A Computational Study of Rare Gas Clusters: Stepping Stones to the

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A Computational Study of Rare Gas Clusters: Stepping Stones to the Solid State Eric D. Glendening and Arthur M. Halpern* Department of Chemistry and Physics, Indiana State University, Terre Haute, Indiana 47809, United States S Supporting Information *

ABSTRACT: An upper-level undergraduate or beginning graduate project is described in which students obtain the Lennard−Jones 6-12 potential parameters for Ne2 and Ar2 from ab initio calculations and use the results to express pairwise interactions between the atoms in clusters containing up to N = 60 atoms. The students use simulated annealing, or the genetic algorithm, to find the globally optimized binding energies and structures of the Ne and Ar clusters. They employ the liquid drop model to extrapolate the cluster binding energies to the solid state and compare the result with the experimental cohesive energy of the rare gas solid. A Windows-based application is provided that allows students to explore the energetic and structural properties of the rare gas clusters. Students encounter the “magic numbers”, for example, N = 13, 55, and others, associated with clusters that have higher-than-expected binding energies arising from enhanced nearest-neighbor interactions. They also estimate the solid density of each element from the size of the model cubic cluster (N = 14) that represents the face-centered unit cell. KEYWORDS: Upper-Division Undergraduate, Inorganic Chemistry, Physical Chemistry, Computer-Based Learning, Computational Chemistry, Noncovalent Interactions, Solids

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properties of the bulk material. Considering the importance of understanding clusters and how they bridge the microscopic and macroscopic domains, it is surprising that only a few articles on clusters have been published in this Journal.4 It is valuable to introduce students to the fundamentals of cluster energetics and structure and how such knowledge can be used to predict bulk phase properties. This project provides the instructor with a framework of ideas and activities that he or she can adapt to guide students to use and understand sophisticated techniques for obtaining such information about Ne and Ar clusters and to apply their results to predict properties of the crystalline solids.

e describe a project that introduces upper-level undergraduate or graduate students to the structural and bonding properties of rare gas clusters. Here, a cluster is defined as an ensemble of atoms having a discrete structure and a well-defined binding energy. In principle, a cluster may consist of as few as two or three atoms, or scores, hundreds, or as many as one can imagine (but presumably less than Avogadro’s number!). However, for practical reasons, this project engages students to study rare gas clusters with the number of atoms, N, between 2 and 60. Rare gas atoms are used in the study because, unlike the situation with metal and molecular clusters, the bonding of these atoms in clusters is much easier to characterize and calculate. In principle, any rare gas system can be explored in this project, but the results of studying Ne and Ar are described. Helium is excluded because of the difficulty in properly accounting for the weak interactions between these atoms (and the quantum complications encountered in the solid state1), and Kr and Xe are excluded because the large number of electrons in these systems makes quantum chemical calculations impractical. The number of research and general-public articles dealing with clustersboth atomic (especially those of metals) and molecularhas vastly increased during the past several decades. This interest stems from a number of reasons, including the novel catalytic and structural properties of metal clusters and nanoparticles and the ability to isolate them in sufficient quantity to study their properties.2 Clusters can be thought of as the state of matter between small, discrete molecules and the bulk phase.3 Many questions arise regarding how large a cluster must be before it begins to manifest © 2012 American Chemical Society and Division of Chemical Education, Inc.

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PROJECT OBJECTIVES, STRATEGY, AND LEARNING OUTCOMES The goals of this project are to calculate solid-state propertiescohesive energy and densityfrom a systematic study of clusters. The components of the project are as follows. Students will (1) assume that the interaction between atoms in Ne and Ar clusters is represented by the pairwise Lennard−Jones (LJ) 6-12 potential; (2) obtain the LJ parameters from ab initio quantum chemical calculations; (3) use a search procedure such as the genetic algorithm5 (GA) to find the lowest-energy cluster geometries; (4) apply the liquid drop model6 (LDM) to obtain the bulk cohesive energies, Ecoh, of the solids; (5) obtain cluster zero-point energies using the harmonic approximation; (6) construct a cluster that is a model of the face-centered cubic unit cell to estimate the solid density; and (7) compare their Published: October 15, 2012 1515

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atoms, and most other post-Hartree−Fock methods underestimate binding energies. Students require some guidance in choosing the internuclear distances to use for their PE scans for Ne2 and Ar2. This is important to manage the computational times required for each point on the PE curve. First, students can be guided to estimate r values for Ne2 and Ar2 at which V(r) = 0 in eq 1 (i.e., σ) and where V(r) is a minimum (the bottom of the well). Students should be asked to show that for the LJ potential V(r) is a minimum at r = 21/6σ. Only a few scan points are needed for r < σ because the repulsive portion of the V(r) is steep; more points are needed to capture the attractive part of the potential well. For the purpose of this project, one approach is to perform a 14-point scan for the Ne2 potential, between 2.7 and 4.0 Å in 0.1 Å increments. For the Ar2 potential, a 15-point scan can be performed between 3.3 and 4.7 Å in 0.1 Å increments. Figure 1 shows the scan results for the Ne2 and Ar2 potentials and the respective regression curves of the data fit to eq 1.

results with the respective experimental solid-state data. The experimental value of Ecoh is obtained from low-temperature vapor pressure or heat capacity measurements and corresponds to the sublimation energy extrapolated to 0 K. All quantum chemistry calculations, including searches of the LJ cluster geometries and energies, and the analysis of the results using the LDM can be performed on a PC using commercially available programs and an application (SAGA)7 that is designed for this project. SAGA is a free program that allows students to search LJ cluster geometries using either a simulated annealing (SA)8 algorithm or GA. Learning effectiveness can be tested through written reports, exams, or oral presentations. The time required to complete the work depends on how the instructor designs the project, the number and size of student groups, and the nature of the computing resources available. Specific details of performing the calculations and other operational comments are provided in the Supporting Information.

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INTERATOMIC INTERACTIONS A fundamentally important part of any theoretical approach to cluster studies is the treatment of interatomic interactions. An interaction potential is sought that accurately accounts for bonding in the cluster and is readily understood and managed by students. For this purpose, only two-body interactions are considered between the N atoms of a cluster and the wellknown and widely used LJ potential adequately represents these interactions. The LJ (6-12) potential is ⎡⎛ σ ⎞12 ⎛ σ ⎞6 ⎤ VLJ(r ) = ⎢⎜ ⎟ − ⎜ ⎟ ⎥ ⎝r⎠ ⎦ ⎣⎝ r ⎠

(1)

where r is the internuclear distance, ε is the well depth, and σ is the value of r at which V(r) = 0. For more precise work, the LJ potential can be extended to include additional terms in (1/r)n, where n = 8, 10, 14, 16.9 SAGA can accommodate such potentials. Although the LJ parameters (and also those for extended LJ potentials, if desired) can be found in the literature,10 the students acquire σ and ε by fitting the points of a potential energy (PE) curve obtained from quantum chemical calculations to the LJ potential, eq 1. In this way, students obtain a sense of “ownership” of the project and also learn about and gain experience in applied computational chemistry. Moreover, students have the satisfaction of estimating a property of a solid-state material from scratch as no experimental data is used. Students obtain the LJ parameters by calculating the energy of a pair of atoms at fixed distances in the vicinity of the PE minimum (relative to two isolated atoms) and fitting eq 1 to these energy values. A compromise is sought between accuracy (computational rigor) and practicality (reasonable resources and calculation times). With these considerations in mind, calculations for Ne2 use a high-level electronic structure methodcoupled cluster with single, double, and perturbative triple excitations, denoted CCSD(T)11and a large basis set, the augmented correlation-consistent polarized valence quintuple-zeta set, aug-cc-pV5Z.12 For Ar2 calculations, CCSD(T) with the quadruple-zeta basis set belonging to the same family, aug-cc-pVQZ is used. The CCSD(T) method is selected for these calculations because the Hartree−Fock approach does not appropriately capture the attractive interactions of rare gas

Figure 1. (Top) Regression fit of the Lennard−Jones parameters (eq 1) to the CCSD(T)/aug-cc-pV5Z scan points for Ne2. The parameters are σ = 2.763(2) Å and ε = 29.4(3) cm−1. (Bottom) Regression fit of the Lennard−Jones parameters (eq 1) to the CCSD(T)/aug-cc-pVQZ scan points for Ar2. The parameters are σ = 3.383(1) Å and ε = 93.8(6) cm−1.

The PE scans can be readily carried out using a one of a number of quantum chemistry programs.13 The Gaussian suite of programs for Windows platforms13a was used for the ab initio calculations reported here. Details for performing Gaussian calculations, extracting the CCSD(T) energies from the output files, and obtaining the LJ parameters from fits to the CCSD(T) curves are provided in the Supporting 1516

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Information. Values of σ and ε from these fits, their standard deviations, and literature values are given in Table 1. A

(SA) and genetic algorithm (GA), are provided to conduct the search for global minima on the PE surface. Because it is difficult to prove that a search of this surface has identified the true global minimum out of a multitude of local minima, it can be instructive to explore the surface using more than one search algorithm. Different students or groups of students engaged in this project might be directed to employ different search algorithms (SA or GA) to judge the extent to which these methods yield common cluster geometries. The results reported here were obtained using the GA method, which is perhaps less widely used than the SA approach. Thus, the GA method for locating the global minimia of the rare gas clusters is briefly presented. The capabilities of SAGA are more fully described in the user’s guide that is distributed with the application.7 GA is an optimization method based on the ideas of natural selection.5 The algorithm maintains a population of clusters, each cluster consisting of N atoms and having a well-defined geometry and, hence, energy given by eq 2. A mating operation is performed on the population in which two “parent” clusters mate to yield an offspring cluster. The parent clusters are randomly rotated and then sliced in half, the upper half of the first parent combining with the lower half of the second to form the child. If the child has lower energy than the cluster of highest energy in the population, then the child replaces that cluster (the high-energy cluster is discarded). High-energy child clusters (“mutants”) are occasionally permitted to enter the population. Such “mutations” allow the population to maintain a degree of diversity that ensures a more complete exploration of the interaction potential surface by the GA. As mating operations are applied iteratively on the population, the population will tend to converge on a cluster of lowest energy, presumably that associated with the global energy minimum.

Table 1. Lennard−Jones σ and ε Values Ne2 σ/Å

Source a

This Work Experimentb

2.763(2) 2.78

Ar2 ε/cm

−1

29.4(3) 24.3

σ/Å

ε/cm−1

3.383(1) 3.405

93.8(6) 83.3

a Values obtained from regressing the PE scan data to eq 1. bValues obtained from the temperature dependence of second virial coefficients of the gases.10

comparison of the LJ parameters in Table 1 reveals good agreement for the calculated and experimental σ values, but somewhat larger well depths (ε) from the calculations. It should be noted, however, that the PE curves obtained from the ab initio calculations are consistent with those reported for other, higher-level quantum chemical calculations.14

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MINIMUM ENERGY GEOMETRIES The total interaction potential energy of clusters containing N atoms is approximated as the sum of all two-body interactions, for which each ij interaction is accounted for by the LJ potential of eq 1. Thus, N

Vint(N ) =

∑ VLJ(rij) i 0), attempts to compensate for the reduced affinity of the surface atoms. Equation 4 may be recast to express the intensive, or unit binding, energy of the cluster by dividing by N. Thus, Eb /N = a v − asN −1/3 + ...

(5)

It is apparent that in this expression of the LDM, the unit binding energy of a cluster approaches the cohesive energy of the bulk solid (i.e., av) in the limit of large N. Note that the LJ potential, incorporated in eq 2, represents the classical binding energy, that is, the difference between the energy of N isolated atoms and the bottom of the cluster potential. This energy is denoted as Eb,e to distinguish it from the (0 K) quantum mechanical binding energy, Eb,0, that references the ground vibrational state of the potential (Eb,e = Eb,0 + zero-point energy). Accordingly, cluster binding energies will be expressed as Eb,e or Eb,0, as appropriate, and the corresponding regression parameters as av,e, as,e or av,0 and as,0. To achieve one of the goals of this project, students will fit their Eb,e cluster data to eq 4 to obtain the regression value of av,e, equate it with Ecoh, and compare it with experimental data. Interpreting the surface energy, as,e, is beyond the scope of this project. The binding energies of clusters from N = 2−60 are obtained, employing the GA as implemented in SAGA. The application of the LDM to the cluster binding energies is first discussed and then extrapolated to the bulk solid. Finally, the cluster geometries are discussed. Figure 2 (upper curve) shows the regression fit of eq 4 to the Eb,e values for the Ne clusters. The regression value of av,e is 232(2) cm−1, which compares with the reported experimental cohesive energy of 160 cm−1.21 Schwerdtfeger et al.,18 using an extended LJ potential for Ne, employed the LDM model to obtain a corresponding value of 219 cm−1. These LDM-based estimates of the cohesive energy are too high, in part, because they neglect zero-point effects. Neon may be regarded as a quantum solid because of the relatively small atomic mass and weak binding energy. To illustrate this point, note that the classical binding energy and the zero-point energy of the Ne dimer are 29.4 and 13.6 cm−1, respectively, based on the ab initio scan of the internuclear potential reported here. Thus, nearly half of the well depth is zero-point energy (in the case of Ar, it is ca. 15%). Students might not have been able to anticipate the problem in using Eb,e values to obtain Ecoh, but once it is pointed out to them they should readily understand

Figure 2. Binding energies of the Ne clusters: (top) plot of Eb,e versus N and (bottom) plot of Eb,0 versus N. The respective regression curves, based on fits to eq 4, are also shown. The regression values of av,e and as,e are 232(2) and 326(5) cm−1 for the upper curve. The regression values of av,0 and as,0 are 161(1) and 242(5) cm−1 for the lower curve, respectively.

the need to work with zero-point-corrected cluster binding energies.

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CLUSTER ZERO-POINT ENERGIES The ability to calculate the zero-point energies of the optimized clusters within the harmonic potential approximation has been implemented in SAGA. A brief description of this calculation may be found in the SAGA user’s guide.7 SAGA reports both the classical and zero-point-corrected energies of the optimized clusters. Equipped with this information, students can now apply the LDM (eq 4) to the zero-point-corrected binding energies of the clusters, Eb,0. The outcome of this analysis is shown in the lower portion of Figure 2. The regression value of av,0 is, as expected, smaller, having a value of 161(1) cm−1, which is in remarkably good agreement with the reported experimental value of 160 cm−1.21 It is appropriate to point out to students that the closeness of this agreement is fortuitous; there are too many approximations (e.g., the use of the LJ potential to account for pairwise interaction energy and the harmonic approximation to obtain zero-point energies) in the cluster energetics and LDM models (viz. eqs 4 and 5) to justify such precise agreement with experiment. The results of the LDM analysis of the cluster binding energies for Ne and Ar and their respective experimental cohesive energies are presented in Table 2. As clusters become very large, their properties may be expected to approach those of the bulk solid, and, indeed, the 1518

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points that lie above the smooth-curve trend (the regression curves based on fitting the Eb/N data to eq 5), for example, at N = 13, for several clusters before N = 30, and at N = 55. These features are not computational artifacts. The GA algorithm converges to global minima having cluster binding energies that are much more precise than the apparent deviation of these points relative to the regression curve. Thus, these points represent clusters that are more stable than expected from the general Eb versus N trend (i.e., eq 4). These effects are also present in the data shown in Figure 2, but are not as noticeable. Such behavior has been well characterized in atomic cluster studies,18 and the values of N at which these effects are observed have been called “magic numbers”.23 The first use of the term “magic number” is attributed to Wigner,24 in reference to those nuclides that had higher stabilities than predicted by the LDM (applied to nuclides). In this context, students will readily recognize a similar phenomenon with the ionization energy trends of atoms in the periodic table, for example, the high ionization energies for elements with atomic (magic) numbers 2, 10, 18, 36, 54, and 86. The magic-number effects pointed out in Figure 3 can be more readily observed by plotting the incremental cluster binding energies versus N. Such a plot is shown in Figure 4 for

Table 2. Volume Energies av of the Ne and Ar Clusters and the Experimental Cohesive Energies Calculated

Experimental

Cluster

av,ea/cm−1

av,0b/cm−1

Ecohc/cm−1

Ne Ar

232(2) 742(5)

161(1) 668(5)

160 647

a

Calculated from eq 4 with the zero-point energy uncorrected. Calculated from eq 4 with the zero-point energy corrected. cData from ref 21. b

application of the LDM6 seems to provide a very reasonable estimate of the cohesive energy of the solid. Another, more instructive way to illustrate this convergence is to examine the unit binding energies of the clusters, that is, Eb,0/N, and to show how this quantity converges to Ecoh in the limit of large N (see eq 5). In Figure 3, the plots Eb,e/N and Eb,0/N versus N for the

Figure 4. Plot of incremental binding energies versus N (eq 6) illustrating the higher stabilities of the “magic number” clusters of Ar.

Ar clusters. The incremental binding energy ΔEb,e of an N-atom cluster is defined as the binding energy of a cluster minus that of the cluster having one fewer atom, viz. ΔE b,e(N ) = E b,e(N ) − E b,e(N − 1)

(6)

Magic numbers, corresponding to energy maxima in Figure 4, appear at N = 7, 13, 19, 23, 26, 29, 32, 36, 38, 43, 46, 49, and 55.25 The magic numbers N = 13 and 55 are perhaps not particularly surprising because the corresponding cluster geometries are fully icosahedral, as shown in Figure 5, having either one complete shell (N = 13) or two complete shells (N = 55) of atoms that surround a central atom. All other clusters,

Figure 3. (Top) Plot of Eb,e/N versus N. (Bottom) Plot of Eb,0/N for the Ne clusters. The smooth curves are calculated using eq 5 and the regresison parameters of Figure 2 and Table 2.

Ne clusters are shown, along with the respective regression curves based on the fits to eq 5. These graphs show that even at N = 60 there is a way to go before the data closely approaches the respective values of av,e and av,0.22

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CLUSTER GEOMETRIES AND MAGIC NUMBERS The discerning student will notice in Figure 3 that the unit binding energies of the clusters (for N > 10) do not represent a smoothly increasing function, as might be expected. There are

Figure 5. Images of the N = 13 (left), 38 (center), and 55 (right) clusters of Ar. The geometries of the 13- and 55-mers are icosahedral; the 38-mer is octahedral. 1519

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for N = 7−60, have similar icosahedral geometries, but with incomplete outer shells of atoms. The only exception is N = 38, for which the cluster has a unique octahedral arrangement of atoms (Figure 5). It is instructive to have students examine several cluster geometries in detail to identify the origin of the magic numbers. We consider the clusters N = 22−24, which have respective incremental binding energies of 150, 177, and 132 cm−1. Figure 6 shows the minimum energy geometries for these clusters in

d=

4M NAl 3

(7)

where M is the molar mass. Note that the density expressed by eq 7 is based on 4 atoms per unit cell, each such cell being encased symmetrically by 26 other such cells in the bulk solid, and the use of the cluster dimension, l, instead of the lattice constant, and thus reflects the assumption stated above. Table 3 shows the results of applying eq 7 to finding the densities of solid Ne and Ar. Table 3. Optimized Values of the 14-mer Cubic Cluster Edge Lengths, l, Lattice Constants, a, and Densities, d

Figure 6. Images of the N = 22 (left), 23 (center), and 24 (right) clusters of Ar. The six atoms that define the optimal binding pocket of the 22-mer are labeled. An atom binds to this pocket in the 23-mer. The 24-mer forms when an atom binds to the four centers labeled in the 23-mer.

Parameter

Ne

Ar

l (cluster model)/Å a (experiment)/Å d (cluster model)/(g cm−3) d (experiment)/(g cm−3)

4.522a 4.464b 1.449d 1.507b

5.431a 5.300c 1.656d 1.782c

a Obtained from MP2/aug-cc-pVDZ-optimized cubic 14-mer. bData from ref 26. cData from ref 27. dValue calculated from eq 7.

The results indicate good qualitative agreement between the calculated and experimental bulk densities. Evidently, the dimensions of the cubic Ne and Ar 14-mers, found from MP2/aug-cc-pVDZ optimizations, do not differ significantly from those of the solid-state unit cells, probably reflecting the weak van der Waals interactions between the rare gas atoms, both in the cluster and in the solid. Students should be made aware of the fact that the cubic 14-mers have slightly higher binding energies than the respective globally optimized isomers obtained from their SAGA calculations. These clusters have 36 nearest (close contact) neighbors while the optimized 14-mers have 45.

similar orientations. The arrangement of atoms in the 23-mer is largely identical to that of the 22-mer, except for an extra surface atom. Comparison of these geometries reveals that the extra atom of the 23-mer occupies what was a pocket on the surface of the 22-mer. This pocket is defined by six atoms, one atom at the base of the pocket and five atoms encircling it. Thus, when an atom binds to the pocket, forming the 23-mer, the atom interacts most strongly with six nearest neighboring atoms. Hence, the incremental binding energy of the 23-mer is rather large. Similar comparison of the 23- and 24-mers reveals that the extra atom of the 24-mer binds to a relatively exposed surface position in which the atom has only four nearest neighboring atoms. Fewer nearest neighbors results in weaker attraction to the cluster and, therefore, a smaller incremental binding energy.

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ASSOCIATED CONTENT

S Supporting Information *

Computational details. This material is available via the Internet at http://pubs.acs.org.

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SOLID DENSITIES Students may wonder if they can obtain the densities of the Ne and Ar bulk solids from their cluster calculations. This objective is complicated by the fact that calculating the appropriate volumes of the clusters is not a straightforward matter, and therefore finding the densities of the clusters is not feasible. Even if the cluster densities were found, the extrapolation to the solid state is not a simple task. Instead, we suggest that students first find out for themselves that the rare gas elements form cubic close-packed (face-centered cubic) solids, create a model cluster that represents this unit cell, and then find its optimized coordinates, from which they calculate the cluster volume. The appropriate model consists of 14 atoms located at the 8 corners and 6 faces of a cube. This cubic cluster (Oh symmetry) is readily set up in Cartesian coordinates and then optimized using an ab initio method, such as MP2 with the aug-cc-pVDZ basis set.12,13 The details for carrying out this calculation are presented in the Supporting Information. If the assumption is made that the edge lengths, l, of these cubic Ne and Ar 14-mer clusters do not appreciably differ from those of the lattice constants, a, of the respective solids, one can estimate the solid density, d, from

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Professors M. Brack and P. Schwerdtfeger for helpful e-mail exchanges. REFERENCES

(1) (a) Guyer, R. A.; Richardson, R. C.; Zane, L. I. Rev. Mod. Phys. 1971, 43, 532−600. (b) Polturak, E.; Gov, N. Contemp. Phys. 2003, 44, 145−151. (2) (a) Gates, B. C. Chem. Rev. 1995, 95, 511−522. (b) Heiz, U.; Bullock, E. L. J. Mater. Chem. 2004, 14, 564−577. (3) (a) Fehlner, T.; Halet, J.-F.; Saillard, J.-Y.; Molecular Clusters: A Bridge to Solid-State Chemistry; Cambridge University Press: Cambridge, 2007. (b) Berry, R. S.; Rice, S. A.; Ross, J. The Structure of Matter, 2nd ed.; Oxford University Press: New York, 2002; pp 347− 348. (c) Jena, P; Castleman, A. W., Jr. Nanoclusters: A Bridge Across Disciplines; Elsevier Science: Amsterdam, 2010; pp 1−36.

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Article

(4) (a) Jelski, D. A.; George, T. F. J. Chem. Educ. 1988, 65, 879−883. (b) Kahn, D.; Viswanathan, R. J. Chem. Educ. 1997, 74, 982−984. (5) (a) Holland, J. H.; Adaptation in Natural and Artificial Systems; University of Michigan Press: Ann Arbor, MI, 1975. (b) Holland, J. H.; ; Genetic Algorithms ; http://www2.econ.iastate.edu/tesfatsi/holland. gaintro.htm (accessed Oct 2012). (c) Deaven, D. M.; Ho, K. M. Phys. Rev. Lett. 1995, 75, 288−291. (d) Deaven, D. M.; Tit, N.; Morris, J. R.; Ho, K. M. Chem. Phys. Lett. 1996, 256, 195−200. (6) (a) Ohanian, H. C. Modern Physics, 2nd ed.; Prentice Hall: Upper Saddle River, NJ, 1987; pp 353−359. (b) Evans, R. D. The Atomic Nucleus; McGraw-Hill: New York, 1955; pp 365−367. (7) SAGA Home Page. http://carbon.indstate.edu/saga (accessed Oct 2012). Mac users can install and execute the application using the underlying Unix operating system. (8) (a) Bertsimas, D.; Tsitsiklis, J. Stat. Sci. 1993, 8, 10−15. (b) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes, 2nd ed.; Cambridge University Press: New York, 1992; pp 436−447. (9) Hajigeorgiou, P. G. J. Mol. Spectrosc. 2010, 263, 101−110. (10) Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; John Wiley & Sons: New York, 1954; p 1110. (11) Levine, I. N. Quantum Chemistry, 5th ed.; Prentice Hall: Upper Saddle River, NJ, 2000; pp 568−573. (12) (a) Levine, I. N. Quantum Chemistry, 5th ed.; Prentice Hall: Upper Saddle River, NJ, 2000; p 492. (b) Woon, D. E.; Dunning, T. H., Jr. J. Chem. Phys. 1993, 98, 1351−1371. (13) (a) Frisch, M. J. et al. Gaussian03W, Version 6.1; Gaussian, Inc.: Wallingford, CT, 2004. (b) Spartan’10, Wavefunction, Inc., Irvine, CA. (c) GAMESS: Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su, S.; Windus, T. L.; Dupuis, M.; Montgomery, J. A. J. Comput. Chem. 1993, 14, 1347−1363. (d) MOLPRO is a package of ab initio programs written by Werner, H. J.; Knowles, P. J.; Knizia, G.; Manby, F. R.; M. Schütz; Celani, P; Korona, T.; Lindh, R.; Mitrushenkov, A.; Rauhut, G.; Shamasundar, K. R.; Adler, T. B.; Amos, R. D.; Bernhardsson, A.; Berning,A.; Cooper, D. L.; Deegan, M. J. O.; Dobbyn, A. J.; Eckert, F.; Goll, E.; Hampel, C. A.; Hesselmann, A.; Hetzer, G.; Hrenar, T.; Jansen, G.; Köppl, C.; Liu, Y.; Lloyd, A. W.; Mata, R. A.; May, A. J.; McNicholas, S. J.; Meyer, W.; Mura, M. E.; Nicklaß, A.; O’Neill, D. P.; Palmieri, P.; Pflüger, K.; Pitzer, R.; Reiher, R.; Shiozaki, T.; Stoll, H.; Stone, A. J.; Tarroni, R.; Thorsteinsson, T.; Wang, M.; Wolf, A. (14) Cybulski, S. M.; Toczyłowski, R. R. J. Chem. Phys. 1999, 111, 10520−10528. (15) Halpern, A. M. J. Chem. Educ. 2012, 89, 592−597. (16) Hoare, M. R. Adv. Chem. Phys. 1979, 40, 49−135. (17) Utreras-Díaz, A. C.; Shore, H. B. Phys. Rev. B 1989, 40, 10345− 10350. (18) Schwerdtfeger, P.; Gaston, N.; Krawczyk, R. P.; Tonner, R.; Moyano, G. E. Phys. Rev. B 2006, 66, 064112. (19) Brack, M. Rev. Mod. Phys. 1993, 65, 677−732. (20) The N2/3 dependence of the number of surface atoms is strictly valid for large N; for example, it is more accurately followed for N between 40 and 60 than between, say, 13 and 40. Equation 4, however, is a smoothly varying function that does not predict anomalous behavior of smaller clusters vis-à-vis larger ones. (21) Kittel, C., Introduction to Solid State Physics, 6th ed.; John Wiley & Sons: New York, 1986; p 55. (22) Using the regression values of av,e and as,e and av,0 and as,0 for the upper and lower curves in Figure 3, the approximate values of N at which Eb,e and Eb,0 equal 90% of the corresponding bulk values are 2700 and 3400, respectively. (23) (a) Brack, M. Sci. Am. 1997, 277, 50−55. (b) Harris, I. A.; Kidwell, R. S.; Northby, J. A. Phys. Rev. Lett. 1984, 53, 2390−2393. (b) Teo, N. K.; Sloane, N. J. A. Inorg. Chem. 1985, 24, 4545−4558. (d) Vafayi, K.; Esfarjani, K. Abundance of nanoclusters in a molecular beam: The magic numbers. 2005, Los Alamos National Laboratory, Preprint Archive, Condensed Matter; arXiv:cond-mat/0506092.

arXiv.org e-Print archive. http://arxiv.org/abs/cond-mat/0506092v1 (accessed Oct 2012). (24) Audi, G. Int. J. Mass Spectrom. 2006, 251, 85−94. (25) The similarity of ΔEb,e values for N = 51, 52, 53, 54, and 55 arises from the fact that in these clusters the respective additional atoms are added to equivalent vertex sites of a newly completed icosahedral geometry. (26) Batchelder, D. N; Losee, D. L.; Simmond, R. O. Phys. Rev. 1967, 162, 767−775. (27) Peterson, O. G.; Batchelder, D. N.; Simmons, R. O. Phys. Rev. 1966, 150, 703−711.

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dx.doi.org/10.1021/ed300235j | J. Chem. Educ. 2012, 89, 1515−1521