Theoretical study on clusters of magnesium¤

Andreas Florian Weigend and Reinhart Ahlrichs*Ko� hn,

L ehrstuhl T heoretische Chemie, Institut Physikalische Chemie, Karlsruhe,fu� r fu� r Universita� t76128 Karlsruhe, Germany. E-mail : reinhart.ahlrichs=chemie.uni-karlsruhe.de

Received 28th September 2000, Accepted 24th November 2000First published as an Advance Article on the web 11th January 2001

The energetic and structural properties of neutral magnesium clusters (n \ 2 to 22 and selected clusters upMgn

to 309) are investigated by means of density functional theory. The small clusters of sizes n \ 4, 10 and 20show enhanced stability, which is in line with the predictions of the shell model. Larger clusters were selectedaccording to the principle of geometric shell closings ; di†erent kinds of shapes and packing types wereconsidered. Among the cluster sizes studied, icosahedral packings are most favourable, which supports theexperimental Ðndings. Weak evidence is found for hcp packings to become most stable for large n. Thecalculated cohesive energies lead to an extrapolated bulk value of 1.4 eV, in reasonable agreement withexperiment (1.51 eV). The electronic shell structure of large clusters is discussed in terms of the shell model,which is found to be reliable for up to 40 electrons, while the stability of higher shells is very dependent on thecluster shape.

I. Introduction

The development of group II elements like magnesium frommolecular species towards the bulk phase is of particularinterest due to the quasi-closed-shell character of the 1Sground state of the atom. The dimer is only weakly boundthrough van der Waals forces with a binding energy of 0.025eV per atom,1 while in the solid state the interaction energyper atom is 60 times higher, 1.51 eV.2 It has been the conclu-sion of various studies3h7 that the convergence of propertieswith cluster size is slow. Thus calculations on large magne-sium clusters seemed desirable.

The theoretical treatment of metal clusters is a highlydemanding task, see e.g. ref. 8. Since these clusters typicallyhave several nearly degenerate electronic states, among thepure ab initio methods, only multi-reference conÐgurationinteraction (MR-CI) and, in special cases, coupled-clustermethods including connected triples corrections [CCSD(T)]will lead to at least qualitatively correct results. For magne-sium clusters, this has been pointed out by Lee, Rendell andTaylor.9,10 However, this seems currently not feasible for thetreatment of more than a few atoms.

Density functional theory (DFT) has been shown to be aviable choice for the calculation of large aluminium clusters.8Also for magnesium, a variety of studies based on DFT havebeen carried out. The molecular dynamics method of Car andParrinello11 has been applied to small clusters up to Mg13 .3h5The same range and additionally the clusters andMg57 Mg69have been investigated using Gaussian basis functions.6,7 Inall cases, the e†ect of core electrons was replaced by e†ectivecore potentials. The local density approximation (LDA) usedin nearly all the DFT studies mentioned above, tends to over-estimate binding energies by more than 70% compared to gra-dient corrected schemes4 though it still gives similar energetictrends.

In this study, we present all-electron DFT calculationsemploying the gradient correction of Becke and Perdew12

¤ Presented at the Third European Conference on ComputationalChemistry, Budapest, Hungary, September 4È8, 2000.

(denoted as BP-86) for neutral magnesium clusters withMgnup to 309 atoms. The goals are the assessment of preferred

packings and cluster shapes as well as the identiÐcation ofislands of stability and the validation of shell models (bothgeometric and electronic). We will proceed as follows : In thenext section we brieÑy review the basic concepts of shellmodels and deÐne the geometry of several cluster typesemployed. Details of present calculations are described inSection III. Subsequently we discuss results for the structuresand energetics for clusters up to for which simulatedMg22annealing runs have been carried out. For the larger clusterswe only consider selected structure types to ascertain the sta-bility as a function of packing and shape with special atten-tion to the validity of the electronic shell model.

II. Theory

A. Criteria for cluster stability

The basic quantity for discussing the stability of clusters is thecohesive energy deÐned asecoh

ecoh\ [[E(Mgn) [ nE(Mg)]/n. (1)

We focus on electronic e†ects and do not include the zeropoint vibrational energy.

In order to rationalize the abundance in mass spectroscopicexperiments, there are two conditions for stability that can beeasily inspected. The Ðrst condition concerns the stability withrespect to unimolecular decomposition

E(Mgm`n

) \ E(Mgm) ] E(Mg

n) for all m, n. (2)

This condition is always met if the cohesive energy increasesmonotonically with n. Second, there is a local stabilitycriterion13

*2 E(Mgn) \ E(Mg

n`1) ] E(Mgn~1) [ 2E(Mg

n) \ 0. (3)

This can be understood as a discretized second derivative ofthe total electronic energy and is often discussed in connectionwith the jellium model that is reviewed in the following sub-section. Condition (3) implies that is stable with respectMg

nto a disproportionation into andMgn~1 Mg

n`1.

DOI : 10.1039/b007869g Phys. Chem. Chem. Phys., 2001, 3, 711È719 711

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B. Electronic shells

A prediction of the most stable electronic structures is provid-ed by the shell or jellium model.13 In this model one assumesthat the clusters are spherical and that the e†ect of the Mg2`cores can be replaced by a uniform positively charged back-ground. One ends up with a model of free electrons in a spher-ical potential ; the most simple form for such a potential is aharmonic oscillator. One gets a highly degenerate shell struc-ture of spherical harmonics (1s)(1p)(2s,1d)(2p,1f )(3s,2d,1g). . . .In this simple model, of eqn. (3) is equal to the energy gap*2Eto the next shell, and the local stability criterion is met for allclusters with a closed shell. This leads to magic electronnumbers 8, 20, 40, 70, 112, 168, 240, 330, . . . . We willnel\ 2,refer to this model as the harmonic shell model.

Another simple choice, the square well potential, leads to asubshell structure where subshells with a high l quantumnumber are lowest in energy. The intermediate case, i.e. a“ roundedÏ square well, can be realized as an anharmonic per-turbation of the harmonic shell model. Such a modiÐcation ofthe harmonic shell model has been given by Clemenger.14 Thelargest gap within one shell occurs between the subshells ofhighest and second-highest angular momentum. This leads tomagic numbers of 18 (1d subshell), 34 (1f subshell), 58, 92, 138,198, 274, . . . .

Electronic shell models invariably have problems for largeclusters ; the HOMOÈLUMO gap decisive for the stability ofclusters vanishes as n~1@3, and details of the geometric struc-ture should become crucial.15

C. Shells of atoms

Elements exhibiting electron deÐcient metallic bonding preferdense packings with a large number of next neighboursandÈfor clusters of such elementsÈa small surface to volumeratio. This leads to the concept of geometrically closed struc-tures and the occurrence of “magic numbers of atomsÏ asreviewed by Martin.16 The topological types of clusterstreated in this work are collected in Table 1, and the mostimportant shapes are displayed in Fig. 1. They are brieÑydescribed in the following.

The highest possible cluster symmetry is shown by icosa-Ihhedra. This structure type will be denoted by icoxx, where xx

Fig. 1 Shapes of several clusters of 54 to 57 atoms.

is the number of atoms. These are most densely packed andhave a small surface to volume ratio. Distances within a shellare larger than between shells (by a factor of 1.05), leading toinner strain. This type cannot be realized in the bulk phasebecause of the Ðvefold symmetry.

Decahedra, decxx, also display Ðvefold symmetry and aredensely packed, but they have exposed atoms as outer rings.By removing these rings one gets truncated decahedra,trdecxx, e.g. for a decahedron consisting of 23 atoms the cor-responding truncated decahedron, trdec13, is obtained byremoving the outer 10 atoms in the equatorial plane. Further-more one can remove the edges parallel to the Ðvefold axisfrom the truncated decahedra, which leads to the so-called“Marks truncated decahedraÏ,17 mtrdecxx.

Octahedral clusters, octxx, that correspond to the fccpacking are free of inner strain since all atoms can have equaldistances to their next neighbours. Closed shells of atoms canbe obtained in two ways. Either one starts with a centralatom; an additional shell is constructed by adding atoms in[110] directions to surface atoms. This leads to cuboctahedralclusters, cuboctxx. Alternatively one can start with a centraloctahedron of six atoms and then proceed as above. Thisleads to truncated octahedra, troctxx, in which the triangularsurfaces of an octahedron are cut to hexagons. Tetrahedra,tetxx, constitute another class of polyhedra that can bederived from fcc packing. For magnesium, their magicnumbers are the same as those of the electronic shell model(“doubly magic Ï). However, tetrahedra have a low averagenumber of next neighbours, e.g. SnnT \ 7.07 for tet56 com-pared to 7.86 for cuboct55, and thus are not expected to bevery stable.

Bulk magnesium has an hcp structure. The closed hcp clus-ters have symmetry and consist of planar hexagonalD3hsheets in an ABAB packing, with the number of atoms in arespective layer given by : A layer n \ (1), 7, 19, 37, . . . and Blayer n \ 3, 12, 27, . . . . The sixfold axis of the bulk disappearswhen translational symmetry is broken. In this paper weinvestigate the cluster sizes hcp13 (3/7/3), hcp26 (7/12/7),hcp57 (7/12/19/12/7), hcp89 (12/19/27/19/12), hcp103 (7/12/19/27/19/12/7) and hcp157 (12/19/27/37/27/19/12).

III. Details of computation, basis sets andaccuracy of the DFT method

All calculations have been carried out with the DFT imple-mentation of TURBOMOLE18,19 using the BP-86 function-al.12 Coulomb interactions were treated within theresolution-of-the-identity (RI) approximation.20 The gridsrequired for the numerical integration of the exchange andcorrelation contributions were of medium coarseness (m3, ref.21) for all clusters. The use of Ðner grids leads to insigniÐcantchanges in calculated quantities. When using m5 instead ofm3, the average MgÈMg distance of is shortened by 0.3Mg140pm and the cohesive energy is lowered by less than 2 meV.For geometry optimizations redundant internal coordinates22were used throughout.

Table 1 Nomenclature and magic numbers for clusters with polyhedral geometries (n O 150)

Polyhedron Abbreviation Magic numbers

Icosahedron ico 1, 13, 55, 147, . . .Decahedron dec 1, 7, 23, 54, 105, . . .Truncated decahedron with quadratic side planes trdec 1, 13, 55, 147, . . .Marks truncated decahedron mtrdec 1, 75, 147, . . .Octahedron oct 1, 6, 19, 44, 85, 146, . . .Cuboctahedron, triangular (111) surfaces cuboct 1, 13, 55, 147, . . .Cuboctahedron, hexagonal (111) surfaces troct 1, 38, 116, . . .Tetrahedron tet 1, 4, 10, 20, 35, 56, 84, 120, . . .Rhombic dodecahedron bcc 1, 15, 65, . . .

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BP-86 isÈapart from one Ðtting parameter in the BeckeexchangeÈa non-empirical functional that has been shown tobe widely applicable in chemistry. It has been pointed out thatnone of the various functionals recently proposed leads to aconsiderable improvement in the prediction of both structuraland energetic properties.23h25 From our experience BP-86 iswell suited for the treatment of metal clusters, e.g. foraluminium8 and sodium26 clusters and, as will be shownbelow, for magnesium clusters.

Basis sets had to be chosen with care. Basis sets for Mg areusually optimized for the 1S state,27 because in most applica-tions magnesium occurs as ionic species for which valanceorbitals have a small occupation and are of minor importancefor these molecular cases. In clusters, on the other hand,Mg

nmetallic bonds between atoms are present. Structure param-eters and binding energies of and withMg4 Mg38Ècalculatedseveral basis setsÈwere compared to estimate the inÑuence ofthe basis set. The basis sets tested may be characterized asfollows :

B1 : SVP, i.e. split valence optimized for the 1S state, plusone p function for polarization.

B2 : TZ.3P, i.e. triple zeta valence plus two p functions, opti-mized for the 3P state.

B3 : TZ.3P] d, i.e. B2] one d function (g \ 0.25).B4 : SV.3P] d, i.e. split valence plus two p functions, opti-

mized for 3P state ] one d function from B3.B5 : (SVP] d)opt, i.e. split valence plus one p and one d

function for polarization, valence shell and polarization func-tions optimized at (tetrahedron).Mg4The results of the basis set study are sampled in Table 2.The energies and structure parameters obtained for B3 to B5

are very similar, whereas those of B2 show smaller di†erencescompared to the former. B1 obviously is not suited for calcu-lations of Mg clusters. Since B5 is more economical than B3and B4, all calculations were performed with B5. The basis setsuperposition error (BSSE) arising with this basis set amountsto approximately 5È6% of the total binding energy. However,carrying out counterpoise corrections is a tedious task for thebinding energies of large clusters. Since the BSSE is a rathersystematic error and since we are interested in relative stabil-ities rather than in very accurate values for cohesive energies,we will neglect this error in the following.

In order to be able to use the RI approximation, an aux-iliary basis set for B5 was optimized (also at B5 and theMg4).corresponding auxiliary basis set are listed in Table 3. Theerrors in the Coulomb energy for amount to 0.42Mg4 (Mg38)(0.23) meV atom~1, which is negligible compared to any othererrors.

In Table 4 we compare for to the structureMg2 Mg4parameters and binding energies obtained with DFT BP-86using B5 with those obtained at other levels of theory andÈasfar as knownÈwith experiment. These small clusters are achallenge for quantum chemistry. The pseudo-closed-shellcharacter of the Mg atom leads to very soft bonds and a lowcohesive energy. It is therefore not too surprising that theagreement of the present results for the cohesive energy andthe bond distance of with experiment and the MP2-R12Mg2method is only qualitative. The results for and areMg3 Mg4in much better agreement with both CCSD(T) and MP2-R12.The bonding distance of deviates less than 1 pm from theMg4most recent CCSD(T) result of Bauschlicher and Partridge.28The very accurate estimates for of 1.14 eV [fromDe(Mg4)

Table 2 Structural parameters (respectively bond length and average bond length SdT ^ corresponding variance s), total binding energyre De ,and BSSE (both in the corresponding equilibrium geometry) for and obtained with several basis setsMg4 Mg38Mg4 Mg38

Basis re/pm De/eV BSSE/eV SdT ^ s/pm De/eV BSSE/eV

B1 SVP 321.1 0.594 0.016 309.5^ 7.4 17.96 0.67B2 TZ.3P 314.6 1.004 0.030 316.0^ 3.9 26.47 0.68B3 TZ.3P] d 309.6 1.171 0.035 313.5^ 3.3 28.50 0.76B4 SV.3P] d 308.1 1.261 0.067 313.7^ 3.1 31.90 2.37B5 (SVP] d)opt 309.4 1.215 0.076 314.0^ 3.0 29.58 1.58

Table 3 (SVP] d)opt Gaussian basis and corresponding auxiliary bases for RI Coulomb Ðt. Contractions are indicated by coefficients di†erentfrom 1.0

Basis Auxiliary basis

l Exponent CoefÐcient l Exponent CoefÐcient

s 4953.833 919 6 0.005 777 896 74 s 4575.625 663 7 1.0745.180 441 54 0.043 124 761 08 s 861.120 740 42 1.0169.216 049 72 0.192 682 169 87 s 257.134 227 79 1.047.300 672 019 0.486 414 391 16 s 100.551 201 40 1.014.461 336 973 0.425 508 940 77 s 43.700 075 808 1.0

s 24.768 174 789 0.087 956 969 98 s 15.737 198 085 1.02.494 094 534 9 [0.551 650 581 28 s 6.564 695 418 6 1.00.878 075 845 3 [0.534 432 948 33 s 2.813 904 863 1 1.0

s 0.100 644 643 3 1.0 s 1.059 225 315 4 1.0s 0.038 005 795 0 1.0 s 0.429 541 489 9 1.0p 98.053 010 494 0.014 480 564 60 s 0.204 701 762 4 1.0

22.586 932 277 0.095 495 750 78 s 0.099 082 193 6 1.06.839 150 984 2 0.307 876 726 51 p 19.472 019 706 0.189 397 088 912.233 284 381 8 0.499 362 928 86 1.409 948 438 7 0.405 847 031 730.716 065 993 9 0.315 034 762 13 p 0.563 979 375 5 1.0

p 0.123 383 375 2 1.0 p 0.225 591 750 2 1.0d 0.172 697 086 4 1.0 d 50.007 124 619 0.240 457 495 88

8.607 042 881 5 0.319 985 139 531.332 622 917 2 0.720 119 191 49

d 0.189 806 991 2 1.0f 0.230 644 653 1 1.0

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Table 4 Comparison of calculated energetic and structure data of (n \ 2, 3, 4) obtained at di†erent levels of theoryMgn

Species Property CCSD(T)a CCSD(T)b MP2-R12c DFT BP-86d Exp.e

Mg2 re/pm (389.1)f 358.0 389.1De/eV 0.057 0.09 0.049

Mg3 re/pm 338.7 (338.7)f 329.1De/eV 0.25 0.33 0.41

Mg4 re/pm 311.0 310.3 (311.0)f 309.4De/eV 1.04 1.14 1.37(1.21)g 1.22

a Values taken from ref. 10. The authors use a [7s6p3d1f] ANO basis set. b Values taken from ref. 28. The bond length is calculated with acc-pVTZ basis, is extrapolated from an extended basis set study. c Values taken from ref. 29. An uncontracted 19s13p8d4f basis set was used.Ded This work. e Parameters of the RydbergÈKleinÈRees potential energy curve of ref. 1. f The authors of ref. 29 use the experimental geometry for

and the geometries of ref. 10 for the trimer and the tetramer. g The authors of ref. 29 note that perturbation theory tends to overestimate theMg2binding energy due to near degeneracy e†ects. Therefore they give an estimate of 1.21 eV based on the CCSD(T) result of ref. 10 plus correctionsfor the interelectronic cusp estimated from the R12 method.

CCSD(T) and basis set extrapolations28] and 1.21 eV [fromCCSD(T) and MP2-R12 combined,29 see Table 4] are inagreement with the DFT value of 1.22 eV (1.14 eV, if thecounterpoise correction is included). As a comment on alter-native functionals, we note that B3-LYP30 underestimates thebinding energy of by nearly 50%, whereas the PBE31Mg4functional shows overbinding by about 20%. There is still aneed for improved functionals for the difficult cases andMg2which are dominated by van der Waals interactions (cf.Mg3 ,also Section IV) ; however, we expect BP-86 to perform verywell for and higher aggregates.Mg4Concerning the MO occupation the problems were similarto that observed for clusters of aluminium;8 occupations hadto be watched during geometry optimizations and had to bechanged if necessary. DFT is of great help in this respect sinceorbital energies can be relied upon to Ðnd the optimaloccupation in connection with the aufbau principle.

To preserve high symmetries, spin polarization can beneglected and an averaged spin density can be used :

oa \ ob\ oav \ ;i

nio iTSi o . (4)

Usually the occupation numbers are for the HOMOni\ 1 ;

shell (respectively HOMO shells in case of near degeneracy) arational number is chosen to obtain the desired number ofelectrons. The e†ects neglected by this treatment may lead tosymmetry breaking by JahnÈTeller distortions and to a lowertotal energy. This was investigated for selected examples.

IV. Results and discussion

A. Small clusters (n = 2–22)Mgn

The principal goal of our study was to investigate the ener-getic properties of these small clusters. For several clusterssimulated annealing techniques32 were applied to obtain newstructures. Higher and lower aggregates could then be createdby removing or adding atoms. Especially around the elec-tronic magic numbers 20 and 40 a variety of(Mg10) (Mg20)structures were considered and simulated annealing was usedextensively. This often leads to several isomers of very similarenergy, which is documented below for selected cases only.Since simulated annealing and subsequent optimization stepshave been carried out in symmetry, all structures obtainedC1should correspond to local minima. This has been checked inseveral cases by force constant calculations, which alwaysgave real frequencies. It should be noted that we constrainedour search to closed-shell cases. Despite these restrictions theenergies of the proposed clusters allow for a discussion ofenergetic trends, since the conclusions are not a†ected if struc-

tures slightly lower in energy were found in addition. Resultsare collected in Table 5 and displayed in Figs. 2 to 5.

to As pointed out in Section III, the DFTMg2

Mg4.

description of the weakly bound dimer is only qualitativelycorrect at best. The potential curve has a very shallowminimum at a rather large bond distance, suggesting a van derWaals type of interaction. forms an equilateral triangleMg3with still a rather low binding energy (0.14 eV per atom, cf.Table 5) and rather long bonds (329 pm). is a tetra-Mg4hedron, and the binding energy per atom (0.30 eV per atom) isdoubled with respect to The bonding distance (309 pm)Mg3 .falls below the experimental bulk value2 of 319 pm. This indi-cates that the bonding situation in can no longer beMg4looked upon as van der Waals like. In SCF and CASSCFcalculations is unbound,9 and the chemical bonding orig-Mg4inates from dynamical correlation only, as has already beennoted by Chiles, Dykstra and Jordan.33 This is at variancewith where sp hybridization is more efficient due to moreBe4compact 2p orbitals. The planar rhombic structure, whichD2his preferred by e.g. sodium and aluminium, is in the case of

energetically disfavoured by nearly 0.6 eV and displaysMg4long bonds.

to The tetrahedron remains the basis for theMg5

Mg8.

aufbau principle for the next larger clusters. is a trigonalMg5bipyramid, i.e. two tetrahedra sharing a common face. Mg6consists of two fused tetrahedra with a common edge. Whenarranging Ðve tetrahedra with common faces in a ring, onearrives at the structure of a pentagonal bipyramid. ThisMg7 ,is the smallest member of the decahedron family. The Mg8cluster is obtained by capping one triangular face of the deca-hedron. The bond distance of the central Mg pair elongates to308 pm. An alternative structure is derived by capping two ofthe rectangular sides of a trigonal prism (cf. This struc-Mg9).ture is 0.26 eV higher in energy.

to is a tricapped trigonal prismMg9

Mg11

. Mg9 (D3h) ;and can be derived by capping the remaining tri-Mg10 Mg11gonal faces of the prism. The jellium-like shell structure oforbitals is very pronounced for these clusters. We Ðnd a dis-tinctive increase in the cohesive energy from toMg8 Mg9 ,which has 18 valence electrons corresponding to a closing ofthe 1d subshell. Judging from its negative value for *2E,however, is not stable with respect to a dispro-Mg9portionation into and Another increase of theMg8 Mg10 .cohesive energy can be observed when going from toMg9the latter possessing a magic number of 20 electrons.Mg10 ,As already mentioned in Section II, the magic electronnumbers coincide with those of tetrahedra. For we ÐndMg10 ,

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Table 5 Calculated cohesive energies and relative stability *E of small Mg clusters (n \ 3È22). All clusters are found to have a totallyecohsymmetric singlet ground state. G denotes the molecular point group

n G ecoh *E Comment

2 D=h 0.0453 D3h 0.138 Equilateral triangle4 Td 0.304 Tetrahedron4 D2h 0.157 ]0.589 Rhombus5 D3h 0.308 Trigonal bipyramid6 C2v 0.319 Concatenated tetrahedraa7 D5h 0.372 Pentagonal bipyramid, dec7

8 Cs 0.398 Monocapped dec78 Cs 0.365 ]0.264 Bicapped trigonal prism9 D3h 0.467 Tricapped trigonal prism

10 C3v 0.529 Tetracapped trigonal prism10 Td 0.489 ]0.396 Regular tetrahedron11 D3h 0.524 Pentacapped trigonal prism12 Cs 0.515 Capped Mg1113 Cs 0.528 Fused Mg10 and Mg313 C1 0.519 ]0.123 Structure of ref. 513 Ih 0.418 ]1.428 ico13 D3h 0.414 ]1.486 hcp13 D5h 0.411 ]1.517 trdec13 Oh 0.399 ]1.686 cuboct

14 C1 0.542 Fig. 3(a)15 Cs 0.576 Fig. 3(b)15 D4h 0.481 ]1.429 Distorted bcc15 Oh 0.434 ]2.134 bcc16 C3v 0.594 Fig. 3(c)17 D4d 0.628 Fig. 3(d)18 C2v 0.639 Fig. 3(e)

19 C3v 0.678 Fig. 3(f )19 D3h 0.668 ]0.199 hcp19 Oh 0.616 ]1.184 oct20 C3 0.706 Fig. 3(g)20 Td 0.614 ]1.839 tet21 C1 0.688 Fig. 3(h)22 C1 0.681 Fig. 3(i)

a Some authors, e.g. those of ref. 3, denote this structure as a capped trigonal bipyramid.

the tetrahedron 0.40 eV above the tetracapped prism.However, there is an interesting relation between these twostructures : as indicated in Fig. 2, one can obtain the cappedprism by rotating the six atoms of one tetrahedron basis by 60degrees around the axis.C3

to The structures of these clusters all containMg12

Mg15

.an kernel (tetracapped trigonal prism) to which furtherMg10Mg atoms are added. consists of an and anMg15 Mg9 Mg10entity sharing four atoms. is an interesting case sinceMg13there are geometric shell closings for several shapes (ico13,hcp13, trdec13, cuboct13). Within this series the icosahedronis most stable, cf. Table 5. However, its energy is 1.4 eV abovethe most stable low-symmetry structure. This is expected fromthe simple shell model since the HOMO occupation of (1f )6should lead to pronounced JahnÈTeller e†ects. The high-

Fig. 2 Calculated equilibrium structures of clusters. The struc-Mg10tural relationship between the tetrahedron and the capped trigonalprism (60 degree twist of the bottom layer of the tetrahedron) is indi-cated.

symmetry structures and of are also strongly(Oh D4h) Mg15disfavoured by 1.4 eV.

to This region contains the magic electronMg16

Mg22

.numbers 34 and 40. shows according to Fig. 4 onlyMg17slightly enhanced stability. The most stable isomer has D4dsymmetry and is displayed in Fig. 3(d). It is a centered quadra-tic antiprism with eight additional ““equatorial ÏÏ atoms whichcap the triangular faces of the antiprism. For the geometricmagic number 19 an octahedron is expected : however, oct19is not the most stable structure for this size. It is nearly 1 eVhigher in energy than the corresponding hcp structure, cf.Table 5. The most stable isomer can be derived from a dis-torted truncated decahedron, i.e. dec23, from which fourcorner atoms have been removed. The cohesive energy of

is a local maximum in our calculations. The corre-Mg20sponding structure reminds one of a distorted icosahedron towhich further atoms have been added, see Fig. 3(g). The corre-sponding regular tetrahedron is energetically disfavoured by1.84 eV. The irregular structures for n \ 21 and 22 wereobtained by adding further atoms to simulated anneal-Mg20 ;ing runs did not yield isomers of lower energy.

Overall trends. All clusters discussed so far are stable withrespect to unimolecular decomposition into smaller fragments(despite the fact that the cohesive energy has a maximum atn \ 20). The local stability criterion, eqn. (3), suggests that thesizes 4, 10 and 20 are most stable. This supports the validity ofthe jellium model in this range.

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Fig. 3 Calculated equilibrium structures of small (n \ 14 to 22)Mgnclusters. The topologies of the smaller cases (n \ 2 to 13) are the same

as in ref. 3 ; they are brieÑy described in Table 5 and in the text.

The cohesive energy increases slowly and reaches as muchas 50% of the bulk value for The binding energy perMg20 .bond (bulk value 0.25 eV) amounts to 0.21 eV already for Mg4and 0.22 eV for suggesting that surface e†ects (atomsMg20 ,with a small number of next neighbours) are responsible forthe further convergence towards the bulk.

In Fig. 5 we have plotted the lowest singlet and triplet exci-tation energies calculated by means of the TDDFTmethods.34 All computed excitation energies are positive,which implies that the DFT solution is not singlet and/ortriplet unstable. Apart from an oscillatory behaviour the ener-gies of the lowest excited states decrease with cluster size. Wefurther notice that the energy di†erence between the Ðrsttriplet and the Ðrst singlet excitation becomes very small. Thedi†erence is due to exchange interactions that vanish as R~1,where R denotes the average electron separation. This reÑectsthe development towards metal-like properties. From Fig. 5 itmay further be conjectured that high-spin ground states couldoccur for larger clusters.

The observed geometric patterns do not exhibit obvioushcp motifs. The structures up to have the same topol-Mg13

Fig. 4 Calculated cohesive energies (upper graph) and secondecohderivatives of the total energy [lower graph, cf. eqn. (3)] for Mgn(n \ 2 to 22).

Fig. 5 Calculated lowest singlet (full line) and triplet (dotted line)excitation energies of the most stable clusters (n \ 2 to 22).Mg

n

ogies as those presented in ref. 3. The small clusters (up tocan be derived from tetrahedra ; a capped trigonal prismMg8)is the dominant motif for clusters up to For the largestMg16 .

clusters studied in this section to we Ðnd dis-(Mg18 Mg22),torted fragments of decahedral or icosahedral structures.

B. Selected larger clusters up to Mg309

For larger clusters a systematic study, e.g. the use of simulatedannealing techniques to determine the most stable isomers, isnot feasible for us at present. We conÐned our investigationsto several cluster types which display a high symmetry, to getan idea of general trends. Since we have obtained relativelysymmetric structures for a variety of the smaller clusters, weare conÐdent that energies are not far o† the global minima.In particular the following quantities were investigated :

(i) cohesive energy,(ii) number of next neighbours deÐned by a cut-o† at 7.0 au

(ca. 370 pm)Èthe Ðrst coordination sphere pm)(dMgMg \ 340is well separated from the second pm),(dMgMg[ 400

(iii) average distance of the next neighbours SdT and corre-sponding variance s.

The results are collected in Table 6, and in Fig. 6 cohesiveenergies are plotted vs. n~1@3, cf. eqn. (5) below. No clearpicture arises from these data. We Ðnd the following clustersclearly above a linear regression line obtained by Ðtting thecohesive energies of all high-symmetry clusters to a linearizedform of eqn. (5) : troct38, oct44, oct140, mtrdec75, ico55 andico147. No preference is found for clusters derived from hcppackings so far.

When comparing the cohesive energies of di†erent types inthe regions of geometric magic numbers, i.e. n \ 54È57 and

Fig. 6 Calculated cohesive energies of selected clustersecoh Mgn(23O n O 309) vs. n~1@3. All structures derived from octahedra are

here denoted as “ fcc Ï, and structures derived from decahedra arelabelled “dec Ï. The line corresponds to a linear regression including allstructure types. Note that both abscissa and ordinate are truncated.

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Table 6 Calculated cohesive energies and structure data (averaged distance between next neighbours SdT and corresponding variance s andecohaverage number of next-neighbours SnnT, cf. Section IV) of selected high-symmetry magnesium clusters. G denotes the molecular point group ;open-shell systems that are calculated with neglect of spin polarization are designated with “av Ï, cf. Section III

n Type G State SdT ^ s/pm SnnT ecoh/eV

23 dec D5h 3A2@ 314.2^ 5.1 6.70 0.65023 dec D5h av 314.4^ 5.4 6.70 0.64226 hcp D3h 1A1@ 316.3^ 11.8 6.92 0.66838 troct Oh 1A1g 314.0^ 3.0 7.58 0.77839 trdec C2v 1A1 312.3^ 10.1 7.59 0.74544 oct Oh 1A1g 313.5^ 7.4 7.64 0.82554 dec D5h 3A2@ 314.9^ 9.5 7.89 0.83054 dec D5h av 314.8^ 9.6 7.89 0.828

55 ico D5d 7A2g 313.0^ 7.2 8.51 0.85755 ico Ih av 312.9^ 7.1 8.51 0.85055 trdec D5h 7A2@ 312.1^ 5.3 7.96 0.82555 trdec D5h av 312.1^ 5.4 7.96 0.81855 cuboct D4h 3A2g 311.3^ 5.0 7.86 0.82355 cuboct Oh av 311.5^ 4.9 7.86 0.823

56 tet Td av 313.9^ 11.0 7.07 0.77357 hcp D3h 1A1@ 312.6^ 6.7 8.00 0.83265 bcc Oh 1A1g 322.6^ 23.1 9.35 0.85975 mtrdec D5h 1A1@ 314.0^ 7.9 8.51 0.91185 oct Oh av 315.3^ 7.5 8.47 0.91885 trdec D5h 1A1@ 313.1^ 6.5 8.57 0.91189 hcp D3h 1A1@ 313.9^ 5.4 8.56 0.915

103 hcp D3h av 316.0^ 14.7 8.68 0.929

105 dec D5h 1A1@ 315.6^ 11.3 8.64 0.933116 troct Oh av 314.6^ 6.1 8.90 0.953140 troct Oh 1A1g 316.3^ 8.7 9.09 1.000147 ico Ih av 314.5^ 6.9 9.47 1.014147 trdec D5h av 315.0^ 5.9 9.06 0.971147 cuboct Oh av 314.5^ 5.2 8.98 0.969153 hcp D3h 1A1@ 314.3^ 6.4 9.10 0.984309 ico Ih av 315.8^ 6.8 10.02 1.083

140È153, one observes a clear preference for icosahedral pack-ings. This is in contrast to aluminium clusters, where icosahe-dra were the least stable isomers ; this may result from thelarger force constants of aluminium, which do not allow forthe great variations in bond distances necessary for icosahe-dral structures. Moreover aluminium prefers the bulk phasepacking (fcc) for clusters larger than about 100 atoms.8

The preference of magnesium for icosahedral structures forlarger clusters is in line with mass spectroscopic investigationsfor (147 O n O 2869).35 The somewhat reduced stabilityMg

nof ico309 (as compared to a linear extrapolation of ico55 andico147) may be considered as a Ðrst hint that the icosahedralpacking eventually becomes less stable for larger clusters.

For the sake of completeness we point out that tetrahedra,which all lead to the magic numbers of the harmonic shellmodel, are of low stability. For tet56 we get eV,ecoh\ 0.77markedly below ico55 or hcp57.

For several open-shell clusters we examined the e†ect ofspin polarization and possible subsequent JahnÈTeller distor-tions. The smallest case is dec23, which has an HOMO(e1@ )2occupation. In the case of triplet coupling, no Ðrst-orderJahnÈTeller e†ect is present and spin polarization lowers thetotal energy by 0.16 eV (the total energy lowerings are notexplicitly documented in Table 6). Enforcing a closed-shelloccupation, the molecular geometry is distorted to but theC2voverall energy is lowered by only 0.02 eV as compared to theaveraged calculation.

The decahedra dec54 and trdec55 show a similar picture.They have a HOMO occupation of and(e1@ )2 (e1@ )2(e2@ )2(e1A)2respectively, both leading to non-degenerate high-spin states.The energy lowering is moderate, 0.08 eV for the total energyof dec54 and 0.44 eV for trdec55.

Larger e†ects are expected whenever degenerate electronicstates lead to considerable geometric distortions. This hasbeen observed for the cases ico55 and cuboct55 of aluminium8

but not for magnesium. The icosahedral structure leads toalmost degenerate HOMOs of and symmetry occupiedgu t2gby six electrons. The lowest average energy state was obtainedfor Due to the near degeneracy, symmetry lowering to(gu)6.leads to a state with an occupation ofD5d 7A2grather than to a triplet state expected from(e2g)2(e1u)2(e2u)2The total energy lowering is 0.44 eV (spin polarization(gu)6.and JahnÈTeller distortion). For the cuboctahedron even nochange in the energy and only minor geometric distortions arefound. We conclude that the spin averaging leads to onlyminor errors in the cohesive energy, especially for larger clus-ters.

By a simple account may be approximated asecohecoh B ecoh, bulk] asurface n~1@3] aedge n~2@3 ] acorner n~1, (5)

which allows for an extrapolation of Neglecting edgeecoh, bulk .and corner e†ects, which depend on the speciÐc cluster type,we use a linear regression for the extrapolation towards thebulk value. This leads to the following results for the extrapo-lated cohesive energy (ecohextr.) :icosahedra (n \ 55, 147, 309) ecohextr.\ 1.37^ 0.03 eV,fcc (n \ 38, 44, 55, 85, 116, 140, 147) ecohextr.\ 1.34^ 0.04 eV,hcp (n \ 26, 57, 89, 103, 157) ecohextr.\ 1.39^ 0.01eV.

The range of cluster sizes considered does not allow one todraw deÐnitive conclusions, but (including all cluster types) anextrapolated bulk value of 1.38^ 0.02 eV emerges from ourresults. This is reasonably close to the experimental value2 of1.51 eV. Calculations for the bulk phase yield 1.64 eV usingthe LDA functional36 and 1.43 eV with a gradient correctedfunctional37 (Becke correction for the exchange part and amodiÐed version of the Perdew gradient correction for thecorrelation part38) ; the latter result compares well with ourextrapolated value.

The average bond length of the larger clusters (n [ 20) is

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Fig. 7 Calculated valence density of states (in arbitrary units) vs.orbital eigenvalues for di†erent cluster shapes of for hcp).Mg55 (Mg57The computed discrete levels were broadened by Gaussians of 0.1 eVFWHM. The numbers denote the sum over valence electrons up to agiven energy.

312È316 pm (except bcc65), 4È8 pm shorter than in the bulkphase (320.9 pm inside a hexagonal layer, 319.7 pm betweenlayers).

Finally we comment on the validity of the shell model ofelectrons. Due to the harmonic model (cf. Section II), onewould expect an energetic structure of shells with 1, 3, 6, 10,15, . . . orbitals (2, 6, 12, 20, 30, . . . electrons), leading to themagic electron numbers 2, 8, 20, 40, 70, . . . . Inspecting thedensity of states (DOS) arising from the valence molecularorbitals of ico55, cuboct55, trdec55 and hcp57, Fig. 7 (for theshapes of the corresponding clusters cf. Fig. 1), one obtains thefollowing picture. In all four cases the Ðrst three shells areclearly separated from each other and also from the highervalence orbitals, yielding the magic numbers 2, 8 and 20. Thispattern is in agreement with the harmonic model and is foundfor nearly all of the larger (n [ 20) clusters treated in thiswork. The next magic number, 40, is visible only for ico55 andhcp57 ; but all four clusters in Fig. 7 show a zero density ofstates when the total number of 34 valence electrons isreached. Furthermore for ico55 one recognizes a zero density

Fig. 8 Calculated valence density of states (in arbitrary units) vs.orbital eigenvalues for the icosahedral cluster The discreteMg147 .levels were broadened by Gaussians of 0.1 eV FWHM. The shellstructure is indicated at the top. The numbers denote the sum overvalence electrons up to a given shell closing. Note that the sphericalharmonics higher than l\ 2 are not invariant under symmetry andIhthus split into several levels.

of states above 58 and 92 electrons. As pointed out in SectionII these numbers (34, 58 and 92 electrons) correspond to sub-shell closings that occur upon anharmonic distortions of thee†ective potential. In all cases the subshells with the highest lquantum number are lowest in energy, which leads directly tomagic numbers of 18, 34, 58, 92, 138, 198, . . . . The density ofstates for ico147, shown in Fig. 8, shows a zero density ofstates above all these numbers, whereas the magic numbersarising from the harmonic model can only be seen for the Ðrstthree shells.

The spherical shell model, however, is not stable withrespect to slight changes of the cluster shape. The clusterscuboct55, trdec55 and hcp57, which are still quite spherical,do not show vanishing DOS above n \ 58 and 92 and lead toquite di†erent DOS pictures than found for ico55. This sup-ports the Ðndings of ref. 15.

V. ConclusionsDFT BP-86 appears to give a reliable account of magnesiumclusters. Binding energies and bond distances are in agreementwith the few reliable data available, i.e. for (from exten-Mg4sive ab initio calculations10,28,29) and for the bulk (fromexperiment2).

The smaller clusters (n O 22) were investigated using simu-lated annealing techniques. They often show irregular struc-tures with an tetrahedron and a capped trigonal prism asMg4a dominant motif ; the structures of to resembleMg18 Mg22fragments of decahedral or icosahedral clusters.

For larger clusters only geometrically closed (high-sym-metry) structures were considered. We Ðnd a preference foricosahedra among the selected cases studied. This supportsthe interpretation of experimental mass spectra for Mg

n(147 O n O 2869) given in ref. 35. Clusters derived from fccpackings are only slightly lower in energy and those derivedfrom hcp packings are still less stable. Extrapolation towardsthe bulk value, however, yields hcp more stable than fcc ; notenough data are available for icosahedral clusters for thispurpose.

Calculations conÐrm the validity of the shell model for upto 40 electrons in agreement with the conclusions for alu-minium clusters.8 Small clusters show enhanced stability whenelectronic shell closings occur ; for larger clusters the shellstructure of only the lowest levels is preserved when the shapeof the cluster changes.

AcknowledgementsThis work was supported by the Deutsche Forschungsgemein-schaft, SFB 195 (“Lokalisierung von Elektronen in makrosko-pischen und mikroskopischen SystemenÏ).

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