University of Rhode Island DigitalCommons@URI Chemistry Faculty Publications Chemistry 1991 The Quantum Mechanics of Clusters: The Low - Temperature Equilibrium and Dynamical Behavior of Rare-Gas Systems Steven W. Rick D. L. Leitner See next page for additional authors Follow this and additional works at: https://digitalcommons.uri.edu/chm_facpubs Terms of Use All rights reserved under copyright. 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 present paper 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 appreciably nonclassical. 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 substantial practical 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 adequacy of 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-body systemsin 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 been utilized 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-body systems.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-point fluc- ciple, lessclouded by uncertaintiesin the fundamental inter- 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 intermediate scaleof clusters permits a variety of theoretical systemsand the extent to which their dynamical behavior techniquesto be brought to bear on 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 properties of clusters were, 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 thesemeth- ods to neon and argon clusters and draws a number of con- ‘) Present address: Theoretische Chemie, PhysikaIish-ChemischesInsti- tut, Im Neuenheimer Feld 253, D-6900 Heidelberg, Germany. clusions concerning the 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 as- signed 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, established Monte 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