Chemical dynamics from the gas-phase to surfaces

: The field of gas-phase chemical dynamics has developed superb experimental methods to probe the detailed outcome of gas-phase chemical reactions. These experiments inspired and benchmarked first principles dynamics simulations giving access to an atomic scale picture of the motions that underlie these reactions. This fruitful interplay of experiment and theory is the essence of a dynamical approach per-fectedongas-phasereactions,theculminationofwhichisastandardmodelofchemical reactivity involving classical trajectories or quantum wave packets moving on a Born– Oppenheimer potential energy surface. Extending the dynamical approach to chemical reactions at surfaces presents challenges of complexity not found in gas-phase study as reactive processes often involve multiple steps, such as inelastic molecule-surface scattering and dissipation, leading to adsorption and subsequent thermal desorption and or bond breaking and making. This paper reviews progress toward understanding the elementary processes involved in surface chemistry using the dynamical approach.

problems in surface chemistry, which may often seem impenetrably complex and challenging.

THE ORIGINS OF CHEMICAL DYNAMICS
Seeds leading to the emergence of the field of chemical dynamics were clearly sown during the whirlwind of progress ushered in by the discovery of quantum mechanics, 29 Figure 1. Theoretical chemistry was well underway as a burgeoning field reaching perhaps its most important milestone, the so-called absolute theory of reaction rates. 34 Chemists now had not only the mathematical theory of chemistry, but in addition, new chemical concepts-for example, PESs and transition states-defined in terms of fundamental physics.
Despite this profound conceptual progress, a cloud hung over this field for decades to come, as only limited means existed to test the approximations introduced by theorists, tests that would be needed to prove that the theories had practical value. By 1933, the Journal of Chemical Physics (JCP) had published its first issue and, reflecting the tentative nature of the young field, its first editor Harold Urey wrote, ". . . The methods of investigation used are, to a large extent, not those of classical chemistry and the field is not of primary interest to the main body of physicists. . . ". 35 The fact was that methods of experimental verification and numerical simulation lagged dramatically behind the conceptual breakthroughs of the quantum era. This was to change soon and JCP would become the vanguard forum showing the successes of converging the methods of physics with problems of chemistry.

THE EXPERIMENTAL BASIS FOR MODERN CHEMICAL DYNAMICS
The second half of the 20th century witnessed a revolution for the atomic scale viewpoint of chemistry, as molecular spectroscopy, 36 the laser, 37,38 molecular beams, 39 and high-power computing 40

Infrared emission spectroscopy
Even before masers 37 and lasers 38 introduced the concept of a population inversion, scientists had evidence of molecules in the atmosphere and in the laboratory with quantum state population distributions far from thermal equilibrium, a glittering signpost for the importance of dynamical processes. Emission spectra from the night sky which could be recorded on photographic plates showed OH(v =6-9) 41 -the same spectra were seen using flash photolysis 36 to drive the reac- 42 Similar methods revealed reactions that produced highly vibrationally excited O 2 43,44 and N 2 . 45 This field accelerated dramatically with the advent of infrared chemiluminescence, 4 which revealed vibrationally excited HCl produced in the reaction of H+Cl 2 . 46 New dynamical concepts also emerged, notably the "Polanyi Rules"-see Figure 2. These state that for exothermic reactions, an energy barrier located early along the reaction path will preferentially produce vibrationally excited products, whereas a late barrier will likely lead to higher product translational energy and require reactant vibration to proceed efficiently. 4,48 Work in this direction also led to clear ideas about how chemical reactions can produce population inversions and a prediction that a chemical laser could work based on a "partial" population inversion. 49 This prediction was realized only a few years later, exploiting the H + Cl 2 → Cl + HCl(v > 0) reaction to produce a vibrational population inversion. 47

F I G U R E 2
The (John) Polanyi Rules: The reaction H + Cl 2 → Cl + HCl(v > 0) produces vibrationally hot HCl, which emits in the infrared. In fact, this reaction formed the basis of the first chemical laser. 47 The PES on the left exhibits an early transition state; one that resembles the H + Cl 2 reactants, where r 1 would represent the H-Cl distance and r 2 the Cl-Cl distance. Reagent translation allows efficient access to the transition state, where a large energy release occurs before the HCl bond is fully formed, snapping the two atoms together and producing HCl vibrational excitation with a bobsled-like trajectory. The late barrier shown on the right requires reagent vibration to be efficient-with reagent translation, the reaction fails as shown. 4

Laser-induced fluorescence and resonance-enhanced multiphoton ionization
The inventor of the ruby laser 38 once described the laser as "a solution seeking a problem." 50 Chemical dynamics had the problems! Due to the ease of constructing gas discharges, c. w. and pulsed "line tunable" lasers using He, Ne, 51,52 Ar or Kr, 53 and N 2 /CO 2 54,55 were the first sources of laser light used in this field. Their application quickly led to a new technique that marked a sea change in chemical dynamics, laser-induced fluorescence (LIF). 56 This technique enabled quantumstate resolved studies that quickly became a major focus of experimentalists. While LIF was first detected as spontaneous infrared emission produced after infrared laser excitation, [57][58][59] its true potential became evident only after a HeNe-laser was used to excite K 2 and this molecule's emission spectrum was photographically recorded behind the Berkeley 21-foot concave grating. 60 Soon after, LIF was detected by a photomultiplier 61 and lifetimes of excited electronic states were derived with a phase shift method. 62 By observing LIF while tuning a dye laser, 63,64 background-free laser excitation spectra were demonstrated-this method was so sensitive that it could be exploited to probe the internal state distribution of reaction products 65 and later even to obtain state-resolved angular distributions of reaction products. 66,67 The pulsed ruby laser provided such high peak powers that multiphoton ionization of atoms [68][69][70] could be demonstrated including resonance enhancement. 71 Applied first to molecules as a 1+1 resonance enhanced multiphoton neutralization of a 1-keV beam of C − 2 , a spectrum was obtained by detecting neutral C 2 as a function of the laser wavelength. 72 Resonance enhanced multiphoton ionization (REMPI) was first demonstrated using Cs 2 . 73 Soon thereafter, the 3+1 REMPI spectrum of NO 74 and the 2+1 REMPI spectrum of I 2 75 were reported followed by REMPI spectra of larger polyatomics. 76,77

Molecular beams
Molecular beams became another vital piece of the experimental tool-box of chemical dynamics-the high-energy physicists' scattering experiments using charged particles and accelerators were being transitioned to the study of ion-molecule 79,80 and even moleculemolecule 39 reactions at low, chemically relevant energies. Reactions could now be studied by crossed molecular beams methods. 81 Figure 3 shows an instrument that detects reaction products resulting from the collisions between molecules in two different beams. The densities of the two beams ensured that within the crossing volume of the beams, at most, one collision could occur and the products could escape collision free.
The data observed with this instrument provided the product fluxmap in the center-of-mass frame of the reacting molecules. Figure 4 shows an experimentally derived flux map for the F + H 2 → HF(v) + H reaction. 82 Several vibrational states of HF could be resolved and their flux-maps obtained. Experimental observations like these provided attractive targets for the developing field of theoretical chemical dynamics.
Applying external electric fields to polar molecules in low J-states produced molecular beams of oriented molecules and observations of steric effects in chemistry. 83 Using pulsed electric fields, deceleration of molecular beams also proved possible, 84,85 improving our ability to study quantum effects in collisions between molecules. 86 But REMPI detection in combination with the power of molecular beams proved pivotal, 87 setting the stage for what was to become one of the most important tools for chemical reaction dynamics, ion imaging. 88

Ion imaging
Ion imaging became possible by applying REMPI to reaction products and detecting them with position sensitive (electron multiplier

F I G U R E 3
The universal crossed molecular beam machine--nicknamed "Hope." The reaction H + Cl 2 → Cl + HCl(v) 2,3,78 as well as many others was observed under "single collision conditions" with this instrument. Laboratory frame scattering data yield the center-of-mass frame product flux-map. See In its initial incarnations, analyzing ion images required that the plane of the ion camera be parallel to a cylindrical symmetry axis of the threedimensional product distribution, a requirement that is often difficult to fulfill. With the advent of "slice imaging" in 2001, 93 this requirement was relaxed and has over time become the method of choice for problems in chemical dynamics. 94,95

COMPUTATION COMPLETES THE DYNAMICAL APPROACH
With these new experimental tools, exquisite observations of elementary chemical reactions became available, 96 but it was only with the advent of high power computing that the field could reach its poten-tial, as cooperation between experiment and theory became increasingly valuable.
The extraordinary growth in the capabilities of computers has thoroughly transformed theoretical chemistry; however, the development of efficient and accurate algorithms has been at least as important.
The calculation of PESs has evolved from empirical and semi-empirical models, to Hartree-Fock theory, 97 to wavefunction-based methods that include electron-electron correlation, such as valence-bond, 98 Møller-Plesset perturbation, 99 coupled-cluster, 100 and configuration interaction theories. 101 In parallel, density functional theory (DFT) 102 has emerged within a "sweet spot," balancing accuracy with affordability for larger molecular systems. Machine learning algorithms now provide efficient and accurate fitting of high-dimensional PESs to ab initio energies. 103 Methods for tracing out the atomic motions governed by PESs have also advanced, from classical mechanics, 104,105 to time-independent quantum scattering theory, 106,107 to time-dependent wave-packet motion. 108,109 Classical mechanics remains the workhorse, especially on-the-fly methods that compute the classical forces as the trajectory proceeds, usually by DFT. 110 The design of algorithms has adapted to hardware advances like parallel computing and the use of graphical processing units as computational engines. 111 Widely available The product flux map in the center of mass frame for the F + H 2 → HF(v) + H reaction. The contour plot reflects the probability for product HF to appear with a specific velocity vector (e.g., dashed arrow). The velocity vectors of the reactants are also shown as vectors, labeled F and H 2 . The dashed line of smallest radius produces HF(v = 3) from both F( 2 P 3∕2 )-mainly forward scattered-and F( 2 P 3∕2 ) labeled v = 3 ′ , which is mainly backward scattered and appears with slightly higher speed. HF(v = 2) appears within a larger radius (dashed) circle and is mainly formed in a rebound reaction, where the HF recoils in the opposite (backward) direction of the incident F-atoms. 82  using an improved semi-empirical PES. 116 Soon, quantum dynamics calculations using time-independent scattering theory on semi-empirical PESs appeared, first with approximations 106,117 and then with numerically exact solutions. 107,118 Quantum mechanical resonances were predicted 119 -peculiar oscillations in the reaction probability's dependence on collision energy that arise because a piece of the wave packet becomes stalled at the transition state. Theory also predicted quantized bottleneck states (QBS), 120 where wave-packet motion through the quantized transition state produces interferences.
Experiments, however, turned out to be tremendously challengingone group used beams of tritium to investigate H + T 2 → HT + T capturing the T-atom products on MoO 3 "buttons" arranged around the reaction zone. By later scraping off the MoO 3 and analyzing the samples with a scintillation counter, angular distributions could be derived. 121 Eventually, angular distributions of scattered products could be seen using electron bombardment ionization detection. 122 This showed that the transition state was short lived and products formed by a rebound mechanism. Soon, nascent low-resolution product speed distributions were obtained. 123 eV, but not at E i = 0.53 eV. 130 Also, some PESs worked better than others. 131 The influence of quantum mechanics on the H 3 reaction is profound.
Applying the Rydberg tagging approach led to direct observation 133,134 of the influence of the predicted QBS-states. 120 Perhaps most spectacular are observations of Berry's phase 135 influencing the reaction.
When three H-atoms are arranged in an equilateral triangular geometry, the two lowest electronic states, 1 and 2 , are degenerate as they are forbidden by symmetry to interact with one another, ⟨ 1 |Ĥ | 2 ⟩ ≡ 0. On the other hand, at all nonsymmetric structures, the two states mix and split. This gives rise to a "conical intersection" marked with an × in Figure 5(a). In fact, conical intersections are very important in chemistry. [136][137][138][139] For the DHH isotopologue shown in Figure 5, reaction can occur via two pathways: the red pathway is a simple abstraction of green H by brown H, while the green pathway involves F I G U R E 5 Quantum interference through a conical intersection: (a) a cut through the HHD PES showing the conical intersection (×) and three transition states (T) that connect three stable arrangements of the atoms. Note the color of the atoms. Direct abstraction visits one transition-state (RED ARROW), while the spiral or roaming reaction visits two (GREEN ARROW). Both paths lead to the same products. (b) The experiment (•) detects reactive flux arriving in the backward scattering direction producing H 2 (v ′ = 2, J ′ = 3) as the incidence energy is scanned. The oscillations are due to quantum interference between the two topological pathways. The red line shows quantum scattering calculations that neglect the phase-shift of (geometric phase) introduced by traversal around a conical intersection. The blue curve accounts for the geometric phase. 132  But beyond this, quantum mechanics requires that when a conical intersection is traversed, the phase of the quantum flux passing on opposite sides of the conical intersection must be shifted by with respect to one another-Berry's phase. 135 Obviously, this affects the interference. 132,140,141 These observations relied on Rydberg-atom tagging, but REMPI-based methods like ion imaging and Photoloc [142][143][144][145][146][147][148][149][150][151] have also been crucial to revealing the dynamics of this system. [142][143][144][145][146][147][148][149][150][151][152][153] The basis of this success and the others that space does not allow us to present is the concept that chemical reactivity involves quantum mechanical motion of nuclei on a Born-Oppenheimer PES.
The remarkable agreement between the predictions of the theory and the observations from experiment earns this concept the name the standard model of chemical reactivity. 8

Classical roaming reaction
The standard model affords the possibility of computing and illustrating the time-dependent motions of individual atoms through a chemically reactive encounter by following, for each atom, either the classical mechanical position or the quantum mechanical expectation value of position. It is even possible to make movies of reactions using calculated trajectories, effectively providing a microscope with time and space resolution far better than will ever be experimentally possible. One of the most inspiring examples of this is the gas-phase roaming reaction, first reported in the unimolecular decomposition of H 2 CO. 154 Following up on suspicions that the reaction H 2 CO → H 2 + CO may proceed by more than one mechanism, 155 ion imaging was applied to obtain speed and angular distributions of specific rotation-vibration states of CO(v CO , J CO ). Figure 6 shows data revealing that when CO is produced with low rotational excitation, H 2 is produced with low speed and high vibrational excitation, and vice versa. Using a six-dimensional

Electronically nonadiabatic dynamics
The reaction of H + + H 2 → H + 2 + H appears superficially simpler than the H 3 reaction-H + 3 has one less electron. However, looks may deceive-this reaction may occur in three ways. Isotopic labeling helps illustrate this. Reacting H + with D 2 may involve ion exchange, producing HD + D + , electron transfer, producing H + D + 2 , or ion exchange with electron transfer, forming HD + + D. We need to extend the standard model to consider the quantum motion of protons influenced by an avoided intersection between the two lowest energy electronic states of H + 3 . In a reactive encounter, the nonadiabatic coupling- Figure 8(d) -is defined as: and controls the probability of a transition between the two PESs.
Here, φ 1 and φ 2 are the adiabatic wave functions and ⃗ R defines the positions of the nuclei. | ⃖⃖⃗ D 12 | is a measure of how rapidly nuclear motion flips the electronic wavefunction from one electronic state to the other. | ⃖⃖⃗ D 12 | is large at the avoided crossing when the incident proton wavepacket is at a distance ∼8Å, ensuring nearly unit probability of a change in adiabatic state (no electron transfer) as the system passes through the crossing. As the wavepacket moves closer, it may branch onto both PESs. Further branching can occur each time wavepackets enter regions of space where | ⃖⃖⃗ D 12 | is large.
The H + 3 reaction inspired a successful approximate method, "trajectory surface hopping," 158 a procedure for integrating the classical mechanical equations of motion of the nuclei on a single adiabatic PES, until a hop to a different PES occurs at random according to probabilities determined from the magnitude of | ⃖⃖⃗ D 12 |. Application of this theory to the H + + D 2 reaction reproduced quite accurately the experimentally measured absolute cross sections 159 for the three reaction channels as a function of energy-see Figure 9.
In the initial version of surface hopping, transitions between PESs occurred only at positions of maximal nonadiabatic coupling. This is unrealistic. Nonadiabatic coupling can be significant over broad regions of space, meaning that wavepackets may enter into strong coupling regions without reaching an avoided crossing. Multiple transitions may occur leading to different pathways with different quantum mechanical phases, resulting in interference effects. 160 Well-defined avoided crossings may not even exist. 161 Theoretical advances to address these issues are continually being developed. In Ehrenfest theory, 162  to unravel the multiple product channels of this reaction. 177 Femtosecond soft X-ray spectroscopy of the electro-cyclic ring-opening reaction of 1,3-cyclohexadiene revealed the ultrafast time scales of the nonadiabatic events. 178 The conical intersection dynamics of the RNA base uracil was studied using a UV pump with stimulated-Raman probe. 179 All of these studies were successfully modeled by surface hopping calculations.

Transition state theory
Chemistry is the science of materials conversion and while thermodynamics tells us which reactions are fundamentally possible, to be of practical importance, we must design chemical pathways to reach the desired products using rapid reactions. This simple argument under-lies the entire field of catalysis and drives much of synthetic chemistry.
One of the major motivations to develop an atomic scale foundation of understanding in chemical dynamics is the desire for a predictive theory of chemical reaction rates. Transition state theory (TST) has filled this need, allowing us to exploit our understanding of chemical dynamics to make quantitative predictions about the speed of a reaction.
In its original formulation, Eyring postulated a special state of the system-the activated complex-that when formed, would with almost complete certainty, go on to products. In a remarkable leap of insight, he assumed that this species would be in thermal equilibrium with the reactants. If one could determine its energy and structure-necessary to obtain its entropy-the activated complex's concentration as well as the speed of passage on to products could be found with statistical mechanics. 34 At the time these ideas were developed, it was difficult to predict theoretically many of these quantities. However, the development of computational chemistry has provided all of the machinery necessary for making these calculations accurately for many gas-phase reactions.
The current formulation of TST prescribes a dividing plane that separates reactants from products such that every trajectory that originates in the reactant region of configuration space and evolves to the product region must pass through the dividing plane at least once.
The TST thermal rate constant is equal to the equilibrium one-way flux through the dividing plane in the direction of reactant to product.
Thus, TST provides an upper bound to the rate constant, since some trajectories might pass through the dividing plane more than once or pass through without leading to product-so, the equilibrium flux will include nonreactive events. The location of the dividing plane is usually chosen at the reaction barrier, but ways that are more sophisticated can be used, including variational TST, 180 in which the location of the dividing plane is chosen to minimize the TST rate. Improvement to TST can be obtained by running classical trajectories to count the number of recrossing events and reduce the TST rate accordingly. 181 This technique is particularly advantageous in cases where the reaction barrier height is high, perhaps many times kT. An effective way to simulate this is to initiate trajectories at the dividing plane, integrate forward and backward in time, and modify the TST rate constant by the fraction of trajectories that underwent recrossings.
TST has been extremely successful and many comparisons between measured and predicted rates have proven its validity. It has been particularly important in atmospheric 182 and combustion chemistry 183 and finds widespread use to predict reaction rates, especially where they are impossible to measure. It is also worth contemplating that TST has led to a deeper understanding of how enzymes work. 184

EXTENDING THE DYNAMICAL APPROACH TO SURFACES
Extending the dynamical approach to problems in surface chemistry may appear impenetrably complex and challenging. The remainder of this review breaks down the complexity and reveals the commonalties to gas-phase dynamics. Progress derives from adapting the basic concepts and many exquisite theoretical and experimental tools of gasphase dynamics to problems at surfaces and inventing new ones based on the spirit of the dynamical approach.
The most obvious challenge facing the extension of the dynamical approach to surfaces is that surfaces are big and they are dense. In a crossed molecular beam experiment, most of the molecules in one beam pass through the other without colliding, ensuring single collision conditions where we can observe elementary reaction steps. In contrast, in a beam-surface scattering experiment, every molecule collides.
While this contributes to strong signals, molecule-surface encounters may involve many collisions. Hence, the challenge of size is not just one of high-dimensionality; rather, we need to disentangle a sequence of We also face new experimental challenges when making the leap from the gas phase to surfaces. Of course, we need ultra high vacuum (UHV) (∼10 −10 mbar) to establish conditions where surfaces remain clean and we need the tools of surface science for cleaning and characterizing the sample. Fortunately, these tools are now commercially available and offer no significant barrier to entering the field.
The truly daunting challenges include the following. We often do not know a priori the reactive site, as adsorption and diffusion as well as the surface heterogeneity are conditions not faced in the gas phase. A theoretician working to understand experiments in the gas phase takes comfort in knowing the stoichiometry of the transition state. In surface reactions, knowing which atoms to include in a model of reactivity may be the first puzzle to solve. Of course, the surface itself behaves like a reactant. Yet, there are few tools available to excite specific motions of the solid to investigate their influence on reactivity-most studies simply vary the temperature. More subtle problems also arise. Gas-phase experimentalists take for granted tools that offer detection sensitivities of as low as one molecule per cm 3 ; common methods in surface science offer adsorbate detection sensitivities of about 10 13 molecules cm −2 or 0.01 of a monolayer (ML).
While the challenges and limitations described above present barriers, they also provide opportunities for great progress through a combination of ingenuity, advancing technology, and theory development.
It is a common occurrence-in fact, many examples can be found in Section 2 of this review-that today's sensation or "miracle experiment" becomes tomorrow's routine calibration measurement. In experimental science, the more we learn, the more we are able to learn. This inevitable improvement of measurement methods should make us optimistic about meeting the challenges.

Vibrational relaxation rates of adsorbates
Lifetimes of vibrationally excited molecules on surfaces have been inferred from infrared lineshapes and measured directly using infrared pump-probe methods. Lifetimes range from milliseconds for CO * (v = 1) physisorbed on NaCl 187,188 to 2-3 ps for chemisorbed CO on metals. [189][190][191][192][193] The long lifetime of CO * (v = 1) on NaCl reflects the fact that more than thirteen phonons must be excited to relax the molecule-CO's vibrational frequency (∼2100 cm −1 ) is much higher than the highest frequency phonon of NaCl (∼160 cm −1 ). The anharmonicity of the PES is so small initially that excited CO vibration does not easily decay to other vibrational degrees of freedom. In fact, relaxation occurs via an electromagnetic interaction independent of the PES-the Sommerfeld ground wave limit. 188 When CO is chemisorbed on metals, the coupling to phonons is no more favorable; therefore, the ps vibrational lifetimes observed suggest that vibrational relaxation via excitation of electron-hole pairs (EHPs) in the metal is highly efficient.
The importance of vibrational relaxation to excite EHPs has been confirmed by a variety of theoretical methods. One of the first used is Fermi's golden rule (FGR) with the jellium approximation for the metallic conduction electrons to study H 2 relaxation near Al, Mg, and Na surfaces. 194,195 Similarly, CO on a Cu cluster was examined using FGR with DFT. 196 A Newns-Anderson Hamiltonian approach showed similar results for CO and CN adsorbed on Ag, Cu, Au, and Pt. 197 Electronic friction methods for CO on Cu(100) employed a local-density friction approximation (LDFA) and also yielded picosecond lifetimes. 198 Recently, pump-probe measurements of the vibrational relaxation of physisorbed CO on Au(111) showed a lifetime of ∼50 ps, much longer than chemisorbed CO 199 -a recent theoretical treatment was consistent with this measurement. 200 Such long vibrational lifetimes for physisorbed species suggest that reactions of vibrationally excited adsorbates 201 may be more important than previously believed.
All of these studies investigated the high-frequency CO stretch.
For other modes and lower frequency stretch modes, phonons can play a significant role in coupling to adsorbate vibrations, as shown by FGR calculations of vibrational lifetimes of all four vibrational modes of CO on Cu(100) using finite-sized Cu clusters at the Hartree-Fock level. [202][203][204] Here, EHP excitation entirely dominated the lifetime of the internal stretch (3.3 ps) and bend (2.3 ps) modes, while phonon excitation significantly contributed to the lifetimes of the CO-surface stretch (22 ps) and the frustrated translational modes (14 ps). 204 All four of these lifetimes are in reasonable accord with experiment. 189,192 The conclusions are that EHP excitations dominate the relaxation of the internal stretch and bending modes, whereas the molecule-surface stretch and frustrated translational modes relax mainly via phonons. It is not known whether these trends hold for other adsorbates or other metal surfaces.

Early molecular beam surface scattering experiments
The success of molecular beam scattering experiments in gas-phase dynamics created a lot of enthusiasm that similar success was possible by the application of these techniques to surfaces. Early experiments faced limitations stemming from the difficulty of combining beam and surface science methods. High-performance molecular beam machines typically had poor vacuum and it was necessary to work with surfaces at elevated temperatures to prevent contamination or to use continuous epitaxial deposition to maintain a clean surface. 205,206 UHV surface science machines retrofitted with rotating detectors for beam surface scattering, usually using effusive beams, 207 209 and treating the substrate atoms as having truncated spherical caps 210 to allow modeling of parallel momentum transfer. This later refinement was particularly necessary to obtain agreement with measurements at hyperthermal incidence energies. 211 Velocity distributions of molecules scattered from clean and wellcharacterized surfaces became available only after combining UHV surface science techniques with state-of-the art molecular beam methods. In one of the first instruments, 212 three differentially pumped supersonic beam sources were directed at the surface, producing molecules with narrow velocity distributions. A differentially pumped mass spectrometer and associated vacuum pumps mounted on a rotating platform sealed with Teflon "tec" seals 213 provided a rotatable detector, which, with the use of a chopper wheel, allowed for the measurement of scattering-angle resolved times of flight (TOF). Rotation of the solid target allowed variation of the incidence angle. The instrument was bakeable, used UHV compatible pumps, components, and materials of construction and had surface science equipment to clean and characterize the target surfaces. Figure 10 shows an example of measured velocity distribution data for Ar scattering from a Pt(111) surface. 214 The iso-flux contour plots shown combine results of TOF measurements at many scattering angles. Cuts of the iso-flux plots give the velocity distribution in the directions normal and parallel to the surface. The spread in velocities in the perpendicular direction (v z ) is clearly larger than that in the parallel direction (v y ) showing the coupling of normal momentum to the surface is larger than that of parallel momentum. The data show the "the law of parallel momentum conservation" that had emerged from interpretations of angular distribution measurements 215 is not correct, although there F I G U R E 1 0 Velocity distribution measurements for in-plane Ar scattering from Pt (111). Iso-flux contours for Ar with an incidence energy E i = 94 meV and angle i = 45 • . Out-of-plane scattering data were also obtained by tilting the surface. 214  The availability of accurate scattering data stimulated the development of methods to simulate gas-surface scattering. Even for scattering of rare gas atoms from clean, perfect surfaces, it is a challenge to employ an accurate gas-surface interaction potential, energy dissipation to phonons, and quantum mechanical effects. In principle, for metal surfaces, the effects of EHP transitions should also be included, but these appear to be relatively unimportant for rare gas atom scattering.
The first advance in theory beyond the cube models was to integrate the classical equations of motion numerically for the rare gas atom and a slab of surface atoms. These studies modeled the gas-surface interaction using empirical potentials with harmonic interactions among the surface atoms. It was found that, for metal surfaces, Lennard-Jones pairwise additive gas-surface potentials produced too much corrugation; presumably, the metallic electron cloud smooths out the corrugation. This problem was addressed by the addition of a background smoothing potential. 217 Initial simulations employed only a small number of surface atoms with frictions and fluctuating forces to define surface temperature, T S and with memory chosen to approximately reproduce the phonon spectrum. 218 As more powerful computers became available, hundreds of surface atoms could be included, with or without frictions and random forces. 219,220 To date, however, there do not appear to be any simulations of the scattering of rare gas atoms from surfaces based on accurate ab initio PESs. (111) 224 for E n = E i cos 2 ( i ) as indicated. Note the emergence of a broad peak at high J as E n is increased. (b) Rotational temperature, T R of the low-J region of the spectrum as a function of E n . The linear increase in T R is evidence that the rotational excitation is the result of transfer of translational motion into molecular rotation (T→R coupling), a signature of direct scattering Simulations based on classical adiabatic molecular dynamics using empirical potentials provide reasonable agreement with measured energy and angular distributions. An example for the direct inelastic scattering of Xe scattering from Pt(111) is shown in Figure 24(a)-(d).

F I G U R E 1 1 (a) Rotational state distributions normalized to degeneracy for NO scattering from Ag
For this system, there are two peaks in the TOF spectrum; one resulting from direct "single bounce" inelastic scattering, and the other from molecules that trap on the surface and then desorb. The general agreement of measured rare gas atom scattered velocity distributions with calculations based on a single PES indicates that electronically adiabatic coupling to phonons dominates and that electronically nonadiabatic excitation of EHPs is not important. At higher energies where E i cos( i ) > 3 eV, the measured energy loss is larger than that calculated from the adiabatic picture, suggesting that nonadiabatic excitation starts to become important, even for rare-gas metal interactions. 221

State-resolved detection of scattered molecules, rotational effects
Probing transitions between a molecule's rotation-vibration states due to collisions at a surface required state-specific detection techniques.
Here, molecular beams deliver rotationally cold molecules in their ground vibrational state moving with controlled and narrow speed distributions, while detecting scattered molecules with LIF or REMPI.
Early experiments done with NO showed strong rotational excitation and no vibrational excitation. [222][223][224] Rotational excitation increased with incidence kinetic energy and a broad nonthermal peak in the rotational state distribution emerged 224 ; see Figure 11(a). Increased rota-tional excitation in the scattered molecules arises from a loss of translation energy (T → R coupling), Figure 11(b). This conclusion was supported by independent measurements for NO scattered from Au (111) showing decreased translational energy of the scattered NO when produced in higher energy rotational states. [225][226][227] The nonthermal rotational states populated at high incidence translational energy were interpreted as a "rotational rainbow," where a specific orientation angle of the molecule with respect to the surface leads to maximum rotational excitation. The rotational rainbow gets its name from the mathematically analogous optical rainbow, where a specific impact distance of a light ray from the center of a water droplet leads to a maximum scattering angle and an intensity maximum at that angle. 228 Classical trajectory and quantum wavepacket calculations based on empirical potentials provide support for this interpretation 229,230 and rotational rainbows have since been seen many times. 229,231-237 Since then, it was possible to see an N-side and an O-side rainbow for NO scattering using oriented beams of NO 238 and to witness a rainbow in formaldehyde, where the rotation about the CO bond axis exhibits a rainbow. 239 Combining TOF with REMPI provides velocities of scattered, stateselected molecules. Knowing the initial and final translational and rotational energy, we can compute the energy transfer to the lattice.
Interestingly, the energy going into the lattice depends on the degree of rotational excitation; molecules that undergo more rotational excitation transfer less energy into the phonons of the substrate, in good agreement with theoretical calculations. 225,240 This anticorrelation of rotational excitation and phonon excitation seems also to be a ubiquitous feature of molecular scattering from surfaces. 9,227,238,241

State-resolved detection of scattered molecules, vibrational effects
Observations of vibrational excitation were reported for REMPI detected NH 3 after its collision with Au(111) surface-umbrella motion in NH 3 gives rise to low-lying vibrational levels that become increasingly populated with increased incidence transitional energy. 242 Observed thresholds tellingly close to the detected state's vibrational excitation energy showed that a minimum of incidence energy was needed to produce each new vibrational states (T→ V coupling). The excitation probability was, furthermore, insensitive to T S . The authors concluded that vibrational excitation was occurring via a direct (single bounce) adiabatic ("mechanical") coupling of the incoming normal motion to the vibrational modes of the molecule.
Efficient vibrational excitation from v = 0 → 1 was also observed in collisions of NO with a hot Ag (111) 243 surface. In contrast to the results for NH 3 , the excitation probability increased exponentially with T S , displaying an Arrhenius-like behavior with an activation energy equal to the NO vibrational spacing. Furthermore, no thresholds were seen in the incidence energy dependence. The authors argued that NO vibrational excitation resulted from an electronically nonadiabatic coupling of NO stretch motion to thermally excited EHPs in the metal and that the coupling increased at higher incidence energies as a closer approach was possible. This interpretation was supported by later measurements of the velocity distributions of scattered NO in the vibrationally elastic and inelastic channels showing the energy for vibrational excitation did not come from translational motion. 244 Since this discovery, the literature has filled with reports of similar observations-systems where hot EHPs excite vibrations include: HCl on Au, 245 CO on Au, 246 and Ag. 247 For NO on Au, it was even possible to see Δv = 1, 2, and 3, each displaying an Arrhenius-like behavior with an activation energy equal to ℏ vib × Δv. 248 In these experiments, the production of NO(v = 3) with its vibrational energy of 0.687 eV occurred in collisions of NO (v = 0) with a hot Au(111) surface at incidence energies of only 0.4 eV. 248 Figure 12 shows absolute measurements of excitation probabilities of NO(v = 1) and NO(v = 2) for NO(v = 0) colliding with Au(111) over a wide range of incidence energies and surface temperatures. 249 These data are particularly valuable for comparison with theories of nonadiabatic energy transfer because the availability of absolute measurements for both single and multiquantum excitation helps distinguish different theoretical treatments of the nonadiabatic coupling. We will return to this topic in Section 5.7.

Vibrational state-to-state scattering
Improved observations of vibrationally inelastic scattering are possible using a state-to-state approach, combining optical pumping of molecular beams with REMPI detection of scattered molecules. While experimentally more complex, this set-up also provides improved TOF capability, simplifying the identification of direct scattering channels versus trapping followed by desorption. Simply by varying the delay between F I G U R E 1 2 Excitation probabilities for NO(v = 0→1, 2) scattering from Au(111) as a function of surface temperature, T S , for two translational incidence energies, E i . Squares show results for NO(v = 0→1) and circles for NO(v = 0→2). The Arrhenius form possesses a prefactor, and an "activation energy" that is equal to the energy gap of the inelastic transition. Note that the prefactor increases with E i , but that the activation energies are independent of E i while being larger for NO(v = 0→2) compared to NO(v = 0→1). This reflects the stronger temperature dependence of the populations of higher energy thermally excited EHPs. 249 Note that the vibrational energy of NO(v = 2), 0.462 eV, is substantially higher than the incidence energy of 0.28 eV seen in one of the experiments shown here, proving that the vibrational excitation does not originate as incidence translational energy the two lasers and separating the laser beams from one another, stateto-state TOF can be performed. 226,227,250,251 IR pumping allows for the production of beams with substantial populations in the v = 1-3 vibrational states 227 and with stimulated emission pumping (SEP), much higher vibrational states can be reached. 252 SEP is the conceptual child of microwave-optical double resonance [253][254][255][256][257] and optical-optical double resonance spectroscopies. 258,259 In SEP, molecules are excited or pumped by one laser to an excited electronic state-subsequently, emission is stimulated by a second laser, "dump"-ing the excited state population back to the ground electronic state. By tuning the lasers so that the stimulated emission goes to a vibrationally excited state, this "pump-dump" approach populates the beam with highly vibrationally excited molecules selected by the frequency difference of the two lasers. In surface scattering, SEP is most often performed with pulsed nanosecond lasers. 260 The first application of SEP to surface dynamics 261 prepared NO(v = 15) and used REMPI to determine the final vibrational state distributions after collision with Au(111)-see Figure 13 Since that time, methods have continued to improve. Spontaneous emission from the intermediate "stepping-stone state" used in SEP produces vibrationally excited states indiscriminately, which is usually an unwanted background. The utility of pump-dump optical excitation improves dramatically when using a "sweep" laser that dissociates the stepping-stone state within a few ns after pump-dump has been performed, removing most of the spontaneous emission and concomitant background. 262 It also proved possible to develop a variant of SEP that was used to produce highly vibrationally excited CO exploiting perturbations. 263 Overtone pumping of HCl and NO to low lying vibrational states was another path to additional data.
We now have rich and extensive data on the inelastic scattering of vibrationally excited molecules colliding with metal surfaces-it is one of the best-studied examples of the failure of the Born-Oppenheimer approximation to describe molecular interactions at metal surfaces. [6][7][8][9] State-to-state data are now available for vibrational relaxation and excitation of HCl, [264][265][266]  While Ag and Au have many similarities, it is noteworthy that Ag possesses a substantially lower work function than Au (4.5 vs. 5.3 eV).
NO vibrational relaxation is believed to occur by an electron transfermediated mechanism involving a transient anion, NO − . If true, it would not be surprising that the work function plays an important role. Systematic control of the work function was achieved using atomically layered films of Ag grown on Au (111). 272 Silver grows layer by layer on Au, hence, only when the nth layer closes does the n+1th layer begin to grow. Evaporating Ag onto Au with a Ag-beam block moving continuously across the Au crystal allows fabrication of an atomically defined edge structure-see inset of Figure 14  Experiments with oriented NO were also developed, where either N or O faces the surface upon collision. 238,273,274 Vibrational relaxation is more efficient when N is oriented toward the surface than away, 274,275 consistent with theoretical predictions that electron transfer is more labile for this orientation. 276

Vibrationally promoted electron emission
Taken together, there is compelling evidence that vibrational relaxation of NO and CO molecules colliding with a metal surface occurs via an electron transfer process. If vibrational energy loss resulted in low energy excitations of many electrons, electron emission could not occur. However, if many quanta of vibrational energy can be channeled to a single electron giving it enough energy to overcome the work function, not only will emission be possible, but it also will be a strong sign that "one electron does all the work," a concept consistent with an electron transfer process. Thus, looking for electron emission and measuring its quantum yield can teach us something very important about the dynamics of the EHP excitation.
When highly vibrationally excited NO with variable incidence vibrational energy was scattered from a surface with 1.6 eV work function, 279 (111) and Ag (111). The multiquantum vibrational relaxation is much stronger on Ag, whose work function (4.5 eV) is substantially lower than that of Au (5.  Figure 16. This directly shows that vibrational energy has been used to excite a single electron. Furthermore, this is not a rare or highly forbidden channel-the yield is large, reaching over 10% for v = 18. An inverse velocity dependence gives further evidence of a transient negative ion formation, since, due to increasing image charge stabilization, a newly formed negative ion must emit its electron before getting too close the surface. 278  The friction and random force satisfy the fluctuation-dissipation theorem, 306 such that the system properly approaches the desired temperature. In principle, the frictional terms include memory of the past evolution of the system. While memory effects may well be important to describe frequency-dependent friction due to, for example, electronic resonances or nonuniform densities of states, to our knowledge, memory effects have not been explored in this context. Rather, in practice, memory effects have been neglected. In this Markov limit, the equations that result for the electronic friction are given by FGR. [194][195][196]203 Note that the friction is a tensor of the components of the atomic velocity vectors, for example, for a diatomic molecule, it is a 6×6 tensor, and in general is not diagonal in either Cartesian or normal mode coordinates. 307 The implementation of FGR sometimes presents numerical difficulties, in part due to representing the electronic continuum by discrete levels. A DFT-based procedure, dubbed orbital-dependent friction (ODF), appears to overcome these issues and provides accurate and stable results. 308,309 ODF in the Markov approximation has been applied to compute lifetimes of the C-O stretch, CO-surface stretch, bend and frustrated translation vibrational modes of CO on Cu, Ag, Ni, and Pt surfaces, obtaining good agreement with experiment, where available. 308 One exception to this is CO on Au, where great care must be taken with electronic structure theory to ensure an accurate description of the physisorption binding in this system. 310 Importantly, the off-diagonal elements of the friction tensor are significant in some cases. ODF with electronic frictions computed on-the-fly for scattering and dissociative chemisorption of H 2 indicated that the major pathway for energy transfer was via the H-H stretching mode. 311,312 Moreover, they observed a dynamical steering due to tensorial friction that influenced the scattering. Overall, however, they observed a minor effect of nonadiabaticity on the probability of chemisorption, 313 in agreement with prior LDFA studies for H 2 on Cu (110) 299 and Ru(0001). 294 Recently, a symmetry-adapted neural network representation of the electronic friction tensor has been developed and promises to render ODF calculations quite practical. 314

Independent electron surface hopping
The experiments shown in Figure 13 (111). 249 The agreement is good. In contrast, an electronic-friction calculation using the same Hamiltonian used in IESH severely underestimates the excitation probability. 249 However, later experiments revealed shortcomings of the IESH calculations, 267 notably that predicted sticking probabilities were too high due to an unrealistically attractive adiabatic PES. 318 This artificially enhanced IESH's predicted probabilities for multiquantum vibrational relaxation at low incidence energy and led to a fortuitous agreement with the experimental observations of Figure 13. When comparisons were made at high incidence energies, where trapping was absent in the theory, IESH predicted too little multiquantum vibrational relaxation. This is at least partly due to the fact that the adiabatic PES used here also has no dissociation channel. Since then, a more realistic PES has been developed, one that is less attractive and allows for dissociative adsorption of NO on Au (111). 319,320 Adiabatic calculations using this PES predict enhanced multiquantum vibrational relaxation, but still substantially less than seen in experiment. A renewed attempt to test IESH using this improved PES is warranted, but this will also require accurate calculations of excited and charge-transfer states. New electronic friction approaches have also been reported 315 and reproduce some of the data seen in the laboratory-but multiquantum vibrational relaxation is still not captured by friction theory as it predicts that only low energy EHPs can be excited.
IESH still appears to be the theory in front-runner status to eventually solve the problem of NO vibrationally inelastic scattering on noble metals and while it is tempting to assign much of the disagreement between simulation and experiment to the input PES used in IESH, other more fundamental issues may prove important. Specifically, the approximations invoked to construct the (N+1) × (N+1) diabatic Hamiltonian were relatively crude and remain untested. Further work to test IESH against available experimental results is clearly needed. fact that H atoms stick to metals with high probability is suggestive that there must be large contribution of EHP excitation to the energy loss, since otherwise, due to the mass disparity, it would not be possible for an incident H atom to lose enough energy to adsorb. Hence, it was long suspected that EHP excitation is needed for sticking of H at metals. 289,322 Furthermore, calculations of the energy loss by the best adiabatic methods predicted an energy loss of order 2%. 323 It appeared that Rydberg-atom tagging would easily provide the energy resolution necessary to detect any extra contributions to the energy loss due to nonadiabaticity. Both experiment and theory could be extended to H and D scattering from Pt, Ag, Pd, Cu, and Ni 296 . In each case, a PES was generated by fitting an EMT function to DFT data, using a genetic algorithm. 324 As with Au, there is excellent agreement between data and simulations for H and D atom scattering from these metals. 296 In all cases, EHP excitation dominates the energy loss-a small mass-dependent contribution to the energy loss from phonon excitation was also identified. Experimental and simulated angular distributions are also in good agreement-they are broad but clearly not due to trapping followed by thermal desorption. The results from metal to metal are so similar that one is tempted to conclude that there is a nearly universal behavior-the dynamics depend mainly on the metal's electron density and weakly on its mass.

High-resolution inelastic scattering-Rydberg-atom tagging
Examination of the classical trajectories used for the simulations revealed that even at the high incidence energies of the Rydberg-atom tagging experiments, sticking is efficient. Analytic expressions were fitted to incidence energy, E in , and angle, in , dependent sticking coefficients, S(E in , in , M), derived from numerical simulations. This provides a practical way to estimate the sticking coefficient for H or D to any metal, by knowing only the metal's mass, M.
where h is the Heaviside step function and the parameters for H are given by:

Influence of coverage on adsorption and desorption
The dependence of desorption rates on coverage can be complex and sometimes mystifying. Figure 22 shows a beautiful example, the isothermal desorption of Xe from W (110). The presence of attractive interactions between adsorbates can have striking effects on desorption rates. Here, there is a sharp change of slope in the rate which arises when a first-order two-dimensional phase transition occurs at = 0.3. 332 The free energies of the two phases are continuous through the transition, but the enthalpies and entropies that control the rate of desorption can be very different. 332

Influence of steps on desorption
Desorption can also take place from different surface sites, for example, from terraces and steps. The step desorption's pre-factor can be several orders of magnitude larger than that of terrace desorption. 333 This is an entropic effect where a step-bound adsorbate is constrained to live in a lower entropy 1D world compared to a terrace bound species. The reduced entropy of the adsorbate dramatically enhances the desorption rate.
Peculiar coverage dependencies can arise when adsorbates are able to diffuse to defects, which commonly bind molecules more strongly than do terraces. 334 Figure 23(a) and (b) shows the energetic landscape and the desorption measurements, respectively. In experiment, the ini-tial rate of desorption is rapid-terrace desorption-until the terrace sites are empty and then desorption decelerates being limited by the rate of step to terrace diffusion. 335

Detecting trapping/desorption
In the previous section, we have seen the results of many beautiful experiments that infer the nature of molecule-surface interactions from measurements of the quantum state, speed, and angular distributions of molecules undergoing direct scattering, conditions where the molecule has no time to reach equilibrium with the solid. Langmuir considered this process of molecular "reflection" already in 1917 and pointed out the importance of and difficulty associated with distinguishing "reflection" from condensation followed by evaporation. 336 With the experimental tools now available, this is readily accomplished.
Using a molecular beam with a narrow velocity distribution, Xe atoms scattered from Pt(111) exhibit distinctly bimodal TOF distributions, the slower fraction trapping and then desorbing, while the faster fraction scatters directly. 337 The angular distribution of direct scattering is also much narrower than that of trapping/desorption. Naturally, the residence time and the speed distribution of trapping/desorption depends on T S . Figure 24(e-j) shows vibrational state-to-state TOF experiments. 310 Here, CO(v = 2) is prepared in a molecular beam just 0.5-mm before collision with a Au (111)  states of CO appears to increase with T S . This system has also been studied by molecular dynamics on an HDNN PES. 338 Inspired by ideas developed in this work, it was possible to show that collisions of CO on Au(111) first pass through a metastable chemisorption well before partially equilibrating in a physisorption well, the lowest energy surface binding site. 310

Detailed balance, or why the desorption rate depends on the sticking probability
The properties of thermal equilibrium can be used to great advantage when studying trapping and desorption. To see this, consider adsorption in the language of TST, where the flux through a "point of no return" dividing plane in the adsorption direction is: where k B is Boltzmann's constant, ( , T) is the chemical potential of the gas, Q 2D is a simplified two-dimensional ideal gas partition function for a noninteracting adsorbate, and h is Planck's constant-see Ref. [339] and Figure 25.  (2) thermal diffusion from steps to terraces followed by thermal desorption. 334 (b) Experimentally observed bi-exponential desorption predicted by this model of CO desorption from Pt (111). 335 The fast component is simple desorption from terraces, while the slow component is a sequential process involving thermal diffusion from steps to terraces followed by desorption from terraces Of course, some molecules that pass through the dividing plane may not trap, but instead bounce back from the surface introducing a recrossing correction that is the sticking probability, P S ( , T). The recrossing corrected TST rate of adsorption then becomes: Note that at equilibrium, the adsorption and desorption rates must balance.
R des = R ads = P S ( , T) F TST ( , T) Remarkably, the rate of desorption is proportional to the sticking probability-this is, at first glance nonintuitive, but it expresses the F I G U R E 2 5 Transition state theory of adsorption and desorption. It is convenient to define the point of no return to be a plane parallel to the surface at large distance. When this is done, the thermal sticking coefficient can be used to correct the recrossing error in TST.
(a) High-energy molecules are less likely to stick. This appears as a recrossing error in TST. Consequently, desorption rates are higher for low energy molecules and the translational temperature of desorbing molecules can be lower than T S . (b) Low energy molecules are more likely to recross and energy distributions of desorbing molecules can be hyperthermal principle of detailed balance. The nonequilibrium dynamics of desorption is encoded in the sticking probability's dependence on incidence conditions, coverage, and temperature. 340 In fact, Equation (5) Measurements of state-specific and velocity-resolved rates of desorption reliably predict their corresponding sticking probabilities. 341,342

Application of detailed balance
The analysis of the velocity distribution of the trapping-desorption fraction for Ar scattered from hydrogen-covered W(100) 343 showed that at low T S the mean energy of desorbed atoms was 2k B T, in accord with a Maxwellian distribution at the T S . This is consistent with a near-unity trapping probability at low gas and surface temperature. At higher surface temperatures, the mean energy of the desorbed atoms was markedly lower than 2k B T, consistent with a decrease of the trapping probability at higher gas and surface temperatures. Furthermore, at low surface temperatures, the trapping-desorption fraction obeyed the cosθ angular distribution required if the sticking probability were unity, but at higher surface temperature, the angular distribution was observed to be broader than cosθ. This indicates that the trapping probability depends more strongly on the component of gas momentum normal to the surface plane than parallel to it, resulting in a larger decrease of normal momentum than parallel. Molecular beam and computational studies of Ar scattered from Pt (111) 344 confirm that momentum in the normal direction is accommodated more rapidly than the parallel component, and that at higher surface temperatures for which the Ar residence time is less than 100 ps, atoms desorb prior to accommodating their parallel component of momentum.
Dissociative adsorption and recombinative desorption of molecules is more complicated and more interesting than that of intact adsorption/desorption. The sticking probability, in principle, can depend on not only the surface temperature and the initial translational energy and angle of approach of the molecule, but also on its initial electronic or spin-orbit state, the initial vibrational state, the initial rotational energy and polarization, and the initial orientation of the molecule.
The design and fruition of extraordinary spectroscopic and molecular beam techniques, coupled with advanced computational modeling, has provided detailed and quantitative knowledge of the dynamics of molecular bond-breaking and making at surfaces. 345 Recently, detailed balance together with an elaborate microkinetic analysis has been used to show that adsorption to a physisorption well may be facilitated by transient chemisorption in a metastable well with stronger molecule surface interactions. 310

DYNAMICS OF REACTIONS AT SURFACES
Even prior to the quantum revolution, Langmuir and others were thinking about atoms and molecules on surfaces. 346 An alternative to this mechanism is that of Eley and Rideal. 350 In this "ER" mechanism, a gas-phase atom or molecule collides at the binding site of a chemisorbed atom or molecule and reacts without coming into thermal equilibrium with the solid. Modern experimental methods readily distinguish LH from ER, since the energy available to the products is typically much higher for ER reactions than for LH 351less chemical energy is lost to the solid 351 -and ER reactions exhibit a "memory effect," 352,353 where the speed, angle, or quantum state of the incident reactant influence the speed, angle, and quantum state of the product. This is obviously not the case for an LH reaction where the reactants equilibrate with the solid before reaction.
Intrinsic to LH is the idea that adsorbed reactants thermalize with the surface and that products form at a speed controlled by thermal reaction and diffusion. Nevertheless, LH reactions can produce hyperthermal products-for example, when two adsorbed H atoms thermally desorb from a copper surface, they must overcome a substantial barrier; the H 2 formed at the barrier has no time to equilibrate with the solid and is ejected from the surface with a great deal of translational (and vibrational) energy. 354 These nonthermal effects lend themselves to state, speed, and angle-resolved experiments that are particularly sensitive to the PES of the reaction in the vicinity of the transition state, allowing the extension of the dynamical approach from the gas-phase to reactions at surfaces.
This section contains highlights of work on nonthermal ER reactions and direct dissociative adsorption, together with an exposition of the problems of measuring and predicting the rates of thermal reactions.
Arguably, the most important goal of the dynamical approach for surface chemistry is to accurately predict thermal reaction rates-see, for example, Ref. [355]. Thermal surface reactions are by far the most common in nature and most required for practical use. Predicting thermal rates represents the true payout for a highly developed theoretical understanding. For this, we need detailed dynamical experiments capable of probing the key features of the PESs of elementary reaction steps, thereby testing the computational methods used to generate them. We also need means to determine reaction mechanisms-LH versus ER for example-but even more basic than that, we must find out which elementary reactions are important, and determine the active sites of those reactions.

H 2 on copper
One of the best understood systems is the reaction H 2(g) Cu (111) ⟷ 2H * , which has been studied in both directions and for different isotopologues and previously reviewed. 8 The reaction has served a similar role for the theoretical development of surface chemistry as the H 3 reaction has for gas-phase reactions. Experiments on dissociative adsorption and associative desorption show that there are two reaction mechanisms, an activated dissociation process that is promoted by both reactant vibration and translation 356 and another, still unexplained reaction inhibited by translational energy but promoted by vibration. 357,358 The translationally activated reaction and corresponding measure-  (111). 363 The reaction has also been studied in this way on stepped surfaces 2H * Cu (211) ⟶ H 2(g) . Unlike many surface reactions, here steps are somewhat less reactive than terraces. 357 Application of the dynamical approach to this reaction involves improving theory to reach agreement with data like that just described, in this case optimizing a Born-Oppenheimer PES, using a semiempirical specific reaction parameter (SRP) functional. 363 The fact that this has proven possible is apparently due to the fact that the most important complications anticipated in moving from the gas phase to metal surfaces-the influence of phonons and EHPs-do not appear to affect this reaction greatly. 364 HCl on gold and silver The situation appears to be much more challenging for the reaction HCl The experimentally derived reaction probabilities were much smaller than predicted by theories using a 6D Born-Oppenheimer PES with quantum wave packets. 366 Some improvement was found using a different functional. 367 Classical AIMD calculations that allow the surface atoms to move also failed to bridge the gap to experiment 368 ; furthermore, the reaction path and barrier height depend strongly on choice of functional. 369 A new PES has been developed with a higher reaction barrier but agreement with experiment is still poor. 265 In related work on the ER reaction: H (g) + Cl * Au (111) ⟶ HCl (g) , AIMD calculations produced HCl vibrational excitation far larger than that seen in experiment. By including LDFA friction, vibrational excitation was reduced 370 and trajectories showed H atoms lose energy before reacting with Cl * , a plausible influence of electronic friction on reactivity. But the HCl vibrational distribution was still much hotter than seen experimentally-compare figure 18 of Ref. [352] with figure 2 of Ref. [370], possibly an indication of the limitations of the LDFA.
While the research on this reaction, so far, is no success story, it is without doubt one of the most interesting for future study.
It has recently been suggested that for reactions where the difference between the surface' work function and the electron affinity of the adsorbate is smaller than 7 eV, DFT-GGA calculations of barrier heights may be unreliable. 371 As seen in Figure 15, it is precisely under these conditions that electronically nonadiabatic coupling between F I G U R E 2 6 State-resolved sticking coefficients, S 0 , for CH 4(g) in the 1 (▴), 2 3 (□), 3  The influence of vibrational excitation on dissociative adsorption was first seen by changing the beam source temperature and seed gas to thermally excite CH 4 vibrations. 380,381 This influence of vibration is more cleanly seen with laser pre-excitation of the molecule, where a molecular beam of methane with controlled translational energy is excited by a c. w. infrared laser with high power and coherence producing selected vibrational states. Auger electron or reflection absorption infrared spectroscopy (RAIRS) is then used to look for the buildup of reaction products on the surface. 375 Studies like this have demonstrated mode specificity 379,382,383 -see Figure 26-and bond selectivity 384 as well as steric effects 385 in chemisorption reactions, highlighting the nonstatistical and complex nature of gas-surface reaction dynamics. 373 These studies have also demonstrated that surface atom motion plays an important role in determining the ease with which the gas-phase molecule surmounts the reaction barrier. 386 It is even possible to reveal the dynamics of reaction at specific surface sites. 387 While the many degrees of freedom of methane make fulldimensional quantum calculations challenging, 388 an approximate 15 DOF quantum method has been demonstrated using a reaction path Hamiltonian, 389 originally developed for gas-phase problems. 390 The "quantum reaction path" method provides an accurate description of the translational motion and nine internal molecular DOFs of methane, and while a vibrationally adiabatic basis set is used, all vibrationally nonadiabatic couplings are included. This method was the first to succeed in capturing mode-specific reactivity like that shown in Figure 26.
It also avoids an artifact of classical mechanics, where zero point energy can flow within the molecule-such effects are difficult to avoid when using QCT and AIMD trajectories. 391 Beyond this, the quantum reaction path method also helped reveal the influence of surface atom motion on the reaction, seen experimentally as a strong surface temperature dependence of the reaction probability. 392 DFT calculations show that the barrier to dissociation is modulated by the out-of-plane motion of the metal atom most intimately involved with the transition state. 393 For methane on Ni, the effect of this on the reaction probability was treated in an approximate way, using a lattice sudden model, 394,395 which effectively averages over the barrier height and momentum distributions along the reaction coordinate produced by thermal motion of the Ni atom. 396 More recently, this approximate scheme has been validated by 8D quantum dynamics calculations on a 14D PES. 397 This and the previous studies all conclude Ni-lattice motion is involved in the reaction.
Observing adsorbed reaction intermediates and products with surface-site specificity can be achieved using RAIRS to detect products, and preparing state-selected reactants in a molecular beam.
Using CH 3 D with either the C − H or C − D bond pre-excited, RAIRS can distinguish adsorbed products-CH * 3 or CDH * 2 -at different surface sites. 387,398 For reactions on Pt (211), at low incidence energies of translation and without vibrational pre-excitation, only dissociation at steps was observed, without isotopic selectivity. However, with vibrational pre-excitation of the C − H bond, only infrared absorption corresponding to CDH * 2 bound at steps could be seen at low incidence energies of translation. At higher translational energy, dissociation at terraces appeared and bonds without vibrational excitation became more reactive. 399 Note that under the conditions of these experiments, CH * 3 diffusion is believed to be unimportant. The authors concluded that the barrier to reaction at steps is at least 0.3 eV lower than at terraces 387 -DFT predicted barriers for dissociation at steps and terraces of 0.55 eV and 0.82, respectively, when using an SRP functional. 400 The dissociation of water on metals has also attracted attention, inspired by vibrational state selected molecular beam measurements of dissociation probabilities for the reaction Ni (111) ⟶ OD * + D * . 401 Here, vibrational efficacy is larger than the translational, probably related to the late barrier to reaction 401 -see the Polanyi rules (Section 2.1). 48 The quantum reaction path method was also applied and after rescaling the barrier height, good agreement with experiment was found. 402 These calculations exhibited bond selective dissociation for HOD and again, a strong influence of surface atom motion. A nine DOF PES was also produced and site specific reactivity could be studied 403 ; a site averaging model was also tested. 404 Scientists have also begun to explore the properties of water dissociation on other surfaces, like Cu, 405,406 AgNi, 407 and Cu/Ni alloys. 408,409 In contrast to these reactions, where the molecule travels over the dissociation barrier on a sub-picosecond time-scale, LH involves newly adsorbed reactants rapidly equilibrating with the solid. Does one then expect vibrational promotion of LH reactions? Recently, the vibrational relaxation lifetime of molecules bound by physisorption interactions has been measured to be ∼50 ps, 199 more than an order of magnitude longer than vibrational relaxation for chemisorbed molecules. 190,410 This helps to explain why it is possible to observe the trapping followed by thermal desorption of a vibrationally excited molecule. 411 These observations suggest that while the LH mechanism involves thermalization of reactant translation and rotation, reactant vibration may relax more slowly and may live long enough to accelerate surface reactions prior to the vibrational energy being lost to the solid.
In an experiment similar to those described in Ref. [375], CH 4 dissociation probabilities on Ir (111) were obtained for selected vibrational states as a function of translational energy. 412 QCT simulations on a PES that had been fitted to 5000 DFT points were performed to obtain the sticking probability dependence on surface temperature as well as incidence translational and vibrational energy of CH 4 . Remarkably good agreement with experiment was found. 412 Further analysis of the trajectories showed that at low incidence energy, adsorbed molecules with unrelaxed vibrational excitation could dissociate with higher efficiency than vibrationally cold molecules. TST rate calculations assuming the precursor-mediated mechanism suggested that vibrationally excited states might react at surface defect sites. 413 It is well known that while molecules initially adsorb to majority sites, diffusion to minority defect sites like steps is often much faster than desorption and reaction rates at these defects can be much higher than at majority sites. This intriguing work suggests the same might be true for vibrationally excited molecules physisorbed to catalytic surfaces.

The influence of nonadiabatic electronic effects on reaction probabilities
The question of how strongly Born-Oppenheimer failure influences surface chemical reactions remains unanswered at this time. One problem is that nearly all theoretical studies have so far been limited to modelling electronically nonadiabatic effects with electronic friction at the level of the LDFA. 293,414 In such calculations, electronic nonadiabaticity is typically small, for example, for 415 (111), whereas an IESH model involving strong coupling via an electron transfer reaction gave good agreement 249 -see Figure 12.
Other experimental evidence that tends to contradict the conclusions of LDFA-based friction studies comes from studies where observed chemicurrents, 418 which were attributed to the reaction of 2H * Au (111) ⟶ H 2(g) , and from permeation studies of the same reaction. 419 Further development of electronically nonadiabatic dynamical propagation algorithms is clearly needed, for example, ODF, 309 and methods beyond those involving a weak coupling approximation. 276,316 We also clearly need studies of a wider variety of reactions.
One example of recent progress in this direction deserves mention. Experiments and theory on the dissociative chemisorption, [420][421][422] inelastic scattering 423 427 This was attributed to the charge redistribution that accompanies the formation and breaking of chemical bonds. 427 Since then, using neural networks to fit DFT data, a full-dimensional PES has been produced. 428 This allowed study of the effect of surface atom motion on the reaction. The same PES was used to make a direct comparison between LDFA and ODF methods-this revealed that the latter achieves better agreement with experiment for reaction probabilities and vibrational energy distributions, although it slightly underestimates translational energy loss. 309 While dissociative adsorption of N 2 on Ru has proven a valuable testing ground for these new theories, we caution that the experimental dissociative adsorption probabilities being compared to exhibit large error bars. [420][421][422] Thermal reaction rates Molecular beams have long been used to measure surface reaction rates. 21 One of the most successful approaches is molecular beam relaxation spectrometry (MBRS) 436,437 -see Figure 27. For a simple reaction like first-order desorption, the molecular beam delivers and replenishes the adsorbate concentration and the time-dependent product signal is used to extract the desorption rate constant. Note that when the sticking probability is independent of coverage and tem- dissociates more rapidly at steps. 454,455 The methods necessary to solve some of these problems became available only after the adaptation of slice ion-imaging to reactive surface scattering 456-458 -see Figure 28-providing numerous advantages for obtaining rates of elementary surface reactions.
The primary advantages of this method come from its ability to measure simultaneously the product density and speed, leading to the product flux, also called the kinetic trace. Consider a simple first-order reaction, CO desorption from a surface, CO * k des (T) → CO (g) , governed by the following kinetic equation: The reaction rate, R des , has units of cm −2 s −1 -a flux. An experiment that automatically provides the flux of desorbing molecules is highly advantageous. Knowledge of the product's speed is also used to correct each ion signal for the time spent flying from the surface to the laser ionization volume, t 2 in Figure 27 for MBRS. 436 A third advantage is that different reactions may produce the same product but with different speeds. Figure 29 shows the results of ion imaging applied to the CO oxidation reaction on Pt (111). 456   Pd. 466,467 When MBRS was applied to CO oxidation on Pd(111), the derived activation energies increased from low to high O * coverage. 438 This was rationalized as having to do with the influence of islands at high coverage that became less important at low coverage. 438 The O * -islands discussed so far exist at equilibrium, and phase diagrams of O * -islands on Pt (111) have been calculated 468  showed that O * 2 dissociation is more likely in the vicinity of O * . 465,470 This set of observations reveals how dissociation kinetics can lead to highly nonequilibrium surface structures, resulting from a dynamic heterogeneity in the adsorption mechanism. In short, a kinetically determined ordering of the adsorbate.

Making measurements on the living catalyst
These studies point out that a catalyst is dynamic, generating active sites due to exposure to reactants, a "living catalyst." 471 This presents another experimental challenge. Since measuring rates of surface reactions requires signal averaging, the catalyst must return to its original state at the end of each measurement to allow for averaging. The dynamic catalyst may make this impossible.
Recently, velocity-resolved kinetics using high-repetition-rate pulsed laser ionization and high-speed ion imaging detection has been achieved. 472 This overcomes the time-consuming scanning of the delay F I G U R E 3 0 Comparison of delay scanning (a) versus high-rep-rate detection (b). In delay scanning, we average many ion images at each delay between the initiating molecular beam pulse and the ionizing laser pulse, generating the kinetic trace. For high rep-rate detection, we record many points in the kinetic trace for each molecular beam pulse, increasing the duty cycle by orders of magnitude, enabling measurements while the catalyst is changing. 472  faster. In addition, this approach overcomes the diffusion problem arising in many molecular beam reactive scattering experiments. 473 To demonstrate the problem, consider that if one starts with an O *saturated Pt surface and doses it with CO pulses; then, O * is removed more rapidly from the part of the surface where the center of the molecular beam is incident. As time passes, most CO 2 formed from CO that adsorbed near the center of the beam where O * is depleted, and diffused to the edges of the beam where O * is still present. Using high rep-rate velocity-resolved kinetics, the diffusional influence could be observed and quantitatively modeled. 472 Exploiting our dynamical understanding of surface reactions to obtain a predictive theory of reaction rates remains yet an unfulfilled goal, as current limitations of the tools needed to make accurate measurements of reaction rates at well-characterized active sites have prevented meaningful tests of rate-theory. However, with experiments and theory both improving, we are increasingly benefitting from the fruitful interplay and we stand at the threshold of bringing this effort to fruition. Comparing to another subfield of complex chemistry provides some perspective. While surely not more complex, the study of heterogeneous catalysis has suffered from a structural deficit compared to enzymology. Methods to identify enzyme active sites-X-ray and electron diffraction-preceded by decades the theory of enzyme rates and dynamics at the transition state. Analogous methods to routinely identify surface active sites lay still in the future. In comparison to enzymology, we are still groping a bit to realize the structure function relationship. Nevertheless, as we learn more about active sites in heterogeneous catalysis through a fruitful interplay of experiment and theory, there is every reason to be optimistic that a predictive theory of rates will become available. After nearly a century, the dynamical approach is the gift that just keeps on giving, finding application to understand increasingly complex chemistry.

OUTLOOK
In this penultimate section, we wish to express our opinions regarding important directions for future research. This review has cited very little research on electronically inelastic scattering-there is very little.
It should be noted that an instrument using a Stark decelerator for surface scattering has been reported 474 and used to investigate electronic quenching of CO a 3 Π on metals with and without adsorbates. 475 This is certainly an interesting direction for the future.
Perhaps of greatest concern to us is the need to compare theory and experiment on the level of rates of elementary reactions. Here, many more rates must be measured accurately and these must be compared to rates calculated from first principles theory. The experimental methods are now available to obtain accurate rates on simple model surfaces and these must be extended to more complex surfaces that reflect new phenomena present in real catalysts. It will only be with the development of predictive theoretical methods, validated by experiment, that we may one day achieve useful microkinetic models of real catalysts.
An important target is an experimental tool for routine use that can reveal the structure of active sites of surface reactions in real catalysts.
Here, the combination of near-field infrared spectroscopy with scanning probe methods 476-478 may be of great importance; however, it will need to work at temperatures relevant to catalysis. Theory can also be crucially important here, for example, providing ab initio equilibrium structures present under realistic catalytic conditions. 479 We also need new theoretical tools. DFT-GGA is still suspect when it comes to calculating reaction barriers. Accurate wave function-based electronic structure theory 480 perhaps employing self-consistent embedding methods [481][482][483] or quantum Monte Carlo 484 could at least provide benchmarks for methods. They also have the potential to offer accurate information on excited electronic states. The approximate methods must themselves also improve. Hybrid functionals applied to GGA densities is one promising direction. 371 A nagging question is to what extent energy dissipation to the solidfor example, due to coupling to EHPs-influences the rates of surface reactions. The influence of friction on reaction rates in liquids and biological systems has been widely demonstrated and quantified, based on the pioneering theory of Kramers. 485 At surfaces, both phonon and EHP dissipation must be considered. For the latter, we require demonstrably correct theoretical methods for electronically nonadiabatic dynamics. Current friction approaches while improving 314 are unlikely to be sufficient as they rely on a weak coupling approximation and are likely not applicable to problems that exhibit strong and localized nonadiabaticity, such as electron transfer. IESH still represents a fruitful avenue of future study and improvement. 316 We also need measurements on a wider variety of systems. Systems for which The importance of quantum effects in surface chemistry needs much more effort. Recently, H atom beams have been produced at energies as low as 0.2 eV and with such narrow speed distributions, they could not be measured by Rydberg-atom tagging. 486 These could deliver a great deal of excellent quantitative scattering data capable of testing quantum mechanical theories of reaction dynamics. Highdimensional quantum calculations, for example, by unifying the power of multi-configuration time-dependent Hartree dynamics 487,488 with that of neural network PESs, 489 appear to be within reach. However, at least until quantum computers become widely accessible, fully quantum mechanical descriptions of catalytic systems are likely to remain intractable. Hence, accurate mixed quantum classical dynamics are extremely important.
Another interesting condensed phase quantum system is CO adsorbed to NaCl. In Section 5.1, we pointed out that CO vibrational relaxation is extremely inefficient. This allows high-resolution infrared emission spectroscopy to be carried out from which two orientation isomers have been identified. 490 Normally, CO adsorbs with the C-atom close to a Na + of the NaCl surface, but an "upside down" isomer has also been observed. 490 Remarkably, experiments reveal well-resolved line spectra. This problem has already drawn interest from theorists developing PESs 491,492 applying quantum dynamical methods. 493 The challenges are formidable, but this system offers a proving ground for testing quantum dynamical methods in the high dimensions of condensed phases, providing a blueprint for exciting experimental and theoretical advances that may lead to fascinating discoveries, and take us systematically toward the dream of predicting and watching the intricate motions of individual atoms during a catalytic reaction.

CLOSING REMARKS AND APOLOGIES
In this manuscript, we have not attempted a comprehensive review of the field of chemical dynamics. Rather, our aim has been to highlight the most important concepts and present selected examples from gasphase dynamics that illustrate how successful the dynamical approach has been and to extend the discussion to encompass the complexity and richness of reactions at surfaces. We have attempted to draw a line from the discovery of quantum mechanics, which inspired chemical dynamics, to the time of this writing in the first days of 2021. This perhaps foolishly ambitious undertaking means we must apologize to all those who do not find their work reviewed here. Beyond limits of time and space, it is hardly possible to be aware of all the significant work occurring over nearly a century. We do hope that those dedicated to the approaches of gas-phase chemical dynamics see here the possibility to extend the reach of their expertise to complex problems in other branches of chemistry.

ACKNOWLEDGMENTS
The authors acknowledge all of the students, post-docs, and coworkers with whom they have been so fortunate to work over the years. Thanks also go to Prof. Igor Rahinov for carefully reading an early draft and providing many helpful comments. The authors also thank Mr. Hartmut Sebesse who prepared many of the figures. Finally, they thank Prof.s Joel Bowman and Arthur Suits for providing animations of roaming reactions and Dr. Alexander Kandratsenka for making movies of H scattering from graphene.

AUTHOR CONTRIBUTIONS
The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.