Computational and Theoretical Chemistry xxx (2013) xxx–xxx

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Computational and Theoretical Chemistry

journal homepage: www.elsevier .com/locate /comptc

Analyzing the properties of clusters: Structural similarity and heatcapacity

2210-271X/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.comptc.2013.06.004

⇑ Corresponding author at: Physical and Theoretical Chemistry, University ofSaarland, 66123 Saarbrücken, Germany.

E-mail addresses: y.dong@mx.uni-saarland.de (Y. Dong), m.springborg@mx.uni-saarland.de (M. Springborg), pangdayong@hotmail.com (Y. Pang), pacomm@ciencias.unam.mx (F.M. Morillon).

Please cite this article in press as: Y. Dong et al., Comput. Theoret. Chem. (2013), http://dx.doi.org/10.1016/j.comptc.2013.06.004

Yi Dong a,b,⇑, Michael Springborg a,b, Yong Pang a, Francisco Morales Morillon a,c

a Physical and Theoretical Chemistry, University of Saarland, 66123 Saarbrücken, Germanyb School of Materials Science and Engineering, Tianjin University, Tianjin 300072, PR Chinac Facultad de Ciencias, Universidad Nacional Autonoma de Mexico, 04510 Mexico, DF, Mexico

a r t i c l e i n f o a b s t r a c t

Article history:Received 13 May 2013Received in revised form 4 June 2013Accepted 5 June 2013Available online xxxx

Keywords:SimilarityHeat capacitySilicon clustersGermanium clustersSilicon–germanium clustersVibrational properties

Two approaches for extracting information on clusters from unbiased structure-optimization calculationson a larger set of cluster sizes are presented. At first, we study how structural similarity can be quantifiedand present an approach that seems to match what subjectively would be expected. Second, we present amethod for calculating the vibrational contributions to the heat capacities of the clusters. As test systemswe have applied the methods to Sin, Gen, and SinGen clusters with up to 44 atoms in total.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

Theoretical studies devoted to the development of a generalunderstanding of the relations between properties on the one handand size and stoichiometries on the other hand of larger systemsare obscured by several aspects. First of all, such studies requirethat the proper structures, i.e., those of the lowest total energy,are identified for a larger range of systems. However, for commonlyused electronic-structure methods the computational costs for cal-culating the properties for just a single structure scale with the sizeof the system, measured in, e.g., the number of electrons of nuclei,to some power that typically is 3 or larger. Therefore, even formedium-sized systems these computational demands can put seri-ous limits on what is possible. Independently of this scaling issue,another, complementary, problem leads to further complications.Thus, it has been shown that the number of nonequivalent minimaon the total-energy surface grows faster than any polynomial inthe size of the system [1]. These problems are well-known and sev-eral strategies have been proposed to meet those, see, e.g., [2–7].

There is, however, a third issue that has been given less atten-tion. Such calculations as the ones mentioned above will provide,

at first, total energies as well as the coordinates of the variousnuclei for a smaller or larger set of sizes and/or stoichiometries.In many cases, also other information like the spatial distributionsand energies of the electronic orbitals is obtained. Thus, it is oftenfar from trivial to reduce this large information to some few keyquantities that describe the development of the properties of thesystems with size and/or stoichiometry in a simple, although con-cise way that ultimately will allow for the development of a moregeneral chemical or physical understanding.

In the present contribution we shall focus on this last issue and,thereby, use a set of clusters as the systems of our interest. Duringthe last years we have developed various descriptors aimed atidentifying general trends in structure and energetics of clusters(see, e.g., [6,7]), but, as we shall argue below, not all those are opti-mal and improvements as well as further descriptors may be(more) useful.

The systems we shall consider are Sin, Gen, and SinGen clusterswith up to a total of 44 atoms. For those, we have previouslyreported results of an unbiased structure optimization study [8].In that study, we used a parametrized, density-functional, tight-binding method in calculating the total energy for a given structurein combination with a genetic-algorithm approach for an unbiasedstructure optimization. The purpose of the present study is not todiscuss the properties of those clusters specifically, neither to ad-dress the accuracy of the computational approach. Rather, we shall

2 Y. Dong et al. / Computational and Theoretical Chemistry xxx (2013) xxx–xxx

use those results as a pool that here is analyzed using the tools thatwe shall describe below.

Specifically, we shall in this contribution present results for anew approach for quantifying structural similarity between differ-ent clusters as well as present results of a recently developed ap-proach for studying heat capacities for the clusters as a functionof their size and the temperature. Since our approach is based onearlier works, we shall in the next section review some of those.Subsequently, we shall in Section 3 describe our theoretical ap-proach and in Section 4 the results. Finally, our results are summa-rized in Section 5.

2. Previous work

2.1. Structural similarity

In earlier works [9,10] we have introduced descriptors, so calledsimilarity functions, that were developed in order to quantifystructural similarity. We shall here briefly outline their basic ideas,also because we will use these for comparison with the new mea-sures and since we will discuss their limitations.

We consider two different systems, A and B, and want to study,whether the system A can be considered more or less similar to apart of system B. An example could be that A is an optimized struc-ture of a cluster with N atoms, whereas B is a (larger) part of a crys-tal or is another cluster, having M atoms. We assume that M P N.

The structure of A is characterized by N positions, R!

A;i; i ¼1;2; . . . ;N, whereas we for B have R

!B;i; i ¼ 1;2; . . . ;M. It is useful

to define the center for each system,

R!

A;0 ¼1N

XN

i¼1

R!

A;i

R!

B;0 ¼1M

XM

i¼1

R!

B;i:

ð1Þ

Subsequently, we define scaled positions relative to the centers,

~rA;i ¼ ð R!

A;i � R!

A;0Þ=uA; i ¼ 1;2; . . . ;N

~rB;i ¼ ð R!

B;i � R!

B;0Þ=uB; i ¼ 1;2; . . . ;M:ð2Þ

uA and uB are pre-chosen, fixed scaling parameters that may be dif-ferent for the two systems.

If M = N we may use the two sets of interatomic distances

dA;ij ¼ j~rA;i �~rA;jjdB;ij ¼ j~rB;i �~rB;jj;

ð3Þ

that we will sort in increasing order, in order to quantify the struc-tural similarity. I.e., we will consider

Q ¼ 1K

XK

i¼1

ðsA;i � sB;iÞ2

S ¼ 1

1þ Q 1=2 :

ð4Þ

Here, K ¼ MðM�1Þ2 is the number of terms in the summation. Finally,

sA,i and sB,i are either the sorted, scaled, interatomic distances, orthe sorted, scaled, inverse, interatomic distances. In the former case,Q may have dominating contributions from atoms that are far apart,whereas similarity among close neighbors is given less importance.In order to try to give more emphasis on the near surroundings ofthe individual atoms, we have, therefore, also considered the casethat Q is calculated from the inverse interatomic distances.

For M = N not very small, K is much larger than the number ofcoordinates, 3M = 3N, that is needed to specify the structure

Please cite this article in press as: Y. Dong et al., Comput. Theoret. Chem. (201

uniquely. Thus, in that case the approach should be able to quan-tify structural similarity. However, for M > N difficulties show up.

The simplest case is that of M = N + 1, which happens when, e.g.,trying to quantify growth patterns, i.e., to quantify whether thecluster with N + 1 atoms is similar to the one with N atoms plusan extra atom. In that case, we modified the approach slightly byconsidering all those M = N + 1 structures that could be generatedfrom the cluster with M = N + 1 atoms by removing a single atomand keeping the positions of the remaining N atoms unchanged.For each of those M = N + 1 structures with N atoms we calculateda similarity function from Eq. (4) and ultimately chosen the largestvalue of those.

Also for M = N + 2 a similar approach can be applied. Then, wewill consider all the MðM�1Þ

2 structures that can be constructed fromthe cluster with M = N + 2 atoms by removing two atoms and keep-ing the positions of the remaining atoms fixed. As above, we calcu-late a similarity function from Eq. (4) and ultimately chose thelargest value of those MðM�1Þ

2 different ones.But if M and N differ by much more than 1 this approach be-

comes increasingly prohibitive, and in the extreme case thatM ?1, which, e.g., is the case when comparing the cluster withN atoms with the crystalline material, the approach above cannotbe used.

As a simplification we, therefore, suggested to base the similar-ity function on the scaled radial distances,

rA;i ¼ j~rA;ijrB;i ¼ j~rB;ij:

ð5Þ

Also these are sorted and S is defined as in Eq. (4). Then, K = N, andsA,i and sB,i are either the sorted radial distances or the sorted, in-verse, radial distances. Then, the two cases distinguish betweenwhether the outermost or the innermost atoms are given mostimportance.

This approach works reasonably well as long as the two struc-tures that are to be compared both are compact. This was the casefor most of the systems we have looked at so far, but if the struc-tures are less compact problems may easily show up. Thus, whencomparing a hollow nanoparticle (for instance, a C60 molecule)with one that is filled with some atoms, the approach would findthat the two structures are markedly different, although intuitionwould tell that they share many features, i.e., the shell. Moreover,if the larger system is, again, the crystal for which M ?1, there isan ambiguity in the definition of the center [cf. Eq. (1)] making thesimilarity function non-unique. Furthermore, in that case, the ra-dial distances would not be able to identify the structural similar-ity between an elongated cut-out of the crystal and the crystalitself. On the other hand, S may predict close structural similarityin cases where intuition would say the opposite. Thus, a squareand a tetrahedron may be found to be identical (by proper choiceof the lengths of the sides).

Other approaches that have been devised to quantify structuralsimilarity are mainly used for comparing structures of the samesize. This is important in the structure-optimization proceduresince it can be used to avoid that similar structures are been trea-ted more than one time. This is, e.g., the case for the approach byLee et al. [11]. Using two different cut-off distances, d1 < d2, theycalculated for each atom the number of atoms, n1, within d1 fromthe reference atom and the number of atoms, n2, with a distancebetween d1 and d2 of the reference atom. Thereby, they get for eachof the two structures to be compared two sets of numbers H1(n)and H2(n), respectively, which give the number of atoms withn1 = n and n2 = n, respectively. Finally, the differences betweenH1(n) and H2(n) is used in identifying structural similarity. In a typ-ical, closed-packed system, n can at most be of the order of 10,meaning that the structural information is reduced to roughly 20

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Y. Dong et al. / Computational and Theoretical Chemistry xxx (2013) xxx–xxx 3

numbers. More serious, however, is that it is not obvious how togeneralize the approach to systems of different sizes.

A similar motivation was behind the approach of Cheng andFournier [12]. As Lee et al. [11], Cheng and Fournier focused mainlyon the coordinations of the various atoms in a given cluster. Fromthose, they extracted the smallest, the largest, and the average va-lue as well as a root of the mean of the squares of the deviationsfrom the average. Besides these four parameters, they used alsotwo based on the moments of inertia. In total, the structural infor-mation is accordingly reduced to just 6 numbers. Again, it is notclear how to generalize the approach to systems of different sizesand, in particular, to identify the case that a smaller cluster is iden-tical to a part of a larger cluster.

Also Rogan et al. [13] introduced a quantity that should quantifythe similarity between two clusters of the same size. In this case,only a single number, related to the moments of inertia, is usedfor each structure. As above, a generalization to clusters of differentsizes seems to be difficult.

Finally, an approach that in spirit is similar to the one we shalldescribe below was recently presented by Helmich and Sierka [14]although the two approaches differ in their mathematical formula-tion. Helmich and Sierka applied their approach for systems withthe same numbers of atoms, although it should be possible to com-pare systems of different sizes, too.

The study of structural similarity is not new but has been a cen-tral issue within bioinformatics for many years, see, e.g., [15–21].For instance, structural similarity has been suggested as oneparameter that can quantify whether two drugs may have compa-rable activities. A very simple way of defining structural similarityis then quantified by observing to which extend the two com-pounds contain the same chemical groups or by comparing theirbonding patterns. For recent examples, see, e.g., [22,23]. However,many other approaches have been developed in bioinformaticsthat are very similar in spirit to the one we shall propose, i.e., basedon superposing the two systems and subsequently minimizing thesum of the squares of the distances between the atoms of the twosystems. Such methods were among the first used within bioinfor-matics for structural alignment [15] and later developments havemade them more accurate than what we shall present but simulta-neously computationally more demanding. This suggests thatthese methods only with difficulties can be applied ‘on the fly’ inan unbiased structure-optimization calculation, where ‘similarity’is used to avoid that a certain part of structure space is studiedmore than once.

2.2. Heat capacities

Heat capacities of size selected clusters have been the subject ofvarious experimental studies. Some of the first experimental stud-ies of the thermodynamic properties of clusters were due to Martinet al. [24] who studied mass-abundance spectra of NaN clusters asfunction of temperature and from those could estimate a transitionfrom a solid-like to a liquid-like behavior for different values of N.

Much more detailed experimental studies were initiated withthe seminal work of the group of Haberland who, in 1998, analyzedthe melting temperatures of NaN clusters as function of N with70 6 N 6 200 [25]. From the caloric curve U(T), i.e., the internal en-ergy as a function of temperature, and from the heat capacity,

CV ¼@UðTÞ@T

; ð6Þ

they could identify the melting temperature as the value of T forwhich CV has a maximum. Since then, this group has continuedthe experimental studies of the thermodynamic properties of NaN

clusters as function of temperature [26–28].

Please cite this article in press as: Y. Dong et al., Comput. Theoret. Chem. (201

Later, also the group of Jarrold has developed an experimentalapproach (multicollision induced dissociation) that allows for theexperimental determination of CV(T) for mass-selected clusters[29–31]. The group has used their method in studying melting ofGaN, AlN, and SnN clusters, including attempting to unravel themelting mechanism of these systems.

As for the experimental studies, the theoretical studies of ther-modynamic properties of clusters have focused on the size depen-dence of the melting of clusters. Most of these studies has beenperformed using molecular-dynamics simulations for specific clus-ters as a function of temperature (see, e.g., [3] and referencestherein). Through this, the caloric curve and thereby the heatcapacity can be calculated and, thereby, phase transitions beidentified.

In a very recent work [32] we presented a new approach forstudying the heat capacity of a whole range of clusters and appliedit for smaller AuN clusters with N up to 20. This approach shall herebe used for the Sin, Gen, and SinGen clusters. In contrast to the stud-ies mentioned above, our approach does not attempt to identifyphase transitions but focuses on the low-temperature behavior.As we shall demonstrate also here, these turn out to contain inter-esting information when studying size dependent properties.

3. Theoretical approach

3.1. Total-energy and structure-optimization methods

The purpose of the present study is not to report results of astructure-optimization study of Sin, Gen, and SinGen clusters. Thiswas done in our earlier work [8] to which the interested readeris referred. However, for the sake of completeness we briefly sum-marize the main aspects of our computational approach.

In order to calculate the total energy as a function of structurewe use the so called DFTB (density-functional tight-binding) meth-od of Seifert and coworkers [33–35]. This method offers a fair com-promise between accuracy and computational speed.

Within the DFTB approach, the total energy relative to that ofthe non-interacting atoms is given as

Etot ’Xocc

i

�i �X

j

Xm

�jm þ12

Xj–k

Ujk R!

j � R!

k

��� ���� �; ð7Þ

where �i is the energy of the ith orbital for the system of interestand �jm is the energy of the jth orbital for the isolated mth atom.Moreover, Ujk is a short-range pair potential between atoms j andk that is adjusted so that results from parameter-free density-func-tional calculations on two-atomic systems as a function of the inter-atomic distance are accurately reproduced. Finally, only the valenceelectrons are explicitly included in the calculations, whereas theother orbitals are treated within a frozen-core approximation.

The elements of the Hamiltonian and overlap matrices are ob-tained from calculations on diatomic molecules. The Hamiltonoperator contains the kinetic-energy operator as well as the poten-tial. The latter is approximated as a superposition of the potentialsof the isolated atoms,

Vð~rÞ ¼X

m

Vm ~r � R!

m

��� ���� �; ð8Þ

and we assume that the matrix element hvm1n1jVmjvm2n2

i vanishesunless at least one of the atoms m1 and m2 equals m. Here, vmn isthe nth atomic orbital centered at the mth atom.

Thus, all information that is needed can be extracted from theproperties of diatomic molecules. These can, in turn, be determinedfrom accurate, density-functional calculations. In order to assessthe accuracy of the approach we performed additional calculations

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4 Y. Dong et al. / Computational and Theoretical Chemistry xxx (2013) xxx–xxx

on the infinite, periodic systems and verified that the obtainedstructures, binding energies, and bulk moduli are accurate.

For an unbiased structure determination we combine the DFTBmethod with genetic algorithms [36,37] that have been found toprovide an efficient tool for global geometry optimizations. Theversion of the genetic algorithms that we are using is as follows.

A population of P initial structures is chosen randomly (theseclusters are called parents) and each structure is relaxed to itsnearest total-energy minimum. By cutting each parent randomlyinto two parts a next set of P structures is obtained by interchang-ing (‘mating’) these two parts and allowing the resulting ‘children’to relax, too. Comparing the energies of the 2P clusters of both sets,those P with the lowest total energies are chosen to form the set ofparents for the next generation. This procedure is repeated formany hundred generations until the lowest total energy is un-changed for a larger number of generations.

3.2. Structural similarity

Our new approach for quantifying structural similarity is moti-vated by what you do in your mind when examining whether twosystems are similar. Thus, at first each of the two structures isscaled, and subsequently one of the two is rotated and translatedso that it maximally overlap with the other structure. As in Seciton2.1, we will let B be the larger system with M atoms and A beingthe smaller one with N atoms. Also in the special case that M = Nwe will use this notation.

In detail, our approach consists of the following steps.

1. For each of the two systems, we use the scaled relativecoordinates of Eq. (2), i.e.,

Please

~sA;i ¼~rA;i; i ¼ 1;2; . . . ;N~sB;i ¼~rB;i; i ¼ 1;2; . . . ;M:

ð9Þ

2. Subsequently, we apply an initial rotation and translationto system A. By repeating this and the following steps fordifferent initial rotations and translations, we hope to beable to find the optimal way of placing structure A on struc-ture B.

~sA;i !~s0A;i ¼ R0~sA;i þ~t0: ð10Þ

R0 is the rotational matrix expressed in terms of a set of Euler anglesand~t0 is the translation vector.

3. For each atom, i, of system A, we identify that atom in sys-tem B, nBA(i), that is closest to the atom i of system A, i.e.,the partner of atom i. nBA(i) is that atom of system B thatmakes

qi ¼ ~s0A;i �~sB;nBAðiÞ

��� ���2 ð11Þ

as small as possible. It may happen that two (or more) atoms of sys-tem A, i1 and i2, have the same partner, i.e., nBA(i1) = nBA(i2). In thatcase, nBA(i1) (nBA(i2)) is kept if the corresponding value of qi1 is smal-ler (larger) than the equivalent value of qi2 , and for the i2th (i1th)atom of system A another partner of system B is found. Thus, notwo atoms of system A will have the same partners.

4. We apply a rotation and a translation to the complete sys-tem A. This changes the positions of the atoms in system Aaccording to

~s0A;i !~s00A;i ¼ R~s0A;i þ~t: ð12Þ

Here, R is a rotation matrix, described through three Euler angles a,b, and c. Moreover,~t is a translation containing the three parame-

cite this article in press as: Y. Dong et al., Comput. Theoret. Chem. (201

ters x, y, and z. Thus, in total, 6 parameters, a, b, c, x, y, and z, definethe displacement of system A.

5. These 6 parameters are optimized so that

3), http

Qða;b; c; x; y; zÞ ¼ 1N

XN

i¼1

~s00A;i �~sB;nBAðiÞ

��� ���2 ð13Þ

is minimized.Instead of optimizing all 6 parameters simultaneously, we use thatfor any a, b, and c, that value of~t that minimizes Q(a,b,c,x,y,z) isgiven by

~t ¼ 1N

XN

i¼1

~sB;nBAðiÞ � R �~s00A;ih i

�~sB;0 � R �~s00A;0 ð14Þ

with

~sB;0 ¼1N

XN

i¼1

~sB;nBAðiÞ

~sA;00 ¼1N

XN

i¼1

~s00A;i:

ð15Þ

By inserting this into Eq. (13) we find that

Q Rða; b; cÞ ¼1N

XN

i¼1

~sB;nBAðiÞ � R � ~s00A;i �~s00A;0� �

þ~sB;0

h in o2ð16Þ

shall be minimized. This minimization can with advantage be doneby using the derivatives of QR with respect to its three arguments.The vector containing those is given by

~rRQ R ¼2N

XN

i¼1

ð~sB;0 �~sB;nBAðiÞÞy � R0 � ~s00A;i �~s00A;0

� �; ð17Þ

where the prime on R implies that the rotation matrix shall be dif-ferentiated with respect to a, b, or c. Then, a simple steepest-des-cent approach can be used (which is what we shall do).

6. It cannot be excluded that through the transformation ofEq. (12), the atoms of system A will be closer to other atomsthan its original partners. Therefore, we repeat step 3 andidentify a new set of partners for the atoms of system A.For this new set we calculate Q of Eq. (13) for the specialcase that a = b = c = 0 and~t ¼ ~0 (i.e., the rotation and trans-lation is not optimized). If this new value of Q is lower thanthe one we have optimized in the previous step, we go backto step 4 for this new set of partners.

7. Otherwise, we may return to step 2 for a new set of initialrotations and translations.

8. Finally, in order to check whether system A is the mirrorimage of system B, the whole procedure is repeated byreplacing, e.g., all z coordinates of the atoms of system Aby their negative values (or, in the special case, that theyall vanish, then the x or y values).

From the smallest value of QR we define finally the similarityfunction as

S ¼ 1

1þ Q 1=2R

: ð18Þ

We emphasize that our approach does not guarantee that we haveidentified the global minimum of QR. More advances approaches forminimizing QR may be applied with, however, the disadvantage thatthe computational requirements may increase even dramatically.For the systems of the present work we found that the present ap-proach is numerically efficient, which is important when attempt-ing to use it repeatedly during a global structure optimization.

://dx.doi.org/10.1016/j.comptc.2013.06.004

Fig. 1. The thick curves show the similarity functions of our new approach, whereasthe thin curves show those of our earlier approach. N is the total number of atoms inthe clusters, i.e., n for the pure Sin and Gen clusters and 2n for the SinGen clusters.With the new approach, the solid curves mark the results based on setting thescaling parameters for Sin, Gen, and SinGen equal to 5.35, 5.85, and 5.85 a.u.,respectively, whereas they were set equal to 5.851, 6.095, and 5.973 a.u., respec-tively, for the results of the dashed curves. With the old approach, only the formerset of scaling parameters was used. Moreover, then the solid curves show theresults based on the radial distances, the dashed curves the results based on theinverse radial distances, the dash-dotted curves those based on the interatomicdistances, and the dotted curves those based on the inverse interatomic distances.In the last case, Q of Eq. (4) has been multiplied by 10.

Y. Dong et al. / Computational and Theoretical Chemistry xxx (2013) xxx–xxx 5

3.3. Heat capacity

In the calculation of the heat capacity we will assume Boltz-mann statistics to be valid and, in addition, concentrate on thevibrational contribution. We shall look at relatively low tempera-tures that, however, are assumed to be so large that the contribu-tion from rotational and translational degrees of freedom to theheat capacity has converged as a function of temperature to the va-lue 5

2 kB or 3kB per cluster for linear and non-linear structures,respectively. Here, kB is the Boltzmann constant. Therefore, forthe sake of a comparison between different temperatures and/orcluster sizes, their inclusion is unimportant. Moreover, since theenergy gap separating occupied and unoccupied electronic orbitalsis of the order of 1 eV for the systems of our interest [8], electronicexcitations are irrelevant here. However, more important may it bethat we only consider the single structure that we have identifiedas that of the lowest total energy. Whether this assumption is jus-tified is beyond the scope of the present work.

This leaves us with the vibrational contribution to the heatcapacity. We apply the Normal Mode Harmonic Oscillator (NMHO)approximation [38] and assume, accordingly, that the structural

Fig. 2. The optimized structures of the SinGen cluster

Please cite this article in press as: Y. Dong et al., Comput. Theoret. Chem. (201

dependence of the total energy Etot, when expanded in a Taylor ser-ies, can be terminated after the 2nd order terms. Then, the vibra-tional frequencies, {xi}, are the square roots of the eigenvalues ofthe dynamical matrix with the elements

Dij ¼1ffiffiffiffiffiffiffiffiffiffiffiffi

MiMj

p fij ¼1ffiffiffiffiffiffiffiffiffiffiffiffi

MiMj

p @2E@qi@qj

: ð19Þ

Here, i and j represent two of the 3N Cartesian coordinates, denotedqi and qj, of the cluster atoms, and Mi and Mj are the correspondingnuclear masses.

In order to calculate the force constants fij we use a finite-differ-ence approximation,

fij ¼@

@qi

@Etot

@qj¼ @

@qj

@Etot

@qi

¼ 12

@

@qi

@Etot

@qjþ @

@qj

@Etot

@qi

!¼ �1

2@Fi

@qjþ @Fj

@qi

!

’ �14Ds½Fiðqj þ DsÞ � Fiðqj � DsÞ þ Fjðqi þ DsÞ � Fjðqj � DsÞ�:

ð20Þ

Ds is a small finite coordinate change, and Fm(qn ± Ds) denotes themth component of the force on the structure which results from ashift ±Ds of coordinate n. As we shall show below, we found thatfor Ds around 0.001–0.01 a.u. the results are essentially insensitiveto variations in Ds.

Finally, from the calculated vibrational frequencies and with theuse of Boltzmann statistics we can calculate the vibrational heatcapacities per atom,

Cvib ¼kB

N

XNVM

i¼1

a2i eai

ðeai � 1Þ2ð21Þ

with

ai ¼�hxi

kBT� Ti

T: ð22Þ

NVM is the number of non-zero frequencies and equals 3N � 6(3N � 5) for non-linear (linear) systems. Moreover, we associateeach vibrational mode i with a characteristic temperature Ti. Ti is re-lated to the temperature (T ’ 2.35Ti) at which the contribution ofthe given mode changes most rapidly as a function of temperature.At T ’ 2.35Ti the contribution of the ith mode equals roughly 64% ofits maximal contribution (which is obtained for T ?1). Finally, forN, T ?1, Cvib ? 3kB ’ 25 J/mol.

4. Results

We shall use the structures of our earlier work on the Sin, Gen,and SinGen clusters [8] as a structure data base in order to illustratethe approaches of the present work. There is, however, some find-ings of our earlier work that are relevant to emphasize here. Thestructure of crystalline Si and Ge, i.e., the diamond structure, isnot at all closed packed and, moreover, the atoms are fourfoldcoordinated. For the clusters we found, however, that many atoms

s with n = 13, 14, 15, and 16 (from left to right).

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Fig. 3. The optimized structures of the (from left to right) Si44, Ge44, and Si22Ge22 clusters.

Fig. 4. The similarity function for the comparison of the structures of the Sin, Gen,and SinGen clusters with a fragment of the diamond crystal. N is the total number ofatoms in the clusters, i.e., n for the pure Sin and Gen clusters and 2n for the SinGen

clusters. The solid curves mark the results based on setting the scaling parametersfor Sin, Gen, and SinGen equal to 5.35, 5.85, and 5.85 a.u., respectively, whereas theywere set equal to 5.851, 6.095, and 5.973 a.u., respectively, for the results of thedashed curves.

Fig. 5. The similarity function for the comparison of the structures of the clusterswith N atoms with those of the same type with N �M atoms. The upper, middle,and lower panel shows the results for the Si, Ge, and SiGe clusters, respectively.M = 1, M = 2, M = 3, and M = 4 for the solid, dashed, dash-dotted, and dotted curve,respectively.

6 Y. Dong et al. / Computational and Theoretical Chemistry xxx (2013) xxx–xxx

had higher coordinations, with coordinations of 5 and 6 being notuncommon in particular for the Sin clusters. Moreover, the struc-tures could in many cases be interpreted as being formed bystrongly bonded building blocks of roughly 10 atoms that wereconnected with each other through few bonds. The overall struc-tures were, thus, quite open and far from being compact.

4.1. Structural similarity

In order to test the performance of our approach for quantifyingstructural similarity we tested it on the optimized structures of Sin,Gen, and SinGen clusters. In all cases, we did not distinguish be-tween the atom types. We made the following comparisons:

Please cite this article in press as: Y. Dong et al., Comput. Theoret. Chem. (201

1. Sin�m with Sin,2. Gen�m with Gen.3. Sin�mGen�m with SinGen,4. Sin with Gen,5. Si2n with SinGen,6. Ge2n with SinGen,7. Sin with diamond crystal structure,8. Gen with diamond crystal structure,9. SinGen with diamond crystal structure.

For the diamond crystal structure we constructed a cubic cut-out of the crystal containing 5 � 5 � 5 units of each 8 atoms. These8 atoms were placed at ð0; 0;0Þ; 1

4 ;14 ;

14

� �; 1

2 ;12 ; 0

� �; 3

4 ;34 ;

14

� �;

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Fig. 6. The characteristic temperatures for (top) Sin, (middle) Gen, and (bottom)SinGen clusters as a function of the total number of atoms, N, in the clusters. For eachvalue of N, one line marks that at least one vibration has that value of thecharacteristic temperature. Negative values correspond actually to imaginarycharacteristic temperatures.

Y. Dong et al. / Computational and Theoretical Chemistry xxx (2013) xxx–xxx 7

0; 12 ;

12

� �; 1

4 ;34 ;

34

� �; 3

4 ;14 ;

34

� �, and 1

2 ;0;12

� �, and the three lattice vectors

are (0,0,1), (0,1,0), and (1,0,0).We considered two different ways of choosing the scaling

parameters of Eq. (2). In one set of calculations, we set these forall Si clusters equal to 5.35 a.u., whereas we for all Ge and all SiGeclusters used a value of 5.85 a.u. These values were found in ourearlier work [8] to be reasonable estimates of the average of thenearest-neighbor and next-nearest-neighbor distances for each ofthe three types of clusters. Alternatively, we used the average valuefor the nearest-and next-nearest-neighbor distance of the crystal-line material. For Si and Ge, these can be determined from the lat-tice constants, i.e., 5.431 and 5.6575 Å for Si and Ge, respectively[40]. For the mixed clusters we chose the average of the valuesfor the pure clusters. This choice was based on the finding thatthe lattice constants of SixGe1�x alloys is almost linear as functionof x [40]. Thereby the scaling parameters become equal to 5.851,6.095, and 5.973 a.u. for Sin, Gen, and SinGen, respectively. For thecrystal we used the average of the nearest-and the next-nearest-neighbor distance, i.e., the average of

ffiffi3p

4 and 1ffiffi2p , which equals

0.5700597.Fig. 1 shows the similarity function as calculated in our previous

study together with the similarity functions based on either the ra-dial distances, the inverse radial distances, the interatomic dis-tances (which then resembles the results of our earlier study[39,8], although slightly different scaling parameters were used),and the inverse interatomic distances. In each case, S ’ 1 suggeststhat the two structures that are being compared are very similar.Thus, any drop in a similarity function in the left panels in Fig. 1suggests that a structural change has taken place at that size. If Sremains low, each cluster has a structure different from that ofthe cluster of the neighboring, lower size, whereas if S after a jumpto a low value becomes close to 1, then a new structural motif hasemerged that is kept for more cluster sizes.

In Fig. 1 we first notice that the two sets of results with the newapproach for different scaling parameters lie almost on top of eachother, implying that this approach is not very sensitive to the valueof these scaling parameters, as long as these are chosen reasonably.Second, we see that S based on the inverse interatomic distancespossesses very little structure even after multiplying Q of Eq. (4)with 10 (so that S in this case covers essentially the same rangeas in the other cases). Accordingly, this choice does not providevery much information. Third, for S based on the inverse radial dis-tances we see very low values for certain cluster sizes, which is dueto the fact that one atom happens to sit very close to the center ofthe cluster, Eq. (1).

For the cases that we compare the structure of the pure Sin orGen clusters with the one with one atom less, i.e., Sin�1 or Gen�1,the former and the present results show a very similar shape,although the variations in the similarity function as calculatedwith our new approach are larger than those of our former ap-proach. As we have discussed in Section 2, the present approachis more restrictive than the earlier one, so that it is to be expectedthat the present similarity function in general takes lower valuesthan the earlier one.

Also for the comparison of SinGen with Sin�1Gen�1, the presentsimilarity function is much more structured than the former one.There is an interesting case of n = 15 (i.e., N = 2n = 30) for whichthe new approach predicts a closer structural similarity, whereasthe former approach did not identify a such. In order to study thiscase in more detail we show in Fig. 2 the structures of the SinGen

clusters with 2n = 26, 28, 30, and 32 atoms. Indeed, this figure doesgive the impression that the clusters with 28 and 30 atoms arestructurally more close than what is the case for the clusters with26 and 28 atoms or for those with 30 and 32 atoms.

Another interesting difference between the present approachand our former one is that in our earlier study we found that when

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comparing the structures of the SinGen clusters with those of thepure Si or Ge clusters with the same number of atoms we foundfor larger n that the structures of the mixed clusters are more sim-ilar to those of the pure Si clusters than to those of the pure Geclusters. In the present study we use slightly different scalingparameters and do not recover this finding. This implies that theformer definitions of similarity functions are much more sensitiveto scaling parameters than the present definitions. In Fig. 3 weshow the structures of those three types of clusters. A directinspection of the three structures does not give the impression thatthe Si44 cluster has a structure that is significantly closer to that ofthe Si22Ge22 cluster than is the case for the Ge44 cluster.

Next, we compare the structures of the clusters with the dia-mond crystal structure. Since the clusters have structures thatare not at all close to being spherical (see Ref. [8]), this comparisonwas not possible in our earlier work. As discussed in Section 2.1,when comparing structures that are not very small and that arevery different in size (which is the case when comparing a clusterand a crystal), a similarity function based on interatomic distancescannot be used. Moreover, for structures that are not close to beingspherical, also a similarity function based on radial distances is notuseful.

The results of this comparison are shown in Fig. 4. It is interest-ing to notice that also in this case the results are independent of

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Fig. 7. The vibrational heat capacities per atom as function of cluster size for different temperatures and for the different systems, as marked in the panels. Each panelcontains five curves, obtained from the five different values of Ds.

8 Y. Dong et al. / Computational and Theoretical Chemistry xxx (2013) xxx–xxx

the precise values of the scaling parameters. Thus, although thecomparison of two larger, identical cut-outs of a crystal structurewith slightly different scaling parameters will yield a similarityfunction below 1, the relative low values in Fig. 4 even for the larg-est clusters are not a result of inadequately chosen scaling param-eters but indeed imply that the structures of the clusters are quitedifferent from fragments of the diamond structure.

Finally, we extend the results of the left panels in Fig. 1 so thatnot only clusters of neighboring sizes are been compared. Thus, wecompare the structure of the cluster with N atoms with those of theclusters with N �M atoms. Here, N = n for the pure Sin and Gen

clusters and N = 2n for the SinGen clusters, whereas M equals 1, 2,3, and 4 for the pure clusters and 2 and 4 for the mixed clusters.The results of this comparison are shown in Fig. 5.

Such a comparison may provide information on more, differentissues. Thus, if, e.g., the clusters grow in some regular way wherebyatom by atom is added to a given core, the similarity function forall values of M will stay close to 1. However, since our structure-optimization approach is based on attempting to identify the glo-bal total-energy-minimum structure for a given cluster size, itmay happen that for certain cluster sizes, the structure of the reg-ular cluster growth is not that of the lowest total energy but,maybe, the 2nd, 3rd, . . . lowest structure. If this happens, only forfew cluster sizes, the similarity function of Fig. 5 would show localminima for different values of N when varying M.

This is, however, not the case. Thus, for, e.g., the Sin clusters, Shas a clear minimum for N = 20 independent of M, implying thatthe structure of the 20-atomic cluster is markedly different fromthose of a larger range of neighboring sizes. Moreover, for N larger

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than roughly 15, the similarity function for a given N has, in mostcases, its largest values for the smallest M and decreases as a func-tion of M. This suggests that the cluster structures are increasinglydifferent for structures of increasingly different sizes, i.e., that noregular growth mechanism can be identified — at least for the sizerange of the present study.

4.2. Heat capacity

A critical issue in the calculation of the heat capacities is the va-lue of the step size, Ds, of Eq. (20). It should be so large that numer-ical truncation errors become insignificant, but simultaneously sosmall that the harmonic approximation remains good. Mathemat-ically, it is known that the 6 (5) smallest frequencies for the nonlin-ear (linear) system should be identically zero corresponding to the3 pure translational degrees of freedom and the 3 (2) rotational de-grees of freedom. Numerically, however, this condition is onlyapproximately fulfilled. Thus, whereas the translational degreesof freedom indeed lead to frequencies very close to zero, this is lessclearly the case for the rotations. Moreover, in the results that weshall present here we found that there is no general value for D sthat is ‘optimal.’ Moreover, even if a Ds can be identified for whichthe rotational degrees of freedom lead to essentially vanishing fre-quencies, it cannot be excluded that other frequencies thereby areless accurately reproduced. In the present work we have chosenthe pragmatic solution of presenting results for different valuesof Ds, thereby obtaining information on the robustness of our find-ings. We have chosen Ds = 0.001, 0.005, 0.010, and 0.015, and0.020 a.u., which we consider values of a reasonable magnitude.

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Fig. 8. The vibrational heat capacities per atom as function of temperature fordifferent cluster sizes. Those are given in the upper left corner of the panels in thevertical order they appear for T = 500 K.

Fig. 9. The optimized structures of the (from left to right) Ge10 and, Ge11 clusters.

ig. 10. The stability function for the different clusters of the present study asnction of their sizes.

Y. Dong et al. / Computational and Theoretical Chemistry xxx (2013) xxx–xxx 9

Moreover, in the calculation of the vibrational heat capacities weshall then simply ignore the 6 (5) lowest frequencies.

The characteristic temperatures of Eq. (22) are the key quanti-ties for the vibrational heat capacities. Therefore, we show at first,in Fig. 6, those for the three different cluster types of the presentstudy. In this figure we have included all frequencies, i.e., alsothose of the translations and vibrations. Thereby, negative valuescorrespond to imaginary frequencies. Thus, we do indeed see thatwe have some sizes for which imaginary frequencies are found. Theresults of the figure were produced for Ds = 0.01 a.u. Changing Ds(not shown) would change both values and sizes of the clustersfor which the imaginary frequencies were found.

The results of the figure show also that the characteristic tem-peratures of the Sin clusters in general are higher than those ofthe SinGen clusters that in turn are higher than those of the Gen

clusters. This difference is a trivial consequence of the differencesin the masses of the atoms. A more careful inspection of the figurereveals that there are cluster sizes for which particularly low char-acteristic temperatures are found. From the discussion in Section3.3 this suggests that these clusters would have particularly largeheat capacities.

Next, we show in Fig. 7 the vibrational contributions to the heatcapacities per atom for the three different systems of the present

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Ffu

study and for temperatures from 10 K to 300 K. In each case, the re-sults for the five different values of Ds are shown and it is seen thatin most cases the results are very close, with the Sin clusters at thelowest temperature being the most pronounced exception. Thus,our approach appears as being robust, in particular when beingaware of possible problems (systems with low characteristic tem-peratures at low temperatures).

The results of Fig. 7 show also that for the Gen clusters, forwhich the highest characteristic temperatures are roughly 500 K,the results for 300 K are qualitatively very close to the T ?1 limit.In addition, in all cases, the heat capacity per atom for the lowesttemperatures show a pronounced size dependency. Thus, for theSin clusters at 30 K, peaks are found for n = 16, 20, 21, 31, amongothers. For Gen at 10 K, peaks are found for n = 9, 11, 17, 20, 29,and 43. And for SinGen at T = 10 K, we have peaks for N = 2n = 14,18, 24, 38, and 42. In Fig. 8 we show the temperature dependenceof the vibrational heat capacity per atom for different cluster sizesas calculated using Ds = 0.01 a.u. Indeed, again, Sin clusters seem tobe those with the most pronounced effects and for those we ob-serve several crossings of the curves, suggesting that these size ef-fects should be experimentally accessible. For the two othersystems of the present study, similar effects are less pronounced.

A careful inspection of the characteristic temperatures in Fig. 6reveals why certain cluster sizes have large vibrational heat capac-ities at low temperatures. It turns out that it is due both to the exis-tence of particularly soft vibrational modes with low frequenciesand to the occurrence of several of those. It is, however, not trivialto identify those clusters through simple inspection. This can beseen by looking at Fig. 9 where we show the structure of Ge10, acluster with a particularly high value of the vibrational heat capac-ity at T = 10 K, and that of Ge11 that has a particularly low value ofthe vibrational heat capacity at T = 10 K.

One may speculate that clusters with particularly large valuesof the vibrational heat capacities at low temperatures are thosethat most easily can take up energy in order to change the struc-

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10 Y. Dong et al. / Computational and Theoretical Chemistry xxx (2013) xxx–xxx

ture and, accordingly, would be the least stable ones. This wouldsuggest a correlation between the low-temperature results ofFig. 8 and the stability function, defined as

D2EðnÞ ¼ Etotðnþ 1Þ þ Etotðn� 1Þ � 2EtotðnÞ; ð23Þ

[with Etot(n) being the total energy of the cluster with n atoms orpairs of atoms], shown in Fig. 10. D2 E(N) has peaks pointing up-wards (downwards) for clusters that are particularly stable (unsta-ble) compared to those with neighboring sizes. However, it is seenthat there is no such correlation, implying that the experimentalstudy of vibrational heat capacities of the clusters can give very use-ful information on their vibrational properties, but hardly on theirenergetic properties.

5. Conclusions

Over the years, many unbiased methods have been proposedand applied to theoretically determine the structures of clusterswithout having to make serious assumptions on their structures.Less attention has been paid to extract useful information fromsuch studies on a larger class of sizes. The purpose of the presentwork was to describe two approaches for obtaining general andqualitative information in a more or less automatic way.

At first, we have proposed an automatic scheme for identifyingstructural similarity between different arrangements of atoms.This could be, e.g., different clusters or a cluster compared with afragment of a larger object like an icosahedron or a crystal. Ourmethod was based on mathematically formulating what usuallyis done when by eye performing a such comparison. Ultimately,our approach turns out to be efficient and provide ‘realistic’measures for similarity, with ‘realistic’ meaning that the resultsagree with what qualitatively would be found by simply inspectingthe structures. This was often found by using our earlier schemes,too, but there are cases where the present approach performs sig-nificantly better.

Second, we present our method for calculating the vibrationalheat capacities of clusters, a property that is experimentally acces-sible. In this case we found that only for the lowest temperaturesinteresting size dependencies can be identified, but also that thesecannot be correlated with stability of the clusters.

Acknowledgements

This work was supported by the German Research Council(DFG) through Project Sp439/23 as well as by the German Aca-demic Exchange Service (DAAD).

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