our results confirm that small amounts of Pd atoms significantly increase the reactivity of the inert Cu(111) surface, although the measured initial sticking probability values were found to be much lower than the ones estimated from the STM work. For a Pd coverage of 1% and E i  ∼ 0.10 eV, the H 2 initial sticking probability was found to be 2 × 10 −3 , increasing to 4 × 10 −3 for a Pd coverage of 10%. This agrees well with a recent molecular dynamics study, in which the initial sticking probability was estimated in 7 × 10 −3 for a Pd coverage of 11% and E i  ∼ 0.15–0.20 eV. Essentially the same values were measured in the whole incident energy range investigated. Therefore, our results support the main conclusion of the molecular dynamics calculations about the mechanism leading to the increase of reactivity of the Pd–Cu(111) surface alloy, which combines a lower activation energy barrier for dissociation of ∼ 0.25 eV on top of substitutional Pd atoms with spillover onto the Cu(111) surface.

The first calculation of reaction dynamics was made in 1931 by H. Eyring  and M. Polanyi for the gas-phase exchange reaction H(g) + H 2 (g) = H 2 (g) + H(g). Here we use Density Functional Theory to explore the dynamics of an analogous exchange reaction, H(ad) + H 2 (ad) = H 2 (g) + H(ad), taking place at coinage metal surfaces; Cu(100), Ag(100) and Au(100). The attacking atom, H(ad), recoils in a preferred direction toward the physisorbed H 2 (ad), as envisioned in other studies of `Surface Aligned Reaction' (SAR). The dynamics of SAR for the above reaction is compared with gas-phase reaction, employing the same initial state geometry in the absence of the surface. The effect of weak physisorption at the metal surface is to approximately halve the bond dissociation energy of H 2 (ad) as compared with H 2 (g). As a result, the surface markedly catalyzes the exchange reaction, decreasing the energy barrier due to increased heat of reaction in the adsorbed state, in accord with the Brønsted, Evans and Polanyi (BEP) relationship.

Results on the scattering of hyperthermal N 2 molecules from bare and N-covered Ru(0001) surfaces are presented. These are compared with Ar scattering from the same surfaces as a reference non-reactive system. In the case of bare Ru(0001) the measured angular distributions are consistent with scattering from an atomically smooth surface after interaction with a repulsive potential energy surface. Broadening of the N 2 angular intensity distribution at high incident energy, which was observed in molecular beam studies, is not apparent in our measurements. This is attributed to the broadening effect being the result of a transient trapping process that is confined to a relatively narrow incident energy range. The presence of N-adatoms on the surface results in atomic-scale surface roughening and blocking of dissociation pathways. Formation of N 2 molecules as a result of abstraction of N-adatoms by incident N-atoms in an Eley–Rideal-type reaction is considered. Support for such a process is derived from comparison of the trends in the angular energy distributions of N 2 and Ar.

In honoring the seminal contribution of Henry Eyring and Michael Polanyi who first introduced the concept of potential energy surfaces (PESs) to describe chemical reactions in gas-phase [Z. Phys. Chem. 12, 279–311, (1931)], this work comes to review and assess state-of-the-art approaches towards first-principle based modeling in the field of gas-surface dynamics. Within the Born-Oppenheimer and frozen surface approximations, the O 2 -Ag(100) interaction energetics are used as a showcase system to accentuate the complex landscape exhibited by the PESs employed to describe the impingement of diatomics on metal substrates and draw attention to the far-from-trivial task of continuously representing them within all six molecular degrees of freedom. To this end, the same set of ab initio reference data obtained within Density Functional Theory (DFT) are continuously represented by two different state-of-the-art high-dimensional approaches, namely the Corrugation-Reducing Procedure and Neural Networks. Exploiting the numerically undemanding nature of the resulting representations, a detailed static evaluation is performed on both PESs based on an extensive global minima search. The latter proved particularly illuminating in revealing representation deficiencies which affect the dynamical picture yet go otherwise unnoticed within the so-called “divide-and-conquer” approach.

The stability of magnetic states and the mechanism for magnetic transitions can be analyzed in terms of the shape of the energy surface, which gives the energy as a function of the angles determining the orientation of the magnetic moments. Minima on the energy surface correspond to stable or metastable magnetic states and can represent parallel, antiparallel or, more generally, non-collinear arrangements. A rate theory has been developed for systems with arbitrary number, N , of magnetic moments, to estimate the thermal stability of magnetic states and the mechanism for magnetic transitions based on a transition state theory approach. The minimum energy path on the 2 N -dimensional energy surface is determined to identify the transition mechanism and estimate the activation energy barrier. A pre-exponential factor in the rate expression is obtained from the Landau–Lifshitz–Gilbert equation for spin dynamics. The velocity is zero at saddle points so it is particularly important in this context to realize that the transition state is a dividing surface with 2 N  − 1 degrees of freedom, not just a saddle point. An application of this rate theory to nanoscale Fe islands on W(110) has revealed how the transition mechanism and rate depend on island shape and size. Qualitative agreement is obtained with experimental measurements both for the activation energy and the pre-exponential factor. In particular, a distinct maximum is observed in the pre-exponential factor for islands where two possible transition mechanisms are competing: Uniform rotation and the formation of a temporary domain wall. The entropy of the transition state is enhanced for those islands making the pre-exponential factor more than an order of magnitude larger than for islands were only the uniform rotation is viable.

Water clusters have attracted a lot of attention as prototype systems to study hydrogen bonded molecular aggregates but also to gain deeper insights into the properties of liquid water, the solvent of life. All these studies depend on an accurate description of the atomic interactions and countless potentials have been proposed in the literature in the past decades to represent the potential-energy surface (PES) of water. Many of these potentials employ drastic approximations like rigid monomers and fixed point charges, while on the other hand also several attempts have been made to derive very accurate PESs by fitting data obtained in high-level electronic structure calculations. In recent years artificial neural networks (NNs) have been established as a powerful tool to construct high-dimensional PESs of a variety of systems, but to date no full-dimensional NN PES for has been reported. Here, we present NN potentials for clusters containing two to six molecules trained to density functional theory (DFT) data employing two different exchange-correlation functionals, PBE and RPBE. In contrast to other potentials fitted to first principles data, these NN potentials are not based on a truncated many-body expansion of the energy but consider the interactions between all molecules explicitly. For both functionals an excellent agreement with the underlying DFT calculations has been found with binding energy errors of only about 1%.

Our computational study presents embedded cluster models of the hydrated electron focusing on its reactivity with the hydrated proton and the nitrous oxide molecule leading to formation of a hydrogen atom in the former case and a nitrogen molecule plus hydroxyl radical and hydroxide anion in the latter case. We show using ab initio calculations combined with the nudged elastic band method for determining minimum energy paths that carefully chosen cluster models with no more than six water molecules embedded in a polarizable continuum are able to capture the essential features of the reactive processes of the hydrated electron.

We report on the Stark deceleration of a pulsed molecular beam of NO radicals. Stark deceleration of this chemically important species has long been considered unfeasible due to its small electric dipole moment of 0.16 D. We prepared the NO radicals in the X 2 Π 3/2 , v  = 0, J  = 3/2 spin-orbit excited state from the X 2 Π 1/2 , v  = 0, J  = 1/2 ground state by Franck–Condon pumping via the A 2 Σ + state. The larger effective dipole moment in the J  = 3/2 level of the X 2 Π 3/2 , v  = 0 state, in combination with a 316-stages-long Stark decelerator, allowed us to decelerate NO radicals from 315.0 m/s to 229.2 m/s, thus removing 47% of their kinetic energy. The measured time-of-flight profiles of the NO radicals exiting the decelerator show good agreement with the outcome of numerical trajectory simulations.

We present a synchrotron for polar molecules consisting of 40 straight hexapoles arranged in a circle with a 50 cm diameter. The mechanical design and alignment procedure as well as the trigger scheme used to switch the voltages applied to the hexapoles are described in detail. The stability of the synchrotron is demonstrated by measurements in which multiple packets are stored for over 13 s, during which they have completed over 1000 round trips and traveled a distance of over one mile. Furthermore, we demonstrate the simultaneous trapping of 26 packets; 13 revolving clock-wise and 13 counter clock-wise, that are injected into the synchrotron by two Stark decelerator beamlines. We discuss the opportunities for using the synchrotron as a low-energy collider.