A comparative theoretical study of stable geometries and energetic properties of small silver clusters

25 February 1994

CHEMICAL PHYSICS LETTERS Chemical Physics Letters 218 (1994) 395-400

A comparative theoretical study of stable geometries and energetic properties of small silver clusters R. Santamaria,

I.G. Kaplan ‘, 0. Novaro

Instituto de Fisica, UNAh4, Apdo. Post. 20-364, 01000 Mexico DF, Mexico

Received 6 October 1993; in final form 2 1 December 1993

Abstract The systematic quantum-mechanical investigation of the stable geometries and some energetic characteristics of neutral silver clusters up to the hexamer is performed by the all-electron spin density approach with non-local corrections included. We compare these results with experimental data and with previous calculations on anionic silver clusters. We find that for the hexamer the ground-state geometry of the neutral cluster differs from the geometry of the anionic. For the neutral silver clusters Ag, the transition from a 2D conformation to a 3D occurs at n > 6. The size effects for the atomic fragmentation energy yield opposite

alternations for neutral and anionic clusters.

1. Introduction The properties of small metal clusters as a function of cluster size have been the subject of many theoretical and experimental investigations in recent years, see for example proceedings [ l-3 ] and reviews [ 4,5 1. The transition metal clusters are of special interest not only because of their practical application in catalysis and photography [6,7] but also because of their theoretical significance in elucidating the role of d electrons in similar systems. There exist many experimental [ 8-121 and theoretical [ 13-241 studied of the coinage metal (Cu, Ag and Au) clusters. In particular for Ag, clusters there are precise calculations published for dimers and trimers [ 14- 16,19,2 11, for tetramers and pentamers [ 17,22,23], and for hexamers [24] by several research groups. Until recently there has been a lack of studies using a single calculation method of the de’ On leave from the Karpov Institute of Physical Chemistry, Moscow, Russian Federation.

pendence of silver cluster properties on the cluster size n. We also point out that in previous calculations of the energetic characteristics of silver clusters, good agreement with experimental data was achieved only after introducing scaling factors, see refs. [ 17,2 11. At the time of writing this article, the publication of BonaM-Kouteckf et al. [ 25 ] appeared in which properties of the silver clusters Ag, with n up to 9 were calculated by a method different from ours. The energy separations among the different stable conformations for isomers with the same n in ref. [ 25 ] essentially differ from the ones obtained in our calculations. This confirms the necessity for further investigation. The probable reasons for the discrepancies obtained will be discussed below. In the present Letter, we give the results of systematic quantum-mechanical calculations of the stable geometries and some energetic characteristics of neutral silver clusters up to hexamers. We compare the results obtained with those of anionic silver clusters previously obtained by the same approach [ 261.

0009-2614/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDZ0009-2614(94)0010-N

396

R. Santamaria et al. /Chemical Physics Letters 218 (I 994) 395-400

2. Method

rections. Computations were performed on a CRAYYMP4/432 by a single processor.

In ref. [26] it was shown that for the case of anionic silver clusters good agreement with experimental properties can be obtained without a scaling procedure using the all-electron local spin density (LSD) method with non-local corrections included (NLSD ) . Henceforth, we apply this method, as was done for anions, to perform the large number of calculations required to establish, among several conformations, the ground-state geometries adopted by the neutral silver clusters of size n, with n= 1, 2, .... 6. In particular, we use the DGauss version of the LSD approach [27,28] and, in addition, because of the highly inhomogeneous electronic charge clouds and strong correlation between core-valence and valence-valence electrons, we have also considered the BeckePerdew non-local correction for both exchange and correlation energies inside the self-consistent field procedure (see ref. [26] and references therein). As shown by Andzelm and Wimmer [ 28 1, inclusion of non-local corrections allows the prediction of bond dissociation energies within 0.1 eV. On the other hand, it has been established that for silver the influence of relativistic effects is small [29]. The search for the most stable geometries was carried out by the gradient method with self-consistent non-local cor-

3. Results and discussion Numerical results for the most stable conformations are presented in Tables 1 and 2; geometries for tetramers up to hexamers are shown in Fig. 1. As follows from Table 1 for the dimer and tetramer, the agreement of our ground-state geometrical parameters with other calculations and experiment is satisfactory. The ionization potential (IP) for the dimer coincides quite well with the experimental one without resource to any scaling procedure. For the tetramer, the agreement between our theoretical vertical ionization potential (IP,) and the experimental one is less satisfactory. The difference might be due to the fact that experimental IP,s of ref. [ 81 represent an upper limit to the true IP,s, with an error bar of about 0.1 eV. The trimer deserves a separate comment. For this cluster, we found a flat potential energy surface which did not allow us to determine with precision the apex angle. For the bond lengths we obtained a value of 2.73 A, while for the apex angle we deduced a tentative value of 74.4”. In previous ab initio investigations on this trimer [ 19 1, the pitfalls of its flat poten-

Table 1 Comparison of our calculations with literature data for silver dimer (upper part) and tetramer (lower part)

re(A)

IP, (W

0. (W

theory

exp.

theory

exp.

theory

exp.

2.66 ’ 2.66 [ 161 2.58 [ 191 2.63 [25] 2.64 [30]

2.53 [31]

1.48 ’ 1.425 [ 191 1.34 1161 1.34 [21] 1.59 [25]

1.65 [32]

7.56 ’ 6.74 [ 171 6.70 [21] 7.64 [25]

7.56f0.02 7.60 [8]

Rhombic geometry ra (A) 2.859 2.862 2.870 2.800 ’ This work.

-

[ 171 [22] [25]

IP, (ev) a (deg)

theory

exp.

56.1 = 57.6 [ 171 55.5 [22] 56.4 [25]

6.33 ’ 6.37 [ 171 5.75 [21] 6.61 [25]

6.65 [8]

[33]

397

R. Santamaria et al. /Chemical Physics Letters 218 (1994) 395-400 Table 2 Ground-state energy and IP, for neutral isomers Ag,,, n = l-6 Geometry ’

&(A&)

linear 4

rhombic, Dr,, planar, Czv linear, Dmh

5

linear, Dooh trigonal, Crv pentagonal pyramid, CS, tripyramidal, Clv hexagonal ring, D,,

Eb(Ag.+) ’ (au)

IP, (eV) talc.

exp. 7.57 [34] 7.56kO.02 [33] 7.60 [ 81

- 5 199.948966 - 10399.952159

-5199.671200 - 10399.614276

1.56 7.56

-20799.936806 (AE=O) - 20799.929623 (AE=O.20 eV) -20799.919199 (AE=O.48 eV)

-20799.704267

6.33

-20799.695939

6.36

-20799.669901

6.78

-25999.716146

6.06

-25999.701248

6.00

-25999.664892

6.36

-31199.710046

6.80

-31199.710022

6.65

-31199.716036

6.06

-31199.665741

6.80

- 25999.93893 1 (AE=O) -25999.921791 (AE=O.47 eV) -25999.898702 (AI?= 1.09 eV)

trapezoidal, Crv pyramidal, Crv

6

(au)

-31199.959795 (AE=O) -31199.954415 (AE~0.15 eV) -31199.938619 (AE=O.58 eV) -31199.915572 (AE= 1.20 eV)

6.65 [8]

5.75kO.03 [12] 6.35 [8]

7.15 [8]

’ For the dimer geometry refer to Table 1, for tetramer up to hexamer geometries see Fig. 1. ’ The prime designates that the geometry of the anionic silver cluster is chosen to be the same as the stable geometry of the neutral cluster, with lengths and angles as given in Fig. 1.

tial and the important effects of f functions were discussed. These are not implemented in the DGauss program and we expect that inclusion off Gaussiantype orbitals should yield better results, see also ref. 1351. Table 2 presents ground-state energies corresponding to the different isomer geometries of Fig. 1. In order to compare with experiment, we calculated vertical ionization potentials as II’,=

- ]&(Ag,)

-MAg,+

)I .

(1)

The prime in the notation designates that the geometry of the cationic cluster was chosen to be the same as the stable geometry of the neutral cluster, with angles and length scales as given in Fig. 1. The symmetry for the most stable structures coincides with those found in previous investigations [ 17,21-23,251 except for the hexamer. If we com-

pare energy separations among the different groundstate isomers, for a given cluster of size n, with those of ref. [ 25 ] we find great discrepancies. We assume the reason is in the one-valence-electron effective core potential used in ref. [ 25 1. In such an approach all d electrons are frozen in a core potential. This leads not only to the neglect of the overlap and hybridization of d electrons with the valence 5s electrons but also eliminates the cluster d-shell polarization present in the configuration interaction scheme. For Cu and Ag clusters this polarization accounts for a large part of the configuration interaction. The 4d and 5s orbitals have the same value at a radius of 1.8 au and they overlap over a significant region. As explicitly shown in ref. [ 14 ] for Agz and Ag, clusters, the freezing of d electrons in a core further results in a significant distortion of the bonding charge density and a change of the binding energy by as much as 0.4-0.5 eV. For

R. Santamaria et al. /Chemical Physics Letters 218 (1994)395-400

398

n=4 Rhombic,

I&,,

Planar,

Cav

/LZq

\r,D*h

n=5 Trapezoidal,

Pymmidal , C,,

CZV

2,869

2,762

-3,226

n=6 Tr igonal , Czv

Pentagonal pyramid , Csv

Tripyramidal

, C2v

2ff)3,0@

2,880

k?XOgOnd

ring,

D6h

{=}

2.+14

2J66

L2,839

Fig. 1. Optimized geometries for the neutral Iisomers Ag,, n= 4-6. All distances in A.

Ag,, Bauschlicher et al. [ 171 presented calculations with d electrons explicitly taken into account. Their ground-state energy for the planar structure is 0.31 eV lower than the bipyramidal structure, while in the calculation by BonaEiC-Koutecky et al. these structures almost have the same energy. For hexamers we obtained the planar conformation as the most stable geometry, while in refs. [24,

25 ] the most stable geometry is the pentagonal pyramid. In this connection we note that in our optimization procedure for the pentagonal pyramid we found only two bond lengths to optimize, while for the planar trigonal structure we found five bond distances to optimize, in contrast to ref. [ 25 ] in which there are two optimized bond lengths for each of these structures. Therefore we believe that a more limited

R. Santamaria et al. /Chemical Physics Letters 218 (1994) 395-400

optimization and a higher symmetry restriction may lead them to underestimate the binding energy of the planar hexamer. The comparison of theoretical IP, values with experiment can help to establish the cluster geometry which is actually observed in the experiments (as was done by comparing the vertical detachment energy VDE for silver anions, see ref. [ 261). Unfortunately, we are faced with the problem of considerable energy differences (as high as 0.6 eV) among experimental results [ 8,12 1. Thus, according to our calculations up to n = 6, the neutral silver clusters have a planar geometry. The same occurs for Na, [ 361. On the other hand, for Li, the transition from a 2D structure to a 3D occurs at n=6 [37] andforBe,evenloweratn=4 [38]. It is well known that the stable structure for the anionic cluster Ag< is linear while the neutral Ag, has a triangular stable geometry [ 161. If we add one electron to a sufficiently large many-electron system it should not result in drastic changes. Comparing the symmetry of stable geometries for anionic clusters (Table II in ref. [ 261) and for neutral clusters (Table 2 in this paper) we observe that they coincide for n=4 and 5. But for hexamers the stable anionic geometry is tripyramidal and not planar as for neutral hexamers. The reason for such behavior is not clear and requires further investigation. Size effects in clusters [ 391 are manifested not only in such spectroscopic features as IP, and VDE, but also in the stability of the clusters. This stability is well characterized by the atomization energy per atom (E,) and the atomic detachment energy (E, ), which are defined by the expressions 3

e,(Ag,)=-(lln)]Eo(Ag,)-nEo(Ag)l

(2)

Ea(Ag,)=-(lln) x[Eo(Ag,)-(n-l)Eo(Ag)-Eo(Ag-)l,

(3)

399

E(eV)

2,oo

0.50 ’ 2

I

I

I

I

3

4

5

6

n

)

Fig. 2. The dependence of the binding energy per atom (t.) and the atomic fragmentation energy (t,) on the number of atoms for neutral and anionic silver clusters.

mer to trimer there is a sharp increase of ea for anions and a small decrease for neutral clusters. This situation is similar to Li, [ 401. Size effects for the atomic detachment energy e, are, however, more pronounced. The value of er alternates with n and this alternation has opposite signs for neutral and anionic clusters. For the neutral clusters the detachment of an atom takes place more easily for clusters with odd n; for anions the same applies for even IZ.

4. Acknowledgement We are grateful to Dr. I. Top01 and Dr. A. Ramirez for helpful discussions and to the DGSCA staff of the UNAM supercomputing center for access to the CRAY-YMP4/432. We also acknowledge CONACyT for financial support under contracts Nos. 920 100 and 920207.

ei (Ag,) = - [Eo(&,)

1>

-Eo(&,-,I-Eo(Ag)

(4) 5. References

e1(Ag,) = - [Eo(AR,)

-Eo(&,,)

-Eo(Ag)

1.

(5)

From Fig. 2 we can appreciate the dependence of ea and t, on the cluster size. For n> 3 the behavior of E, ( Ag, ) and ea ( Ag; ) is similar, but in going from di-

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