Quantum Mechanics of Hydrogen on Nickeland Palladium Clusters

M. A. Gomez!, B. Chen!, David L. Freeman2, and J. D. DolI!

! Department of Chemistry, Brown University, Providence, RI 02912, USA2 Department of Chemistry, University of Rhode Island, Kingston, RI 02881, USA

Abstract. Within the broad class of metal-hydrogen systems, clustersare of particular importance. Their high surface to volume ratio makesthem ideal candidates for catalytic applications. Surface and bulk stud­ies have shown that transport and vibrational spectroscopy of hydrogenare very sensitive to substrate structure. The wide variety of geometriesexhibited by clusters offers a noteworthy opportunity to examine the ef­fect of substrate geometry on hydrogen. Further, hydrogen's small massand uniquely large isotopic variation gives rise to a number of intrin­sically quantum mechanical effects. For example, inverse isotope effectshave been observed for hydrogen chemisorption on palladium clusters.Comparing classical and quantum mechanical Monte Carlo methods, theeffects of quantum mechanics on cluster structure, population distribu­tion, vibrational spectra, and rates for hydrogen motion are discussedfor a single hydrogen on nickel and palladium clusters.

1 Introduction

The role of hydrogen on metal clusters is important both from a technological anda fundamental point of view. Technologically, their high surface to volume ratiomakes them ideal candidates for catalytic applications. Since many applicationsinvolve hydrogen, an understanding of the properties of metal-hydrogen clustersis of special significance.Fundamentally, hydrogen's small mass and uniquely large isotopic varia­

tion gives rise to intrinsically quantum mechanical effects. For example, Fayet,Kaldor, and Cox [1] notice an inverse isotope effect in the rate of chemisorptionof H2 and D2 on palladium clusters. Such inverse isotope effects are also observedin the palladium bulk and the surface of Pd( 111) [2, 3]. Further, since clustersprovide a wide variety of geometries, they offer a noteworthy opportunity toexamine the effect of substrate geometry on the behavior of hydrogen.In a previous study [4] we examined the properties of small nickel and palla­

dium clusters that contained a single hydrogen atom. This study revealed thatquantum mechanical structures differ from classical ones for some palladium clus­ters with a hydrogen atom. The differences result from an inversion in energyordering of isomers when zero-point effects are taken into account.

In this chapter, we discuss the effect of quantum mechanics on structure onlarger nickel and palladium clusters. For the Pd7H, we also discuss non-zero

J. Jellinek (ed.), Theory of Atomic and Molecular Clusters© Springer-Verlag Berlin Heidelberg 1999

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temperature population distributions, vibrational frequencies, and rates for hy­drogen entering the cluster. This chapter is arranged as follows: Section 2 brieflydescribes and justifies the model potential used in this work. Section 3 discussesthe classical structure of nickel and palladium clusters with a hydrogen atom.Section 4 explains how quantum mechanics changes the zero temperature struc­ture. The next sections comprise a detailed case study of Pd7H. In particular,non-zero temperature effects on isomer population distributions for this clusterare discussed in Sec. 5. Section 6 explains the quantum mechanical effects onthe vibrational frequencies of Pd7H. Next, Sec. 7 discusses quantum mechanicaleffects on rates for hydrogen entering the seven atom palladium cluster. Finally,in Sec. 8 we present some concluding remarks.

2 Potential

Throughout this study the embedded atom method (EAM) of Daw and Baskes [5]is used. While it is an incomplete description of the present cluster systems, us­ing the same potential in the current work as in previous surface studies [3, 6, 7]enables comparison of hydrogen in cluster and surface environments. Althoughwe have chosen not to use them, we note that more recent EAM parameteri­zations for nickel and hydrogen systems have been published by Rice et al. [8],and Wonchoba and Truhlar [9]. In addition, Jellinek and collaborators have ex­tensively studied deuterium/nickel systems with the Voter-Chen potential [10]as well as other many-body potentials [11]. Fournier, Stave, and DePristo havealso studied deuterium/nickel systems with Corrected Effective Medium (CEM)methods.[12] Curotto, Matro, Freeman, and Doll used an extended Huckel the­ory potential to study mono and di-hydrogenated nickel clusters.[13] Gronbeck,Tomanek, Kim, and Rosen studied the effect of hydrogen on the melting of pal­ladium clusters using a many body alloy potentiai.(14] Further, Wolf et al. havedeveloped more recent H/Pd EAM potentials [15]. We emphasize that none ofthe methods in the present study depend on the details of the potential. Morecomplete estimates of the microscopic interactions can and will be used as theybecome available.

Even though EAM [5] is fit to bulk data, previous research shows that EAMis robust and yields at least qualitative data for the structure and binding en­ergies of nickel clusters. Sequential nickel binding energies obtained by Staveand DePristo [12] using computationally more intensive empirical potentials arecloser to the experimental results of Lian et al. [16] than those predicted withEAM [17]. However, the patterns in these energies are qualitatively similar [4].Most of the structures predicted by EAM [4, 11, 17] and CEM [12] are consistentwith those inferred by Parks et al. [18J from N2 absorption data. In the EAMpotential [5] used in the present study, the exceptions are NiB and Ni 14 .

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3 Classical Structure

Classical minima are found using simulated annealing [19], and quantum anneal­ing [20] on random configurations. About 200 random configurations are usedfor simulated annealing and quantum annealing. As a further check, the simplexmethod [19] is used on between 200 and 600 structures constructed with theminima of the previous size cluster plus an a new atom in a random position.For the purposes of the present discussion, the lowest potential energy struc­ture found for a given system by these searches is referred to as the "classicalglobal minimum." While appreciable care has been exercised in locating theseminima, experience with such problems makes it clear that it is generally im­possible to know with total certainty when the absolute potential minimum hasbeen located.The classical global minima for NinH with n=11-16 are shown in Fig. 1. Hy­

drogen prefers to bind on the outside of nearly completed coordination shellsand inside coordination shells that are just starting. Enlarging a tetrahedral siteto accommodate a hydrogen in nearly complete coordination shells compressesadjacent tetrahedral sites and breaks the symmetry of the cluster. In the incom­plete shells, hydrogen binding usually occurs inside tetrahedral sites that do nothave neighboring tetrahedral sites.The global minima for the smaller nickel clusters are reproduced in Fig. 2 for

comparison. A comparison of global minimum structures of nickel clusters withand without a hydrogen atom reveals rearrangements of the structure. Fig. 3shows the structure of bare nickel clusters of seven to ten atoms. The globalminimum potential energy structures of the seven and ten atom nickel clusterswith the hydrogen rearrange to incorporate a larger binding site for the hydrogen,namely an octahedral site. In both of these cases, the second lowest minimumfor the bare cluster has an octahedral site [4].The global minima for PdnH with n=11-15 have the same basic structures

as the nickel clusters except PdllH. The global minimum of PdllH is shown inFig. 4. In this case, hydrogen is at the outside edge of the tetrahedral site. Itshould be noted that while hydrogen is outside the tetrahedral site for clustersof 12 and 13 palladium atoms just as in nickel, it is just barely outside. The factthat palladium can sometimes accommodate hydrogen inside while nickel cannot is not surprising. Hydrogen readily percolates [21] through bulk palladiumbut must be "pounded" into nickel [9, 22]. This suggests that palladium canaccommodate hydrogen in its smaller binding sites far better than nickel can.The lattice constant in palladium is 3.89 Awhile in nickel it is 3.52 A. Thedifference indicates that the lattice structure is more open in palladium than innickel. A similar trend is seen in the clusters. Palladium clusters generally formlarger sites than nickel.This tendency for palladium to accommodate hydrogen better than nickel

is also seen in the smaller palladium structures [4]. The global potential energyminimum structures for palladium clusters with a hydrogen atom are the sameas those for nickel clusters with the exceptions of Pdn H with n =7, 9, and 10which are shown in Fig. 5.

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Fig. I. NinH global minima for n=1l-16

313

Fig.2. Global minimum energy structures of NinH for n=4-10. For purposes of iden­tification, the H atom is shown without bonds to the nickel atoms.

314

Fig. 3. Global Minimum energy structures for nickel clusters of sizes 7 to 10 atoms.

4 Quantum Mechanical Influence on Ground StateStructure

Normal mode analysis, which reflects both potential energy minima and localcurvature, provides a very useful tool for predicting the quantum mechanicalenergy ordering of different isomers. As seen in our previous paper [4], zeropoint energy effects can alter the energy ordering of isomers of some palladiumclusters with a hydrogen atom. For example, normal mode analysis predictsthat clusters Pd7H and PdlOH have different classical and quantum mechanicalstructures at zero temperature [4]. Diffusion Monte Carlo confirms this [4]. Thebasic quantum mechanical structure at zero temperature for Pd7H and Pd10His that of the corresponding nickel clusters shown in Fig. 2.In this chapter we apply this analysis to larger clusters. The addition of

zero-point energy to these clusters and their close lying isomers reveals thatordering in energy for most clusters remains the same. However, for Pd llHand

315

11Fig. 4. Global minimum energy structure for Pd ll H

Ni16H, the ordering is changed. While the global minimum structure for Pd l1 Hhas the hydrogen inside the tetrahedral site as shown in Fig. 4, there is a closelying isomer with the hydrogen outside the tetrahedral site just as in the nickelclusters shown in Fig. 1. When harmonic zero-point energy effects are taken intoaccount, the energy of Pd l1 H with hydrogen on the inside at zero temperaturegoes from -33.439 eV to -32.70 eV. The energy of Pd l1 H with hydrogen outsidethe cluster goes from -33.438 eV to -32.71 eV. Normal mode analysis, whichtakes into account not only the potential energy of the cluster but also thecurvature of the potential, indicates that when quantum mechanical effects areincluded hydrogen prefers binding outside Pd l1 .

The first inversion in energy of nickel clusters with a hydrogen atom occursfor Ni16H, the classical structure of which is shown in Fig. 1. The energy forthe cluster with hydrogen bound inside the tetrahedral site at zero temperaturegoes from -57.002 eV to -55.71 eV. By contrast the energy for Ni 16H withhydrogen bound outside the cluster at zero temperature goes from -56.984 eV

316

Fig. 5. Global Minimum energy structures for palladium clusters of sizes 7, 9, and 10atoms with a hydrogen.

to -55.73 eV when zero point energy is taken into account. Hence, normal modeanalysis predicts that the structure of Ni16H at zero temperature is the one shownin Fig. 6.

5 N on-zero Temperature Quantum Mechanical StructureDistributions

The study of cluster isomerization offers valuable insight into the nature of clus­ter dynamics [23, 24]. The relative populations of the stable structures are exam­ined using the Path Integral Monte Carlo (PIMC) method coupled with quench­ing techniques [25]. The Monte Carlo configuration is quenched every 1200 movesvia very fast simulated annealing [19].Figure 7 presents the probability that the Pd7H explores non-ground state

structures given that it is started in the ground state configuration at varioustemperatures. This probability is labeled probability of isomerization. The onsetof isomerization is seen to occur at around 500 K. At lower temperatures, the

317

Fig.6. Minimum harmonic ground state energy structure for Ni1SH

system remains localized in the vicinity of the initial isomeric forms. The prob­ability of isomerization increases with temperature and approaches a constantvalue of 0.82 at temperatures higher than about 800 K. The onset temperaturemay be biased by the "quasi-ergodicity" problem. The low quantum tunnelingrate prevents the observation of transition between isomers during the finitelength of simulation. The strong temperature dependence of the quenching pop­ulation indicates the activated nature of isomerization. Care must be exercisedto assure proper low temperature sampling [26].Figure 8 shows the temperature dependence of the isomer distribution for

Pd7H. As is shown there, the population of the ground state isomer approaches100% for temperatures lower than 500 K. As the temperature increases, theatoms become more and more mobile. As a result, other isomer structures start toappear. Only the isomers with energies in the vicinity of the ground state energyplaya significant role at lower temperatures where the quantum ground statestructure dominates. It should be noted that the classical global minimum is notthe ground state structure. This is revealed both by normal mode analysis anddiffusion Monte Carlo [4]. Classically, Pd 7H contains a pentagonal bipyramidalpalladium skeleton with a hydrogen atom in one of its tetrahedral sites. Whenzero-point energy effects are incorporated, a palladium skeleton of a cappedoctahedron with hydrogen in the octahedral site is preferred. Both the classical

318

0.9 I I I I I

t> 0.8 ,-----.'- I -

:s I0.7 l- I -.c I

.8 I0.6 '- I -e I

I~ 05 - I -~ I

.~ 0.4 - I -l

·a 0.3 - I -/

ICI)0.2 - / -

j II

0.1 - / -I

I I I I I I0.00 200 400 600 lKlO 1000 1200

Temperature (K)

Fig.7. Temperature dependence of isomerization probability of PdrH. The initialstructure is the quantum ground state structure.

global minimum structure and the quantum mechanical ground state structureare shown in Fig. 9. At higher temperatures, quantum mechanics becomes lessimportant. Ultimately, the classical Boltzmann distribution in which the classicalglobal minimum dominates is reached.

6 Quantum Mechanical Effects on VibrationalFrequencies

The cluster, Pd7H, is studied to show the effect of anharmonicities and quantummechanics on frequencies. As seen in Fig. 9, the quantum and classical zerotemperature structures for Pd7H differ. In order to focus on the effect of quantummechanics on frequencies, both our classical and quantum spectra calculationsuse the same structure, namely the quantum mechanical ground state structure.First, we consider the zero temperature classical frequencies (i.e. the normalmode frequencies). Next, we look at non-zero temperature classical frequencieswhich are obtained from a molecular dynamics trajectory. Finally, we compareand contrast these with quantum mechanical frequencies.The zero temperature classical frequencies are the normal mode frequencies

of the system. The quantum mechanical ground state structure for this clus­ter (Figure 9) has hydrogen in a Cav environment. Consequently, the hydrogen

319

10

...............................,....

............

-,~-------------~_ ,..... -:.---:-.-- .. ., .• _.. .. 2S

SOO 600 700 800 900 1000 1100

Temperature (K)

...........

..................•.

..............••••••..

400

0.8

....8 0.6

~~

0.4

0.2

0.0200

Fig.8. Isomer distribution of Pd7H as a function of temperature. The open circlesrepresent the quantum ground state structure. The closed circles represent the classicalglobal minimum structure. The lower two curves are the isomers with next closestenergies.

Fig. 9. The two lowest energy isomers of the seven atom palladium cluster with a hydro­gen atom arc shown. The classical global minimum is on the left. However, zero-point.energy effects show that the structure on the right is representative of quantum me­chanical ground state structure. This is confirmed via Diffusion Monte Carlo.

320

vibrational motions are the same for the cluster and the surface, a symmetricAl state and a doubly degenerate E state. Normal mode analysis reveals thathydrogen does contribute significantly to these three modes. The vibrationalfrequencies for hydrogen in the quantum mechanical ground state structure forPd7H are 1591 cm- l for the doubly degenerate E modes, and 1228 cm- l for theAl mode. The Pd-Pd frequencies range from 80 to 402 em-I.Comparing normal mode frequency estimates with classical spectra provides

a qualitative measure of the anharmonicities in the potential. The classicalspectra at 100 K shown in Fig. 10 is obtained by taking the Fourier transformof the hydrogen position autocorrelation function. The vibrational frequenciesfor hydrogen in the quantum mechanical ground state structure for Pd7H are1572 cm- l for the doubly degenerate E modes and 1213 cm- I for the Al mode.Comparison with normal mode frequencies suggests that there are anharmonic­ities in the system. Since the frequency is lowered more in the E modes, thesemodes may be more anharmonic than the Al mode.

Classical Spectra for Pd7H at T=lOOK

200015001000500

I I I

---

-

-

-

J~~-

~ ~oo

1000

6000

2000

3000

4000

5000

Frequency (em-I)

Fig. 10. Classical hydrogen vibrational spectra for Pd7Hat 100 K. Results are obtainedby Fourier transforming hydrogen position autocorrelation function data.

The quantum mechanical frequencies of the Pd7H cluster at 100 K are shownin Fig. 11. The spectra are obtained by inverting the imaginary time hydrogenposition autocorrelation function via maximum entropy methods [27]. The po­sition autocorrelation function is calculated using a Fourier Path integral ap­proach [25]. The solid curve and dotted curve correspond to the doubly degen-

321

erate E state, and the dot-dashed line is the spectra for A1 state. The calculatedfrequencies are lower than both the normal mode frequencies and the classicalfrequencies at 100 K. Quantum mechanically, the anharmonicities in the poten­tial are probed more thoroughly since the zero-point energy causes the moleculeto move in higher energy regions of the potential. The classical cluster wouldneed a higher temperature to probe the same regions. The amount of shift ismost notable for E state, about 15% from the normal mode frequency. The shiftfor A state is much smaller by comparison, only about 2% from the normalmode frequency. The smaller anharmonic effect for A1 state is a consequence ofits smaller zero-point energy contribution. The frequencies of the E and the A1

modes are 1355 cm -1 and 1200 cm -1, respectively. The ordering is still the samebut the difference in frequency is significantly smaller than the correspondingclassical result.

0.09

'5 0.08

'§ 0.07

~ 0.Q6

S 0.05:a~

0.Q4

~ 0.03

'i 0.02

~ 0.01

0.000 1000 1500 2000 2500 3500

Frequency (cm-I)

Fig. II. The quantum mechanical spectra at lOOK are shown for the doubly degenerateE state (solid and dotted curves) and the A state(dot-dashed curve).

7 Quantum Rate Effects

As noted in the introduction, inverse isotope effects are observed for hydrogen dif­fusion in bulk palladium and palladium clusters [1,2,3]. The hydrogen diffusion

322

studies of Rick, Lynch, and Doll suggest that the reason for the inverse isotopeeffect is highly constrained transition state sites [3]. The quantum mechanicalzero temperature structure of the Pd7H cluster has very similar geometry to thePd(111) surface sites. In particular, there are three-fold hollow, octahedral, andtetrahedral sites in sequence along an axis. As a result of the similarity in thesites of Pd7H and the Pd(111) surface, this cluster is a reasonable starting pointfor the study of unusual quantum rate effects in palladium clusters.

As was the case in discussing quantum effects on cluster structure, a zero­point analysis is a convenient starting point for the discussion of quantum effectson thermal rates. Figure 12 shows the potential profile of hydrogen entering thePd7H isomer. From Fig. 12, we see that the classical barrier going from the threefold hollow surface site to the octahedral site is 0.110 eV while the barrier in thereverse direction is 0.436 eV. Zero point energies modify these results, yielding0.100 and 0.560 eV, for the two barriers respectively. While too simplistic tobe highly accurate, the present normal mode analysis serves to indicate thatquantum mechanical effects can act to either increase or decrease a reactionbarrier.

z

-1').0

-1')2

-1') 4

s; -1').6~ -1').8

0' -20.0

8 -202>-20.4

-20.6

-208

-21.0

-212-3 :5 -3 -25 -2 -1.5 -1 -0.5

z (A)

o 0.:5 1~

Fig. 12. Potential energy profile of hydrogen entering the cluster. At the barrier, thepotential energy as a function of x and y with z at the barrier is shown. The upwardcurvature indicates that zero point energy effects need to be taken into account at thebarrier.

323

Transition state theory provides a useful framework with which to extend theprevious analysis of quantum effects on rates. Classically, the transition statetheory rate is given by,

1 (2kT) 1/2kI:: = 2 1l"m < b(x - q) >A (1)

where x is the position and q is the transition state. The average of the deltafunction is done over the Boltzmann distribution in the reactant system, A.Quantum mechanically, Miller, Chandler, and Voth [28J have shown that rate canbe written approximately in the same form as Eq. 1 with the centroid variable,xc, replacing the position, x. The average of the delta function is done over thequantum distribution in site A.We performed the averages required by the classical and quantum rate calcu­

lations using Voter's displacement vector Monte Carlo method (DVMC) [3, 28,29J. The DVMC method is a technique for computing ratios of partition func­tions. These calculations predict that the rate constant for hydrogen enteringthe seven atom palladium cluster at 300 K are 0.95 ± 0.03 pS-1 classically and0.4 ± 0.2 ps-1 quantum mechanically. As with the previous surface studies [3],quantum mechanics is acting to reduce the rate of hydrogen transport into the"sub-surface" palladium regions.

8 Conclusion

This chapter has shown how quantum mechanics can affect structure, popu­lation distribution, vibrational spectra, and rates. Structures of similar classi­cal potential energies can have significantly different ground state energies as aconsequence of different local curvatures in the potential. Population distribu­tions at low temperatures are significantly affected by these zero point energyeffects. Quantum mechanically, vibrational frequencies corresponding to veryanharmonic modes shift significantly to lower frequencies. Finally, quantum me­chanics can lower rates when the the transition state is more highly constrainedthan the reactant state.

9 Acknowledgments

This work was supported in part by the National Science Foundation throughgrants CHE-9411000 and CHE-9625498. One of us (MAG) also wishes to thankboth the National Science Foundation and the Cooperative Research FellowshipProgram of AT&T for graduate fellowship support.This research was sponsored in part by the Phillips Laboratory, Air Force

Material Command, USAF, through the use of Maui High Performance Comput­ing Center (MHPCC) under cooperative agreement number F29601-93-2-0001.The views and conclusions contained in this document are those of the authorsand should not be interpreted as necessarily representing the official policies or

324

endorsements, either expressed or implied, of Phillips Laboratory or the U.S.Government.

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