roi baer, yehuda zeiri and ronnie kosloff- influence of dimensionality on deep tunneling rates: a...

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Influence of dimensionality on deep tunneling rates:

A study based on the hydrogen-nickel system

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 26 April 1996

The tunneling of subsurface hydrogen into a surface site of a nickel crystal is used to investigate deep

tunneling phenomena. A method to calculate tunneling lifetimes based on an energy and time filter is devel-

oped, enabling converged lifetimes differing by 14 orders of magnitude. It is found that the reduced dimen-

sional approximation always overestimates the tunneling rate. The vibrational adiabatic correction improves

dramatically the one-dimensional calculation but nevertheless overestimates the cases of deep tunneling. The

isotope effect is studied, pointing to experimental implications. S0163-18299651332-9

I. INTRODUCTION

The tunneling motion of hydrogen atoms in metals is animportant process with a range of practical applications, suchas hydrogen embrittlement, catalysis, and fuel storage.1

Moreover, tunneling draws fundamental interest since it ex-hibits dominant quantum effects at low temperatures.2,3 Ex-perimentally, for tunneling processes the Arrhenius rate lawbreaks down, exhibiting a sharp crossover to a temperature-

independent rate,6 as predicted by Wigner in the 1930s.4,5In molecular dynamics, tunneling phenomena are extreme

examples where classical treatments fails.2,3 Since quantumcalculations scale exponentially with dimension, approxima-tions for the tunneling motion based on a reduced dimension-ality description have been developed. It has beenestablished6 that defining a tunneling coordinate and thenfreezing the perpendicular degrees of freedom causes severeerrors in the estimates of tunneling rates. Instead, variousstudies have employed a reaction path approach,710 which isessentially an adiabatic approximation suitable when the per-pendicular degrees of freedom can be treated as fast andthe reaction, tunneling coordinate as slow. The approxi-

mation amounts to adding the ground-state energy of theperpendicular oscillators to the reaction path potential, andwill be denoted as the vibrationally adiabatic approximationVAA.

Schatz et al.11 have published a comparison of full quan-tum three-dimensional 3D calculations of proton transferreactions vs VAA-type transition-state theory with tunnelingtreated semiclassicaly VAA/ST. They find large departuresof the calculated reaction rates, up to a factor of 2.5 at roomtemperature. Garrett et al.12 claim that this discrepancy is notnecessarily indicative of the quality of the approximation andcan also be attributed to differences in potential-energy sur-faces or to errors resulting from the numerical implementa-

tion of the quantum method. For more conclusive results asimpler solution is necessary.

Motivated by new experiments on the chemistry of bulkhydrogen,1316 the dynamics of a hydrogen atom in a nickelfcc crystal is studied. The atom is located in a subsurfaceinterstitial site from which it tunnels, through a 0.65-eV bar-rier, to a threefold hollow site on the 111 metal surface.

In this system, degrees of freedom can be divided intotwo types: those directly belonging to the tunneling atom,and those of the heavy metal atoms which comprise thecrystal vibrational heat bath. The masses associated with

each class of degrees of freedom DOF are very different,and it is therefore assumed that the hydrogen atom is onlyweakly coupled to the motion of the crystal. The full 3Dhydrogen quantum results are compared with reduced dimen-sional 2D and 1D calculations, as well as with the 1D vibra-tional adiabatic approximation 1D-VAA. The treatment ofthe crystal DOFs as well as the possibility of exciting themetal electrons will be addressed in a future publication.

II. METHOD OF CALCULATION

The tunneling rate is related to the imaginary componentof the complex eigenvalue of the Hamiltonian belonging to

an asymptotically outgoing-only eigenstate.The tunneling eigenvalue spectrum is discrete and com-

plex: nEnin/2, n0,1,2, . . . , where En is the energyof the state and n the tunneling rate. The tunneling statesare improper eigenstates of the Hamiltonian, since the nega-tive imaginary part of the eigenvalue imposes a correspond-ing imaginary value of the outgoing momentum in the as-ymptotic channel, leading to the divergence of the eigenstateat infinitely large distances. In order to regularize thesestates, several methods have been developed, such as thecomplex scaling method17,18 and others.19

An alternative regularization method is achieved by a lo-calized negative imaginary potential NIP in the

asymptotes.20 The NIPs are used to impose an outgoingboundary condition on the eigenstate calculations. By aug-

PHYSICAL REVIEW B 15 AUGUST 1996-IIVOLUME 54, NUMBER 8

540163-1829/96/548/52874 /$10.00 R5287 1996 The American Physical Society


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menting the system Hamiltonian H to HHiVNIP , a

nonhermitian Hamiltonian is obtained, and the tunneling

states become the right eigenstates. Once a functional form

for the imaginary VNIP is chosen with parameters

A k,k0,1, . . . ; the actual parameter values should be opti-

mized for each tunneling eigenvalue separately. The param-

eter values are selected according to an insensitivity crite-

rion: (dEn /dA i)0, as has been used in the complex scaling

methods.17 In practice, the spectrum is found to be weakly

dependent on the potential parameters.20

A. Algorithm

The algorithm for calculating tunneling states follows two

basic steps. The first step consists of locating the approxi-

mate real part n of the energy of the tunneling. The second

step consists of repeatedly applying a filter g(n) to a seed

state, until convergence to tunneling states is achieved. Inboth steps the potential and wave function are represented on

on an equally spaced grid.21 The calculation is set on a grid

employing the Fourier method.22,21

STEP 1. The first step consists of calculating the bound

states of the well. Bound states are obtained by adding anartificial binding potential in the asymptotic region. Then,the low-lying bound eigenstates n are searched for, using afilter-diagonalization scheme.23,24 These states constitute anapproximation to the tunneling states in the well with a realeigenvalue ofn , approximately equal to En , the energies ofthe tunneling states.

STEP 2. Next, the vibrational eigenstates n are used as aseed from which the tunneling state n is filtered out. Whenthe filter g

(n) see next subsection for details, operates on

n , it enhances the components that have long lifetimes andare close in energy to n . Repeated application of the filterresults in the purified discrete tunneling state n , since boththe continuum states of the Hamiltonian short lifetimes andother tunneling states different energies are filtered out.

B. Energy-lifetime filter

The energy-lifetime filter used has the following form:

gEe iEH

sinEH

EH, 2.1

Where, is a real positive parameter. Note that if is largeenough, the operator g

(E) becomes identical to the Greens

operator G(E) : limg(E)(EH

)1G(E) . The

limit exists since all the eigenvalues of H have negative

imaginary parts. In practice, is much smaller than 01 so

that the limit is not achieved for low-lying tunneling states.The effect of the filter on a seed state n can be seen by

expanding it in terms of the eigenstates of H:nk1

N a kkda()(), where n are the the

tunneling eigenstates of H and () are the continuumeigenstates. Upon application of g

(n) ,

gnn

k1

N

a keink

sinnk

nkk

dae in

sinn

n, 2.2

the dual operation of the filter is observed: the integral overthe continuum vanishes, if is chosen such that ()1for all ; and the discrete sum over the tunneling eigenstates,enhances the component of energy closest to n .

The Greens operator in this calculation is expanded by aChebychev polynomial series,22,25 which, when truncated toa finite number of terms N yields g

(E), where N determines

in a simple linear relation.

C. Checking the convergence of the tunneling state

A necessary condition for a state to be an eigenstate of

H is that the dispersion is zero: 2()H 2H 20. This criterion is used to check the conver-gence. The filter is applied to a given trial state as manytimes as needed to obtain 2 smaller than a predefined value.Specifically, with double precision of 16 digits accuracy,

21017 was chosen.Once this criterion is met, convergence is further checked

by comparing two different methods of calculating the tun-neling rate. First, the rate is calculated from , which is halfthe imaginary part of the eigenvalue. Second, the rate is cal-

culated from J, the integral of the flux flowing through thesurfaces encircling the localized negative imaginary poten-tials NIPs. Since the NIPs are the only amplitude sinks,the two rates should be equal. The difference J wasfound to be very sensitive to the quality of the calculation.

The method was first tested by comparing to analyticalresults. The calculations have also been tested against thesensitivity to the NIP parameters. Very small sensitivity hasbeen found.20 The accumulation of round-off errors was alsochecked by comparing the double precision calculations toquadruple precision ones. It was found that round-off errorsare small even for the very low tunneling rates even lowerthan 103 sec1). It was also found that for all but the

lowest tunneling rates, the grid spacing can be determined bythe criterion22 x2 Pmax, where Pmax

22m(VmaxVmin). In

the case of very low tunneling states a denser grid was re-quired for convergence.

III. HYDROGEN IN NI111: SUBSURFACE-SURFACE

TUNNELING RATES

The adiabatic potential surface, describing the hydrogenflow to the surface threefold-hollow site was calculated usingthe embedded-diatomic-in-molecules method of Truonget al.26 For a single hydrogen atom this approach is similar tothe embedded-atom method of Daw and Baskes.27

The full three-dimensional calculation was used as abenchmark for the other, reduced dimensional calculations.One- and two-dimensional calculations were performed onthe minimum energy path cuts of the 3D potential. Finally,

R5288 54ROI BAER, YEHUDA ZEIRI, AND RONNIE KOSLOFF


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1D calculations corrected for the zero-point energy of modesperpendicular to the tunneling direction were also performedVAA. The perpendicular zero-point energy for each pointin the reaction coordinate was calculated numerically. Thereaction coordinate in this model has no curvature.

The tunneling spectrum, is shown in Fig. 1. The first pointto be noted is that the lifetimes span about 14 orders ofmagnitude. The 2D and more so the 3D spectrum show acomplicated nonmonotonic pattern. In general, tunneling

rates of similar energies can differ by orders of magnitude. Incontrast, the 1D results show a monotonic increasing patternof tunneling rates vs energy, typically two orders of magni-tude higher than the maximum 3D results of similar energy.The adiabatic zero-point correction VAA dramatically im-proves the naive 1D calculation. Nevertheless for thelow-lying states, rates of similar energies differ by a factor of7.

The thermal rate of the depletion of the subsurface hydro-gen can be calculated from the tunneling spectrum by Bolt-zmann averaging over the rate constants. Figure 2 shows thethermally averaged tunneling rates calculated using the re-sults shown in Fig. 1.

It is apparent from the figure that the 3D calculation ex-hibits the lowest thermal rate. For low temperatures the tun-neling rate is dominated by the vibrational ground state andis therefore nearly temperature independent. The crossover totemperature-dependent tunneling is at 190 K for the 3Dcalculations, while it is at approximately 210 K for the 1D/VAA calculations. Below this temperature the rates of thefull 3D and 1D/VAA differ by a factor of 7, while for highertemperature, the rates seem to settle to a factor of 1.7. Thelow-temperature 2D calculation has similar tunneling rates tothe 3D one, probably due to the high cylindrical symmetry atthe bottom of the potential well. At higher temperature thetwo rates diverge, reaching factors of 56. The 1D calcula-tion, exhibits tunneling rates almost two orders of magni-tudes too high.

To check the strong isotope effect expected in tunneling,similar calculations were performed for deuterium. For the

lowest-energy states, the deuterium tunneling rates calcu-lated by the 1D/VAA approximation are higher by a factor of80 compared to the full 3D calculations. At higher vibra-tional states the ratio becomes smaller down to a factor of1.7. The deuterium calculations also predict a crossover tem-perature of 190 K 3D calculations and 210 K 1D/VAA.

For deep tunneling the isotope effect is enhanced dramati-cally, as shown in Fig. 3. The figure compares the thermalprogrammed depletion of subsurface hydrogen and deute-

rium. Here the surface temperature is increased at a constantrate of 2 K/sec, and the rate of the depletion of subsurfacesites is simulated.

It is clear that hydrogen leaks out at a significant rate evenat low temperatures, peaking at 210 K, while deuterium hasnegligible leakage at low temperatures with a maximumdepletion rate at 250 K. In hydrogen, the 1D/VAA vibra-tional ground-state tunneling rate is too high, resulting in afast depletion even at low temperature, with no high-temperature peak, while for deuterium, this approximation

FIG. 1. The tunneling resonances of a Ni-subsurface hydrogen

atom as it tunnels to a surface threefold hollow site. The discretetunneling rates are shown as a function of their energy for 3D,

1D-VAA, 1D, and 2D calculations. For convenience, the spectra are

shifted to a common zero-point energy.

FIG. 2. The tunneling rate of a hydrogen atom as a function of

inverse temperature for the different models considered. The insertshows the ratio of the different rates as compared with the 3D

calculations.

FIG. 3. The calculated thermal programmed depletion spec-

trums of subsurface hydrogen H and deuterium D for a surface

heating rate of 2 K/sec. The results shown are for the full 3D and

the 1D/VAA calculations dashed.

54 R5289INFLUENCE OF DIMENSIONALITY ON DEEP . . .


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does much better peaking only 5 K lower than the full 3Dcalculation. The shape of the depletion curves depend criti-cally on the rate of heating. At 90 K, where subsurface hy-drogen experiments were performed,15 the present calcula-tions predict that subsurface hydrogen will emerge to

produce a monolayer within approximately 1 min while thesame process for deuterium will take 4 h. The large isotopeeffect may be the reason that deuterium is easier to stabilizein the subsurface state and thus easier to treat inexperiments.15

IV. CONCLUSIONS

The dynamics of hydrogen on metal surfaces has been asubject of intensive study2830 In this context the issue oftunneling is of fundamental importance for all activated pro-cesses. Since multidimensional quantum calculations are ex-tremely expensive, reduced dimensional approximations areof the utmost importance. In this study three-dimensionaltunneling dynamics have served as a benchmark in under-standing the influence of reduced dimensionality. The mainconclusions are the following: i The tunneling rates aredramatically sensitive to naive reduction of dimensionalityand this leads to overestimation of the tunneling rates byorders of magnitude. ii The tunneling rates vary monotoni-cally with energy for 1D calculations, while the 3D tunneling

rates exhibit a complicated irregular dependence. iii Theadiabatic VAA approximation dramatically improves the 1Dcalculations but still overestimates the rate for deep tunnel-ing states. For higher energy the accuracy of the VAA ap-proximation improves. iv The tunneling rates of hydrogen

are quite distinct from the rates of deuterium, typically threeorders of magnitude higher. This fact predicts that subsurfacedeuterium will be easier to stabilize.

The reduced dimensional approximations fail for two rea-sons. The first is the variation of the perpendicular zero-pointmotion along the reaction path, which is corrected by theVAA. The second is the overestimation of the coherencebetween the two sides of the barrier in all the reduced dimen-sional calculations. This leads to the final conclusion thateven the 3D calculations performed overestimate the tunnel-ing rate.

ACKNOWLEDGMENTS

This research was patially supported by the German IsraelFoundation. It was also partially supported by the IsraelAtomic Agency commision and the Israel Council for HigherEducation. The Fritz Haber Research Center is supported bythe Minerva Gesellschaft fur die Forschung, GmbHMunchen, FRG.

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