Dynamical and Structural Properties of the Ni4 Cluster

phys. stat. sol. (b) 213, 283 (1999)

Subject classification: 61.46.+w; 64.70.Dv; S1.2

Dynamical and Structural Properties of the Ni4 Cluster

M.H. GuÈven and M. EryuÈrek

Physics Department, Zonguldak Karaelmas University, 67100 Zonguldak, TurkeyFax: 378-2574181; e-mail:[email protected]

(Received September 22, 1998; in revised form February 11, 1999)

Microcanonical molecular dynamics simulations are used to determine the structures of NiN(N = 4, 5, 6) clusters and to study the dynamical and structural properties of the Ni4 cluster. Theinteraction between the atoms is modeled by an empirical potential. The geometries and the re-lated parameters are compared with the previous literature. The Ni4 cluster is heated up to 2000 Kand during the phase transition a stable isomer of the cluster is observed. The melting temperatureof Ni4 clusters is tried to be determined from the dynamical displays.

1. Introduction

Phase changes in clusters are one of the active areas of cluster science. Theoretical stu-dies have been carried out for Lennard-Jones systems [1,2], ionic [3], covalent [4] andmetal clusters [5,6] using molecular dynamics and Monte Carlo simulations. Clustersplay a central role in a variety of natural phenomena (e.g. in the upper atmosphere andin the soil) and in processes of environmental concern, as well as in technologies relatedto heterogeneous catalysis, microelectronics, materials design and fabrication, etc.

Small clusters have limited degrees of freedom, therefore they can be described andunderstood in purely dynamical terms. A melting-like transition in a cluster can beviewed as a transition from a highly restricted and correlated motion of the constituentparticles, characteristic of a solid, to their uncorrelated large amplitude or even diffu-sive motion, characteristic of a liquid [7].

Truly adequate potentials do not exist for metal clusters at present and semiempiricalor empirical potentials are being used successfully in theoretical studies to understandthe melting behaviour of these clusters [8].

In this study, the minimum energy geometries of NiN (N = 4, 5, 6) microclusters areobtained by the simulated annealing method. The solid-to-liquid phase transition in theisolated Ni4 microcluster is studied by using constant energy, microcanonical moleculardynamics (MD) simulation. In the simulation, an empirical model potential proposedby ErkocË is used [8,9], which contains two- and three-body atomic interactions and themelting behaviour of the cluster is presented.

The potential and the computational procedures are given in the next section, the resultsand discussion are presented in Section 3. The conclusions are summarized in Section 4.

2. Theoretical Background and Computational Procedure

The potential energy function (PEF) of the cluster as a function of its configuration ismodeled by the ErkocË potential which has the following form [8,9]:

F � F2 � BF3 ; �1�

M.H. GuÈven and M. EryuÈrek: Dynamical and Structural Properties of the Ni4 Cluster 283

where F2 is the total two-body (pair potential) interaction term, more explicitly,

F2 �Pi<j

U�ri; rj� �Pi<j

U�rij� �Pi<j

Uij ; �2�

Uij � Ar0

rij

� �2n

eÿ2��rij=r0�2 ÿ r0

rij

� �n

eÿ��rij=r0�2" #

; �3�

where rij = |ri ±± rj| is the distance between the particles i and j. The F3 is the totalthree-body interaction potential expressed as linear combination of the pair energiesformed by the three particles. It is given explicitly by

F3 �P

i<j<kW�ri; rj; rk� �

Pi<j<k

W�rij; rik; rjk� �P

i<j<kWijk ; �4�

Wijk � B�Uijfijk �Uikfikj �Ujkfjki� �5�with

fijk � eÿ�r2ik�r2

jk�=r20 ; fikj � eÿ�r

2ij�r2

jk�=r20 and fjki � eÿ�r

2ij�r2

ik�=r20 : �6�

The potential parameters for nickel were determined for this PEF as: A = ±±8.28 eV,r0 = 2.20 � 10±±1 nm, n = 2.89247 and B = ±±1.290 433. Since B has a negative value, thethree-body potential has a positive contribution to the total interaction energy. Thepresent PEF satisfies the bulk cohesive energy and the crystal stability condition fornickel. It also gives the f.c.c. structure as most stable [9].

Static and dynamic quantities of the cluster can be calculated from the averages ta-ken along the trajectory.

The calculated static quantities are:(i) Temperature (T) of the cluster

T � 2 Ekh ik 3N ÿ 6� � ; �7�

where Ek is the total kinetic energy, N is the number of atoms in the cluster and k isthe Boltzmann constant. The symbol h i implies averaging over the entire trajectory.(ii) Relative root-mean square (rms) bond length fluctuation (d)

� � 2N N ÿ 1� �

Pi<j

�hr2iji ÿ rij

�2�1=2

hriji : �8�

(iii) Instantaneous temperature (Ti) of the cluster

Ti � 2hEkisk�3N ÿ 6� : �9�

The symbol h is implies averaging over 1000 steps.The calculated dynamic quantities are:

(i) Mean square displacement

hr2�t�i � 1N

PNi�1h�r�t� ÿ r�0��2i ; �10�

where the ensemble average is over 4000 independent time origins.

284 M.H. GuÈven and M. EryuÈrek

(ii) Velocity autocorrelation function C(t),

C�t� �

Pnt

j�1

Pni�1

vj�t0j � t� v t0j

ÿ �Pnt

j�1

Pni�1

v2i �t0j�

; �11�

where vi is the velocity of atom i and nt is the number of the different time origins t0j.(iii) Power spectrum of C(t), i.e. the power spectrum

I !� � � 2�10

C t� � cos !t dt ; �12�

which is the Fourier transform of C(t).The crucial property of small clusters is their unique ground state geometrical struc-

tures which are governed by local chemical bonding instead of the long-range order asin the bulk [10,11]. The structures change with size and can be different from the bulk.This leads to new electronic, magnetic and other properties. Unfortunately, there is noexperimental technique to determine the structures of the clusters directly. They are toosmall for diffraction techniques and too large for spectroscopic techniques.

The geometrical structures of small Ni clusters were probed via molecular adsorptionof nitrogen on their surfaces by Parks and coworkers [12]. They proposed cluster struc-tures for all clusters in the 3 to 15 atom size range except Ni4 and Ni11. The nitrogenuptake for Ni4 is consistent with virtually any structure from tetrahedral to planar orbutterfly, or from planar to linear. The proposed structures for Ni5 and Ni6 are triangu-lar bipyramidal and octahedral, respectively.

From the theoretical point of view the d-states of Ni offer a particular challenge forthe theory. The number of states and their character require extensive computationalresources [11]. Realistic calculations were done by Reuse and Khanna [11] using a line-ar combination of atomic orbital±molecular orbital approach within the density func-tional formalism and the electronic structure, geometries and magnetic moments ofsmall NiN (N = 2 to 6, 8, 13) clusters were reported. In the study, Reuse and Khannatried a compact D2d and a square geometry for Ni4. The calculated binding energies forD2d and square geometries were equal and it is concluded that these geometries com-pete for the definition of the ground state of the Ni4 cluster. They found that a triangu-lar bipyramid is more stable than a square pyramid for Ni5. For Ni6, the octahedron istried, which showed minimal distortions from the perfect geometry.

Stave and DePristo [13] used the corrected effective medium theory to determine thebinding energies and geometrical structure of NiN (N = 4 to 23) clusters and reportedthe geometry of Ni4, Ni5 and Ni6 as tetrahedral, triangular bipyramidal and octahedral,respectively. Rey and coworkers [14], using constant energy molecular dynamics simula-tions, studied the binding energies and melting behaviour of NiN, PdN, AuN and AgN

clusters (N = 2 to 23) on the basis of the embedded atom model and the second-mo-ment approximations to the tight-binding method. In their study, the structures withminimum energy were obtained by using the steepest descent method and they de-duced the same basic geometrical configurations for Ni clusters as reported by Staveand DePristo in the size range N = 4 to 23 atoms.

Estiu and Zerner [15] studied the electronic structure and magnetic properties ofsmall NiN clusters (N = 4, 5, 6, 8, 13) at self-consistent-field multireference configura-

Dynamical and Structural Properties of the Ni4 Cluster 285

tion interaction of singlet level using the intermediate neglect of the differential overlapmodel Hamiltonian parametrized for spectroscopy. They studied tetrahedral and squareplanar structures for both the interatomic bulk distance (2.49 �A) and the optimizedgeometries. They found that optimization of the geometry has a similar structural effecton both the square planar and the tetrahedral Ni4 clusters decreasing the interatomicdistances to 2.298 and 2.310 �A, respectively. They also showed that the D4h structuresare more stable than the Td one, whereas high spin multiplicity characterizes both sym-metries. In their calculations for Ni5, the C4v structure is 2.9 eV more stable than theD3h one for the interatomic bulk distance. Optimization of the geometry decreases theinteratomic distance in the C4v structure but the geometry of the D3h structure cannotbe optimized without distortion. For Ni6 they centered their study on structures of Oh

symmetry.Nayak and coworkers [10] calculated the equilibrium geometries, the binding ener-

gies of NiN clusters (N � 23) by using an empirical many-body potential which is ta-ken from the work of Finnis and Sinclair [16] and molecular dynamics simulation. Inthis work simulated annealing with a different approach was used to locate the abso-lute global minimum structure. Their findings are tabulated in Table 1. Note that intheir publication the ab initio binding energy result for Ni4 is misprinted. Nayak andcoworkers have recently reported the equilibrium geometries, energetic and electronicstructure, vertical ionization potential, and magnetic properties of Ni clusters contain-ing up to 21 atoms using a combination of classical molecular dynamics simulationand first principles molecular orbital theory. They used first principles DMOL soft-ware and found that tetrahedron geometry is more stable than a square geometry forthe Ni4 cluster. Note that binding energies are overestimated in the LSDA level oftheory [17].

In our study, the minimum energy geometries of the NiN (N = 4, 5, 6) clusters areobtained as tetrahedral, triangular bipyramidal and octahedral, respectively, by the si-mulated annealing method, where the functional controlling descent is the total energy.The binding energies per atom, bond lengths and the geometries which are calculatedfrom the model potential are compared with the other reported values in Table 1. Ourresults are in good agreement with the previous studies. Therefore we used the ErkocËpotential to study the structural and dynamical properties of the Ni4 cluster.

The structural and dynamical properties of the Ni4 cluster are extracted from micro-canonical molecular dynamics simulations. Hamilton's equations of motions were solvedby the ErkocË potential for all the atoms in a cluster on a grid of total energies usingHamming's modified fourth-order predictor±corrector propagator with a step size of5 � 10±±16 s. The cluster was prepared with zero initial total linear and angular momen-ta. Trajectories of length of 2.5 � 106 steps were generated on a grid of total energieslarge enough to observe the solid to liquid-like transition in the cluster. The total en-ergy in the individual runs was conserved within 0.003%.

The simulations were started from the minimum energy geometry, and the clusterwas gradually heated up to 2000 K for sixteen different constant energies. The finalconfiguration of the previous run defined the initial positions of the atoms for the nextrun at a different energy. The initial values of the velocities for every new run wereobtained by scaling the final velocities of the preceding run to give the desired value ofthe new total energy. The coordinate and momentum components of each trajectorywere used to calculate the static and dynamic quantities of the cluster [1].


3. Results and Discussion

The variation of d as a function of T is displayed in Fig. 1, in which d rises up to 695 Kand then it levels off. This evolution of the pattern is similar to that observed in Ni12

and Ni13 clusters as well as certain size Lennard-Jones and ionic clusters [7]. The abruptchange in d indicates that the cluster undergoes a phase change but this should besupported by other criteria. The caloric curve is one of them, see Fig. 2. The melting-like transition is associated with variations in the slope of the caloric curve. It changesat two energy values: E1 = ±±2.07 eV/atom and E2 = ±±1.99 eV/atom. According to thesevariations the dynamics of the cluster is investigated in three different energy ranges,i.e., Etot < E1, Etot > E2 and E1 � Etot � E2. Ti values, which are obtained at the ener-gies indicated by arrows in Fig. 2, are displayed in Fig. 3.

The instantaneous temperature is a measure of the short time average kinetic energyof a cluster and its time dependence is an informative characteristic of a state of the


Ta b l e 1Calculated binding energy, bond length and geometry of NiN (N = 4, 5, 6) clusters

binding energy(±±eV/atom)

bond length (�A) geometry

N 4 5 6 4 5 6 4 5 6

present work 2.14 2.42 2.67 2.34 2.501)2.332)

2.41 Td D3h Oh

CEM [18] 1.65 1.87 2.08 2.27 2.271)2.212)

2.23 Td D3h Oh

CEM [13] 1.87 2.10 2.33 1.88 ÿ 2.20 Td D3h Oh

MD/MC CEM [13] 1.75 1.95 2.13 2.32 ÿ 2.36 Td D3h Oh

TBM1TBM2EAM1EAM2 [14]

1.901.602.902.00

2.101.803.102.30

2.251.903.202.40

ÿ ÿ ÿ

Td

Td

Td

Td

D3h

D3h

D3h

D3h

Oh

Oh

Oh

Oh

LCAO-MODFF [11] 2.34

2.342.83 3.27

2.121)2.742)2.10

2.231)2.282)

2.321)2.352) D2h

squareD3h Oh

Finnis-Sinclair ModelPot. [10] 2.77 2.90 3.03 2.20

2.221)2.282) 2.25 Td D3h Oh

optimized geometryvalues used inINDO [15]

ÿ ÿ ÿ 2.302.30

2.30 2.36 D4h

Td

C4v Oh

Exp. [12] ÿ ÿ ÿ ÿ ÿ ÿ ÿ D3h Oh

DMOL [17] 4.083.93

4.43 4.67 2.242.15

2.29, 2.24 2.28 Td

squareD3h Oh

1) edge length of the polygon in the equatorial plane2) distance between the axial (or apex) atom and the equatorial atom

cluster at a given total energy. As the total energy increases, the pattern of the graphchanges gradually from a constant (Fig. 3a) to moderately fluctuating, but singlebranched (Fig. 3b) and then to resolvable multibranched (Fig. 3c) and eventually tohighly fluctuating with no individual branches resolvable (Fig. 3d).

Fig. 3a corresponds to a low energy rigid cluster, Fig. 3b to a rigid cluster, but softerone, Fig. 3c to a cluster which is undergoing structural changes, Fig. 3d to a high energyliquid-like cluster in which constituent particles move diffusively. The arrow in Fig. 3cindicates the minimum in which the isomerization takes place.hr2i as a function of t is displayed in Fig. 4 for three different total energies corre-

sponding to the cases (a), (b) and (d) of Fig. 2. The diffusion coefficient is proportionalto the slopes of the curves, and is expressed as

D � 16

dhr2�t�idt

: �13�

When the cluster is rigid-like, atoms do not diffuse (corresponding to the case ofFig. 4a). Diffusion takes place at energies corresponding to the case of Fig. 4c, i.e.,when the cluster becomes liquid-like, at particular intermediate energies (correspondingto the case of Fig. 4b), the diffusion motion only begins to develop. The mean square

displacement and the diffusion coef-ficient in case of Fig. 4b are averagesover different structures of the clus-ter which are represented by the dif-ferent branches in the correspondinginstantaneous temperature display.


Fig. 1. The variation of d as afunction of T

Fig. 2. Caloric curve

(i) The solid-like region (Etot < E1)

The distributions of instantaneous temperatures at two different total energies are dis-played in Fig. 5. The smooth curves are the result of a least squares fit to the sum oftwo Gaussians. In Fig. 5a, the distribution shows a sharp peak corresponding to theenergy value of Fig. 3a in which Ti values are almost constant, i.e. the structure is rigid.Fig. 3b shows that there is an increase in the amplitude of the fluctuations which is theresult of softening in the structure and Fig. 5b supports this result, but the structure isstill rigid.

In Figs. 6a and b the normalized velocity autocorrelation function and its Fouriertransform, i.e. the power spectrum are displayed, respectively. As it is seen from Fig.6a, the correlation between the particle velocities continues for a long time. The powerspectra show sharp peaks, which means the cluster oscillates at certain frequencies. InFig. 6b the power spectrum I(w = 0) = 0, which reflects the fact that the particles do notmove diffusively, verifies the rigid structure.


Fig. 3. Variation of the instantaneous temperature (Ti) versus time, (a) for a rigid cluster withEtot = ±±2.13 eV/atom; (b) for a rigid cluster but softer one with Etot = ±±2.09 eV/atom; (c) for acluster which is undergoing structural changes with Etot = ±±1.99 eV/atom. The arrow indicates theminimum in which the isomerization takes place; (d) for high energy liquid-like cluster withEtot = ±±1.89 eV/atom

Fig. 4. Variation of r2 versus time tfor Etot = ±±2.09 eV/atom (a),±±2.05 eV/atom (b), and ±±1.97 eV/atom (c)

(ii) The liquid-like region (Etot > E2)

Fig. 3d displays the instantaneous temperature as a unimodal but broad one. The distribu-tion of Ti covers a wide range of temperatures contrary to the previous case (see Fig. 7).


Fig. 5. The distribution of Ti for Etot = ±±2.13 eV/atom (a) and ±±2.03 eV/atom (b). The frequencyis the number of occurrence of Ti

Fig. 6. a) The normalized velocityautocorrelation function forEtot � ÿ2:07 eV/atom. C(t) is calcu-lated for 4000 independent time ori-gins. b) The power spectra obtainedfrom the velocity autocorrelationfunction

The mean square displacementfor this region is displayed forEtot = ±±1.97 eV/atom in Fig. 4c. Itsslope linearly increases with timeimplying a finite diffusion coeffi-cient.

The velocity autocorrelationfunction is displayed in Fig. 8awhich has shallower reflection mini-

ma than that of the rigid form and as it is seen from the figure, long time correlations arelost. In the power spectrum, the fact that I(w = 0) 6� 0 (see Fig. 8b) indicates the diffusivemotion in the cluster. The curves indicate the liquid-like behavior of the cluster.

(iii) The region between the two phases (E1 � Etot � E2)

In Fig. 3c, the instantaneous temperature displays a bimodal form which also impliesisomerization, i.e., structural changes begin to develop in the solid±liquid phase change.


Fig. 7. The distribution of Ti forEtot = ±±1.89 eV/atom

Fig. 8. a) The normalized velocityautocorrelation function forEtot = ±±1.93 eV/atom. C(t) is cal-culated for 4000 independent timeorigins. b) The power spectra ob-tained from the velocity autocorre-lation function

The distributions of Ti for three differ-ent total energies are displayed inFigs. 9a, b and c. The presence of twopeaks in these figures indicates the bi-modal form of the distributions. InFig. 9a the frequency of the first peakis less than the second one and as thetotal energy of the cluster increases,the frequency of the first peak also in-creases and that of the second one de-creases, but the first peak never oc-curs higher than the second one. Thetemperature bandwidth of the twopeaks also broadens with the increasein the total energy of the cluster.

In Figs. 10a and b, C(t) and I(w)graphs are displayed, respectively, atEtot = ±±2.03 eV/atom. It is observedfrom Fig. 10a that the velocity auto-correlation decreases and in thepower spectrum I(w = 0) is nearlyequal to zero. The power spectrumcurve, I(w), is lower but broader thanthat of the rigid-like case.

Microclusters have lots of stablestructures corresponding to an abso-lute minimum and local minima ofthe potential energy surface which are

separated by potential barriers. If the kinetic energy of the cluster increases, it canmove to a neighboring minimum and can be temporally trapped in one of the minima.It vibrates for some periods and then it returns to the previous region or moves intothe region around another minimum [5,7]. Ti corresponding to an energyEtot = ±±1.99 eV/atom, indicates a minimum as shown by arrows in Fig. 3c. The coordi-nates of the cluster in this minimum are used in simulated annealings to obtain theminimum energy geometry and a distorted rhombus (two of the atoms are in the sameplane, the other two are tilted from the plane) as a geometry of an isomer is obtainedand its binding energy is calculated as ±±2.07 eV/atom (see Fig. 11).


Fig. 9. The distribution of Ti,a) for Etot = ±±2.05 eV/atom,b) for Etot = ±±2.03 eV/atom,c) for Etot = ±±1.99 eV/atom

The stability of this isomer istested by starting the simulationsfrom this geometry and the clusteris gradually heated up to 300 K.The cluster at this temperature un-dergoes again a structural changewhich is shown in Fig. 12 and it re-turns to its most stable structure[19].

4. Conclusions

The size dependent structure ofsmall atomic clusters and theirmelting process is not only of scien-tific interest, but also has sometechnological implications [20 to22]. They exhibit interesting elec-

tronic, magnetic and chemical behaviour. For transition metals, the problem is morecomplicated than the others as mentioned in Section 2.

A new method has been developed to determine the caloric curve of size-selected freesodium cluster ions by Haberland and coworkers [20,21]. They conclude that the sizedependence of the melting point temperature and latent heat is probably a complicatedinterplay between geometric and electronic structure, presenting a challenge for theory.


Fig. 10. a) The normalized velocity auto-correlation function for Etot = ±±2.03 eV/atom. C(t) is calculated for 4000 indepen-dent time origins. b) The power spectraobtained from the velocity autocorrela-tion function

Fig. 11. The geometry of the isomer, a) from top view, b) from side view

Even though model potentials do not have explicit spin terms, since their parametersare obtained by fitting to bulk and surface properties of the element, they implicitlycarry information about the magnetic state of the cluster. If they give the correct geo-metry for the ground state they can be used to study dynamical or structural propertiesof small clusters. The validity of the potential is tested by computing the structures ofNiN (N = 4, 5, 6) clusters by using this potential. Besides structures, we compared bondlengths and binding energy values with the available data. As seen in Table 1 the agree-ment is very good.

Small clusters have a lower melting point (Tmelt) than bulk material due to the factthat most of atoms are located on the surface. To determine the melting point of theNi4 cluster, the Ti versus t graph of the energy which corresponds to the energy comingjust after E2 in our simulations, is examined and it is observed that it shows liquid-likecharacter. This energy value corresponds to 1103 K in the caloric curve and at thistemperature d � 0.1 which satisfies the Lindemann criterion. Therefore, the meltingtemperature of the Ni4 cluster is determined as (1103 � 50) K. This value is approxi-mately 36% less than that of the bulk value.

In the previous section the behaviour of the Ni4 cluster in three different energyregions is discussed. The cluster in the Etot < E1 region shows rigid-like character, inthe Etot > E2 region shows liquid-like character, and in the energy range E1 < Etot < E2,from Ti versus t graphs, shows well separable branches corresponding to local minimaof the potential energy surface. This is the result of the structural change of the cluster,i.e., isomerization. An isomer of the Ni4 cluster is obtained which is very stable asdiscussed in the article. The geometry of the isomer is distorted rhombohedral. Thebinding energy differences between these two geometries are very small, but for theground state the tetrahedral structure is the preferred one.

Acknowledgements We would like to thank Prof. Dr. SÎ akir ErkocË and Dr. Ziya B.GuÈ vencË for their comments. This work is supported by ZKU Research Fund Projectunder project contract 96.111.003.14.


Fig. 12. Variation of Ti, versus time, of the cluster whose geometry is given in Fig. 11, forEtot = ±±2.057 eV/atom

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Dynamical and Structural Properties of the Ni4 Cluster