Roi Baer, Yehuda Zeiri and Ronnie Kosloff- Hydrogen transport in nickel (111)

8/3/2019 Roi Baer, Yehuda Zeiri and Ronnie Kosloff- Hydrogen transport in nickel (111 )

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Hydrogen transport in nickel 111

Roi BaerDepartment of Physical Chemistry and the Fritz Haber Research Center, The Hebrew University, Jerusalem 91904, Israel

Yehuda ZeiriDepartment of Chemistry, Nuclear Research Center-Negev, Beer-Sheva 84190, P.O. Box 9001, Israel

Ronnie KosloffDepartment of Physical Chemistry and the Fritz Haber Research Center, The Hebrew University, Jerusalem 91904, Israel

Received 13 November 1996; revised manuscript received 21 January 1997

The intricate dynamics of hydrogen on a nickel 111 surface is investigated. The purpose is to understandthe unique recombination reaction of subsurface with surface hydrogen on the nickel host. The analysis isbased on the embedded diatomics in molecules many-body potential surface. This potential enables a consis-tent evaluation of different hydrogen pathways in the nickel host. It is found that the pathway to subsurface-surface hydrogen recombination involves crossing a potential-energy barrier. Due to the light mass of thehydrogen the primary reaction route at low temperature occurs via tunneling. A critical evaluation of tunnelingdynamics in a many-body environment has been carried out, based on a fully quantum description. The

activated transport of subsurface hydrogen to a surface site, the resurfacing event, has been studied in detail. Itis shown that the tunneling dynamics is dominated by correlated motion of the hydrogen and the nickel hosts.A fully correlated quantum-dynamical description in a multimode environment was constructed and employed.The surrogate Hamiltonian method represents the nickel host effect on the hydrogen dynamics as that of aset of two-level systems. The spectral density, which is the input of the theory is obtained via classicalmolecular-dynamics simulations. The analysis then shows that the environment can both promote and hinderthe tunneling rate by orders of magnitude. Specifically for hydrogen in the nickel host, the net effect issuppression of tunneling compared to a frozen lattice approximation. The added effect of nonadiabatic inter-actions with the electron-hole pairs on the hydrogen tunneling rate was studied by an appropriate surrogateHamiltonian with the result of a small suppression depending on the electron density of nickel. The resur-facing rates together with surface recombination rates and relaxation rates were incorporated in a kinetic modeldescribing thermal-desorption spectra. Conditions for a thermal-programmed-desorption peak which manifestthe subsurface-surface hydrogen recombination were found. S0163-18299709216-3

I. INTRODUCTION

Hydrogen has a strong affinity for nickel, leading to theuse of nickel as a heterogeneous catalyst for hydrogenationreactions. When a hydrogen molecule approaches the nickelsurface it readily dissociates and adsorbs.1 When atomic hy-drogen approaches the surface it primarily adsorbs on thesurfaces, but a minority is absorbed into the crystal bulk.2,3

Once adsorbed or absorbed, the hydrogen may wander on the

crystal surface4 6 or plunge into the bulk. This motion in-volves crossing potential-energy barriers which separate thevarious interstitial and adsorption sites. Thus hydrogen mo-tion in nickel involves activated transport which, at low tem-peratures, is controlled predominantly by tunneling.7,8

When a cold nickel crystal containing hydrogen is heated,molecular hydrogen is ejected into the gas phase.2,3,9,10 Fornickel crystals which have been exposed to molecular hydro-gen, recombinative desorption takes place at high tempera-tures, peaking at 350 K. However, for crystals which havebeen exposed to atomic hydrogen, a recombination is ob-served at lower temperatures of 180200 K, in addition tothe peak at 350 K. These findings suggest that low-

temperature desorption is due to a recombination of subsur-face and surface atomic hydrogen species.

The present study was initially aimed at elucidating thedynamics involved in surface-subsurface hydrogen recombi-nation, and to seek theoretical support for this special hydro-genation mechanism.10 It was soon appreciated that unravel-ing the reaction mechanism would require a completeassessment of a variety of hydrogen dynamical processes inthe nickel host. The light mass of hydrogen emphasizesquantum effects. The dominant phenomenon is tunneling. Togain insight into the process, a critical evaluation of theoret-

ical approaches to many-body tunneling phenomena is re-quired. The failure of simple approximations has led to thedevelopment of theoretical tools for the assessment of tun-neling in dissipative environments. The odyssey of thepresent study finally returned to the original problem ofsubsurface-surface recombination, indicating the conditionswhich enable the existence of such a process.

Previous theoretical efforts for hydrogen on nickel havefocused primarily on determining the adsorption probabili-ties on a frozen crystal.11,12 Theoretical treatments of otherhydrogen-metal interactions have also concentrated on ad-sorption, the assumption being that detailed balance can ex-plain the experimental desorption results.1316 This approach

can, in principle, yield desorption-product state distributionswhen the multimode character of the metal host is not

PHYSICAL REVIEW B 15 APRIL 1997-IIVOLUME 55, NUMBER 16

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coupled to the reaction transition state. This is probably thecase for the H 2 /Cu system.

17,18 However, for recombinationof a subsurface hydrogen with a surface hydrogen, the reac-tion transition state is strongly influenced by the motion ofthe metal atoms, and detailed balance considerations cannot

yield a simple description.Most of the theoretical effort has been devoted toH 2/Cu adsorption, which has become the benchmark systemfor activated gas-surface dynamics. Detailed and accurateadsorption/desorption experiments, performed by Rettner,Michelsen, and Auerbach19,20 for H 2/Cu, have stimulated ex-tensive theoretical work.21,22 A few of the experimental fea-tures are understood, among them vibrational-assisted stick-ing and rotational effects. These studies show that thedynamics of adsorption, even with a few degrees of freedom,is complex, where steering effects and barrier locations havea strong impact on the dynamical behavior.

The benchmark system for nonactivated adsorption is the

hydrogen/Pd system, for which adsorption probabilities werecalculated by Gross23 for the six degrees of freedom of thetwo hydrogen atoms. This work showed that surface corru-gation has a role in determining adsorption probabilities. De-sorption state distributions were calculated by Gross for theexperimental temperature.

One of the important conclusions from the various theo-retical investigations is that reduced dimensionality treat-ments predict higher probabilities compared to those of fulldimensionality, for example, the resurfacing of bulk hydro-gen defined as the transport from a subsurface to a surfacesite.8 In this case the reaction-path one-dimensional 1Drates differed by two orders of magnitude from the full 3D

rates. This is due to the extreme sensitivity of the tunnelingprocess to any feature of the system. A similar discrepancy,between 2D and 3D results, is observed for Pd surface pen-etration by atomic hydrogen.24 Sticking probabilities also ex-hibit a large sensitivity to reduced dimensionality.22

The system under study is the low-temperature dynamicsof hydrogen within the metal bulk and on its surface. Thedifficulty in setting up a dynamical framework for the in-volved processes stems from the combination of the many-body aspect of the interactions, and from the quantum natureof hydrogen dynamics at low temperatures. Though directcalculations of up to six dimensions are possible,23 they arenot sufficient to capture the role of other degrees of freedom

in the system. Two types of many-body effects need to beaddressed: First, the interaction of an adsorbate with the pho-non modes of the crystal motion, and second, the nonadia-batic interaction with the conduction electrons of the metal.

Vibrational line shapes of adsorbed hydrogen, originatingfrom nonadiabatic interactions, have been investigated byPersson and Hellsing.25,26 For heavier molecular species,such as CO on Cu, the nonadiabatic and phononic line broad-enings were addressed by Head-Gordon and Tully,27,28 usingmolecular-orbital theory and classical dynamics.

Hydrogen diffusion on metals influenced by many-bodyforces has been studied by the canonical variationaltransition-state theory with small curvature tunnelingcorrection.29,7,30 These approaches are reasonably adequatefor overbarrier diffusion. For low temperatures the quality ofthe approximation depends on the validity of the neglectednonadiabatic curvature couplings. The present work develops

an approach to the problem of diffusion in metals. One of itsbenefits is that it is fully quantum mechanical. The approxi-mations used by this method are different than those used bythe small curvature reaction path methods.31

Explicit multimode bath dynamics is also treated using

path-integral approaches.32

Recently, direct real-time path-integral approaches33,34 have been developed, but these havenot yet been used for analyzing atomic diffusion on metals.Path centroid transition state methods3537 supplemented bymolecular-dynamics simulations3842 were recently used forassessing the low-temperature diffusion rates of atomic hy-drogen on nickel. Such indirect schemes are complementaryto the direct propagation method employed here. For diffu-sion in a perfectly periodic environment a direct band motionanalysis has been employed.4345 However, for the tempera-tures addressed here, such a treatment is inadequate, due tothe destruction of coherent tunneling caused by crystal vibra-tions and nonadiabatic interactions.46,47

Energy loss to phonons in gas-surface scattering, and itsassistance to sticking, has also received much attention. Moststudies have employed classical dynamics,48,49 which is suit-able for heavier elements than hydrogen.50,51 Some semiclas-sical methods52,53 and quantum-classical treatments5456

have also been used. A pure quantal calculation for inelasticenergy transfer and sticking probabilities was developed byStiles, Wilkins, and Persson57 and Jackson.58 The approach isbased on a one-phonon model, suited for direct, short-timeinteractions, where the simultaneous excitation of severalphonon modes may be neglected.

The tour through the different approaches to hydrogen-metal encounters eventually has to return to the original

problem. The unique subsurface-surface reaction mechanismamounts to a direct encounter between bulk hydrogen andsurface hydrogen atoms steered by the nickel atoms. It seemsthat only such a mechanism is able to explain the low tem-perature peak in the thermal-programmed-absorption TPDexperiments. A dynamical model able to explain these resultshas to include explicitly dynamics lattice.

The rate-limiting step in such a model is the transport ofthe subsurface hydrogen to the metal surface which will betermed resurfacing. It is known from experiment that hydro-gen contained in a cold metal resurfaces on a time scale ofhours.59 This is in contrast to the resurfacing time calculatedby using a frozen lattice which employed the embedded atom

potential60 EAM, found to be of the order of seconds.8 Thediscrepancy between the experiment and the calculation of23 orders of magnitude suggest that either the barrier struc-ture predicted by the EAM is wrong, or that resurfacing ratesare highly affected by either nonadiabatic effects or the crys-tal motion or by both. As will be shown in this study, thedissipative forces indeed are strong enough to explain thediscrepancy.

The most obvious role of the phonon motion is its influ-ence on the barrier height and width. The motion enhancesthe tunneling, and is known as phonon-assisted diffusion.Such an effect can naturally be taken into account by a sud-den approximation, exploiting the nickel-hydrogen largemass ratio. It will be shown that the simplest approach ofadiabatic separation leads to gross errors in estimating thetunneling rates.

The phonon motion can also induce a loss of coherence

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which acts to deplete the tunneling rates.32 The relative im-portance of phonon assistance and hindrance must be ascer-tained quantitatively. In this study the surrogate Hamiltonianmethod is used to treat the many-body effects,61 see Appen-dix A for a short description. Approximately three different

phonon baths are necessary to define the interaction withthe resurfacing hydrogen atom. The combined effect of thesebaths reduces the tunneling rate by as much as two orders ofmagnitude.

The surrogate Hamiltonian is also employed to study thenon-adiabatic electron-hole pair excitations. For this specificsystem the reduction of the tunneling rate is small.

The final process analyzed separately is the recombinationof two hydrogen atoms leading to H 2 desorption. Only alimited study is possible due to the high dimensionality ofthe process. The temperature dependence of the tunnelingrates shows a universal character where, at low temperature,the rate is temperature independent, and where, above a

crossover temperature, it changes to an Ahrenius relation.Finally all these processes are combined in a kineticmodel simulating the temperature program desorption ex-periments. Qualitative conditions where the subsurface hy-drogen recombines with a surface hydrogen at lower tem-perature than two surface hydrogen can be found.

II. POTENTIAL ENERGY OF THE H/NI111 SYSTEM

In the intricate dynamics of hydrogen with a nickel crys-tal, the potential-energy surface plays a key role. Thepotential-energy surface is the basic entity from which theforces are derived in a molecular-dynamics MD simulation.

It also governs the wave-function dynamics in a quantumcalculation. The main difficulty for the MD simulation taskis the large number of participating atoms which require acalculation of the forces for all atoms. This means that com-putations of large systems are possible only if the evaluationof the potential-energy surface PES is simple and eco-nomic.

Over a decade ago, Daw and Baskes60 proposed an ap-proach, the EAM, for calculating the PES of hydrogen inter-acting with transition metal surfaces i.e., Ni and Pd. TheEAM is a semiempirical method based on the density-functional theory.62 In the EAM, the total energy of the sys-tem is described by

Ei

Fi h ,i 1

2 ij i jR i,j , 2.1

where Fi(h ,i) is the embedding function of atom i in theelectron density of the neighboring atoms (h ,i), and i ,j isa pairwise electrostatic repulsion term which is cast into ananalytic function characterized by a number of parameters.These parameters, together with the embedding functions,are determined semiempirically by requiring that character-istics of the solid-hydrogen system be reproduced by theEAM potential. The quantities used are lattice constant, elas-tic constants, sublimation energy, vacancy formation energy,and the energy difference between the bcc and fcc phases.The EAM results in a very close agreement between thecalculated and experimental values for H/Ni and H/Pd sys-tems. In addition to its accuracy, the EAM is especially

suited for MD simulations since calculation of the forcesrequires only a small additional effort to the calculation ofthe potential.

More recently, Truong, Truhlar, and Garrett63 developed asimilar approach termed embedded diatomics in mol-

ecules EDIM. The approach was used for the investiga-tion of single, two, and three hydrogen atoms interactingwith a nickel surface. The agreement between these calcula-tions and other theoretical as well as experimental studieswas very satisfactory. For a single hydrogen interacting withthe metal surface, the EDIM is similar to the EAM. There area number of differences related to the functional form of theelectrostatic repulsion terms. Moreover, in the EDIM ap-proach there is a distinction between the screening constantsused in the calculation of electron orbitals located on surfaceand bulk atoms. For a larger number of hydrogen atoms,PESs are calculated by the diatomics in molecule formalismwhere the H-surface ground and excited potential are calcu-

lated by using an EAM-type procedure as for a single Hatom.In the present study, the EDIM approach was adopted. It

was used to evaluate the interaction potentials in all theclassical- and quantum-mechanical studies described below.The parameters were adopted from Ref. 63. The potentialalong a cut leading from the subsurface to surface site wascompared to an ab initio calculation based on a GaussianDFT method. The differences were less than 0.02 eV. Thepotential is designed to give the correct asymptotic descrip-tion of the H 2 molecule. In the following two subsections abrief description of the the H/Ni111 and H 2/Ni111potential-energy surfaces will be discussed.

A. PES for a single hydrogen atom near Ni111

The Ni111 surface exhibits four high symmetry adsorp-tion sites: on-top, bridge, and two threefold sites. Thesethreefold sites differ by the position of the second-layernickel atoms. The fcc threefold site has a nickel atom rightbelow it, while the hcp threefold site also termed threefoldhollow site has a nickel atom only in the third layer Fig. 1.

The EAM and EDIM Refs. 60 and 63 calculationsshowed, in agreement with experimental results, that thelargest H/Ni111 binding is obtained for the threefold hol-low hcp site. The differences between the binding of thehydrogen to this site and the other high-symmetry sites was

quite small in the range of 0.030.3 kcal/mole. Larger dif-ferences were obtained for the vibrational frequencies asso-ciated with these four adsorption sites.

The present study focuses on the dynamics associatedwith the emergence of subsurface hydrogen onto the metalsurface a phenomenon termed resurfacing. A cut of thePES of a hydrogen atom in a plane normal to the surface andalong one of the parallel directions is shown in Fig. 2. Herethe negative-Z values correspond to the subsurface positionsof the hydrogen atom. As can be seen in the figures, the moststable subsurface site for hydrogen is 1 below the threefoldhollow site. The binding energy at this octahedral subsurfacesite is approximately 0.5 eV smaller than at the threefoldhollow site on the surface see the inset in Fig. 2. Thus thesubsurface sites are metastable potential-energy wells. Whenthe zero point energies are taken into account this differenceincreases.

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There exist other subsurface bulk sites for the hydrogenatom, the one below the threefold fcc surface site being anexample. However, the octahedral subsurface site is the moststable position for the H atom below the surface and in the

bulk. This agrees well with the experimental data and withother calculations.2,10,3

Inspection of the PES in Fig. 2 shows that the subsurfacehydrogen at the octahedral site can resurface by followingtwo main paths. One is the direct pathway leading upwards

to the threefold hollow site, and the other is via a neighbor-ing tetrahedral subsurface site, which finally surfaces at thethreefold fcc site. Along the first route, the H atom has toovercome an energy barrier of approximately 0.6 eV. On thesecond path the energy barrier is approximately 1.6 eV. Such

a large difference in barrier heights suggests that the onlypractical way for the hydrogen to resurface is via the directroute leading from the octahedral to the threefold hollowsite.

The thermal motion of the lattice atoms may result in aslight opening of the hexagonal structure, which tempo-rarily lowers the energy barrier for the surfacing motion.This possibility is shown in Fig. 3, where a cut of the mini-mal potential-energy path along the perpendicular Z direc-tion is shown. When the surface Ni-atom triad are symmetri-cally displaced, increasing their mutual separation, thebarrier for hydrogen surfacing is significantly reduced. Notethat the stability of the metastable potential well is also in-

creased as a result of this motion. These energetical aspectsof the lattice motion will be addressed in the sections con-cerning the hydrogen atom dynamics see Secs. V and VI.

B. PES of the interaction between two hydrogen atoms

and Ni111

The binding energy of two hydrogen atoms to two neigh-boring threefold sites on the surface is approximately 2.6 eVper H atom. Hence, in the event of recombinative desorptionof two hydrogen atoms, 5.2 eV of energy has to be suppliedto break the H-surface bonds, while only about 4.7 eV aregained in the creation of the H 2 bond. As a result, the de-sorption process is endothermic by approximately 0.5 eV. A

cut in the six-dimensional PES describing a recombinativedesorption process of two surface hydrogen atoms is shownin Fig. 4.

In Fig. 4 the PES as a function of the H-H separation rand the H 2 center-of-masssurface-normal distance Z are

FIG. 1. The Ni111 surface and the positions of the hydrogenatom. The surface hydrogen in the picture is located in a threefoldhollow hcp site.

FIG. 2. The EDIM-based po-tential energy surface of hydrogenin nickel as a function of the dis-tance from the metal surface (Z)

and parallel to the surface coordi-nate (X). The low potential pas-sageways intertwining amonghigh-potential-energy mountainsof the metal atom cores are clearlyobserved. The inset focuses on themost stable subsurface site, thethreefold octahedral site, which ismetastable with respect to the sur-face binding hcp and fcc bindingsites. The H atom tunnels from itto the surface through a potential-energy barrier. The potential dif-

ference between two adjacent con-tour lines in the figure is 0.1 eV.



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shown. The endothermicity of the desorption process, as wellas a potential barrier for adsorption of 0.17 eV along thereaction path, can be clearly seen. These two factors explainthe relatively high temperature 350 K needed to desorbhydrogen from a Ni111 surface. It should be noted that theNi111 surface is quite flat; hence the energy barriers asso-ciated with surface diffusion between adjacent sites aresmall.

When subsurface hydrogen is present in the metal, there

arises another route for hydrogen recombination: a reactionamong surface and subsurface hydrogen atoms. This possi-bility was conjectured by Johnson et al.2 after observing, inthe presence of subsurface hydrogen, a TPD peak at 180 K.This lowering of the desorption temperature can be ex-plained by the fact that subsurface hydrogen is less stablethan surface hydrogen by about 0.6 eV including the per-pendicular mode zero-point energies, thus reducing oreliminating completely the endothermicity of the reaction.However, the reaction must be direct to prevent quenchingby dissipative effects. Therefore, the two hydrogen atomsmust be neighbors.

One case that deserves attention is the nearest neighborreaction, where the surface atom is situated in the threefoldhollow site directly above the subsurface hydrogen atom. Acut of the potential-energy surface for this type of reaction isshown in Fig. 5. It is seen that this reaction is possible only

by the crossing of a 1.2-eV potential barrier. This high bar-rier is expected to yield a high recombination temperature,and, therefore, this pathway can be discarded when consid-ering the temperature decrease in the TPD peak due to sub-surface hydrogen.

The other possible reaction path of subsurface-surface re-combination reaction involves a surface hydrogen atom ini-tially located in the nearest threefold fcc site. The energetics

of this reaction are demonstrated by using a sequence of fourcuts of the potential-energy surface, as shown in Fig. 6.

The PES cuts in Fig. 6 represent the variation of energy asa function of the H-H separation in a direction parallel to thesurface and the position of the subsurface hydrogen along thesurface normal Z2. Negative values ofZ represent subsurfacepositions. Each of the four cuts corresponds to a differentheight of the surface H above the surface plane (Z10.7,0.9, 1.1, and 1.7 . These results clearly show that the re-action path is curved, with a desorption barrier of approxi-mately 0.65 eV. As will be shown in the next sections, sig-nificant tunneling rates of subsurface hydrogen can beachieved at the experimental desorption TPD temperatures.

The relative magnitudes of the energy barriers for desorp-tion along these two pathways suggest that the dominantroute is the second one. Moreover, the occupation of thethreefold hollow sites on the surface by adsorbed hydrogens

FIG. 3. Top: Views of the nickel atoms at their lattice pointsright and of an open nickel triangle left. Below: The correspond-ing reaction path potentials for hydrogen. These paths have beenadiabatically corrected for the perpendicular mode zero-point ener-

gies. The open configuration was obtained from a MD simulation at90 K corresponding to the lowest barrier in a 10-ps simulationperiod.

FIG. 4. A two-dimensional potential-energy surface contourmap for the recombination of two surface hydrogen atoms. Thenuclear separation between hydrogens is r , and the center ofmass distance from the nickel surface is Z . A flat orientationalangle of the H-H axis from the surface has been chosen. The doubleadsorption well of the two hydrogen atoms is 5.2 eV deep. Thereaction path leads to the H 2 molecular well 4.72 eV in depth.Transversing a sketchy reaction path, the system encounters a high

potential barrier, and exhibits an overall desorption endothermicity.This explains the high-temperature needed to desorb surface hydro-gen in a TPD experiment ( 350 K. The potential difference be-tween adjacent contour lines is 0.2 eV.



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prevents the emergence of subsurface H through the octahe-dral site onto the surface. Thus the surface hydrogen at thehcp sites serves as a cap to the resurfacing process. Furtherdiscussion of the dynamics associated with this desorptionprocess will be presented in subsequent sections.

III. MOLECULAR-DYNAMICS SIMULATIONS

Insight into the intricate dynamics of hydrogen in thenickel crystal is gained by molecular-dynamics simulation.However, the MD simulations cannot by themselves describethe complete story of hydrogen resurfacing and recombina-tion. The phenomenon is a low-temperature one and, due tohydrogen tunneling, requires a quantum-description. Yet aquantum-mechanical calculation becomes prohibitively ex-pensive when the many-body aspect of the crystal motioncoupled to the quantum tunneling is considered. Therefore, acombined strategy is used in which information gained from

the classical MD simulation is incorporated into a reduceddimensional quantum calculation, as will be described inSecs. V and VI. Tunneling rates are exponentially sensitiveto the potential barrier height, which is determined by thepositions of the adjacent nickel atoms. Thus, for calculatingthe resurfacing rates, it is essential to determine the distribu-tion of barrier heights during the thermal motion of the crys-tal. The results of such a simulation are shown in Secs. V Aand VI.

A. Molecular-dynamics setup

In all MD simulations described in this study, the nickel

solid is represented by a slab of six movable layers, 24 atoms

FIG. 5. A two-dimensional potential-energy surface contourmap of the recombination of a subsurface hydrogen octahedralsite with a surface hydrogen atom at a threefold hollow site. Thenuclear separation between hydrogens is r and the center of massdistance from the nickel surface is Z . The H-H axis is perpen-dicular to the surface in this cut. The potential difference betweenadjacent contour lines is 0.2 eV.

FIG. 6. A sequence of two-dimensional potential-energy sur-faces for the recombination of sur-face and subsurface hydrogenatoms. The surface hydrogen atom

position Z1 is fixed at a 0.7 ,b 0.9 , c 1.1 , and d 1.7 ,above a fcc threefold surface site.The lateral nuclear coordinate ofthe second originally subsurfaceH atom is X, and its distance fromthe nickel surface is Z2. Notice ind the appearance of the H2potential-energy well. The poten-tial difference between adjacentcontour lines is 0.1 eV.



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each, together with two identical but fixed layers, which aresituated below the six mobile layers. Periodic boundary con-ditions were imposed along the directions parallel to the sur-face. This slab, representing the solid surface, was attachedat the bottom layer to 24 fictitious particles representing the

heat dissipation into the rest of the solid. The motion of thefictitious particles was governed by a Langevinequation.64,65,48,66,67 Integration was performed by a third-order predictor corrector method,68 using 1-fs time steps.Such a short time step was required in the hydrogen penetra-tion calculation because of the large accelerations of the Hatom as it approaches the solid surface from the gas phase.

B. Hydrogen-atom penetration into the solid

One of the experimental procedures to introduce hydro-gen into the bulk metal is by exposing the crystal to a flux ofhydrogen atoms.2,10,3 The process was simulated by MD cal-

culations with the purpose of examining its efficiency. Theinitial conditions for the simulations sampled thermal hydro-gen atoms moving toward a threefold hollow or toward athreefold fcc site on the surface. In each simulation 250 trajectories were calculated where the termination conditionswere as follows: 1 the hydrogen atom was inside the solid,with a kinetic energy of less than 0.5 eV; 2 the hydrogenatom was above the surface, with a kinetic energy of lessthan 0.7 eV; 3 the hydrogen atom was scattered back to thegas phase; and 4 the trajectory time exceeded 25 000 inte-gration steps 25 ps.

Conditions 1 and 2 above correspond to bulk and sur-face hydrogens, respectively. Universally in all trajectories

calculated, the hydrogen atom is captured by the metalphase. As the hydrogen atom approaches the solid, it be-comes accelerated by its strong attraction to the surface. Thetranslational energy gained by this acceleration (2.6 eVallows the hydrogen to cross the energy barrier between thesurface and bulk (0.9 eV easily. After the hydrogen atompenetrates the solid it executes a large number of collisionswith the Ni atoms before it reaches termination conditions.The large difference between masses of hydrogen and nickelrestrict energy transfer from the hydrogen atom to the bulk.

Four sets of calculations were performed in which thetarget site on the surface and the incidence angle polarangle of the hydrogen were varied. The probability of the

hydrogen atom penetrating the solid and remaining in thebulk, Probin, was found to be independent of the target siteon the surface. The probability of forming subsurface andbulk hydrogen was 0.43 for the threefold hollow and 0.42 forthe threefold fcc. In addition, a very weak dependence ofthese probabilities on the incidence angle in was found.The change of from 0 to 30 yielded penetration prob-abilities of 0.45 and 0.43 for the two target sites, respec-tively. These changes in Probin are smaller than the error inthe simulations, hence Probin is practically independent of. The conclusion from these simulations is that Probin isrelatively insensitive to the nature of the target site and theincident angle of the H atom. Since none of the hydrogenatoms were scattered back to the gas phase, the probabilitiesfor finding the hydrogen adsorbed to the nickel surface aregiven by the fraction needed to complete the above prob-abilities to unity.

Inspection of the time evolution of the actual trajectoriesshows that the hydrogen atom always penetrates into thesubsurface region. The rate of energy loss is very slow due tothe mass mismatch, and, therefore, the hydrogen atom under-goes many collisions before it comes to rest in a locally

stable site. This site may be either on the surface or in thebulk. It should be noted that most of the trajectories wereterminated due to the fourth termination condition. Thus thevalues of Probin presented above serve only as upper limitsfor the bulk hydrogen formation probability. The order-of-magnitude discrepancy between the calculated andmeasured2 Probin values indicates that a large fraction ofthe hydrogen atoms defined in the simulation as bulk hydro-gen will emerge on the surface at later times. In addition, theaccumulation of bulk hydrogen may also result in a reductionof Probin values due to a blockage of the pathway for fur-ther hydrogen penetration. This effect was not examined inthe present calculations. In the simulations described above

the only dissipation mechanism considered was energy lossto the phonon bath of the solid.Another possible route of energy dissipation is the exci-

tation of metal electrons. To examine the importance of thischannel of energy loss, additional calculations were per-formed in which electronic friction and the correspondingrandom force were included in the hydrogen equations ofmotion. The electronic friction was computed according tothe method of Li and Wahnstrom.38,42

The probability of forming subsurface and bulk hydrogenfor the two types of target surface sites was found to be 0.54for the threefold hollow site, and 0.31 for the threefold fccsite. These results show quite a different behavior than those

described above. For the threefold hollow site, Probin in-creased by about 25%, indicating that the more efficient en-ergy loss of the penetrating hydrogen atom results in an in-crease of subsurface bulk population. On the other hand,for the threefold fcc site a large reduction in the probabilityof forming bulk hydrogen was observed. This marked de-crease in the magnitude of Probin for this site is due to thelarge frictional forces induced on the penetrating hydrogenatom during its passage through the high electron densityregions near the subsurface Ni atom.

We conclude by emphasizing the importance of electronicexcitations during the penetration of hydrogen into a metal.For the relatively high kinetic energies involved, the rate of

energy dissipation is linear with the hydrogen kineticenergy.69,70 As a result, the electrons are able to dissipateenergy of fast penetrating hydrogen effectively, causing trap-ping. The influence of electronic friction on the tunnelingmotion will be considered in Sec. VII.

IV. RESURFACING DYNAMICS I:

ADIABATIC HYDROGEN TUNNELING

IN A FROZEN NICKEL LATTICE

By examining the potential-energy surface, it becomesobvious that the resurfacing, and therefore the recombinativereaction, involves tunneling. Specifically the tunneling routeis of a hydrogen atom from an hcp subsurface site to thesurface. It is well established that tunneling is extremely sen-sitive to almost any parameter of the system, and hence re-quires a very careful analysis. The first step in such an analy-



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sis is to construct a reliable numerical procedure whichallows accurate calculations for extremely small tunnelingrates. It is instructive to write the full Hamiltonian of theproblem as

HH sH pH eV spV se , 4.1

where H s is the adiabatic Hamiltonian of the hydrogen. As-suming a frozen lattice, H p is the lattice Hamiltonian, com-posed of the nickel vibrational modes, and H e is the Hamil-tonian of the conduction electrons of the metal. The lattice

and conduction electron Hamiltonians, are both coupled tothe hydrogen motion by the potentials V sp and V se .The problem of the hydrogen resurfacing rate is first ana-

lyzed by assuming a frozen lattice, and assuming that thedynamics is electronically adiabatic: V spV se0. Thisanalysis constitutes a reference for further calculations wherethese couplings are taken into account Secs. V, VI, and VII.

The principal problem in calculating tunneling rates is theexperimental time scale of the order of seconds compared tothe basic vibrational subpicoseconds time scale. This factimposes an extreme accuracy requirement on the tunnelingrate computation. The method developed for the purpose hasan accuracy spanning 14 orders of magnitude. The method is

based on a Fourier grid representation of the wave functions.A time-energy filter extracts the individual tunneling eigen-states from the metastable well eigenstates, acting as initialguesses. Outgoing boundary conditions are imposed by theuse of imaginary potentials localized in the asymptotes. De-tails of the method are described in Ref. 8. A typical tunnel-ing spectrum, showing the lifetime of the tunneling states asa function of energy for resurfacing of hydrogen and deute-rium, is shown in Fig. 7.

Two points should be emphasized. There is a finite tun-neling rate from the ground state, meaning that the subsur-face site is metastable, stabilized only when all surface sitesare occupied. The other point is that, for each temperature,the tunneling rate is determined by very few eigenstates rep-resenting the balance between exponential increase of thetunneling rate with energy and exponential decrease due toBoltzmann weighting. This fact has also been pointed out by

Wonchoba, Hu, and Truhlar.7 Comparing hydrogen to deu-terium in Fig. 7, it is observed that the tunneling from theground state of deuterium is two orders of magnitude lowerthan for hydrogen, and that the density of tunneling states ishigher.

The validity of the reduced dimensionality approximationfor the tunneling rates was checked in Ref. 8. It was foundthat these approximations overestimate the tunneling rate byorders of magnitude. The best of the examined approxima-tions was found to be the vibrationally adiabatic approxima-tion VAA, which overestimates the tunneling rate for lowtemperatures by a factor of 5. These deficiencies should betaken into consideration when the influence of other effectson the tunneling rates are evaluated.

V. RESURFACING DYNAMICS II:

PHONON-ASSISTED TUNNELING

Lattice motion has a profound influence on the tunnelingdynamics. In the following sections an incremental analysisis performed, starting from a simple model in this section,which will be elaborated upon in the next sections.

Hydrogen tunneling rates are exponentially sensitive tobarrier height and shape. The analysis in Secs. II and IIIshowed that the lattice modes can lower the barrier by open-ing the nickel-surface-atom triad on the H-atom tunnelingroute see Fig. 3.

Simple approximations which includes this effect are de-sirable. An elementary model consists of a single oscillatorrepresenting the thermal bath and its coupling to the barrierdynamics. A further adiabatic simplification has been at-

tempted, based on the favorable mass ratio between the hy-drogen and nickel atoms. The adequacy of the model and theadiabatic approximation are assessed in the following sub-sections.

The first step, using MD simulations, is to analyze thevariation of the barrier height due to thermal motion of thesolid. This analysis determines the heavy oscillator param-eters, as is shown in Sec. V A. Once the oscillator parametersare determined, an assessment of the model is made in Sec.V B. For further comparison, the same data are used to con-struct a bath of phonon modes for which the dynamics issolved by the surrogate Hamiltonian method.61

A. MD simulations for characterizing barrier motion

Concentrating on crystal-hydrogen coupling in the barrierregion, a Hamiltonian, replacing that of Eq. 4.1 is con-structed:

HHs Pz ,Z P 2

2M

1

2M2R 2V Z ,R . 5.1

This model is aimed at studying the effects of crystal motion,so the nonadiabatic effects are neglected. The lattice motionis represented by a single harmonic oscillator with coordi-nate R, mass M, and frequency . The hydrogen motion isrestricted to one dimension along the tunneling path Z. Theeffect of the two additional degrees of freedom was takenapproximately into account by the VAA.8,7175 In this model,the coupling of the hydrogen and the lattice reduces to

FIG. 7. Tunneling spectrum of subsurface hydrogen and deute-rium moving to a surface hcp site for a frozen lattice. Also shown isthe 1D VAA approximation for hydrogen.



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V Z ,R fZ bRAeRR1, 5.2

where bR represents the basic coupling, linear in the heavyoscillator coordinate, with force constant b . The functionf(Z) localizes the coupling to the barrier region at Zo . Theextent of localization is determined by the parameter ,

fZeZZo2/22. 5.3

The exponential term in Eq. 5.2 represents the H-Ni repul-

sion at large extensions. For every crystal configuration R,the potential V(Z ,R ) represents the correction to the hydro-gen adiabatic potential Vs(Z). It is convenient to define thebarrier height h b(R) as the difference between the potentialat the top of the barrier and that at the bottom of the subsur-face well. Near R0 the barrier height h b is a monotonicascending function of R. As the oscillator moves to the

negative-R direction, the barrier for tunneling is lowered,while it is elevated as the heavy oscillator displacement isextended to the positive-R directions.

To characterize the coupling features namely, the param-eters b and ), MD simulations are used. The change in theadiabatic potential V is reduced to a single parameterit ischaracterized by the change in the barrier height h b . Thisquantity is calculated for different configurations of the solidatoms as dictated by their thermal motion. The calculationsalso account for the presence of subsurface hydrogen in themetastable well. The MD simulations were initiated with athermalization phase which lasted 10 ps. During the simula-

tion, the mass of the hydrogen atom was artificially increasedto 5000 amu in order to prevent its migration to other sites.After the thermalization stage the system evolution was fol-lowed for an additional 10 ps. At each time step of the simu-lation during this second stage, the barrier height h b wasregistered. The estimation ofh b was performed by determin-ing the potential energy for a set of hydrogen atom positionsalong a line that passes from the subsurface site to the three-fold hollow surface site along the Z coordinate.

The values of h b result from thermal fluctuations at aspecific hydrogenic site in the nickel crystal. A similar pro-cedure employing a MD simulation was used to record thebarrier height of two hydrogen atoms, where one atom is in a

neighboring surface site. At the lowest barrier height, thepotential of the metastable well was examined along the re-action path. Comparison to the equilibrium potential curve isshown in Fig. 3. Typical variations of h b due to the thermalmotion of the metal atoms, at 90 K, are shown in Fig. 8.

These results correspond to h b variations during the sec-ond stage of the simulation as noted above, the initial 10 psserved as a thermalization period for the system. Inspectionof these results shows a rapid variation in the value of h bwith an occasionally dramatic change in magnitude. The dis-tribution of the the h b values is shown in Fig. 9.

Examining Fig. 9, a near-Gaussian distribution emerges,with an average h b of 0.624 eV and a variance of 0.079 V.Similar calculations were performed for a wide range of tem-peratures. For all temperature values examined the barrierheight distribution, was Gaussian-like. The variation of, asa function of surface temperature, is shown in Fig. 10. The

FIG. 8. Clocking the barrier motion. The barrier height h bshown in Fig. 3 is recorded as a function of time, during a

molecular-dynamics MD simulation of the lattice motion. A seg-ment of 10 ps is shown.

FIG. 9. The distribution function of barrier heights h b obtainedfrom the MD simulation. The average barrier was found to be 0.62eV. The distribution width for the simulation temperature 90 K is0.079 eV.

FIG. 10. The variance in the barrier distribution in eV as afunction of temperature.



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functional dependence of on the temperature nearly linearincreases fivefold in the range of 20KTs160 K.

These results indicate that the magnitude of h b changesappreciably due to the thermal fluctuations of the surfaceatoms. The principal frequencies are determined by calculat-ing the spectral density ofh b(t) see Appendix C, shown inFig. 11.

Analyzing the figure, the frequencies of the barrier fluc-tuations span the range of 60300 cm1, with the dominantmodes at 200 cm1. These frequencies are small in compari-

son to the H-atom perpendicular vibration in the subsurfacewell of 970 cm1. One expects that this rather large fre-quency ratio is sufficient to allow the neglect of correlationsbetween the two modes. As shown below, this is not the casewhen tunneling motion is involved.

The presence of the surface hydrogen only slightly nar-rows the spectral density of the barrier height. The influenceof a surface hydrogen occupying a neighboring threefoldhollow site on the spectral density is negligible.

B. Lattice represented by a single-mode oscillator: Assessment

The oscillator representing the lattice dynamics can either

be coupled locally to the most sensitive region of the poten-

tial or spread out. It is essential therefore to examine tworather extreme regimes. In the local coupling regime thecrystal motion only influences the barrier itself, and is char-acterized by a small- parameter. The extended couplingregime is characterized by larger , and allows an outspreadeffect of the crystal motion on both the barrier and the meta-stable well. The two-dimensional potential surfaces corre-sponding to these coupling regimes are shown in Figs. 12and 13.

The extended coupling regime has a closer similarity tothe MD simulations. This can be seen in Fig. 3, where thelowering of the barrier height is accompanied by the lower-ing of the metastable well potential. The parameters used forthe two coupling regimes are given in Table I. The mass was

chosen to represent a single nickel atom. The frequency ofthe oscillator was taken to be close to the maximal spectraldensity frequency see Fig. 11. The strength of the coupling,the parameter b , was adjusted to the variance found for thebarrier height fluctuations Fig. 10, assuming that the heavyoscillator is in thermal equilibrium. The potential energycontour map of the two coupling regimes is shown in Figs.12 and 13. The tunneling rates corresponding to the fullycorrelated 2D model of Eq. 5.2 were calculated using themethod described in Ref. 8.

It is now appropriate to use this model for examining thevalidity of neglecting correlations between the two modes ofvery different frequencythe hydrogen motion and the lat-

tice vibration. For this purpose, the tunneling rate is calcu-lated in an uncorrelated model called the static barrier ap-proximation. This model averages 1D tunneling rates J(R)with respect to the static heavy-oscillator displacements R,

FIG. 11. The spectral density of the barrier height, determinedfrom a MD calculation, for a single H atom in a hcp subsurface sitegray filling. The other line corresponds to the spectral density ofthe barrier h b when another hydrogen atom resides on the surface ina nearby threefold fcc site.

FIG. 12. Flux of a 0.35-eV tunneling state, superimposed on thepotential-energy surface of the local heavy oscillator model. Thepotential difference between adjacent contour lines is 0.14 eV.

FIG. 13. Flux of a 0.26-eV tunneling state, superimposed on thepotential-energy surface of the extended coupling regime. The po-tential difference between adjacent contour lines is 0.14 eV.

TABLE I. Parameters of the single heavy oscillator models seeEqs. 5.1, 5.2, and 5.3 for definition of symbols.

Parameter Local coupling Extended coupling

M 110 000 110 000 cm1) 190 190Zo bohr 0.25 0.5 bohr 0.5 1.1A eV 0.136 0.0 bohr1) 8.0 0.0B Ht bohr1) 0.049 0.049



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J T dR PR ,T

JR eE, 5.4

where the probability density of the heavy harmonic-oscillator displacement at temperature T isP(R,T)exp(R2/22)/2 and 2coth(/2kBT)/

M. In Appendix B the static barrier approximation is stud-ied, and Eq. 5.4 is systematically derived. Its derivation isbased on two assumptions: the adiabatic approximation,where the kinetic couplings the derivatives of the hydrogeneigenstates relative to the slow degree of freedom are ne-glected, and an additional assumption that the hydrogen vi-brational energies are independent of the heavy oscillatordisplacement. This last assumption seems plausible, at leastfor the local coupling regime, where the coupling onlyslightly modifies the metastable well shape see Fig. 12.

Using the potential energy of Eq. 5.2 with the param-eters in Table I, the tunneling rates for various metastablevibrational states were calculated. The results are shown inFigs. 14 and 15, where the fully correlated 2D tunnelingrates are superimposed on 1D static-barrier rates, for variousheavy-oscillator displacements. The tunneling rate of hydro-

gen in the extended coupling regime is shown in Fig. 14. The2D tunneling states fall into lifetime bands. It is also seenthat the energy of the 1D states is dependent on the oscillatordisplacement (R), signaling poor performance of the staticbarrier approximation. This is mainly due to the extendedcharacter of the coupling. A similar figure is shown for deu-terium in a local coupling regime. Here, due to the localcoupling, the energy of the tunneling states is almost inde-pendent of the deuterium location.

It is observed in both figures that the fully correlated 2D

model yields tunneling rates corresponding to large negativeoscillator displacements. This hints that the uncorrelatedstatic barrier approximation is inaccurate. Further evidenceof the inadequacy of the static barrier approximation is re-vealed in the flux map of the full exact tunneling state.Examples are shown in Figs. 12 and 13. It can be seen thatthe tunneling flux is located mainly in the low barrier saddlepoint, even though this means very large and rare stretches ofthe heavy oscillator. Moreover, the tunneling flux representsa correlated motion of the hydrogen and the heavy oscillator.

Thermally averaged rates for the two coupling regimesare compared in Fig. 16. The isotope effect is also consid-ered. It is clearly seen that the static approximation grossly

underestimates the tunneling rate by orders of magnitude.The only combination where the static approximation is ap-plicable is for low energy, local coupling, and light mass. Inany other combination, the static calculations are very inac-curate for the estimation of tunneling rates. For deuteriumthis approximation is even poorer due to the smaller massmismatch. It is also seen that, for higher temperatures, thetwo methods predict a different temperature dependence,namely, distinct activation energies.

An important, experimentally measurable quantity is thecrossover temperature. It marks the onset of temperature in-dependent resurfacing rates. The correlated 2D models typi-cally have lower crossover temperatures than the static bar-rier models. Also, deuterium coupled to the heavy oscillatorhas lower crossover temperatures than hydrogen. In a previ-ous publication,8 a crossover temperature of 210 K was re-ported for the 1D/VAA approximation of hydrogen. Here, as

FIG. 14. Hydrogen tunneling spectra in the static barrier ap-proximation for a variety of barrier heights black squares. Super-imposed is the 2D spectra for hydrogen tunneling coupled to aheavy oscillator black dots. The coupling in this calculation is theextended coupling, shown in Fig. 13.

FIG. 15. The same as Fig. 14, but for deuterium, and for localcoupling as shown in Fig. 12.

FIG. 16. Comparison of thermally averaged tunneling rates forresurfacing of hydrogen from an octahedral subsurface site. Calcu-lations based on the local and extended coupling regimes areshown, comparing the two-dimensional model calculations with the

static barrier approximation. Also shown are the local coupling re-gime calculations for deuterium.



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can be seen in Fig. 16, the coupling to a heavy oscillatoryields a markedly lower crossover temperature of 140 K forhydrogen local coupling and 155 K extended couplingcompared to 130 K for deuterium. This narrowing of theT-independent range of the tunneling rates is in good quali-tative agreement with experimental and other theoretical ob-servations of quantum effects related to surfacediffusion.46,40,41,7

The overall effect of the barrier motion in this model is toenhance the tunneling rate with respect to the frozen lattice

calculation by one and three orders of magnitude for thelocal and the extended coupling regimes, respectively. Thiseffect is known as phonon-assisted tunneling.29 The conclu-sion of this section is that neglecting the correlations of dif-ferent degrees of freedom, even when they seem to havedifferent time scales, is usually inappropriate for studyingtunneling dynamics.

C. Single oscillator vs bath of oscillators

In studying the influence of the bath, the next question iswhether a single oscillator coupled to the barrier is an ad-equate model even when fully correlated. In order to study

this, the tunneling rates of the single-oscillator model arecompared to rates calculated by a more elaborate model: amultiphonon bath spanning the spectral density of barriermotion. In Sec. VI an even more elaborate system bath cou-pling will be studied.

Treating the hydrogen dynamics under the influence of amultiphonon bath is done by the use of the surrogate Hamil-tonian method. This method replaces the highly structuredphononic bath by a simpler bath while conserving the spec-tral density. A detailed description of this method is found inRef. 61, and a short summary is given in Appendix A. Thespectral density is estimated by analyzing the MD correlationfunctions as described in Appendix C. The estimated spectral

density is shown in Fig. 11. It is convenient to employ ananalytical approximation of the spectral density J(). This isdone by decomposing the full spectral density function to asum of three Gaussians:

J n1

3

A nen

2/2n2. 5.5

The parameters used here are given in Table II. The fre-quency range represented was set at 060 cm

1 tof300 cm

1.The coupling function f(Z) see Eq. A4 was chosen

with parameters of the extended coupling regime as shown inTable I, and the tunneling states were calculated using thesurrogate Hamiltonian method.61 It was found that the result-ing thermal tunneling rate is satisfactorily converged withthe use of a bath consisting of five two-level-system modes.

The thermal rates are shown in Fig. 17, and are compared tothe 2D single heavy-oscillator rates for the local and ex-tended models.

It is clear that the thermal tunneling obtained by the mul-tiphonon bath model has a markedly different behavior ascompared to the single-oscillator one-mode model. Thetemperature-independent regime spans a much larger range,where the crossover temperature is approximately 200 Kcompared to 140155 K for the single-oscillator models. Forlow-temperature tunneling, the phonon bath rate is betweenthat of the two single-oscillator regimes. The extended cou-pling regime is much more efficient in enhancing the tunnel-ing than the phonon bath, even though both have been builtfrom the same MD data using the same extension . Thisshows that the effect of phonon-assisted tunneling is reducedwhen a single oscillator is replaced by a bath of phonons.Such a conclusion is also observed by looking at the Arrhen-ius regime, where the activation energy for a single oscillatoris lower than that of the bath.

Experimentally, the resurfacing time is of the order ofhours.2 This fact is in discrepancy with the calculated time ofseconds. This discrepancy cannot be overcome by the factthat the VAA was shown to overestimate the tunneling by afactor of 5.8 It is possible that the potential used is not accu-

rate enough due to the extreme sensitivity of tunneling. Re-cent variational transition state calculations with EDIM po-tential produce even higher rates in the range of the single-oscillator calculation.30 The suppression of tunneling by themultimode bath compared to the single-oscillator model sug-gests that an additional tunneling hindrance mechanism ex-ists in this case. This conjecture is backed up by theoreticalestimations showing that the phonon vibrations can decreasetunneling rates see Ref. 32 for a comprehensive discussionand extensive reference list. It therefore seems necessary toreexamine the coupling of the tunneling dynamics to the lat-tice.

VI. RESURFACING DYNAMICS III:LATTICE REPRESENTED BY SEVERAL BATHS

The tunneling dynamics requires spatially long-range co-herence. This coherence can be destroyed if uncorrelated

FIG. 17. Comparison of the thermal tunneling rate of resurfac-ing hydrogen for three models: the single-oscillator local regime,extended coupling regime, and multiphonon bath regime.

TABLE II. Parameters of the Gaussian decomposition of J().

Parameter n1 n2 n3

A n eV 1.0 2.8 1.3n cm

1) 110 200 260n cm1) 22 15 22



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bath dynamics influences different parts of the potentialalong the tunneling route. To study this possibility, a modelis constructed by coupling the primary tunneling degree offreedom to several multiphonon baths. The surrogate Hamil-tonian method61 is employed, enabling a description of tun-neling coupled to several dissipative baths of phonons. Thesebaths reflect in a realistic manner the hydrogen-lattice inter-action.

The Hamiltonian of Eq. 4.1 becomes the Hamiltonian of

a primary system interacting with phonon baths,

HTV sZi ,k

i ,kbi ,k bi ,k

i

fiZk

Vi ,k bi ,kbi ,k. 6.1

Here T represents the kinetic-energy operator for thehydrogen-atom motion, and V s(Z) is the corresponding po-tential energy represented by the average reaction path.There are several phonon baths, indexed by i , each described

by a sum of free-phonon-energy terms, indexed by k, wherei ,ki ,k is the energy of the kth phonon mode of the ithbath the ground-state energies of the phonons are ne-glected, and n i ,kb i ,k

b i ,k is the occupation number, where

b i ,km i ,ki ,k/x i,kip i ,k/m i ,ki ,k, and m i ,k, x i ,k, andp i ,k are, respectively, the mass, coordinate, and momentumof the ( i ,k)th oscillator of the bath.

The spectral density is calculated for various positions ofthe hydrogen atom along the reaction path potential using aMD simulation. Analysis of the MD data discussed in Ap-pendix C results in a spectral density which is dependent onthe position of the hydrogen atom, as shown in Fig. 18.

A model consisting of three independent phononbaths issufficient to describe the spectral density

J,Zi

fiZ2Ji, 6.2

where Gaussian functions in Z are used to localize the pho-non interaction:

fiZeZZoi

2/2i2. 6.3

The local spectral densities are also cast into sums of Gaus-sians in localized in energy:

Jik

Vi ,k2i ,k

1

2

A i ,ei,

2/2i,2

.

6.4

The parameters describing the spectral densities are extractedfrom a molecular dynamics simulation, and are summarizedin Table III.

The first bath ( i1) is localized on the barrier peak po-sition like the bath of Sec. V C, and has a wide frequencyrange. The second bath ( i2) is localized in the metastablewell minimum, with frequencies centered around 250cm1. The third bath with low frequency is localized on the

repulsive slope.The calculation was performed using three bath modes foreach of the three baths. Calculations with a larger number ofmodes have confirmed reasonable convergence. The calcu-lated thermal tunneling rates are shown in Fig. 19, and com-pared with the frozen lattice rates and with the single bathrates of Sec. V C.

The striking feature is that the composite bath, consistingof three different phonon baths, imposes a drastic reductionof the tunneling rate compared to the enhancement caused bythe single-phonon bath with coupling localized on the bar-rier. The rate is decreased by two orders of magnitude, and itplunges below the frozen lattice rate.

Thus the overall effect of the phonons is to hinder hydro-gen tunneling. This observation is in contrast to surface dif-fusion and resurfacing calculations of Truhlar andco-workers,29,30 for which a mechanism of phonon-assistedtunneling is reported see Ref. 61 for a discussion of surfacediffusion on nickel. The three-phonon-bath model resurfac-ing rates are still higher than experimental resurfacing ratesby two orders of magnitude.

FIG. 18. Spectral densities of the system-bath interaction, taken

at various locations of the hydrogen atom on the reaction path.These results are based on a molecular-dynamics simulation at 90K.

TABLE III. Parameters of the spectral densities.

i Zi i A i ,1 i ,1 i ,1 A i ,2 i,2 i ,2bohr bohr Ht cm1) cm1) Ht cm1) cm1)

1 0 0.8 0.025 197.5 21.9 0.021 254.6 26.32 2.5 1.56 0.01 254 26.3 03 0.945 0.325 0.0125 94.3 16.7 0



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Based on these results, two fundamental types of bathinfluence on tunneling can be recognized. They can generallybe distinguished by the symmetry of the coupling functionf(Z) with respect to the location of the barrier peak taken atZ0). If f(Z) is an even function of Z, the phonon vibra-tions cause the potential on both sides of the barrier to oscil-late in-phase. In this case the delicate long-range coherenceof the hydrogenic wave function is not disturbed, and thetunneling is assisted by the occasional lowering of the barrier

height. On the other hand, if f(Z) is an odd function of Z,the potential surface on the phonon motion causes the twosides of the barrier to oscillate out of phase, and this destroysthe long-range coherence of the hydronic wave function.This results in a severe hindrance of the tunneling rate.

VII. RESURFACING DYNAMICS IV:

NONADIABATIC EFFECTS

The exponential sensitivity of the tunneling process toalmost any dynamical factor requires a serious considerationof nonadiabatic effects caused by the interaction between thehydrogen atom and the conduction electrons of metals.76,77

The nonadiabatic effect depends on the coupling parametersof the specific system. The present system can be character-ized by a composite Ohmic bath in which the spectral den-sity is linear in the excitation energy.

The dynamics of coupling the primary system to nonadia-batic interaction has been developed using the surrogateHamiltonian method.61 Here the general scheme is applied tothe resurfacing phenomena. The thermal tunneling rate wascalculated as a function of the electron density, and is shownin Fig. 20. It can be seen that the nonadiabatic interactionshave, in general, a damping effect on the tunneling rate,which increases with the metal electron density. The upperbound of the damping effect is approximately a factor of 3for the temperature range shown. The crossover temperatureis reduced slightly by about 5 K at the higher electron den-sities considered. From the EAM calculation the electrondensity is directly obtained, yielding a density parameter of

rs2.1 for the subsurface hcp site. It is, however, not clearwhat is the exact value of the electron density which needs tobe used in the model, since part of the interaction betweenhydrogen and the electron-excitations is accounted for justby using the adiabatic potential. Thus the calculation wasperformed for several values of the electron density, enablinga parametric study. The results are shown in Fig. 20. Basedon these calculations, it is concluded that, for this process,the nonadiabatic interactions reduce the resurfacing rates ofhydrogen, by a factor of approximately 23.

VIII. DESORPTION OF HYDROGEN FROM NICKEL 111

The final step of the hydrogen odyssey in nickel is therecombinative desorption. The reaction rates are determinedby calculating the width of the recombination resonances.The potential surface shown in Fig. 4 was used. The generalstrategy is to view the reaction as a unimolecular dissociation

H2MH2M. 8.1

The method of calculation is similar to the technique of es-timating tunneling rates described in Ref. 8. The first step isto calculate the eigenstates of the desorption well by block-

ing the exit channels to both the molecular species and sur-face diffusion. In this fashion, 48 vibrational eigenfunctionsof the well were calculated, up to and over the desorptionbarrier height. Next, the exit channel to the molecular specieswas unblocked and a negative imaginary potential wasplaced at the far molecular asymptote. Desorption eigen-states were then calculated by filtering the states with outgo-ing only boundary conditions using the Greenian filteringoperator, in an identical manner to the calculation of tunnel-ing eigenstates shown in Ref. 8. The first ten states werebelow the threshold for reaction. The eigenvalues of the con-secutive 38 states are shown in Fig. 21. It is seen that twokinds of desorption mechanisms exist. One is due to tunnel-

ing through the small adsorption barrier, while the other cor-responds to over-the-barrier Feschbach resonances. The mi-croscopic rates were thermally averaged and the resultingthermal desorption rates are shown in Fig. 22.

FIG. 19. The tunneling rate for three adiabatic approximations:

a frozen lattice, a phonon bath consisting of a phonon bath coupledto the barrier, and three phonon baths coupled to the full extent ofthe reaction coordinate.

FIG. 20. The thermal tunneling rate of resurfacing hydrogen,

shown for three metallic electron densities and for the adiabatictunneling.



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The thermal rate for the relevant temperature range can befit by using a double Arrhenius curve, as discussed in Sec.IX C. This limited dimensional approximation results in acrude upper estimate of the recombination rate. The influ-ence of lattice dynamics was not analyzed for this reaction.

IX. KINETIC MODEL FOR THE TPD SIGNAL

In order to compare the calculated recombination TPDrates to experimental data, the individual rates have to becombined into a kinetic model. The main assumptions of themodel are summarized as follows:

1 Hydrogen readily adsorbs on the nickel surface. Mo-lecular hydrogen dissociates, and the hydrogen fragment canoccupy four types of surface sites, two of which are meta-stable on-top and bridge. The stable sites are the two three-fold sites.

2 Atomic hydrogen, coming from the gas phase, readily

enters the bulk and thermalizes in interstitial sites. The moststable bulk sites are those directly underneath the threefoldhollow surface sites.

3 The recombination of two surface hydrogen atoms isendothermic, by about 0.5 eV, and the desorption process

is activated by a barrier of0.67 eV.4 A hydrogen atom in the subsurface site can resurfaceby tunneling through a 0.6-eV barrier to a vacant threefoldhollow site. The thermal rate of this process has been exam-ined using different models and levels of approximation.

5 When a hydrogen atom occupies a surface threefoldhollow site, it blocks the exit of a subsurface hydrogen atomsee Fig. 5, raising the resurfacing barrier to 1.2 eV. Twoconclusions follow from this observation:

i A high occupation number of hcp sites causes a de-crease in the resurfacing rate. This is called the capping ef-

fect.ii A direct reaction mechanism, between a subsurface

hydrogen atom and a surface on-top hydrogen, cannot ex-plain the low-temperature bulk-surface recombination. Thisconclusion is strengthened by recent experimental measure-ments, which show that bulk-surface recombination productshave a wide angular distribution.78

6 The subsurface hcp site is metastable with respect to asurface site. The energy difference is about 0.4 eV in poten-tial energy and 0.23 eV in zero-point vibrational energy, to-taling to 0.63 eV. It follows that, immediately after resurfac-ing, the hydrogen is very hot, and interacts rapidly with asurface hydrogen located in a neighboring site.

7 The relaxation rate of a hot hydrogen atom is prima-rily due to nonadiabatic electron-hole pair excitations.61

These points are used to construct a kinetic model for thebulk-surface recombination process. In this model thesubsurface-surface reaction is broken into two kineticallydistinct stages. First, a resurfacing stage, where excited hy-drogen is formed after tunneling out of the subsurface. Nextthe hot hydrogen finds a thermalized surface hydrogen andreacts with it. The possibility of reaction among two hothydrogens is negligible, since the hot hydrogens relax ex-tremely fast relative to their formation rate. A further simpli-fication in the model is that the difference between fcc andhcp sites is neglected due to their fast equilibration via sur-face diffusion.61,7,41 The kinetic equations describing thecombined process become:

dB

dtKbB1SS*,

dS*dt

KbB1SS*S*k*S*S,

9.1

dS

dtS*k*S*S2kS2,

TTot,

where all occupation numbers are per surface site, and B isthe occupation number of subsurface hydrogen, S is theoccupation number of thermalized surface hydrogen atomsH, S* is the occupation number of hot resurfacing hydro-

FIG. 21. The desorption spectrum of surface hydrogen based onthe PES of Fig. 4. The potential-energy minimum of molecularH 2 is marked. The first three states of lowest energy are tunnelingstates through the 0.17-eV adsorption barrier. The other states areover-the-barrier resonances. Lines connecting the resonance pointsare a guide for the eye.

FIG. 22. The thermal recombination rate of surface hydrogen.Also shown is the double Arrhenius curve used to fit the data.



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gen atoms H *), Kb is the thermal resurfacing rate constant, is the relaxation rate constant of excited hydrogen, k* isthe thermal rate constant of the reaction H*HH2, andk is the thermal rate constant of the reaction HHH2, and is the surface heating rate, taken as 2 K s1 in accord withthe experimental value.

The TPD signal is given by the desorption rate

dG

dtk*S*SkS2, 9.2

where G is the number of desorbed H 2 molecules per sur-face site. In the following sections the rate constants are

estimated, based on the different calculations described inthe present study.

A. Resurfacing rate Kb

Estimating the tunneling rates is a difficult task due to theexponential sensitivity and the strong dependence on the di-mensionality of the process. Table IV summarizes the hier-archy of approximation used for the hydrogen tunneling re-surfacing rates. For completion the TST calculation of Ref.30 is included in the table, which is two orders of magnitudefaster than the frozen lattice calculation at 0 K. The tunnelingrates as a function of temperature for the different approxi-

mations are summarized in Fig. 23.It is clear from this figure that the different models predictrates differing by five orders of magnitude for the same pro-cess. The tunneling rate constants were fit to a universal form

K TKoexp EakBTaTa

6T1

1 TaT

n

1 TaT

.9.3

This expression contains four parameters: i Ta1.2Tc ,where Tc is the crossover temperature; ii Ea is the activa-

tion energy of the Arrhenius part of the rate constant tem-perature dependence; iii Ko is the temperature independentrate; and iv n is a parameter, describing the curvature of thecrossover region. The most elaborate phonon model, using

three baths Sec. VI, estimates the temperature-independenttunneling rate as Ko0.05 s

1, the crossover temperature asTc185 K and Ea0.87 eV. However, the nonadiabatic in-teractions shown in Fig. 23 exhibit an additional decrease inKo by a factor of 2 and ofTc by 5 K. In Ref. 8 it was shownthat the reduced dimensional VAA causes an overestimationof Ko by a factor of 5, and of the crossover temperature Tcby 10 K. Based on these considerations the following param-eters were chosen for the universal function representingKb : Tc170 K, Ko0.005 s

1, Ea0.87 eV, and n5.1.

B. Relaxation rate

The resurfacing hydrogen atom has an excess energy of0.63 eV with respect to a surface hydrogen at its groundstate. This is due to the 0.4-eV difference in the bindingenergies of surface and subsurface hydrogen atoms with anadditional 0.23 eV, the difference in zero-point energy, andthe surface hydrogen. This excess energy allows the hot hy-drogen to wander freely on the surface. The relaxationmechanism causes loss of the hot hydrogen energy due to

FIG. 23. The thermal tunneling rate of resurfacing hydrogen,shown for the various model calculations. The boxes indicate thecrossover temperature.

TABLE IV. The different estimations of the hydrogen-resurfacing rates crossover temperatures and activation energies.

Adiabatic Number of Lattice Electronicpotential H DOFs motion nonadiabaticity Rate Hz a Tc K Ea eV

1 EDIM 1 frozen neglect 103 200 0.47

2 EDIMVAA 1D frozen neglect 0.5 200 0.573 EDIM 3D frozen neglect 0.1 205 0.54 EDIMVAA 1D 1-mode/Local Coup./Stat. Bar. neglect 0.3 200 0.275 EDIMVAA 1D 1-mode/Local Coup. neglect 1 140 0.266 EDIMVAA 1D 1-mode/Extended Coup. neglect 7101 156 0.217 EDIMVAA 1D 1-multiphonon Ext. Coup. neglect 101 194 0.268 EDIMVAA 1D 3-multiphonon-baths neglect 5102 185 0.159 EDIMVAA 1D frozen e-hole bath 2102 190 0.3210 Ref. 30 TST b partition function neglect 104 130 0.33

aRate at zero temperature.bTST-tunneling-corrected variational transition state theory.



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electron-hole pair excitations. The vibrational line broaden-ing of the transverse mode was calculated in Ref. 61, and itwas found to be 23 cm1, yielding the relaxation rate2101241012 s1. A similar value is obtained for thetranslational energy relaxation due to the nonadiabatic

friction.61

Hence a value of

2

1012

s1

is assumed inthe present kinetic model.

C. Recombination rate k

The surface reaction of the hydrogen recombination wasdiscussed in Sec. VIII. The rate constant is well fit by adouble Arrhenius function

k TA 1eE1 /kBTA 2e

E2 /kBT, 9.4

where the following parameters are used: E10.6 eV,E20.8 eV, A 11.510

10 s1, and A24.11011 s1.

D. Hot hydrogen recombination rate k*The recombination reaction is activated, and therefore en-

hanced by the excitation of one of the participating hydrogenatoms. For this reason the excess energy of the resurfacinghydrogen can promote the reaction. This enhancement is es-timated by lowering the activation energies E1 and E2 in thethermal-reaction-rate expression Eq. 9.4 by Exc . Next, thekinetic prefactor is enlarged. This is due to the fact that thecorrelated motion of the bulk and surface atoms can have astrong effect. The analysis in this study has shown that suchcorrelated effects, for low probability processes can enhancethe rates by orders of magnitude. An example for this is seenin Sec. V B, where correlation effects enhance the tunnelingrates by three orders of magnitude. Since a detailed calcula-tion is prohibitively expensive at this stage, the prefactors aretaken as parameters, their value increased by factors of 10and 100 compared to the thermalized reaction prefactors.Summing up, the rate constant is given by Eq. 9.4 with thefollowing parameters: E10.026 eV and E20.17 eV,

A 11.51011 and 1.51012 s1, and A24.110

12 and4.11013 s1.

E. TPD signal

Analysis of the kinetic model of Eqs. 9.1 shows that thenumber of excited hydrogen atoms on the surface, S*, is

created by a relatively slow process, and destroyed by a fastprocess. This prohibits any attempt to solve Eqs. 9.1 di-rectly. However, S* is at all times extremely small, and thuschanges slowly. Setting dS*/dt0 in Eq. 9.1 yields theset of equations:

dB

dtKbB1SS*,

dS

dtS*k*S*S2kS2,

9.5

S* KbB1S

KbBk*S,

TTot.

This set of coupled equations is numerically integrated. Thestability and accuracy has been checked by monitoring theidentity 2GBSBo where Bo is the initial bulkcontent of hydrogen. It is assumed that the surface is initiallyclean.

The TPD simulated results with the initial bulk contentBo1.5 and initial temperature To80 K are shown in Fig.24. Two spectra for the two sets of prefactors used to esti-mate k* are displayed. In both cases, two peaks are ob-

served. However, the magnitude of the low-temperaturepeak, centered at 170 K, is very sensitive to the reaction-rateconstant used. The relaxation rate of hot hydrogen is fast andthe surface is rather quickly filled, so that at a temperature of190 K the creation of excited hydrogen is almost halted com-pletely. This is a result of the capping effect. At higher tem-peratures, around 200 K, the surface-surface recombinationsets in, and the surface is quickly depleted. This enablesformation of more hot hydrogen atoms, and the two mecha-nisms of desorption work simultaneously. At 210 K, the re-surfacing is so fast that the bulk is emptied almost instantly,and the hot hydrogen recombination stops. This causes adrastic decrease of the TPD rate seen as the sharp peak at

215 K. This peak is not seen when the hot hydrogen reactionis slow. By the time the temperature reaches 230 K, thesurface coverage is depleted enough for the recombinationreaction to slow down, and the TPD signal gradually drops.

The TPD experiment also shows two peaks,2 at approxi-mately 200 and 350 K, respectively. The peak at 200 K is oflarger amplitude than the one at 350 K. This is attributed tothe complete depletion of the bulk at low temperature. Dur-ing this depletion a large fraction of the hydrogen desorbs,while the rest remains thermalized on the surface. At highertemperatures, around 350 K, the remaining surface speciesdesorb.

Qualitatively, the dynamical calculations and the kinetic

model presented confirm the conjecture that the low-temperature peak observed experimentally is due to thesubsurface-surface hydrogen reaction. However, quantita-tively the calculated results do not fit the experimental data.

FIG. 24. The calculated TPD spectra for bulk hydrogen desorp-tion under the proposed kinetic model. The filled line correspondsto the case where the prefactors of the excited reaction rate con-

stants are increased by a factor of 100 relative to those of the ther-mal reaction. The solid line corresponds to an increase of a factor of10.



18/23

The location of the surface reaction TPD peak is at 230 Kcompared to a higher experimental peak at 350 K. This dis-crepancy is probably due to the reduced dimensionality ap-proximations which overestimate the reaction rates. Dissipa-tion effects, in analogy to their influence on the resurfacing

process, can also hinder the recombination reaction.As for the low-temperature peak, the location is close tothat observed in the experiments 180 K; however, the ex-perimental results indicate that most of the bulk hydrogen isdepleted at this temperature, which, due to the capping effectcan only be explained by a very efficient reaction betweenthe hot hydrogen and a desorbed hydrogen. It is seen that,increasing the reaction rate by a factor of 100, allowing forcorrelation effects still cannot explain the efficient depletionof bulk sites. One possible source for this discrepancy is theeffect of occupied neighboring subsurface sites. Thehydrogen-hydrogen repulsion in the bulk can increase thehydrogen metastability in the subsurface site, making the

emitted hot hydrogen atoms even hotter. This mechanism isable to explain the efficient reaction rate, and shows that thefirst TPD peak has the flavor of a transient macroscopicphase transition.

X. OVERVIEW

The long journey through the different possible routes ofhydrogen in nickel has been aimed at unraveling the story ofsubsurface-surface recombination of hydrogen. The final pic-ture is still sketchy, and most details are missing. However,the framework that has been set supports the experimentalevidence for the interesting chemical mechanism involving

bulk hydrogen.2,78 The final kinetic model contains condi-tions for the appearance of an additional low-temperatureTPD peak, attributed to the existence of subsurface hydro-gen.

However, the most interesting stories of the odyssey havebeen collected on the way. The combined process is ex-tremely involved. Rationalizing it requires disassembling toindividual events. The unifying factor for all events was theEDIM potential used for all processes studied. This potentialenabled the identification of stable metastable subsurface andsurface hydrogen species. This identification allows us tobreak up the process into two main parts: the resurfacing ofsubsurface hydrogen, and recombination desorption.

The resurfacing step conceptually seems simple, a directtransport from one potential well to a lower one in the energysurface well. A detailed analysis has revealed a convolutedpicture, dominated by tunneling. The extreme sensitivity oftunneling to any dynamical or environmental factors can leadinvestigation astray. As is evident from Table IV, nine dif-ferent approximations for the resurfacing process have beenanalyzed. The rate calculated by each of the approximationdiffers from the previous one by orders of magnitude. Whenchecked, rigorously simple models fail. One source of erroris due to cutting down the dimensionality of the problem.This has been the conclusion of a previous paper,8 whichshow the failure of reduced-dimensional approximations fortunneling dynamics. Even the best variationally corrected 1Dmethod, which includes the perpendicular zero-point energy,overestimates the tunneling at low temperature by a factor of5.

Even more involved is the influence of the lattice motionof the heavy nickel atoms on the hydrogen tunneling dynam-ics. The most obvious effect is the casual decrease of thebarrier height by the fluctuating lattice. The investigationstarted on the nave assumption that an adiabatic separation

of the fast hydrogen from the heavy nickel degrees of free-dom is possible. A single-mode representation model of thebath dynamics was constructed to test this hypothesis. Theresult was a complete failure of the adiabatic separation. Thetunneling dynamics was found to be highly correlated. Thelesson learned is that an approximation that is absolutelyreasonable for describing majority events of the hydrogendynamical processes, such as the vibrational motion in thepotential well, is inappropriate to describe the minute eventsof tunneling.

The importance of the correlated tunneling motion gaverise to the suspicion that the naive description of the bath bya single mode is misleading. It was expected that the many

uncorrelated bath modes could destroy the system bath cor-relation and interfere with the tunneling motion. In order tobe able to study such a process, a dynamical description ofthe bath was developed: the surrogate Hamiltonian approach.

The surrogate Hamiltonian approach replaces the infinitemany-body lattice dynamics by a representative finite Hamil-tonian composed of two-level systems. The success of themethod relies on the observation that, for finite time, a com-plete resolution of the energy spectrum is not required. Bysampling the spectral density of the bath in an appropriatefashion, the method systematically converges. To obtain thespectral density a molecular-dynamics simulation of the lat-tice motion was performed based on the same EDIM poten-

tial. As expected, the multiphonon bath assists the tunneling,but with a reduced effect compared to the single mode bath.

Tunneling dynamics with a fluctuating bath coupled onlyto the barrier region can be classified as phonon-assisted tun-neling. In contrast, shaking the potential well of subsurfacehydrogen via the phonon dynamics has been shown to pro-hibit tunneling.32 For hydrogen in nickel, both effects areoperative. The balance between enhancement and suppres-sion caused by phonon dynamics is delicate. Only a quanti-tative, elaborate approach can determine the total phononicimpact on hydrogen tunneling. For the resurfacing dynamicsan analysis using the the spectral density calculated along thereaction path has shown that tunneling suppression over-

comes the enhancement.Electronic nonadiabatic effects are are also able to sup-

press the tunneling. Electronic friction is well established forhigh velocities. Computations comparing tunneling rates ona single or two-crossing Born-Oppenheim potential-energysurface shows extreme tunneling suppression in the nonadia-batic case.79 Nevertheless, calculations using the surrogateHamiltonian method show only a small effect consideringthe electron density in nickel obtained from the EAM calcu-lation.

The tunneling dynamics suggests that there should be asignificant isotope effect where the lighter isotope is faster.Conflicting effects can reverse this trend. The tight bottle-neck for resurfacing leads to a large adiabatic correction dueto perpendicular zero-point energy which favors the heavyisotope. The coupling to the phonon bath is also influencedby the mass ratio. Since these conflicting effects are different



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for different temperatures, a comparative isotopic study isuseful.

Only a comprehensive study of all possible effects on theresurfacing dynamics allows a balanced estimation of theconflicting effects. Quantita

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Roi Baer, Yehuda Zeiri and Ronnie Kosloff- Hydrogen transport in nickel (111)

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