A density functional study of small (AlN)x clusters: structures, energies, and frequencies

Chemical Physics 271 (2001) 283±292

www.elsevier.com/locate/chemphys

A density functional study of small (AlN)x clusters: structures, energies, and frequencies Ch. Chang a,*, A.B.C. Patzer a, E. Sedlmayr a, T. Steinke b, D. S ulzle a a

Zentrum f  ur Astronomie und Astrophysik, Technische Universitat Berlin, Hardenbergstrasse 36, D-10623 Berlin, Germany b Konrad±Zuse±Zentrum fur Informationstechnik Berlin (ZIB), Takustrasse 7, D-14195 Berlin, Germany Received 27 April 2001

Abstract Structural properties of energetically low-lying stationary points of small aluminium/nitrogen clusters with unity stoichiometric ratio (AlN)x …x ˆ 1; 2; 4; 6; 12† have been investigated by theoretical density functional techniques employing the Becke±Perdew-86 gradient corrected exchange correlation functional. A large number of singlet and triplet stationary points representing local minima and transition structures of (AlN)x are completely characterised. We report energies, equilibrium geometric parameters, selected harmonic vibrational wave numbers along with corresponding absorption coecients. Stability and geometric aspects of (AlN)12 are discussed in detail by introducing a measure of sphericity. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 33.15. e; 31.15.Ar Keywords: Ab initio calculations; Molecular electronic structure; Properties of molecules

1. Introduction Representing a link between the isolated molecules and the bulk phase clusters of aluminium and nitrogen Alx Ny have drawn increasing attention during the last years. This is mainly because the solid of AlN with its unusual speci®c physical properties is of great interest to microelectronic, ceramic, material, and surface sciences [1±5]. There are three recent experimental studies on gaseous aluminium nitrides involving Knudsen cell mass spectrometry and infrared matrix isolation spectroscopy [6±8] showing the existence of a large diversity of neutral and charged Alx Ny species. *

Corresponding author. Fax: +49-30-314-24885. E-mail address: [email protected] Chang).

(Ch.

Theoretically the electronic states of the diatomic molecule AlN are well understood [9±11]. For small cluster systems Alx Ny (x; y 6 4) there are a few computational investigations employing theoretical approaches at various levels [12±16]. It is the aim of this contribution to reconsider, consummate, and extend the theoretical studies on aluminium/nitrogen clusters with unity stoichiometric ratio, which corresponds to the composition of the solid, for some even values of x from 2 up to 12. 2. Computational aspects Starting from an arbitrarily chosen initial geometric con®guration a non-local algorithmic search strategy was employed to locate the stationary points on the lowest singlet and triplet potential

0301-0104/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 ( 0 1 ) 0 0 4 3 9 - 6

284

Ch. Chang et al. / Chemical Physics 271 (2001) 283±292

energy surfaces as a ®rst step. This procedure (VESH), which is described in detail elsewhere [17], makes use of the intrinsic electronic structure energy scheme of the molecular system under consideration. It is initiated from a selected geometric starting con®guration and progresses via successive vertical spin and/or charge state shifts with interposed optimisation steps. In this way a great manifold of stationary points not only on the ground state potential energy surface but also on excited state surfaces can be established. All (AlN)x species to be promulgated in this contribution were treated this way, except for the case x ˆ 12 where the initial con®guration was chosen on geometric grounds. The search was performed at a rather inexpensive level of theory DFT/BP86/3-21G(d) (density functional techniques, DFT; Becke±Perdew-86, BP86) resulting in a large number of local minima and transition state structures on various spin state surfaces. The energetically lowest of them were then re®ned, fully optimised, and characterised at the DFT/BP86 [18,19] level of theory using the standard medium sized all-electron split valence 6-31G(d) basis set. Previous investigations [20] showed that this level of approximation is a reasonable compromise between computational cost and desired accuracy with regard to atomisation energies needed as input data for future determination of thermodynamic potentials. In the frequency calculations the vibrational wave numbers m~k were computed in the harmonic approximation. The integral absorption coecients Ak for strictly harmonic motion are obtained from the electric dipole moments and the normal coordinates of the kth vibrational mode. Using the atomisation energies Dat calculated from the determined total electronic energies Etot of the molecule and separated atoms the experimentally observable quantity D0 is then given by D0 ˆ Dat Ezp , where Ezp is the zero point vibrational energy. For all computations we made use of the G A U S S I A N 9 8 system of programs [21]. 3. Results and discussion Although the electronic structure of the monomer AlN has been well studied it is worthwhile to

begin an investigation on (AlN)x clusters by a short recapitulation, chie¯y in order to compare the results with those of very accurate theoretical calculations and experiment and thereby assess the quality of the level of approximation used in this work. The comparison of the results summarised in Table 1 shows that the DFT/ BP86/6-31G(d) level of theory aligns well in between the other much more elaborate theoretical approaches. The search strategy mentioned in the previous section yielded 13 low-lying stationary points for the dimer (AlN)2 and seven for the tetramer (AlN)4 distributed within an energy range of about 8 eV. Among them are all those which have been reported in the past and additional ones of hitherto unknown structures that ®ll in the gaps (see Table 2 for a comparison). For the larger clusters (AlN)6 and (AlN)12 ®ve cage-like structures were identi®ed as local minima. For convenience the ®nal structures are labelled by M K…P†, where K counts the species of every (AlN)x system in energetic order, M indicates spin multiplicity, and P point group symmetry. An additional asterisk marks a transition state structure with one imaginary frequency. Tables 3±10 summarise the energetics and geometric parameters of all species. The di€erent equilibrium geometric arrangements that arise from these stationary points are illustrated in Figs. 1 and 2. The calculated IR spectra of the (AlN)x 1 1 isomers scaled to the maximal values Amax of the integrated absorption coef®cients Ak are depicted in Fig. 3 showing that each isomer has at least one prominent line. Fig. 4 illustrates the atomisation energies per AlN unit of all investigated systems. The upper right limit in the diagram is to tend to the value of solid bulk AlN (11.5 eV) [23], i.e. the cohesive energy as a function of cluster size converges to the value of the solid material. For the largest cluster size so far considered about 80% of the bulk value are reached. For both systems (AlN)2 and (AlN)4 planar structures are clearly favoured. The energetic lowest species of (AlN)2 , 1 1…D2h †, is a plane lozenge with a rather acute angle of 36.2° which has been well described by other authors and probably is

Ch. Chang et al. / Chemical Physics 271 (2001) 283±292

285

Table 1 Comparison of results for the 3 P and 1 R‡ states of AlN Reference/method

State

D0 (eV)

m~ (cm 1 )

 re (A)

Pelissier and Malrieu [10] MRCI/multi-f

3

P R‡

2.42 3.794a

772 1267

1.83 1.68

Langho€ et al. [11] MRCI/multi-f

3

P R‡

Dat ˆ 2:20

746 919

1.813 1.693

Gutsev et al. [12] CCSD(T)/WMRb

3

P R‡

2.45

756 985

1.7909 1.6667

This work DFT/BP86/6-31G(d)

3

P R‡

2.87 4.04a

752 940

1.805 1.688

P R‡

2:86  0:39 3.782a

746.9 930

1.7864 1.65

1

1

1

1

Experimental data Chase [22] Stull and Prophet [23] a b

3 1

With respect to dissociation into Al…2 P† ‡ N…2 D†. Widmark±Malmqvist±Roos ANO basis.

Table 2 Comparison of (AlN)2 and (AlN)4 stationary points with reported results from the literature Isomer

This work

Ref. [13]

Ref. [14]

Ref. [16]

Ref. [7]

Ref. [15]

(AlN)2 1 1…D2h † 3 2…D1h † 3  3 …C1v † 1 4…C2v † 3 5…D2h † 3 6…C1v † 3 7…Cs † 3  8 …C1v † 1 9…Cs † 1 10…C1v † 1 11…C1v † 3 12…D1h † 1 13…D1h †

LM LM TS LM LM LM LM TS LM LM LM LM LM

SP

LM LM

LM TS TS

LM LM

LM

(AlN)4 1 1…D4h † 1 2…Td † 3 3…Td † 3 4…D4h † 3 5…D2h † 1  6 …C1v † 1 7…D2h †

LM LM LM LM LM TS LM

LM

LM TS

SP SP SP(C1 )

LM

SP SP SP

LM

SP

LM SP(D2d ) SP(C2v ) SP SP

LM: local minimum; TS: transition structure; SP: stationary point, not explicitly characterised.

the global minimum of the system. This is supported by the fact that many of the optimisation

paths of the search algorithm ended up at this stationary point indicating that it is a wide and deep basin of attraction. Among the other isomers, all higher in energy, there is another rhombic form with wider angles, a C2v species, two crooked Cs forms, and all conceivable linear arrangements. The three linear transition structures 3 3 …C1v †, 3  8 …C1v † of (AlN)2 , and 1 6 …C1v † of (AlN)4 are merely included in order to compare them to already reported species (cf. Table 2). For (AlN)4 the lowest species, 1 1…D4h †, is a planar ring-like system. By about 0.5 eV higher in energy there are two structures viz. 1 2 and 3 3 which though having Td symmetry do not represent perfect cubes. All their faces are slightly distorted as well as bent giving rise to a dihedral angle within the quadrangles. It is noteworthy that the two higher energy species 3 5…D2h † and 1 7…D2h † do exist as local minima whereas the other two possibilities where all Al and N atoms are swapped turned out to be higher order transition states. For the two isomers of lowest energy of (AlN)2 the calculated IR spectra of structures 1 1…D2h † and 3 2…D1h † show prominent peaks at 558 and 611 cm 1 . Frequency calculations for the three possible isotopomers i.e. Al2 14 N2 , Al2 14 N15 N, and Al2 15 N2 reveal very distinct isotopic shifts for the these species (see Fig. 5). As expected, with increasing mass the peaks are shifted to smaller wave numbers. In

286

Ch. Chang et al. / Chemical Physics 271 (2001) 283±292

Table 3 (AlN)2 : DFT/BP86/6-31G(d) energies and prominent spectral featuresa BP86 Etot (hartree)

Isomer 1

1

3

3

1…D2h † 2…D1h † 3  3 …C1v † 1 4…C2v † 3 5…D2h † 3 6…C1v † 3 7…Cs † 3  8 …C1v † 1 9…Cs † 1 10…C1v † 1 11…C1v † 3 12…D1h † 1 13…D1h †

Ag Rg 3 ‡ R 1 A1 3 B1u 3 ‡ R 3 00 A 3 R 1 0 A 1 ‡ R 1 ‡ R 3 ‡ Rg 1 ‡ Rg

594:3444366 594:3400447 594:3199289 594:3151258 594:2786907 594:2626406 594:2551973 594:2525894 594:2152796 594:1780232 594:1629001 594:1433673 594:0998609

Ezp (eV)

Dat (eV)

m~ (cm 1 )

Ak (10

0.20 0.20

12.76 12.64 12.09 11.96 10.97 10.53 10.33 10.26 9.25 8.23 7.82 7.29 6.10

558 611 52i 1991

206 198

0.20 0.19 0.22 0.16 0.17 0.25 0.20 0.13 0.15

a For local minima only wave numbers m~k with an Ak > 50  10 wave number is stated.

8

cm2 s

Table 4 (AlN)2 : equilibrium geometriesa 1

1…D2h † 2…D1h † 3  3 …C1v † 1 4…C2v † 3 5…D2h † 3 6…C1v † 3 7…Cs † 3

3  1

8 …C1v † 9…Cs †

1

10…C1v † 11…C1v † 3 12…D1h † 1 13…D1h † 1

particle

1979 754 829 1

cm2 s

1

particle 1 )

164 50 372 74, 57 50 381 50 153 56

are given. In case of a transition state the imaginary

Table 6 (AlN)4 : equilibrium geometriesa 1

r12 ˆ 2:066 r12 ˆ 1:889 r12 ˆ 2:640 r13 ˆ 2:204 r12 ˆ 1:869 r12 ˆ 2:344 r12 ˆ 1:843 a123 ˆ 176:1

a213 ˆ 36:2 r23 ˆ 1:211 r23 ˆ 1:945 r34 ˆ 1:159 a134 ˆ 93:6 r23 ˆ 1:846 r23 ˆ 1:726 a234 ˆ 134:5

r34 ˆ 1:865 r12 ˆ 1:787 a123 ˆ 167:9

r23 ˆ 1:713 r23 ˆ 1:758 a234 ˆ 131:4

r14 ˆ 1:810 r34 ˆ 1:756

r12 r12 r12 r12

r23 r23 r23 r23

r34 ˆ 1:160 r34 ˆ 1:672

ˆ 2:199 ˆ 1:701 ˆ 1:787 ˆ 1:769

1

1944 719, 1004 330i

8

ˆ 1:799 ˆ 1:752 ˆ 2:501 ˆ 2:554

1

1…D4h † 2…Td †

r12 ˆ 1:760 r12 ˆ 1:912 d1423 ˆ 167:2

a123 ˆ 101:8 a123 ˆ 81:3

a234 ˆ 168:2 a234 ˆ 98:1

r34 ˆ 1:151 a132 ˆ 68:0

3

3…Td †

a123 ˆ 83:1

a234 ˆ 95:0

r34 ˆ 1:159 r34 ˆ 1:783

r12 ˆ 1:920 d1423 ˆ 174:5

3

4…D4h † 5…D2h †

r12 ˆ 1:777 r12 ˆ 1:777 a345 ˆ 92:1

a123 ˆ 108:3 r23 ˆ 1:787

a234 ˆ 161:7 r34 ˆ 1:873

1 

6 …C1v †

r12 ˆ 1:710 r45 ˆ 1:708 r78 ˆ 1:671

r23 ˆ 1:722 r56 ˆ 1:687

r34 ˆ 1:691 r67 ˆ 1:716

1

7…D2h †

r12 ˆ 1:700 a345 ˆ 95:6

r23 ˆ 1:789

r34 ˆ 1:877

3

a  angles aijk , and dihedral angles Interatomic distances rij in A, dijkl in deg; the geometric meaning of these parameters can be inferred from Fig. 2.

 and angles aijk in deg; the geoInteratomic distances rij in A metric meaning of these parameters can be inferred from Fig. 1. a

Table 5 (AlN)4 : DFT/BP86/6-31G(d) energies and prominent spectral featuresa BP86 Etot (hartree)

Isomer 1

1

1

1

1…D4h † 2…Td † 3 3…Td † 3 4…D4h † 3 5…D2h † 1  6 …C1v † 1 7…D2h † a

A1g A1 3 A2 3 B2u 3 B1u 1 ‡ R 1 Ag

1188:8047430 1188:7884575 1188:7849154 1188:7497753 1188:6384768 1188:5602043 1188:5209063

Ezp (eV)

Dat (eV)

m~ (cm 1 )

Ak (10

0.53 0.55 0.53 0.46 0.42 0.47 0.44

28.67 28.23 28.13 27.18 24.15 22.02 20.95

992

126 50 50 50 50

For local minima only wave numbers m~k with an Ak > 50  10 wave number is stated.

8

2

cm s

1

particle

173 931 9i 671 1

8

cm2 s

1

particle 1 )

130

are given. In case of a transition state the imaginary

Ch. Chang et al. / Chemical Physics 271 (2001) 283±292

287

Table 7 (AlN)6 : DFT/BP86/6-31G(d) energies and prominent spectral featuresa BP86 Etot (hartree)

Isomer 1

1

1

1

1…D3d † 2…D2h † 3 3…D3d † 3 4…D2h † a

A1g Ag 3 A1g 3 B1u

1783:4127520 1783:3741332 1783:3730300 1783:3564188

Ezp (eV)

Dat (eV)

m~ (cm 1 )

Ak (10

0.95 0.91 0.86 0.87

48.61 47.55 47.52 47.07

622, 626, 558, 649,

123, 141, 141 90, 58, 50 53, 57 94, 131, 269

For local minima only wave numbers m~k with an Ak > 50  10

8

cm2 s

1

particle

1

802, 806 692, 921 606 860, 890

8

cm2 s

1

particle 1 )

are given.

Table 8 (AlN)6 : equilibrium geometriesa 1

1

3

3

1…D3d †

r12 ˆ 1:849 a123 ˆ 106:7 d6134 ˆ 180:0

r17 ˆ 1:954 a234 ˆ 132:3 d6312 ˆ 169:5

2…D2h †

r12 ˆ 1:910 a12…11† ˆ 79:9 a143 ˆ 82:2 d12…12†…11† ˆ 169:8

r34 ˆ 2:109 a2…11†…12† ˆ 99:7 a432 ˆ 90:9 d8954 ˆ 168:9

3…D3d †

r12 ˆ 1:860 a123 ˆ 108:9 d6134 ˆ 180:0

r17 ˆ 1:961 a234 ˆ 130:6 d6312 ˆ 173:6

a16…12† ˆ 81:9 d1465 ˆ 166:4

a617 ˆ 96:7 d1278 ˆ 161:1

r14 ˆ 1:836 a321 ˆ 85:7

a214 ˆ 100:0

a16…12† ˆ 86:6 d1465 ˆ 170:9

a617 ˆ 92:7 d1278 ˆ 167:4

4…D2h †

r12 ˆ 1:920 r34 ˆ 2:070 r14 ˆ 1:871 a12…11† ˆ 83:1 a2…11†…12† ˆ 96:8 a365 ˆ 87:2 a654 ˆ 97:1 a543 ˆ 84:3 a436 ˆ 90:7 d1423 ˆ 171:0 d12…12†…11† ˆ 174:0 a  angles aijk , and dihedral angles dijkl in deg; the geometric meaning of these parameters can be inferred Interatomic distances rij in A, from Fig. 2.

Table 9 (AlN)12 : DFT/BP86/6-31G(d) energy and prominent spectral featuresa BP86 Etot (hartree)

Isomer 1

a

1…Th †

1

Ag

Ezp (eV)

3567.2810236

Only wave numbers m~k with an Ak > 50  10

2.03 8

cm2 s

1

particle

1

Dat (eV)

m~ (cm 1 )

Ak (10

8

cm2 s

109.61

649 860 890

94 131 269

1

particle 1 )

are given.

Table 10 (AlN)12 : equilibrium geometrya 1

1…Th †

r16 ˆ 3:291 r12 ˆ 1:875 r13 ˆ 2:757 r15 ˆ 1:807 a123 ˆ 94:7 a213 ˆ 42:7 a156 ˆ 126:7 a561 ˆ 26:1 a615 ˆ 27:2 d2134 ˆ 161:3 d3128 ˆ 143:9 d2186 ˆ 166:6 d…11†156 ˆ 133:0 a  angles aijk , and dihedral angles dijkl in deg; the geometric meaning of these parameters can be inferred Interatomic distances rij in A, from Fig. 2.

the case of mixed N isotopes the line of the 1 1…D2h † isomer splits into two whereas the 3 2…D1h † species shows a single peak re¯ecting the location of the short 14 N15 N bond near the centre of mass of the molecule. This should allow for the experimental

detection of these two isomers by infrared matrix isolation spectroscopy. To our knowledge clusters of (AlN)x with x > 4 have not yet been theoretically considered. In contrast to (AlN)4 there are no low-lying planar

288

Ch. Chang et al. / Chemical Physics 271 (2001) 283±292

Fig. 1. Geometries of Al2 N2 isomers. The corresponding geometric parameters are listed in Table 4.

structures. The AlN units tend to form small cages in which the positions are alternatingly occupied by Al and N atoms. This is plausible since the systems thus avoid close interactions between like atoms which would give rise to destabilisation and hence lead to dissociation or species of higher energy. It follows that for perfectly alternant cage-like species with unity stoichiometric ratio Al:N the underlying polyhedral structures must consist of faces all of

which having an even number of vertices. Because of the chemical properties of Al and N there is still another restriction viz. all vertices are to be only trihedral i.e. having just three nearest neighbours. However, upon optimisation, as can be seen from the two lowest structures of (AlN)6 , 1 1…D3d † and 1 2…D2h †, the faces do not keep their ideal geometric shapes (six-prism, double cube) but rather distort and swerve thus loosing planarity giving rise to several extra dihedral angles (cf. Table 8). The overall irregular polyhedron, however, keeps its convexity and point group symmetry. This e€ect could already be observed for the two structures 1 2 and 3 3 of (AlN)4 having Td symmetry. Note that the species 1 2…D2h † and the lowest triplet structure 3 3…D3d † have nearly equal energies (see Table 7). The system (AlN)12 is particularly interesting because of its analogy to boron/nitrogen cage clusters considered in the past as ®rst candidates for inorganic fullerenes [24,25]. Because of the relatively great di€erence between the atomic radii  rB ˆ 0:88 A  [26]) it of Al and B (rAl ˆ 1:43 A, cannot immediately be concluded that AlN and BN clusters would necessarily exhibit the same geometric features based solely on size and the isoelectronic principle. The structure of (AlN)12 indicates that aluminium/nitrogen fullerenes may have similar structures for the same cluster size as predicted and observed for boron/nitrogen cages for which structures and building principles have been intensively studied [27,28]. Resuming the discussion already alluded to in the section on (AlN)6 the most reasonable starting geometry for (AlN)12 is a perfect truncated octahedron which is composed of six squares and eight hexagons. Upon optimisation, however, the faces of the perfect body loose their planarity as well as distort in several complex ways. Chemists would use the notion of ``ring puckering''. The (BN)12 analogue was predicted to have a very similar structural appearance [24], which we veri®ed at the DFT/BP86/6-31G(d) level of theory. Alternatively, it is very instructive, to look at the structure of (AlN)12 from a pure geometric point of view. Although this somewhat mathematical approach does not really contain physical, structural information, as e.g. not all edges in the contorted polyhedron can be explained as bond

Ch. Chang et al. / Chemical Physics 271 (2001) 283±292

289

Fig. 2. Geometries of (AlN)x x ˆ 4; 6; 12 isomers. The corresponding geometric parameters are given in Tables 6, 8, and 10.

lengths, it provides nonetheless a picture of molecular shape. From this point of view the optimised species 1 1…Th † of (AlN)12 is a convex polyhedron the convex hull of which is made up of three di€erent types of triangles (cf. Fig. 6). The convex hull of a ®nite set of points in k dimensions S ˆ fx1 ; x2 ; . . . ; xn g  Rk is the smallest convex set containing S. Thus

( conv…S† :ˆ

n X iˆ1

) n X ki x i ki P 0 ^ ki ˆ 1 : iˆ1

Computations of the convex hull were based on the so-called incremental algorithm [29]. During the optimisation the six formerly perfect squares distort in such a way as to form 12 isosceles of type D…bbd† where b ˆ r12 and d ˆ r13 (cf. Table 10), whereas the eight hexagons give rise to

290

Ch. Chang et al. / Chemical Physics 271 (2001) 283±292

Fig. 3. Calculated (DFT/BP86/6-31G(d)) IR spectra for the (AlN)x 1 1 isomers scaled to the maxima Amax of the integrated absorption coecients Ak (10 8 cm2 s 1 ).

Fig. 4. (DFT/BP86/6-31G(d)) atomisation energies Dat of the (AlN)x clusters per unit AlN. As indicated by the small indices, singlet and triplet states are depicted separately for each cluster size. Transition states are marked with dotted lines.

24 triangles of type D…abc† and eight equilateral triangles of type D…ccc† where a ˆ r15 and c ˆ r16 as shown in Fig. 6. The whole set of 44 triangular

Fig. 5. Characteristic features of the calculated (DFT/BP86/631G(d)) IR spectra for Al2 N2 isotopomers of 1 1…D2h † (Ð) and 3 2…D1h † (


).

Fig. 6. Convex hull illustration of the Al12 N12 1 1…Th † polyhedron composed of three di€erent types of triangles. The four distinct edges …a; b; c; d† involved are marked.

facets are distributed over the surface of the body in such a way as to conserve the overall point group symmetry of Th . Each of the four di€erent

Ch. Chang et al. / Chemical Physics 271 (2001) 283±292

types of edges …a; b; c; d† gives rise to one of the four dihedral angles dijkl (see Table 10). Moreover all the Al atoms occupying 12 vertices lie on a  The same holds common sphere (RAl ˆ 2:779 A).  thus altrue for the 12 N atoms (RN ˆ 3:019 A), lowing for two di€erent vertex environments. It is interesting to consider the spherical aspect of this polyhedron arriving at another stability criterion apart from minimal energy and high symmetry. It can be reasoned that a certain amount of asphericity is an indication for higher strain within the framework. A spherical shape, on the other hand, distributes the strain as evenly as possible through minimising the contribution to strain energy. Note, however, that the most spherical arrangement need not necessarily imply high symmetry. So there is a competition between these three criteria. A pure geometric way to de®ne a measure of sphericity r is to compare the surface area of the polyhedron with that of a sphere of equal volume. A straightforward calculation then leads to the expression s 2 3 36pVp rˆ 3 Ap

0 6 r 6 1;

where Ap and Vp are the surface area and the volume of the polyhedron respectively. For a sphere, of course, having least surface area for any volume r ˆ 1. Taking the species 1 1…Th † of (AlN)12 we obtain a value of r ˆ 0:944751 which should be compared to that of a perfect truncated octahedron where r ˆ 0:909918. This indicates that by distorting the systems becomes more spherical by about 3%. This behaviour can also be observed for the well known C60 fullerene molecule. The perfect truncated icosahedron has a sphericity of r ˆ 0:966622 whereas the real C60 structure with two alternant di€erent edge lengths (a ˆ 1:458 and b ˆ 1:401 [30]) within the hexagons has r ˆ 0:966819. In this case, however, all faces remain planar the distortion being just an elongation within the hexagon edges. Apart from geometric aspects Al12 N12 is a highly stable system not only with respect to dis-

291

sociation into the corresponding atoms (D0 about 108 eV (cf. Table 9)) but also with respect to other fragmentation pathways resulting in (AlN)x containing clusters, for which the enthalpies of reaction at 0 K, Dr H …0†, are predicted to be strongly positive. Using the well known data for Al(g) and N2 (g) [22] the decomposition reaction Al12 N12 …g† ! 12 Al…g† ‡ 6 N2 …g†

Dr H …0†  49 eV

supposed to be the evaporation process of the solid material [31], turns out to be extremely unlikely. The enthalpy of this reaction per reaction product, about 3 eV, is of the same order of magnitude as the corresponding atomisation enthalpy per particle of gas phase C60  7 eV [32].

4. Concluding remarks Clusters of (AlN)x exhibit a rich structural diversity of energetically low-lying isomers. For x ˆ 2, 4, and 6 the singlet and triplet potential energy hypersurfaces were investigated in some detail applying the VESH procedure. Among the stationary points found are all those which have been reported in the past and additional hitherto unknown structures. With increasing cluster size cage-like species become more favourable than planar arrangements. Besides stability and symmetry aspects a geometric measure of sphericity gives additional information about molecular shape. Most interesting is the case of x ˆ 12 for its extraordinary stability and high sphericity. We conclude that (AlN)x systems are possible candidates for inorganic fullerene-like cage structures.

Acknowledgements The calculations were performed on the SGI workstation cluster of the Institut f ur Astronomie und Astrophysik, TU Berlin and on the CRAY computers of the Konrad±Zuse±Zentrum f ur Informationstechnik Berlin (ZIB).

292

Ch. Chang et al. / Chemical Physics 271 (2001) 283±292

References [1] L.M. Sheppard, Am. Ceram. Soc. Bull. 69 (1990) 1801. [2] F.A. Ponce, D.P. Bour, Nature 386 (1997) 351. [3] T.B. Jackson, A.V. Virkar, K.L. More, R.B. Dinwiddie, R.A. Cutler, J. Am. Ceram. Soc. 80 (1997) 1421. [4] F.J. Moura, R.J. Munz, J. Am. Ceram. Soc. 80 (1997) 2425. [5] S.M. Bradshaw, J.L. Spicer, J. Am. Ceram. Soc. 82 (1999) 2293. [6] G. Meloni, K.A. Gingerich, J. Chem. Phys. 113 (2000) 10978. [7] L. Andrews, M. Zhou, G.V. Chertihin, W.J. Bare, Y. Hannachi, J. Phys. Chem. A 104 (2000) 1656. [8] B.D. Leskiw, A.W. Castleman, C. Ashman, S.N. Khanna, J. Chem. Phys. 114 (2001) 1165. [9] M. Pelissier, J.P. Malrieu, J. Mol. Spectrosc. 77 (1979) 322. [10] S.R. Langho€, C.W. Bauschlicher, L.G.M. Pettersson, J. Chem. Phys. 89 (1988) 7354. [11] G.L. Gutsev, P. Jena, R.J. Bartlett, J. Chem. Phys. 110 (1999) 2928. [12] S.K. Nayak, S.N. Kanna, P. Jena, Phys. Rev. B 57 (1998) 3787. [13] R.W. Grimes, Phil. Mag. B 79 (1999) 407. [14] B.H. Boo, Z. Liu, J. Phys. Chem. A 103 (1999) 1250. [15] J.J. BelBruno, Chem. Phys. Lett. 313 (1999) 795. [16] A.K. Kandalam, P. Pandey, M.A. Blanco, A. Costales, J.M. Recio, J.M. Newsam, J. Phys. Chem. B 104 (2000) 4361. [17] D. S ulzle, Ch. Chang, Int. J. Mod. Phys. C 10 (1999) 1229. [18] A.D. Becke, Phys. Rev. A 38 (1988) 3098. [19] J.P. Perdew, Phys. Rev. B 33 (1986) 8822. [20] Ch. Chang, A.B.C. Patzer, E. Sedlmayr, T. Steinke, D. S ulzle, Chem. Phys. Lett. 324 (2000) 108. [21] M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, V.G. Zakrzewski, J.A.

[22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

Montgomery, R.E. Stratmann, J.C. Burant, S. Dapprich, J.M. Millam, A.D. Daniels, K.N. Kudin, M.C. Strain, O. Farkas, J. Tomasi, V. Barone, M. Cossi, R. Cammi, B. Mennucci, C. Pomelli, C. Adamo, S. Cli€ord, J. Ochterski, G.A. Petersson, P.Y. Ayala, Q. Cui, K. Morokuma, D.K. Malick, A.D. Rabuck, K. Raghavachari, J.B. Foresman, J. Cioslowski, J.V. Ortiz, B.B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. Gomperts, R.L. Martin, D.J. Fox, T. Keith, M.A. Al-Laham, C.Y. Peng, A. Nanayakkara, C. Gonzalez, M. Challacombe, P.M.W. Gill, B.G. Johnson, W. Chen, M.W. Wong, J.L. Andres, M. Head-Gordon, E.S. Replogle, J.A. Pople, G A U S S I A N 9 8 , Gaussian Inc., Pittsburgh, PA, 1998. M.W. Chase, J. Phys. Chem. Ref. Data, Monograph 9, NIST-JANAF Thermochemical Tables, fourth ed., ACS, AIP, NSRDS, Gaithersburg, 1998. D.R. Stull, H. Prophet, JANAF Thermochemical Tables, second ed., NSRDS-NBS 37, US Nat. Bur. Std., Washington, 1971. D.L. Strout, J. Phys. Chem. A 104 (2000) 3364. S.S. Alexandre, M.S.C. Mazzoni, H. Chacham, Appl. Phys. Lett. 75 (1999) 61. M.C. Day, J. Selbin, Theoretical Inorganic Chemistry, Reinhold, New York, 1962. G. Seifert, P.W. Fowler, D. Mitchell, D. Porezag, Th. Frauenheim, Chem. Phys. Lett. 268 (1997) 352. P.W. Fowler, T. Heine, D. Mitchell, R. Schmidt, G. Seifert, J. Chem. Soc. Faraday Trans. 92 (1996) 2197. B. Chazelle, Discrete Comput. Geom. 10 (1993) 377. K. Hedberg, L. Hedberg, D.S. Bethune, C.A. Brown, H.C. Dorn, R.D. Johnson, M. de Vries, Science 254 (1991) 410. T.Ya. Kosolapova (Ed.), Handbook of High Temperature Compounds, Hemisphere, New York, 1990. W.V. Steele, R.D. Chirico, N.K. Smith, W.E. Billups, P.R. Elmore, A.E. Wheeler, J. Phys. Chem. 96 (1992) 4731.