The Quantum Mechanics of Clusters: the Low -Temperature Equilibrium

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1991 The Quantum Mechanics of Clusters: The Low - Temperature Equilibrium and Dynamical Behavior of Rare-Gas Systems Steven W. Rick

D. L. Leitner

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Citation/Publisher Attribution Rick, S. W., Leitner, D. L., Doll, J. D., Freeman, D. L., & Frantz, D. D. (1991). The Quantum Mechanics of Clusters: The Low- Temperature Equilibrium and Dynamical Behavior of Rare-Gas Systems. Journal of Chemical Physics. 95(9), 6658-6667. doi: 10.1063/ 1.461536 Available at: http://dx.doi.org/10.1063/1.461536

This Article is brought to you for free and open access by the Chemistry at DigitalCommons@URI. It has been accepted for inclusion in Chemistry Faculty Publications by an authorized administrator of DigitalCommons@URI. For more information, please contact [email protected]. Authors Steven W. Rick, D. L. Leitner, Jimmie D. Doll, David L. Freeman, and D. D. Frantz

This article is available at DigitalCommons@URI: https://digitalcommons.uri.edu/chm_facpubs/29 The quantum mechanics of clusters: The low-temperature equilibrium and dynamical behavior of rare-gas systems Steven W. Rick, D. L. Leitner,a) and J. D. Doll Department of Chemistry, Brown University, Providence, Rhode Island 02912 David L. Freeman and D. D. Frantzb) Department of Chemistry, University of Rhode Island, Kingston, Rhode Island 02881 (Received23 April 1991; accepted22 July 1991) We consider in the presentpaper the quantum-mechanicaleffects on the equilibrium and dynamical behavior of low-temperature rare-gasclusters. Using a combination of ground-state and finite-temperature Monte Carlo methods,we examinethe properties of small (2-7 particles) neon clusters. We find that the magnitude of the equilibrium quantum-mechanical effectsin thesesystems is significant. The presentstudies also suggestthat the low-temperature dynamics of theserare-gas systems is appreciablynonclassical.

I. INTRODUCTION untestablenumerical approximations.The rare-gasthermo- Studiesof the equilibrium and dynamical properties of dynamic studiesof Lee, Barker, and Abraham’ and the mo- clustersconstitute a significant and increasingly active area lecular dynamics studies of Berry et aL3 are examplesof of current research.Interesting becauseof their rich and var- such classical investigations. ied phenomenology,clusters are also of substantialpractical For many problemsclassical mechanical studies are ap- importance. They are intimately involved, for example, in propriate and yield physically relevant predictions. For oth- such important processesas nucleation, thin-film growth, er problems, however, quantum-mechanicaleffects are sig- and catalysis.Beyond their intrinsic merit, however,clusters nificant, casting doubt on the adequacyof purely classical also serveas convenient prototype systemsin the analysisof approaches.There now exist a broad class of both zero- and other, more complex condensedphase and interfacial prob- finite-temperature equilibrium quantum-mechanicalmeth- lems.Both conceptually and practically, clusters bridge the ods that are applicableto many-bodysystems in generaland gap betweenfinite and bulk systems.Cluster investigations to cluster studies in particular. A number of applications of also provide an important common ground for few- and thesemethods to cluster problemshave beenreported. Such many-bodytechniques and serveto clarify the transition be- applications have included numerical path-integral tech- tweenfinite and extendedsystem behavior. niques;4 variational,’ diffusion, and Green’s-function As reviewed in detail elsewhere,’ a variety of experi- Monte Carlo methods;6and basisset methods.’ mental techniquescan and have beenutilized in cluster in- The presentwork is a preliminary step toward the gen- vestigations.This rich experimentalbase provides the neces- eral study of the quantum-mechanicaleffects on the dynam- sary foundation for the sustaineddevelopment of this topic. ics of many-bodysystems. The particular physical context of From an applications viewpoint, cluster systems are the presentstudy is the low-temperature dynamics of small convenienttestbeds for the developmentof numerical meth- rare-gasclusters. Our work is motivated in part by the recent ods. In smaller clusters the microscopic force laws are typi- investigations of Beck et al.” who examined the quantum- cally better characterizedthan the correspondingforce laws mechanicaleffects on “melting” in small neonsystems. That for generalbulk systems.Comparisons of theoretical predic- study found that the effectsof quantum mechanicson melt- tions and experimentaldata in such systemsare thus, in prin- ing were significant. In particular, the large zero-pointfluc- ciple, lessclouded by uncertaintiesin the fundamentalinter- tuations found in small neon clusters raise questionscon- actions than would generally be the case. Moreover, the cerning the qualitative character of these low-temperature intermediatescale of clusterspermits a variety of theoretical systemsand the extent to which their dynamical behavior techniquesto be brought to bearon a common problem thus can be adequatelydescribed by classical methods. facilitating the testing and developmentof what are hopeful- Both finite- and zero-temperature methods are em- ly more generally useful methods. ployed in the presentwork. Thesemethods are summarized The basic theoretical tools availablefor the study of the briefly in Sec.II. In this brief review we concentrateprinci- equilibrium and dynamical propertiesof clusterswere, until pally on finite temperature and diffusion Monte Carlo ap- recently, classicalin nature. Monte Carlo and molecular dy- proachesand refer the interestedreader to recent work’ that namicsmethods have been used to probethe equilibrium and focusespurely on the cluster ground-state properties for a dynamical properties of such systems.Beyond the assump- more complete description of the variational Monte Carlo tion of classical behavior, such methods are free of further method. Section III describesthe application of thesemethods to neon and argon clustersand draws a number of con- ‘) Present address: Theoretische Chemie, PhysikaIish-ChemischesInsti- tut, Im Neuenheimer Feld 253, D-6900 Heidelberg, Germany. clusions concerningthe nature of the low-temperature clus- b IPresent address: Department of Chemistry, University of Lethbridge, ter dynamics. The conclusions of the present work are Lethbridge, Alberta, Canada TlK 3M4. summarizedin Sec.IV.

6656 J.Chem.Phys.95(9),1 November1991 0021-9606/91/216658-10$03.00 @ 1991 American Institute of Physics Rick eta/.: Quantum mechanics of clusters 6659

II. METHODS tries and energiesthat connect thesestable potential minima In the present study we will be concerned with single- have been discussedrecently by Wales and Berry.” component rare-gasclusters. We will assumethat the inter- Knowledge of the relative populations of these stable actions in such systemsare describedby a simple superposi- structures as a function of temperature offers valuable in- tion of pair potentials, with the pair interaction taken here to sight into the nature of the cluster dynamics.‘* Such rela- be the familiar LennardJones potential, tive populations can be obtained, for example, by quench studies based on randomly chosen equilibrium configura- Y(r) &[(-y-(;)6]. (2.1) tions.13 In such an approach each configuration is assigned a label corresponding to the particular “parent” In this expressionE and (Tdenote the usual well depth and structure found by a steepest descents potential-energy sizeparameters. In the presentstudies these parameters were quench that starts at the configuration in question. The tem- taken to be (35.6 K, 2.749 A) for neon and ( 119.4K, 3.405 perature dependenceof the resulting quench populations is A) for argon, respectively. This empirical force law is ade- useful both classically and quantum mechanically in the quate for our current purposes and, moreover, its use will characterization of the cluster dynamics. also allow us to draw on the results of a number of previous, Although not the principal focus of the present work, Lennard-Jones based investigations. We emphasize, how- equilibrium Monte Carlo cluster studies provide sufficient ever, that the theoretical methods used in the present study information to approximate the rate parameters associated are neither restricted to pairwise additive interactions in gen- with the interconversion between the various stable cluster eral nor to Lennard-Jones forms in particular. Although not structures. For classical systems, establishedMonte Carlo the principal focus, we will also briefly consider helium clus- transition-state methods are available for this purpose and ters in the presentwork. For those applications, the helium- are discussedin detail elsewhere.14Analogous quantum-me- helium interaction was assumedto be of the Aziz form.” chanical tools basedon centroid path integral methods de- The characteristics of the stable isomers predicted by scribed in the work by Gillan” and in that by Voth, minima in the cluster interaction potential as well as the Chandler, and Miller.16 nature of their interconnecting transition statesare valuable In the present work we will be interested in the proper- in the characterization of the cluster dynamics. Such infor- ties of finite clusters whosemicroscopic force laws are of the mation is available for many of the smaller clusters.” De- type describedearlier. Sincewe are particularly interested picted in Fig. 1, for example, are the four stable isomers and in possiblequantum mechanical effects,we will make useof the associated potential energy minima for a seven atom direct comparisonsbetween the predictions of classical and Lennard-Jones cluster. Associated transition-state geome- quantum-mechanical methods. We summarize these methods briefly and refer the interested reader to the published literature for more complete descriptions of the methods in- Ne-7 Isomers volved. Classical mechanical methods have been applied to the study of the equilibrium and dynamical behavior of clusters by a number of investigators. These methods, as well as a number of recent applications, are reviewed by Berry et aL3 Once the microscopic force law is specified, Monte Carlo molecular dynamics and related methods produce estimates of the equilibrium and dynamical behavior free of untestableapproximations beyond the assumption of classical behavior. Thesemethods have beenused, for example,to study cluster structure and to explore the magnitude of finite-size effects. They have also been used to study cluster -16.505 -15.935 precursors of familiar bulk phenomenasuch as melting and diffusion.” More recently, nonlinear methods have been usedto characterizethe onsetof irregular and chaotic behavior in the dynamics of these rare-gasclusters. l8 In recent years,practical quantum-mechanicalmethods for the study of both zero and finite-temperature many-body systemshave beendeveloped. The finite-temperature techniques, in principle, contain the zero temperature results as a special limiting case. Although important advances have been made that extend the practical limits of these finite- -15.593 -15.533 temperature approachesto rather low temperatures,” it is still generally useful to have available efficient, specialized techniquesdesigned specifically to study quantum-mechani- FIG. 1. Shown are the four stable structures for a classicalseven-atom Len- cal ground-state problems. nard-Jones cluster. The energies (in units of the Lennard-Jones well depth) are shown with each structure. One simple,but effectivezero-temperature method is

J. Chem. Phys., Vol. 95, No. 9,1 November 1991 6660 Rick et&: Quantum mechanics of clusters the variational Monte Carlo technique.” In this approach numerousexamples are discussedby Ceperleyand Alder.*’ Monte Carlo methods are used to evaluatethe variational We briefly summarizecertain technical issuesspecific to the estimateof the total energycorresponding to somespecified present applications in the Appendix. Although not uti- trial wavefunction. The expectationvalue of the Hamilton- lized in the present work, Green%-functionMonte Carlo ian over this trial wave function providesan upper bound to methods26are also broadly applicableto the calculation of the quantum-mechanicalground-state energy. This bound- ground-stateproperties of many-bodyquantum-mechanical ing property makes it possibleto “optimize” the trial func- systems. tion by demandingthat the associatedenergy be minimized In the presentwork DMC methodsare utilized to obtain with respect to any adjustable parameters present in the what are effectively numerically exact estimates of the wave function. Since the integrals in the variational Monte ground-state properties of various quantum-mechanical Carlo methodare estimatedby statistical rather than analy- clusters. We have found theseindependent estimates of the tical methods,the form of the trial function can bechosen on ground-stateproperties to bevaluable, limiting checksof the the basis of physical suitability rather than mathematical results of more general, finite-temperature methods. The convenience.Recent applications of thesevariational meth- present DMC applications utilize one feature that merits odsto cluster problemsas well aspractical extensionsto deal brief discussionand validation. As noted earlier, the birth with excited states have been reported by Whaley ef and death processesin Eq. (2.2) are governedby the term uZ.~*~’ We utilize traditional variational methods in the [ V(X) - E] q. If the potential is unboundedfrom below presentwork to obtain estimatesof the ground-stateenergies (such as would be the casefor a Coulomb interaction), this and wavefunctions for rare-gasclusters of various sizes.The term can lead to an infinite birth rate. This practical diffi- wave functions used in the present study are describedin culty is typically avoided through the use of “importance detail by Rick et ~1.~Generically, the variational forms used sampling” methods in which Eq. (2.2) is used to derive a in the present work are of the form of a product of pair modified diffusion equation for the product of a trial wave functions. In the caseof helium and neon,the pair functions function and its exact counterpart. In the resulting modified areof the type utilized by Whaley et al.’ with somemodifica- diffusion equation, the birth and death processesare gov- tions to introduce multiple length scalesin the caseof neon. erned by the differencebetween the “local energy” of the More elaborate“ shadow” trial functions9.22V23are also used. trial wavefunction and the constantE. This quantity is typi- For the argon clusters, the wave functions are designedto cally smaller than the correspondingterm in Eq. (2.2) and reflect more nearly the structure of the classicalsystem. thus the fluctuations in the population induced by the birth Although the variational method describedabove is fre- and death processesin the modified diffusion equationtend quently quite useful, it is often desirableto utilize numerical to be smaller than those in Eq. (2.2). The importance sam- methods that are arbitrarily refinable.One such method is pling results are formally independentof the choice of the the diffusion Monte Carlo (DMC) technique.24This meth- trial functions. In the present applications, however, the od exploits the isomorphism betweenthe time-dependent qualitative nature of the ground state is an issueof primary Schriidinger equation (written in imaginary time) and the interest. We havechosen, therefore, to useEq. (2.2) directly diffusion equation. Statistical methods designed for the to avoid any possibility of introducing a bias concerningthe treatment of diffusion problems can thus be invoked to nature of the ground statethrough the choice of a trial wave “solve” the original Schrodinger equation. The essenceof function. We note in this regard that the Lennard-Jonesin- theseDMC methods can be seenby consideringthe Schrii- teractions are bounded from below, and thus the infinite dinger equationfor a particle of massy moving one-dimen- growth in the direct Monte Carlo approachis not an issuein sional potential V(x). The “time’‘-dependentequation in the presentwork. As will be discussedin Sec. III, we have imaginary time (7 = it Hz) becomes confirmed the reliability of this direct DMC approachfor a variety of nontrivial cluster problems. -=---w^ -+? de2fl [v(x>--lrc, (2.2) In addition to the ground-statemethods describedear- dr 2~ dx” lier, we have utilized numerical path-integral methods to wherethe constantE hasbeen subtracted from the potential examine the finite-temperaturequantum-mechanical prop- for convenience.Disregarding its physical origin for the mo- erties of the various rare-gassystems under investigation. ment, Eq. (2.2) describesthe “diffusion” of particles whose The detailsof the Fourier path-integraltechniques used here diffusion constantis #/2,~ in the presenceof particle sources and their application to cluster problemsare describedelse- and sinks. The strength of thesesources and sinks are gov- where.4*27The essentialfeature of such numerical path-inte- ernedby the secondterm on the right-hand sideof Eq. (2.2)) gral approachesis that they expressequilibrium properties [ V(x) - El+. As discussedby Ceperleyand Alder,2s the of the quantum-mechanicalproblem as classical-like aver- solution of Eq. (2.2) is dominated for large valuesof r by agesin an enlargedconfiguration space.That is, the quan- the lowest energysolution to the Schriidingerequation. The tum-mechanicalaverages emerge as higher-dimensionalan- typical strategyfor obtaining the ground-statewave function alogsof the original classicalproblem.28 Once formulated in and propertiesfrom Eq. (2.2) is thus to devisea statistical this manner,the samenumerical Monte Carlo methodsthat procedurein which the “birth,” “diffusion,” and “death” were utilized to solve the original classical problem can be processesare suitably modeledand examinethe solution in usedto solve its quantum-mechanicalanalog. Since the nu- the large r limit. The details of these DMC methods, a merical methodsinvolved arerelatively insensitiveto dimen- complete description of the technical issuesinvolved, and sionality, the introduction of the additional degreesof free-

J. Chem. Phys., Vol. 95, No. 9,1 November 1991 Rick et&: Quantum mechanics of clusters 6661 dom necessaryto account properly for the quantum-me- TABLE I. Total energies(in units of e) for the LennardJones model of the chanical featuresof the problem posesno specialdifficulties. Ne, and Ar, clusters computed by direct diffusion Monte Carlo (DMC), by discrete variable representation (DVR) methods and by variational Nonetheless,it is generally worthwhile asa practical manner Monte Carlo (VMC) techniques.The statistical uncertainties in the DMC to minimize the number of these auxiliary degreesof free- results are indicated in parentheses. dom. To this end we have usedpartial averagingmethods to acceleratenumerical convergencein the present work.29 In Cluster DMC DVR VMC the partial averaging approach the contributions of very - 1.7174(5) - 1.7178 - 1.7046 short length-scale fluctuations to the path-integral results Ne, Ar3 - 2.5532(4) - 2.5535 - 2.5374 are approximated using low-order cumulant methods. The threshold for the use of cumulant methods is adjustable. In its most primitive form where all quantum-mechanical fluctuations are approximated, the partial averaging approach essentially reduces to the Feynman-Hibbs effective poten- argon trimers computed by diffusion Monte Carlo methods tial method. Unlike that approximation, however, the par- and by the discrete variable representation (DVR) ap- tial averagingmethod can be systematically refined. We note proach describedrecently by Leitner et al.’ The generally in passingthat the partial averagingmethod has proved par- excellent agreement of the independently obtained results ticularly useful recently in the treatment of the polaron prob- again suggeststhat the direct DMC method is reliable. A lem.30 detailed discussion of the discrete variable results for the As one last technical point, exceptfor the studiesinvolv- various rare-gastrimers is presentedelsewhere. ’ We note ing helium, exchangeeffects appear to play a relatively mi- here that the ground-state energyof the neontrimer is appre- nor role for the systemsin the present investigation. Conse- ciably abovethe classical ground-state energy of - 3~. In quently, all path-integral calculations in the present work fact, this energy evenlies abovethe minimum classicalener- have been performed assumingBoltzmann statistics. gy of the linear trimer ( - 2.0316). A classical three-atom neon system with a total energy equal to the quantum-me- III. RESULTS AND DISCUSSION chanical ground-state energy would therefore have suffi- We begin our study of the quantum-mechanical effects cient energy to permit the “isomerization” of two equivalent in the neon clusters with a simple order-of-magnitude esti- triangular structures via a linear transition state. This situa- mate. Approximating the Lennard-Jones potential by a qua- tion is in contrast to the case of Ar, whose ground-state dratic form whose frequency is chosen to match the curva- quantum-mechanical energy ( - 2.566) is below the classi- ture of the Lennard-Jones interaction near its potential cal threshold for isomerization. minimum, an elementary harmonic treatment predicts a val- Also displayed in Table I are the results of variational ue of - 0.49256for the ground-state energy of Ne, dimer. Monte Carlo calculations for the neon and argon trimers. That the zero-point energy representsroughly a 50% modi- Thesevariational calculations are readily extendedto larger fication of the classicalbond strength suggestsat the outset rare-gas clusters and a number of such calculations have that quantum-mechanical effects are likely to be significant been reported. The results in Table I employed trial wave for the ground-state energeticsof these systems.Moreover, functions described in detail by Rick et aL9 These vari- at this relatively high energy the system will be probing re- ational estimates are in reasonableagreement with the re- gions of the potential energy outside the harmonic region sults obtained by the two other methods. Using the same thus making appreciableanharmonic effectsalso likely. type of trial function that was utilized for the neon and argon The above suggestions concerning the anharmonic, systems,we have also examinedthe ground state of the heli- quantum-mechanical nature of the neon clusters are con- um trimer. For this system we have utilized the Aziz poten- firmed by direct diffusion Monte Carlo calcula- tial” rather than the Lennard-Jones form for the helium- tions. DMC calculations predict, for example, that the ex- helium interactions. The present variational Monte Carlo act ground-state energy in Ne, is - 0.5667 + 0.0003~, a helium trimer wave function yields an estimate of value that is appreciably above the classical value. We note - 0.0109 f 0.0001~for the helium trimer energy.This is in that anharmonic effects lower the ground-state energy by good agreementwith the estimate of PandharipandeetaI., ” approximately 15% relative to the simple harmonic estimate who obtained a value - 0.0107~for the helium trimer using discussed earlier. This direct DMC result for the neon Green’s-function Monte Carlo methods. As is evident from dimer agreeswell with the value of - 0.56696obtained in- its very large zero-point energy, the quantum-mechanical dependently from a numerical integration of the relevant effects for the helium trimer are substantial. one-dimensionalradial Schrodinger equation. Of central interest in the presentdiscussion is the degree We find similar quantum-mechanical results for the of localization in the ground-state wave functions. Intuitive- neon trimer. Consideration of the trimer system is useful in ly, we would anticipate that the quantum-mechanicaleffects that the various DMC and path-integral results obtained can in helium would be large and that the ground-state wave be verified independently by other methods. Furthermore, function of the helium trimer would be extensively deloca- the wave functions for this systemare relatively easyto visu- lized. Conversely, we would expect that the wave function alize and provide important insights into the extent of quan- for the larger rare-gastrimer systemssuch as argon and be- tum mechanical delocalization. yond would be effectively localized in the vicinity of the clas- Table I lists the ground-state energiesof the neon and sical ground-state structures. These limiting expectations

J. Chem. Phys., Vol. 95, No. 9,1 November 1991 6662 Rick eta/.: Quantum mechanics of clusters are confirmed in Fig. 2. These plots display the Y 1 wave Following the aforementioneddiscussion, we might an- functions of Ref. 9. The coordinatesin Fig. 2 are the usual ticipate that the extent of localization in the neon trimer is Jacobicoordinates (r,R,qS), whereR is the distancebetween intermediatein character,lying somewherebetween that of two atoms, r is the distancebetween the center of massof helium and argon. That the ground-state neon energy is thosetwo atomsand a third, and 4 is the anglebetween unit above the classical isomerizationthreshold might suggest vectorsalong r and R. We seein Fig. 2(a) that the ground- that the neon wave function would, in fact, be delocalized. state helium trimer wave function is extensively delocalized We seein Fig. 2 (c), however, that the neon ground-state and that thereis appreciableamplitude in the vicinity of both wavefunction is strongly localized in the vicinity of the clas- the triangular and linear helium configurations.From Table sical structure with relatively little amplitude in the vicinity I we seethat the energy of the ground state of the argon of the linear configuration.This exampleand related results trimer liesbelow the classical“ isomerization” threshold,the presentedbelow for the seven-atomneon cluster makesit energyof the linear trimer system.It is thus not surprising clear that we must exercisecaution whendeciding questions that the correspondingground-state wave function for the of localization and delocalization.In particular, it is inap- argonsystem [Fig. 2(b) ] is strongly localized in the vicinity propriate to decidethis issueby simply comparing the total of the classical(triangular) ground-stategeometry, with es- energywith classicalisomerization thresholds. sentially no amplitude in the linear trimer region. The relative magnitudesof the quantum-mechanicalef-

4 ‘;(y 1

6 R/u

R/o 2

. 1 - 2

0’ -I 0 1 2 3 4 5 0 1 2 r/o rlo

4 (b)

3 RI---- r R/u A FIG. 2. Shownare contour plots of the variationalwave functions described in Ref. 9 for three atom rare-gassystems. Plots are for He, [Fig. 2(a) 1,Ar, 2 [Fig. 2(b) 1,and Ne, [Fig. 2(c)] wavefunctions. Coordinatesaretheusual Jacobicoordinates (c.f. Fig. 2(b)) withC$ = 90”The. two relevant configurations are the equilateral triangle (R = 2r/v?) and the collineargeometry (r=O).

I I 0 I 0 1 2 r/o

J. Chem. Phys., Vol. 95, No. 9,1 November 1991 Rick &al.: Quantum mechanics of clusters 6663 fects on the ground-stateenergies of larger clusters are simi- 3.0 lar to thosefound in the two- and three-atomsystems. Figure N--Q 3 presentsthe DMC estimatesof the ground-stateenergies for neon clusters ranging from two to sevenparticles. For -25 comparison,we also presentin Fig. 3 harmonic approximations to the various ground-stateenergies. We seefrom Fig. 3 E(T) -1.s ------3.u ------that thesesystems, like the neondimer, are characterizedby appreciable,anharmonic quantum-mechanicaleffects. . . * I . With increasingtemperature, the quantum-mechanical -35 system begins to probe successively higher lying excited states. The interplay of these states is at the heart of the -4.0 + system’s quantum dynamics. For that reasonwe now turn I I.0 1.0 2.0 : our attention to an examination of the behavior of our clusters at finite temperatures. Figure 4 displays the average cluster energiesof 3-5 atom neon clusters as a function of temperaturecomputed using Fourier path integral methods. The agreementbetween the limiting behavior of the low- temperature, distinguishable particle path-integral results and the separatelycomputed, DMC Bose ground-statere- sults suggeststhat the effects of particle statistics on the ground-stateenergies are minimal in thesesmall neon clusters. For comparison,we also presentin Fig. 4 the DMC and harmonic estimatesof the ground-stateenergies of the various clusters. Total energiesin the path-integral calculations wereobtained using direct temperaturedifferentiation of the partition function ( Tmethod) in conjunction with gradient partial averagingtechniques. The improvement in the rate of convergenceof the path-integral calculations provided by this partial averaging procedure is significant, as demon- FIG. 4. Energiesof the Lennard-Jonesmodel of 3-5 atom neonclusters asa function of temperature. Energies are in units of the Lennard-Jones well strated in Table II. We note in passingthat although the depth. Shown are the Fourier path integral Monte Carlo estimates (solid total energiesproduced by path-integral calculations are circles), the correspondingdiffusion Monte Carlo estimatesof the ground- stateenergies (solid line), and harmonic estimatesof the ground-stateenergy (dashedline).

E(N) - NC-~. Ch3, DMC and Hsrmonic guaranteedto convergeto the exact results as the number of Fourier coefficientsis increased,the results are not guaran- teed to convergeto from a particular direction. Using the path-integral and DMC methods discussed earlier, we have examinedthe energeticsof larger neon systems. The path-integral and DMC energiesobtained using the interaction potential describedin Sec.II for Ne, are displayed in Fig. 5. For comparison, we also show in Fig. 5 0 Eq-dm the classical Monte Carlo energiesas well as the known var- + IQ- hmnic ious thresholds for isomerization reported by Wales and . EC

TABLE II. Total energiesfor the Lennard-Jones model of the Ne, cluster computed with and without partial averaging (PA) for various numbers of Fourier coefficients (k,,, ) at a temperature of 2 K. Energiesare in units of the Lennard-Jones well depth. Estimates of the statistical uncertainties in the various results are shown in parentheses.Results shown are computed using the “Tmethod” (seetext).

k max PA no PA

FIG. 3. Energies of neon clusters as a function of the number of cluster 2 - 9.26(2) - 13.99(3) atoms (M. Energies shown are in units of the Lennard-Jones well depth 4 - 9.49(2) - 12.97(2) and correspond to the classical mechanical ground states (solid circles), 8 - 9.62(2) - 11.84(2) diffusion Monte Carlo estimatesof the quantum-mechanical ground states 16 - 9.68(2) - 10.99(2) (open circles), and approximate quantum-mechanicalresults based on sim- 32 - 9.64(4) - 10.28(5) ple harmonic estimatesof the zero-point energy (crosses).

J. Chem. Phys., Vol. 95, No. 9,1 November 1991 6664 Rick eta/: Quantum mechanics of clusters

E(T) - Ne-7 1W

. ’ l ..o*.. 8

-10.0 - % Isomer So

0 2.0 3.0 4.0 5.0 6.0 7.0

T(K)

-2o.o- . . - 1 . . 1 . . . ’ . FIG. 6. Distribution of quench results performed on seven-atomclassical 0.0 2.0 4.0 6.0 neonclusters as a function of temperature.The four curvescorrespond the four stable isomersin Fig. 1 whoseenergies are (in Lennard-Jonesunits) - 16.505(opendiamonds), - 15.935(soliddiamonds), - 15.593(open circles), and - 15.553 (solid circles). Between one and ten thousand quencheswere performed at each temperature on configurations selected randomly from a Boltzmann distribution via Monte Carlo methods. FIG. 5. Energiesof the Lennard-Jonesmodel of the seven-atomneon cluster as a function of temperature.Energies are in units of the Lennard-Jones well depth.Shown are the Fourier path integral Monte Carlo estimates(solid circles), the correspondingclassical estimates (open circlescxxmected by a solid line), the diffusion Monte Carlo estimateof the ground-stateenergy of appreciablecluster isomerization.As will be emphasized (solid line), and the classical mechanical estimatesof the energiesof the againbelow, the strong temperaturedependence of the clas- transition statesthat interconnect the stable structures shown in Fig. 1 re- sical quench populations provide us with evidenceof the ported in Ref. 11. activatednature of the associatedrearrangement dynamics. As discussedpreviously, rate parametersfor the rearrangement kinetics could, in fact, be obtained for the rearrange- Berry.” Severalthings are evidentfrom Fig. 5. We focusthe ments using Monte Carlo transition-state-theory meth- discussionfirst on the classicalresults. From Refs. 3 and 8 ods.I4 we know that the seven-atomclassical Lennard-Jonessys- The increasingfluctional characterof the classicalclus- tem undergoesa rapid transition from a topologically or- ter in this temperaturerange is alsoreflected in other system dered to disordered system in the temperature range of quantities. For example,it can also be seenin the self-diffu- k, T/E = 0.10to 0.15, a rangethat correspondsto 3.6-5.3 K sion constantsshown in Fig. 7. As discussedelsewhere,3 for neon.From Fig. 5 we can seethat this is the temperature the mean-squaredisplacements in finite systemsoften dis- rangein which the system’senergy classically begins to ex- play a linear time dependenceover a range of intermediate ceedthe known rearrangementthresholds. With increasing times. This linear dependencecan be usedto define “diffu- energythe system’sstructure becomesmore and more fluc- sion constants” in thesefinite systems.Here thesediffusion tional, first exploring the anharmonicregion in the vicinity constantswere computedusing a hybrid Monte Carlo-mo- of its lowestenergy structure, and ultimately probingregions lecular dynamics approachin which a reasonablenumber correspondingto successivelyhigher lying isomers. (typically 1000) of cluster initial configurations were cho- The temperature dependenceof the classical cluster sen randomly via the usual Metropolis procedure from a rearrangementscan be made more explicit by the use of canonicalensemble at the desiredtemperature. Particles in quenchtechniques.13 Shown in Fig. 6 are the fractions of theseinitial configurationswere assigned momenta, also se- configurationsselected randomly from a Boltzmann distri- lected randomly from a Boltzmann distribution,32 and the bution at a giventemperature that quenchto the four stable, resulting phase-spaceconfigurations were then propagated seven-atomstructures. We seefrom theseresults that at low- forward temporally using standard molecular dynamics er temperaturesonly configurations in the vicinity of the techniques.Detailed discussionsof the diffusion in thesesys- lowestenergy isomer play a significantrole. With increasing tems and discussionsof other possibleapproaches for its temperature,however, configurationsthat quench to non- study have beendescribed recently by Adams and Stratt33 ground-stateconfigurations emerge, a signatureof the onset and by Beck and Marchioro.34

J. Chem. Phys., Vol. 95, No. 9,1 November 1991 Rick eta/.: Quantum mechanics of clusters 6665

1.4 , 1

(a)

1.3

SIU

1.2 - J

1.1 -

1 2 3 4 5 6 7 8

T (W

FIG. 7. Shown are valuesof the self-diffusion constant as a function of tem- 1.0 I , I I 1 perature for a seven-atomneon cluster. The solid circles are the results of 0.8 1.0 1.2 1.4 1.6 the Monte Carlo-molecular dynamics calculations described in the text. TJS The solid line, intended principally as a visual reference,is a cubic fit to the data. 2.5, 1 I

We now turn our attention to the quantum-mechanical results in Fig. 5. The most striking thing evident in those 5 results is that the ground-stateenergy in Ne, is already well above the relevant classical rearrangementthresholds. That 2.c is, a classical Ne, system with the sametotal energycontent as the quantum-mechanicalground state has energetically stu allowed pathways availableto it that connect the regionsof its potential-energysurface that quenchto all of the various stable isomers. As was the case with the neon trimer dis- 1.5 cussedearlier, however,a simple comparisonof the system’s total quantum-mechanicalenergy to its various classical rearrangementthresholds is not the relevant test for localization. In the present case,for example, we find the ground state of the Ne, system to be localized in the vicinity of the classical structure. This localization is evident in Fig. 8 (a) 1.0' I t I I and is in contrast to the situation with the He, systemdepict- 0.8 1.2 1.4 1.8 ed in Fig. 8(b). The explanationof this localization is that r&/s in a many-dimensionalsystem, there may exist degreesof freedom that are effectively uncoupled from the “reaction FIG. 8. Contour plots of the variational Ne, [Fig. 8(a) ] and He, wave coordinate” involved in the isomerization process.Conse- functions [Fig. 8(b)] of Ref. 9. Coordinatesare thosedescribed in detail in quently, the zero-point energy in these “background” de- Appendix 1 of Ref. 9. The coordinates involved representan overall scale greesof freedom is unavailableto the isomerization dynam- factor (s) and an isomerization coordinate ( rd5). This isomerization coor- ics. As describedelsewhere,9V35 it is thus more appropriateto dinate connects a pentagonal bipyramid structure [ r4s/s = sin(3?r/lO) ] describethe rearrangementdynamics in terms of a “vibra- and a cappedoctahedral structure ( r.,Js = 1) via a saddle ( r4s/s = 3). We seethe Ne, function is localized near the classical minimum-energy struc- tionally adiabatic” model rather that the simple classicalpic- ture (pentagonal bipyramid) while no such localization is evident in the ture. We close by noting the excellent agreementin Fig. 5 He, wave function. betweenthe path-integral results at low temperaturesand the DMC ground-stateenergy. As with the classical results, the temperature dependenceof the quenchpopulation derived from an equilibrium tail, both the quenchesfrom the classical and quantum-me- quantum-mechanical distribution offers insight into the chanical results display a qualitatively similar temperature cluster dynamics. To this end Fig. 9 comparesthe isomer dependence.In particular, both results indicate that the re- distributions resulting from quenchesof the equilibrium arrangementrates are strongly temperaturedependent. We classical and quantum-mechanicaldistributions for Ne, as also note that the population of the lowest-energyisomer functions of temperature.Although slightly different in de- approaches100% in the quantum-mechanicalcalculations,

J. Chem. Phys., Vol. 95, No. 9,1 November 1991 6666 Rick eta/: Quantum mechanics of clusters

-16

-i7 I I

-15 , ,

-16 % Isomer 5o

-17 1 I

-15 I I I

-16

-17 I 0 0 1WOOO 2QbQnc 2.0 3.0 4.0 5.0 6.0 7.0 FIG. 10. Shown are quench results for seven-atom,quantum-mechanical 'UK) neonclusters. In thesecalculations a path-integral Monte Carlo sequenceof configurationsis generatedat the specifiedtemperature (4 K) and the resulting configurationsare quenched.The three panelscorresponding to dif- FIG. 9. A comparisonof classicaland quantum-mechanicalquench results ferent initial cluster geometries,here taken to be the configurationsof the for seven-atomneon clusters. Notation is that of Fig. 6. Quantum-mechani- higher lying classicalisomers (c.f. Fig. 1). The panelsshow the energies(in cal resultsare connectedby dashedlines and the correspondingclassical units of the Lennard-Joneswell depth) of the final quenchedconfigurations resultsare connectedby solid lines. after a specifiednumber of path-integralMonte Carlo sweeps.Quenches are performed after every 10 sweeps.The results indicate that at this temperature the systemmakes frequent transitions between the variousstable classical isomers. an indication of a localizedground state.We confirmedthis localization by performing similar quench studies on the high quality Ne, ground-statewave function reported in of zero- and finite-temperatureMonte Carlo methods.Al- Ref. 9. Quenchesbased on that shadowwave function in- though not as dramatic asin the caseof helium, we find the dicatethat approximately99.95% of the configurationspace magnitudeof the quantum-mechanicaleffects on the equi- densityis associatedwith the classicalground-state isomer, librium propertiesof small (2-7 atom) neonand argon clus- in agreement with the aforementionedresults.

We can seeadditional evidenceof the activatedcharac- -15 ter of the seven-atomcluster rearrangementdynamics for I neon in Fig. 10. There we have displayed the results of quenchstudies performed on configurationsfor Ne, selected randomly from a quantum-mechanicaldistribution at 4

K. Although configurationsin the vicinity of the lowest- -17 L J

energyisomer predominate,at this temperaturethe system -15 accessesconfigurations correspondingto all stable struc- 1 tures.Transitions between these stable structures are clearly I III evidentin Fig. 10.As the temperatureis reduced,however, suchtransitions become less common. Furthermore, at low- I

er temperaturesconfigurations artificially started in the -17 1 1

quenchvicinity of the higher-energystructures have a ten- -15 , dencyto cascadetoward the lowest-energystructure. Such a I cascadeis illustrated in Fig. 11 for a temperature of 2 K. We seethere that at this relatively low temperature,the clusters,having found their way to the vicinity of the lowest-

energystructure, effectivelynever visit other minima in the -17 1 I potential energy. 0 WOO0 200000

IV. SUMMARYAND CONCLUSIONS FIG. 11. As in Fig. 10 with T= 2 K. In each casethe cluster tends to cascadeto a geometrythat quenchesto the lowest energyclassical isomer. The presentwork has consideredthe equilibrium and As seenin the figures,once the vicinity of the lowest energyclassical struc- dynamical behaviorof neonand argon clustersat low tem- ture has been reached,transitions to configurationsthat quench to other perature.These issues have been probed using a combination isomersare quite rare at this temperature.

J. Chem. Phys., Vol. 95, No. 9,1 November 1991 Rick eta/: Quantum mechanics of clusters 6667 ters to be appreciable.We find, for example,that the quan- ‘J. Jellinek,T. L. Beck, andR. S. Berry, J. Chem. Phys. 84,2783 ( 1986); R. tum-mechanicalground-state energies in theseclusters often S. Berry, J. Jellinek, and G. Natanson, Phys. Rev. A 30,919 (1984); H. Davis, J. Jellinek, and R. S. Berry, J. Chem. Phys. 91,495O(1987); T. L. exceedsthe classical thresholds for cluster rearrangement. Beck, J. Jellinek, and R. S. Berry, J. Chem. Phys. 87,545 (1987). Even with this appreciablezero-point energy, however, we ‘D. L. Freeman and J. D. Doll, Adv. Chem. Phys. 70, 139 ( 1988), and find that the ground-statewave functions in these systems referencestherein; U. Landmann, P. Sindzingre, M. L. Klein, and D. M. are strongly localized about the structure of classicalground Ceperley, Phys. Rev. Lett. 63, 1601 (1989). ’ M. V. Rama Krishna and K. B. Whaley, Phys. Rev. Lett. 64,1126 ( 1990). state. In related findings, finite-temperaturequench studies 6R. Melzer and J. G. Zabolitzky, J. Phys. A 17, L565 ( 1984). reveal that the cluster isomerization in these systems is ‘D. M. Leitner, R. S. Berry, and R. M. Whitnell, J. Chem. Phys. 91,347O strongly temperaturedependent, a result that suggeststhat ( 1989); D. M. Leitner, J. D. Doll, and R. M. Whitnell, J. Chem. Phys. 94, the associateddynamics is activated in character. 6644 (1991). ‘T. L. Beck, J. D. Doll, and D. L. Freeman, J. Chem. Phys. 90, 5651 (1989). ACKNOWLEDGMENTS 9S. W. Rick, D. L. Lynch, and J. D. Doll J. Chem. Phys. (to bepublished). IoR. A. Aziz, F. R. W. McCourt, and C. C. K. Wong, Mol. Phys. 61,1487 The authors would like to thank ProfessorsJ. E. Adams, (1987). T. L. Beck, R. M. Stratt, and K. B. Whaley for numerous, ” D. J. Wales and R. S. Berry, J. Chem. Phys. 92,4283 ( 1990); F. H. Stil- stimulating conversationsconcerning the diverse and inter- linger and D. K. Stillinger, J. Chem. Phys. 93, 6013 ( 1990). esting behavior of clusters. J.D.D. wishesto thank Brown “F. G. Amar and R. S. Berry, J. Chem. Phys. S&5943 (1986). I3F. H. Stillinger and T. A. Weber, Phys. Rev. A 25,978 ( 1982). University and the Department of Chemistry for its gener- 14J.Doll, J. Chem. Phys. 73,276O (1980); 74, 1074 (1981). oussupport of this work. D.L.F. acknowledgesthe donorsof 15M. J. Gillan, J. Phys. C 20, 3621 (1987). the Petroleum ResearchFund of the American Chemical “‘G. A. Voth, D. Chandler, and W. H. Miller, J. Chem. Phys. 91, 7749 Society for partial support of this work. (1989). “J. Jellinek, T. L. Beck, and R. S. Berry, J. Chem. Phys. 84,2783 ( 1986). “D. M. Leitner, R. S. Berry, and R. M. Whitnell, J. Chem. Phys. 91, 3470 APPENDIX (1989). I9E. Y. Loh, Jr., J. E. Gubematis, in Electronic Phase Transitions, edited by As describedin Sec. II, the DMC method exploits the W. Hanke and Y. V. Kopaev (North-Holland, New York, 1991). isomorphism betweenthe Schriidinger equation in imagi- “W. L. McMillan, Phys. Rev. A 138,442 (1965); J. P. Valleau and G. M. nary time and the diffusion equation. Using this isomor- Torrie in Statistical Mechanics, Part A, edited by B. J. Beme (Plenum, phism, one “solves” the quantum-mechanicalproblem nu- New York, 1977), and referencestherein. ” M. V. Rama Krishnaand K. B. Whaley, J. Chem. Phys. 93,6739 (1990). merically by using random-walk methods originally “S. A. Vitiello, K. Runge, and M. H. Kalos, Phys. Rev. Lett. 60, 1970 designedfor the simulation of ordinary diffusion processes. (1988). As discussedelsewhere,25 the DMC method is exact only in 23S. A. Vitiello, K. Runge,and M. H. Kalos, Phys. Rev. B 42, 228 ( 1990). the limit that the time step used in modeling the temporal *4J. B. Anderson, J. Chem. Phys. 63, 1499 (1975); 65,412l (1976); P. J. Reynolds, D. M. Ceperley, B. J. Alder, and W. A. Lester, Jr., J. Chem. evolution of the diffusion processtends to zero. In practice, Phys. 77,5593 (1982). acceptablevalues for the time step were establishedby per- *‘B. Alder and D. Ceperley, Science231,555 ( 1986). forming calculations for a rangeof parametersand requiring x D. M. Ceperley and M. H. Kalos, in Monte Carlo Methods in Statistical that the associatedsystematic error in the calculation be Physics, edited by K. Binder (Springer, Berlin, 1979), pp. 145-197. 2’J. D. Doll. T. L. Beck. and D. L. Freeman, Adv. Chem. Phys. 73, 61 smaller than the associatedstatistical error in the calcula- (1990), and references’therein. tions. This range varied with the system under study, but ‘* D. Chandler and P. G. Wolynes, J. Chem. Phys. 74,4078 ( 1981). was typically 100a.u. (2.4~ lo- I5s) or smaller in the pres- “‘J. D. Doll, R. D. Coalson, and D. L. Freeman, Phys. Rev. Lett. 55, 1 ( 1985); R. D. Coalson,D. L. Freeman, and J. D. Doll, J. Chem. Phys. 85, ent neonstudies. For example,step sizes of 100,200,and 400 4567 (1986). a.u. producedDMC estimatesto the total energyof Ne, of “C Alexandrou, W. Fleischer, and R. Rosenfelder,Phys. Rev. Lett. 65, - 0.5667(3)~, - 0.5662(4)~, and - 0.5668(4)~, where 2615 (1990). the numbersin parenthesesindicate the statistical uncertain- ” V. R. Pandaripandie,J. G. Zabolitzky, S. C. Pieper, R. B. Wiringa, and U. Helmbrecht, Phys. Rev. Lett. 50, 1676(1983). ties in the associatedtotal energies. 321nthe present diffusion calculations the total angular momentum was restricted to a value of zero. “‘J E. Adams and R. M. Stratt, J. Chem. Phys. 93, 1332(1990); 93, 1358 ’ For a cross section of recent activity, see,for example, Adv. Chem. Phys. (1990); 93, 1632 (1990). 70 (Part 2) (1988). “‘T. L. Beck and T. L. Marchioro, J. Chem. Phys. 93, 1347 ( 1990); Phys. ‘J. K. Lee, J. A. Barker, and F. F. Abraham, J. Chem. Phys. 58, 3 166 Rev. A 42,5019 (1990). ( 1973); F. F. Abraham, Homogeneous Nucleation Theory (Academic, “B. C. Garrett and D. G. Truhlar, Int. J. Quantum Chem. 29,1463 (1986); New York, 1974), Chap. 9. D. G. Truhlar, J. Chem. Phys. 53,204l ( 1970).

J. Chem. Phys., Vol. 95, No. 9,1 November 1991

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