Phenomenological Shell Model and Aromaticity in ...

Phenomenological Shell Model and Aromaticity in Metal Clusters Tibor Höltzla,b Tamás Veszprémi,b Peter Lievensc and Minh Tho Nguyena,* a)

Department of Chemistry, and Insitute for Nanoscale Physics and Chemistry-INPAC, Katholieke Universiteit Leuven, B-3001 Leuven, Belgium b)

Department of Inorganic and Analytical Chemistry, Budapest University of Technology and Economics, H- 1521Budapest, Hungary c)

Laboratory for Solide State Physics and Magnetisms, and Insitute for Nanoscale Physics and Chemistry -INPAC, Katholieke Universiteit Leuven, B-3001 Leuven, Belgium

(Revised Version: 7 December 2009)

1. Introduction Study of pure and doped metal clusters became an interesting and rapidly progressing research topic in the last decades due to their structural richness, unexpected stability and reactivity patterns, the variation of these properties with respect to the number of the constituting atoms and also because of the observed aromaticity in certain species. Similarly to an organic compound,1 the aromaticity of a metal cluster is showed by the coexistence of different properties like equalized bond lengths and bond orders, the thermodynamic stability, specific reactivity, magnetic behavior, and also closed electronic structure. The electron structure of a given compound ultimately determines its properties, therefore its aromaticity. In chemistry the electron structure of a molecule is often represented by the electron configuration: the occupation of different molecular orbitals. Several simple, but very useful models exist to interpret and also to predict the observed electron configuration of different class of compounds. Therefore the necessary conditions to obtain a closed electronic structure and a possible aromaticity can also be formulated. In fact, simple electron count rules can be used in designing aromatic compounds and also in explaining the observed aromaticity of different

*) Email: [email protected]

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molecules. Electron count rules are based on the observation that the stability of annulenes, alternate as a function of the number of carbon atoms and consequently the number of itinerant electrons. These rules give the number of electrons necessary to obtain an aromatic structure. The most well known rule is the (4n + 2) rule, which is based on the qualitative Hückel model. Accordingly, an annulene is aromatic in its lowest lying singlet state if the number of π electrons equals to 4n+2, where n is a non-negative integer. Of course the degree of the aromaticity depends on the chemical composition of the substance considered, as well as on its chemical bonding. Benzene is the prototype aromatic compound with six π electrons satisfying the 4n + 2 rule. Hydrocarbons such as the cyclopentadienyl anion (C5H5-), cycloheptatrienyl cation (C7H7+) or cyclooctatetradienyl dication (C8H82+) also have six π electrons and are all aromatic. The analogous borazine2 (B3N3H6) is aromatic,3 as it has the same number of π electrons as the previous examples. However its electronic structure is not as homogenous as benzene due to the large electronegativity difference between nitrogen and boron atoms. Nevertheless, the 4n + 2 rule remains a useful tool in the determination of the aromaticity. In contrast, the 4n rule states that annulenes are antiaromatic when the number of the π electrons equals to 4n, where n is a positive integer. As in the Hückel model, only the connection between the atoms is taken into account the 4n and 4n + 2 rules remain valid in the case of rings where s electrons are itinerant. For example, Li3+ has a perfect D3h symmetry and 2 itinerant s electrons. Hence it is expected to be aromatic. The Hückel model can also be applied to three dimensional systems, such as fullerenes,4 where electron count rules can also be formulated. In the case of icosahedral fullerenes, Hirsch et al. have formulated such a rule,5 but other electron counting rules also exist.6 As in fullerenes the itinerant electrons are delocalized over a nearly spherical surface, these rules account for spherical aromaticity.7 Annulenes are π aromatic because the orbitals responsible for this character have one nodal surface in the ring plane. Metal clusters constitute a rich source of more exotic types of aromaticity, such as σ aromaticity8 whose related compounds have no nodal surface in the ring plane. Several σ aromatic clusters exist, and in some cases a δ aromaticity is also observed. Spherically aromatic clusters are also found.

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Electron configuration of metal clusters with itinerant electrons is represented in terms of the phenomenological shell model. The main assumption of this model is that the itinerant electrons are confined in a box according to the cluster shape, and these determine the properties of the given cluster to a great extent. This model was developed to explain the observed stability patterns of sodium clusters and has been successfully applied in other elements (like Li, Al, Cu) and properties (like ionization energy, electronaffinity). Also, it was formulated for different cluster shapes and also for doped metal clusters. In this chapter we aim to demonstrate that the aromaticity of metal clusters can be interpreted in terms of the PSM, which can be used to formulate the criteria to obtain a closed electronic structure in different cluster shapes. Therefore the PSM provides the different electron counting rules with a general framework, and helps to determine how these rules change upon a change of the geometry or a chemical doping of the cluster. This model allows the electronic structure of metal clusters to be interpreted within such a general framework. We show that for example the 4n + 2 rule of the closed-shell electronic structure of annulenes is a special case of the PSM model. Therefore in the first part of this chapter we shortly introduce the PSM from a chemists point of view. Then we summarize the different criteria of aromaticity and show how this phenomenon can be evaluated in metal clusters. Finally, the phenomenon of aromaticity and the application of PSM in some concrete examples taken from our current research work are presented. Although the examples include different alkaline and doped coinage metal clusters, the general principles can also be applied to other cases including the alkaline-earth metals, aluminum and other elements, that have itinerant electrons in their clusters. 2. The Phenomenological Shell Model (PSM) The PSM was originally developed in nuclear physics to interpret the experimental findings such as the existence of unexpectedly stable nuclei when the number of protons and neutrons equal to certain “magic numbers”.9 The 1963 Nobel Prize in Physics was awarded to Maria Goeppert-Mayer and Hans Jenssen for the development of the shell model of the atomic nuclei. This model was applied to metal clusters at the birth of the cluster science to rationalize the observed stability patterns of small alkaline metal clusters. It was found that the alkaline metal clusters are not just small pieces of metals whose properties converge monotonically towards the bulk as their size increases.10 In contrast to the expectations, certain alkaline metal clusters were found to be considerably more stable than the others. For example, high abundance was found in - -

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the mass spectra of the NaN clusters, when N = 8, 20, 40, 58,...11 The high abundance indicates the high stability of the corresponding clusters. These magic numbers N were observed in clusters of other elements such as Li, Cu,12 or Au in their cationic or neutral forms. Magic numbers were also found in the case of pure and doped aluminum clusters.13 The PSM was then formulated to explain the exceptional stability of these cluster sizes. It should be stressed that another successful approach applied to metal clusters is the jellium model,14 which is often mistakenly considered as the PSM. An important difference between the two models is that while the jellium model treats the electron-electron interaction selfconsistently in a positive background potential, the PSM is an one-electron approximation using a (not necessarily homogenous) confining potential. Thus in order to use the PSM, we solve the Schrödinger equation for the one-electron in a box problem using different box shapes. To formulate a model for clusters the following approximations were made. The shape of the stable clusters is spherical, and the itinerant valence electrons are confined by the attractive potential of the nuclei and the electron-electron interaction is neglected. Overall, independent electrons are confined in a spherical box. At a zeroth-order approximation, the confining potential is equal to zero inside the box and infinite on the boundary and outside. The Schrödinger equation for this case is solvable and it yields the well known spherical harmonics as a solution. This is similar to that of the hydrogen atom but with two noteworthy differences. In the hydrogen atom, the central Coulomb potential restricts the angular quantum number for each principal quantum number. It is obvious for chemists that for example 1d orbitals do not exist. In the above electron-in-a-box problem, no such restriction is imposed. Second, the ordering of the energy levels of a cluster differs from that of atoms, and depends on its chemical composition and also on its shape. Throughout this chapter we will denote the atomic orbitals by small case letters, while capital letters stand for the single electron levels of PSM and also the molecular orbitals (MOs) assigned to them. The ordering of the energy levels of this simple model is displayed in Figure 1. In the free atoms, extended stability is expected when the electronic shells or sub-shells are closed. It is obvious that the very high ionization energies of the noble gas are due to their closed s and p shells. However also the alkaline-earth metals show increased ionization energies as compared to their neighbors, which is due to their closed s sub-shells. Analogously, in the alkaline metal clusters, extended stability is expected when the number of - -

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the valence electrons correspond to a shell closing such as 2 (1S2), 8 (1S2 1P6), 20 (1S2 1P6 1D10 2S2), 40 (1S2 1P6 1D10 2S2 1F14 2P6) and so on. Accordingly, this simple model, based on the itinerant electrons confined in a spherical box, can explain the origin of the magic numbers.

a)

b)

c)

Figure 1 Energy levels using the phenomenological shell model for spherical clusters: a) Homogenous background potential, b) Electronegative dopant atom in the center, and c) Electropositive dopant atom in the center.

Thus a simple question arises: based on this model, at which size can we expect a stable alkaline-earth metal cluster?. As each alkaline-earth metal has two itinerant valence s electrons, stable clusters are expected for clusters constituting of 4, 10, 20, ... atoms, which was indeed confirmed by experiments. Although the configuration 1S2 1P6 1D10 corresponds to a closed electronic structure, 18 was not observed as a magic number in homogenous clusters. This is due to the fact that both 1D and 2S levels are close in energy, an extended stability is therefore expected when the latter is also filled. Another legitimate question is as to whether we can extend the applicability of this model to other elements? This question is equivalent to that asking which electrons are itinerant in each type of the elements. It is obvious that in the alkaline and alkaline earth metals, the valence s electrons are itinerant. In other elements, the valence s electrons of the metals are in general - -

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itinerant, and so do the p electrons of aluminum. Things are getting more complicated in the case of the d and f elements. As a general rule, the d electrons of the left part of each row of the periodic Table are itinerant, while those of the right part are not. However the degree of the itinerancy also depends on the chemical environment, bonding, hybridization, charge and spin state. Therefore this property has to be checked in each specific case. This is similar in the case of the f elements. Overall, the d electrons of the coinage metals (copper, silver and gold) and the zinc, cadmium,… are not delocalized. As coinage metals have a (x-1)d10 (x)s1 electron configuration, each atom thus contributes one itinerant electron. The number of the itinerant electrons in different elements is indicated in Table 1. Table 1 Number of itinerant electrons in some elements considered. Element

Number of

Cinfiguration of

itinerant electrons

Itinerant electrons

Li, Na, K

1

s1

Be, Mg, Ca

2

s2

Cu, Ag, Au

1

s1

Hg

2

s2

Al

3

s2p1

Sc

3

s2d1 (in most cases)

Thus the number of itinerant electrons is the sum of the contributions from the constituting atoms, enlarged or reduced by the total charge of the cluster. As such, alkaline metal cluster cations like NaN+ are stable for N = 9, 21, 41, ... A magic number of 20 was observed for example for Au20 (from both experimental and theoretical methods),15 and Cu20 (using theoretical method).16

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The superatom concept was inspired by the above analogy between the electronic structure of the metal clusters and the free atoms.17 This concept was best illustrated in the Al13 cluster by Khanna et al.17 As each Al contributes 3 itinerant electrons, the total number of 39 is less by one than that of the nearest closed electronic shell. Therefore it is expected that the cluster will be stabilized by an additional electron, and the anion Al13- is indeed highly stable. This is supported by the noteworthy large electron affinity of Al13. Thus, based on such behavior, a superhalogen character was attributed to this cluster. Also, in the case of the Al13- anion, a perfect icosahedral symmetry was found. Its nearly spherical shape supports further the validity of the PSM. More recently, magnetic superatoms were also proposed.18 It was pointed out that a general way of achieving stable, endohedrally doped cluster is to take a coinage metal cage and dope it with an appropriate atom to get a closed electronic structure.19 It is well known that the ordering of the atomic orbitals in hydrogen-like atoms is (1s2 2s2 2p6)..., while as shown above, the shell orbitals follow the (1S2 1P6 1D10...) energetic ordering in the case of homogenous metal clusters (composed only from one type of element). The difference between the ordering of energy levels is due to the different nature of the confining potential. In the case of a homogenous cluster, it is approximately uniform as shown in Figure 2.1 a, whereas it is strongly attractive, due to the Coulomb interaction, in the hydrogen-like atoms. This shows that the ordering of the orbitals can be tuned by changing the confining potential in metal clusters.20 This is a point where the richness of chemistry comes into play. Adding an electronegative dopant to the center of a cluster makes the potential attractive, as showed in Figure 2.1-b along with the schematic representation in a spherical cluster.20a As shown, the electronegative dopant stabilizes the S and to a smaller extent the P orbitals, which have high concentration in the center as compared to the other orbitals such as 1D. Overall, when the dopant in the center becomes more electronegative than the other constituent of the cluster, the magic numbers of 10, 20, 40... are observed. A magic number of 10 was observed for example in Zn8, Na9Au, Na6Pb, K8Mg and Na8Zn.21 An electropositive dopant located at the center of a cluster induces an opposite effect: it destabilizes the centrally concentrated orbitals and the magic numbers of 2, 8, 18, 34... are observed.16 That is the case for example in Cu16Sc+ (see details in a following paragraph). A

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magic number of 10 was also observed in Au18Cu+, Au16Al+, Au16In+, Au16Sc+, Au16Y+22 and Au12W.23 It is interesting that in the latter, the d electrons of tungsten are also itinerant. Up to now we have considered only the (nearly) spherical cluster shapes. Indeed, when all the three P or all the five D orbitals are filled with 6 or 10 electrons, respectively, the electronic structure becomes closed, and a nearly spherical cluster shape is found. However a different picture emerges when partially occupied quasi-degenerate orbitals are involved, such as 5 electrons occupying the three P orbitals or 9 electrons being on the five D orbitals. Such electronic structure can lead to a Jahn-Teller distorted, lower symmetry structure. Overall when the P, D, F, ... orbitals are only partially occupied, a lower symmetry cluster shape is expected. This situation was analyzed more quantitatively by Clemenger,24 who applied the Nilsson model25 of atomic nuclei to metal clusters (called the Clemenger-Nilsson model, CNM). In this model, not only spherical but also elliptical cluster shapes are allowed, and from which two types can be distinguished (x, y and z are diameters): 

Prolate cluster shape: when x = y < z



Oblate cluster shape: when x = y > z

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Figure 2 Energies of the shell orbitals with respect to the distortion towards prolate or oblate cluster shapes. The optimized energies of the Clemenger-Nilsson model are marked with small points for the different electron numbers. Orbital correlations for cylindrical planar and linear clusters are also shown. In homogenous clusters, 18 is not a magic number and indicated only schematically in this figure. Similar to the spherical model, the electrons in CNM are non-interacting and are confined according to the cluster shape in a spherical or an elliptical box. The energy of the different levels with respect to the distortion of the cluster shape towards prolate or oblate form is shown qualitatively in Figure 2. A more quantitative diagram can be found in ref. 24. As expected, the oblate shape stabilizes the orbitals which have no xy nodal plane (like Px, Py, Dxy, Dx2-y2) and destabilizes the orbitals which have such plane (like Pz, Dz2). The prolate shape results in an opposite effect. This clearly indicates that the conditions for a closed electronic structure are different for elliptical clusters from those in spherical shape. Note that in the original CNM model18 corresponds only to a nearly spherical cluster. The CNM can go one step further. The energy of the system having a given number of electrons is optimized with respect to the diameters with the restriction that x = y so that an elliptical shape is obtained. By this way, the optimal cluster shape is obtained for a given number - -

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of electrons. These are depicted qualitatively in Figure 2. As expected, the optimal cluster shape is a perfect sphere in the cases of 2, 8, 18,... electrons. In other cases a more or less distorted oblate or prolate cluster shape is found as the distortion of the energy levels lowers the energy of certain orbitals. The presence of semi-occupied P, D, F,... orbitals results in a distortion from the cluster shape, which should be considered when determining the number of the electrons corresponding to a closed electronic structure. This is most apparent in a highly distorted oblate cluster shape. As shown in Figure 2, a highly distorted oblate cluster shape can be regarded as a two dimensional disc. Hence to determine the relevant qualitative electron configuration, the electrons in a two dimensional cylindrical box model can be used.26 In this case, the qualitative shape of the resulting orbitals is also displayed in Figure 2. Let note the noteworthy similarity to the benzene molecule. This is not surprising as in benzene, the π electrons are itinerant, and it has a nearly cylindrical shape (D6h symmetry). Of course, in benzene the nodal character of the orbitals has also to be taken into account. This shows that the Hückel model can be considered as a special case of the PSM and the 4n + 2 rule can also be used to explain the stability and to predict a possible aromaticity in metal clusters. The PSM was also applied in rectangular and triangular shape planar clusters.26 One of the advantages of the PSM is that it provides a consistent framework, which is capable to interpret the electronic structure, stability and aromaticity of different type of clusters present in different shapes. Also, the orbitals can be labeled consistently. In the 2D cylindrical clusters, the ordering of the orbitals remained almost unchanged as 1S {1Px 1Py} {1Dxy 1Dx2-y2}, where the curly bracket notes a (quasi)-degeneracy. In general, it does not mean that for example the 1Pz orbital does not play a role in planar clusters; in some cases, this orbital can also be fully occupied. This is the case of the aromatic Hg46- cluster which is present in certain alkaline-rich amalgams (Na3Hg2).27 It is noted that this cluster is aromatic due to the p AOs. The PSM can also be applied to describe the orbitals involved in this cluster. It has 1S2 {1Px 1Py} 1Dx2-y22 2S2 1Pz2 1Dxy2 electronic configuration. A lift of degeneracy of the in-plane D orbitals is a consequence of the D4h point group symmetry. A linear cluster can be regarded as resulting from a highly distorted prolate cluster shape. In this case, the energetic ordering of the orbitals is 1S 1Pz 1Dz2 ... as shown in Figure.2. As this

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class of compounds is less important with respect to the aromaticity, it will not be discussed in detail here. 3. Concept of Aromaticity and Metal Clusters The discovery of an aromatic character of compounds containing the Al42-

28

and

antiaromatic character of Al44- anion,29 has stimulated the studies of the aromaticity of many similar and other types of metal clusters.30. Several reviews are already available on the subject (ref. 31 presents the all-metal aromaticity and antiaromaticity, whereas ref. 32-33 discuss the aromaticity in transition metals). To the best of our knowledge, this chapter is however the first report discussing the aromaticity of metal clusters in connection with the PSM. Let us first summarize some relevant aspects. As mentioned in Introduction, although the aromaticity plays a central role in chemistry, this important concept lacks a precise and generally applicable definition.32,34 Benzene, the basic aromatic compound, was isolated by Faraday. Later, more importantly this compound was found to be considerably more stable than all the other chemical species with C6H6 formula. The chemical structure of benzene was dreamed by Kekule, as the story became familiar probably to all chemists. Later several “aromatic” compounds were found such as pyridine or pyrrole. Nowadays, the aromaticity of these compounds is not questioned despite their unpleasant odor, which can hardly be regarded as “aromatic” in the original sense. What is aromaticity then? As stated above, there is actually no unique, generally applicable definition for it. Instead, the coexistence of several properties of a given compound points toward its aromaticity. Several of these properties can be evaluated, which thus gives the possibility to quantify the degree of aromaticity. Based on the individual properties, many aromaticity indices can be defined. From the point of view of metal clusters, the most important properties of aromatic compounds can be summarized as follows. a) Stability. The meaning of stability was recently reviewed.35 This term refers to three different phenomena. First, aromatic compounds are stable thermodynamically, as they have exceptionally low energy among their isomers. It is well known that benzene is the most stable isomer among the C6H6 isomers, and so does triphosphabenzene on the C3P3H3 potential energy surface. Second, they are stable kinetically as they do not easily isomerize. Third, they are less reactive - -

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than their isomers and their preferred reactions preserve the aromatic moiety of the compounds. Indeed, benzene and prefer the electrophylic substitution reactions. For metal clusters, stability usually means that the given cluster size has higher abundance in the mass spectra than the other sizes. Hence it refers to the thermodynamic stabilitystability, which can be quantified by different approaches. One of the most generally applicable method is the computation of the energy change in an isodesmic reaction36 (or a bond separation reaction). This is a working reaction where the number of the chemical bonds of each type (like carboncarbon formal single and double bonds) is the same on both sides of the reaction. However the presumably aromatic compound stands only in the left hand side. The example below shows a classical isodesmic reaction for benzene, and an analogous reaction for the scandium doped copper cluster (Figure 3)

Figure 3. Isodesmic reactions of benzene and Sc-doped copper cluster. Construction of such a reaction (and more refined versions) can relatively easily be done for organic compounds, but rather difficult for metal clusters. This is because there are no formal single or multiple bonds, at least in the clusters where the valence electrons of the constituting atoms are itinerant. Thermodynamic stability of clusters is usually quantified using the second difference of the energy (ΔΔE) with respect to the size of the cluster: ΔΔE = 2*EN – EN+1 - EN-1 where EX (X = N, N+1, N-1) is the energy of the most stable cluster containing X atoms. This quantity was found to correlate with the intensity in the mass spectrum of the relevant clusters. One can recognize however that this is also the energy of the following reaction:

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(X+1) + (X-1) → 2 (X) where X stands for the most stable cluster of X atoms. The reaction is analogous to the isodesmic reaction described above. Depending on the chemical composition of the cluster and its geometrical form, the analogy is not always satisfied. According to the definition of an isodesmic reaction, the above equation can be isodesmic when the (N+1), (N-1) and (N) clusters have similar structure. An example is shown in Figure 3 for the Cu16Sc+ cluster, which can be considered aromatic as it has a closed 1S2 1P6 1D10 electronic configuration. On the other hand, the two compounds on the righthand side have no such closed electronic structure. Stability of an aromatic compound can also be probed by its large HOMO-LUMO gap as compared to corresponding non-aromatic species. b) Closed electronic structure of the conjugated electrons. It is well known that the π electrons of benzene are conjugated, and their chemical bonding cannot be represented by a single Lewis resonance structure. Due to the conjugation, the π electrons are itinerant and thus plays a key role in the aromaticity of this compound,. Similarly, the itinerant electrons are responsible the aromaticity in metal clusters. According to the 4n + 2 rule, the criterion for aromatic conjugation is that the number of itinerant electrons equals to 4n + 2, where n is a nonnegative integer. This rule is valid not only in planar annulenes but also in σ-aromatic systems with nearly cylindrical shape.37

Figure 4. Selected resonance structures and chemical bonding in benzene and Au5Zn+.

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As we have seen, an analogous model can also be formulated for metal clusters. Similarly to the Hückel model of annulenes, an electronic configuration is closed when the number of itinerant electrons equals to 4n + 2. In Au5Zn+, the number of itinerant electrons amounts to 6 (one electron from each gold atom plus two electrons from the central zinc and less one because of the positive charge).38 The pentagonal ring structure Au5Zn+ with central zinc atom was found to be aromatic. According to the model, we have six electrons participating in the chemical bonds, but distributed in ten bonded pairs of atoms (five gold-gold plus five gold-zinc bonds); thus the average bond order is 6/10 = 0.6. As seen in Figure 4, no single Lewis-structure can be drawn for such a situation. As each chemical bond in a Lewis structure formally involves 2 electrons, only three bonds can be drawn in each resonance structure. Thus, the electronic structure of the doped gold cluster can be described by many resonance structures. It was observed that the electronic structure of different alkaline (Li3+, Li42+) and alkaline-earth (Mg42+) metal clusters can be described by comparable resonance structures39 (the conjugated electronic structure being marked by a circle). An analysis of the Wiberg indices (computed at the BP86/SDD level of density functional theory) shows that the actual electronic structure of Au5Zn+ is even more complicated than that represented in Figure 4. There is a weak chemical bond (with order of 0.27) between the central zinc and each gold atom. Also the same bond order is found between the adjacent gold atoms. In addition, there are weak chemical bonds (with order of 0.18) between the non-neighbor gold atoms. In fact, this corresponds to a very high degree of conjugation. The total valence of Zn is 1.37, while it is 1.18 for Au, which is in line with the assumption of PSM that each gold atom contributes one electron to the shell orbitals. Let us note that Au5Zn+ has a more stable triangular shape isomer, which is also aromatic. In the isoelectronic species (with respect to the shell model), Au6 shares some common properties with Au5Zn+ such as a triangular form ground state and a closed electronic configuration.40 c) Equalized bond lengths and bond orders. Aromatic compounds in general have equalized bond lengths and bond orders, which often leads to a high symmetry form. This term however has to be used with caution as this does not imply that an aromatic compound should have the highest attainable symmetry. Benzene is a planar compound with a D6h point group, and not an octahedron which could also be possible. Obviously the planarity of the molecular backbone gives a favored overlap in the π system. Moreover the compound in an octahedral form corresponds to a 1S21P4 electronic configuration with semi-occupied degenerate Px, Py and Pz - -

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orbitals. Thus it is expected to undergo a Jahn-Teller distortion. Closed electronic structure and consequently aromaticity can only be attained in the planar D6h arrangement. In metal clusters the symmetry is not always perfect. Instead of high point group symmetry we talk about nearlyspherical or nearly-cylindrical shape. The ring isomer of Au5Zn+ is not fully planar, which is probably due to the fact that the Au5 ring is too small to accommodate a zinc atom. c) Magnetic properties. As aromatic compounds have itinerant electrons, they exhibit characteristic response to external magnetic field. If a planar aromatic compound is placed in an external magnetic field perpendicular to the ring plane, the itinerant electrons begin to rotate and the resulting ring currents are diatropic. The ring current can be computed and visualized explicitly, or measured indirectly. One of the widely applied criteria is based on the computation of the nucleus independent chemical shift (NICS) at the center of the ring.41 NICS is the negative of the trace of the magnetic shielding tensor at the given point of the space. Negative NICS value indicates a diatropic ring current, hence an aromatic character. NICS value close to zero points out non-aromatic compound, while it is positive in antiaromatic cases. This index gives a possibility to quantify the aromaticity: the more negative the NICS the more aromatic the compound. Moreover it can conveniently be computed as a regular NMR property. However this index has to be applied with much care, as effects other than the ring currents can also influence it. For example in a three-membered rings (like the cyclopropenyl cation), NICS is highly perturbed by the σ system (which has nothing to do with the molecular aromaticity in this case42). A straightforward remedy is to put the reference dummy atom (or physically speaking the reference neutron) above the ring (and consequently away from the σ system) by one or two angstroms, which are denoted by NICS(1)43 and NICS(2), respectively. Another refinement for the planar rings is that only the out-of-plane tensor component holds information about the ring currents. This tensor element at the center of the ring is called NICSzz.44 As the shielding tensor is the sum of the contributions from molecular orbitals, several dissected NICSs can be defined. Thus, the CMO-NICS is computed for each canonical molecular orbitals45 (or Kohn-Sham orbitals), and the LMO-NICS is computed using different MO localization schemes. The total NICS value is the sum of different dissected NICSs. This gives rise to a possibility to determine the contributions of different orbitals. In the triangular isomer of Au5Zn+, NICS = -18.3 and NICS(1) = -8.0 at the center of the rings.38 The same quantities are - -

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9.6 and -11.3 in the case of benzene using the same level of theory, respectively. In the fivemembered ring Au5Zn+, NICS(1) = -14.2. Although this value is strongly affected by the central zinc atom, it demonstrates the aromaticity of Au5Zn+. Also, it was showed recently that the magnetic properties confirm that the electron delocalization plays an important role in the stabilization of the Si5Li7+ molecular star.46 We have mentioned above the π-aromaticity in planar systems, which is the most studied and well understood class of this topic. Other types of aromaticity also exist, which play an important role in metal clusters.45,47 The σ-aromaticity implies that the orbitals responsible for it have no node in the ring plane. This was first observed in the 1,3,5-tridehydrobenzene cation (Figure 5),45 which is formally derived from benzene by removing three hydrogen atoms and one electron. According to the previous discussion, it is understood that the π-system (and hence the π-aromaticity) remains intact, and only the σ-electron system has changed. The three dangling bonds are now occupied by 2 electrons. The Hückel model indicates that this system has a closed electronic configuration. It also has high symmetry and the desired magnetic properties; hence in parallel with the π aromaticity, an σ aromaticity exists in the mean time. Accordingly, this cation is qualified as doubly aromatic. Analogously, the cation Li3+ has two s-electrons in a D3h geometry and is also σ aromatic.48 The isoelectronic (relative to the shell model) Cu3+ cation is also σ aromatic due to its two valence s orbitals.49 σ-aromaticity due to the s orbitals was also observed in the cyclo-MnHn (M = Cu, Ag, Au) clusters.32,50 Recently, a triple aromaticity (σ, π and δ) was postulated for Hf351 (Figure 5), and a π and δ double aromaticity was suggested for Ta3O3-.52 Now a question is which electrons cause the aromaticity. In other words, which atomic orbitals participating in the molecular orbitals are responsible for the aromatic behavior. While in benzene they are obviously the π orbitals, the question turns out to be more difficult for metal clusters. For example, the Mo3O92- and W3O92- clusters are probed to be σ aromatic due to the d atomic orbitals.53 Three dimensional d aromaticity was suggested in the case of Td and pseudoOh symmetry M4Li4 and M6Li2 (M = Cu, Ag and also Au in the latter case) clusters.54 The different types of aromaticity (σ, π and δ), and the originating orbitals are reviewed in ref. 32.

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Figure 5. 1,3,5-tridehydrobenzene cation and Hf3. Several resonance structures exist for the cation, which makes the delocalization of the π system and also the σ system possible. This results in a double aromatic character.

Some examples for 3D spherical aromaticity can be found in different fullerenes.Error! Bookmark not defined.

It is well known that the π orbitals of the carbon atoms in fullerenes are

perpendicular to the surface and these electrons are itinerant, similarly to the benzene. Hence the electronic structure of these compounds can be modeled using a Hückel-type model, and the condition to obtain a closed electronic structure and aromaticity can also be formulated as previously done by Hirsch et al. Their rule is valid for icosahedral fullerenes. We will see that spherical aromaticity is a more common feature of metal clusters. 4. Examples of Aromatic Metal Clusters a) Planar Clusters. While pure gold clusters Aun are planar up to 12 constituent atoms (n = 12),55 copper clusters have a clear-cut preference for 3D structures.56 The tendency is more pronounced in the scandium doped derivatives (CuNSc) as scandium prefers a high coordination number. The inherent phenomenon is analyzed hereafter for Cu16Sc+.57 Therefore it comes as a surprise that the planar Cu7Sc is about 0.1 eV more stable than the 3D isomers. The geometry of the ground state Cu7Sc cluster is shown in Figure 6. It consists of a sevenmembered cycle of copper atoms with a central scandium. The overall form has a perfect D7h point group symmetry. As seen in Table 1, the 3s electrons of copper and scandium and the 3d electron of scandium are itinerant. This yields together 10 itinerant electrons, which formally satisfy the 4n + 2 rule. The shape of this cluster is nearly cylindrical, hence, from the PSM view, a 1S2{1Px2 1Py2}{1Dxy2 1Dx2-y22} electronic configuration is expected (the curly brackets representing degeneracy).

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17

The densities of states (DOS) can be regarded as the energy spectrum of the molecular orbitals broadened by Gaussians with a norm of 2 (as each orbital being occupied by 2 electrons). The partial density of states (pDOS) are computed only from certain atomic orbitals, and this can be used to visualize the composition of the molecular orbitals involved. The pDOS computed from the valence s AOs of Cu7Sc are illustrated in Figure 6.

(a)

(b)

(c)

Figure 6. (a) geometry of Cu7Sc. (b) Total and partial (computed from the s atomic orbitals) densities of state of Cu7Sc. The assigned shell orbitals are also indicated on this figure. (c) Electron localizability indicator isosurfaces of Cu7Sc.

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The densities of states can also be used as an aid to identify the orbitals corresponding to the PSM. In Cu, the 4s valence electron is expected to be itinerant, thus in Cu7Sc they occupy the MOs where the pDOS from valence s orbitals have high contributions to the total DOS. The assigned shell orbitals are in perfect agreement with the expectations from PSM, as depicted in Figure 6. The electronic structure of Cu7Sc is further analyzed using the electron localizability indicator (ELI).58 This is similar to the widely used electron localization function (ELF) approach, but the ELI can be exactly decomposed into contributions from MOs (giving the partial ELI, pELI). Based on the ELF or ELI, the real space can be partitioned into domains which hold chemical informations, namely, cores, chemical bonds and lone pairs. The ELI = 1.30 isosurface of Cu7Sc is shown in Figure 6c. The blue regions represent the cores of the copper atoms, whereas the green region indicates the scandium core. The red localization domains are caused by the central scandium atom, as it cannot be observed in the isoelectronic species Cu73-, which has the same number of valence electrons as Cu7Sc. The yellow domains having a banana shape, represent the chemical bonds. According to the pELI computed from the orbitals shown in Figure 6, the shell orbitals are responsible for these localization domains and thereby responsible for the chemical bonding in this cluster. Cu7Sc is in many aspects analogous to benzene: thermodynamic stability compared to its isomers, high symmetry, planar geometry, and closed electronic configuration. These properties indicate an aromatic behavior. To analyze it further we also computed the NICSs. Due to the central scandium, NICS is not applicable in its original definition, and also NICS(1) is strongly perturbed. Therefore we have computed the NICSs in a line starting from the center of the ring towards a copper-copper midpoint. Only the zz tensor component is plotted as it holds information on aromaticity. These distributions and the contributions from different orbitals are shown in Figure 7.

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19

NICS(1)zz 100.00 Total Valence

80.00

Valence sigma Valence pi

60.00

Shell model

40.00

Midpoint of Cu-Cu

20.00

0.00 0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

-20.00

-40.00

-60.00

-80.00 d(Å)

NICS(2)zz 15.00 Total Valence

10.00

Valence sigma Valence pi

5.00

0.00 0.00

Shell Model

Midpoint of Cu-Cu 0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

-5.00

-10.00

-15.00

-20.00

-25.00 d(Å)

Figure 7. The NICS(1)zz and NICS(2)zz distributions from the center of the ring towards the medium point of a Cu-Cu bond in Cu7Sc.

While the total NICS(1)zz is positive only near to the central scandium, which is due to the shielding effect of its core electrons, the NICS(2)zz has a relatively large negative value. This finding, together with the previously mentioned properties, suggests that Cu7Sc has an aromatic character. We have computed the contributions of the σ- and π- orbitals (the former having no nodal plane, whereas the latter having one nodal surface in the ring plane) and also from the shell orbitals. As pointed out by the NICS distributions, the π orbitals are anti-aromatic, while the σ orbitals have a highly aromatic character. Overall this is a σ-aromatic cluster, and its shell - -

20

orbitals are responsible for its aromaticity (the 1S orbital being mainly responsible for the aromaticity with its high CMO-NICS(1) value of -10.9 ppm). Experimental and theoretical studies have shown that the Au6 ring can perfectly host a dopant transition metal atom in its neutral and anionic state.59 The Au6Y- anionic cluster is isoelectronic, with respect to valence electrons, with Cu7Sc, and hence the same electronic configuration of 1S2{1Px2, 1Py2}{1Dxy2 1Dx2-y22} is observed.60 Au6Y- has a perfect D6h point group, and a detailed investigation showed that it is indeed aromatic. On the other hand, its neutral state can be described with the 1S2{1Px2, 1Py2}{1Dxy2 1Dx2-y21} configuration, which results in a Jahn-Teller distortion due to the partially occupied quasi-degenerate orbitals. A NICS analysis emphasized that the neutral state Au6Y is antiaromatic. The Jahn-Teller distortion is manifested in a strongly fluxional behavior, which is in line with the observed broadened infrared absorption bands. The Au6Sc cluster behaves similarly to the isovalent Au6Y. The role of the first-row transition metal dopant atom on the spin state and the symmetry of the neutral clusters was systematically investigated.61 The D6h point group symmetry was observed in Au6Mn, and as a consequence this cluster could be considered as aromatic. However in contrast to the previously reported cases, the latter cluster is magnetic with a high spin quartet ground state. For a description of aromaticity of this cluster, we need to go beyond the PSM and in addition the effect of the d atomic orbitals of manganese has to be taken into account.62 The qualitative model on Figure 8 describes the observed properties of this Mn-doped cluster.

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Figure 8. Schematic MO diagram of Au6Mn and its formation from Au6 and Mn. The z axis is perpendicular to the plane of the figure.

The model considered takes into account the mixing of the shell orbitals of the Au6 ring and the atomic orbitals of the central manganese. The shell orbitals of the Au6 ring are the same as discussed earlier in the nearly-cylindrical clusters. When the manganese atom is inserted into the ring, its originally degenerate d atomic orbitals are split according to the new symmetry. This is analogous to a crystal field or ligand field splitting. In the final step, the split AOs of manganese combine with the shell orbitals. The combination of the 1S orbital of Au6 and the s(Mn) AOs gives rise to the 1S and 2S shell orbitals by positive and negative linear combinations. It is noted that in the separate Au6 ring and manganese atom, two electrons occupy the 1S and the s(Mn) orbitals. However in the Au6Mn cluster, the 2S orbital is unoccupied, and instead the electrons occupy the d orbitals. There is considerable mixtures between the d(Mn),xy - Dxy and the d(Mn),x2-y2 and Dx2-y2 orbitals, respectively. Thus the electronic configuration of Au6Mn can be described as 1S2 {1Px2 1Py2 }{1Dxy2 1Dx2-y22}{dMn,xz1 dMn,yz1 dMn,z21}. This explains its observed quartet ground state. As its shell orbitals are occupied by 10 electrons, the cluster is expected to be stable. However the observed magnetic moment of the central manganese is larger than - -

22

expected from the previous model, and a substantial amount of down-spin electrons occupy the Au6 ring. This shows that in order to describe properly its electronic structure, two configurations have to be taken into account as shown in Scheme 1.

Scheme 1

Structure a in Scheme 1 is described above, while structure b is derived from a by moving two up-spin electrons from the 1Dxy and 1Dx2-y2 to the corresponding d orbitals of Mn. The electronic structure of Au6Mn is thus a mixture of the two above configurations. This is supported by the fact that the computed magnetic moment of Mn lies between that of the configurations a and b. Also, the participation of configuration b helps explaining the down-spin population of the Au6 ring. In the latter configuration, electrons are removed from the shell orbitals, and this leads to a decrease in aromaticity. In configuration a 10 electrons are on the shell orbitals (denoted by capital letters in Scheme 1; of course, they also have smaller contributions from the d orbitals of manganese), hence the Hückel’s 4n + 2 rule is satisfied. The electronic structure is analogous to that of Cu7Sc. This again suggests an aromatic behavior. However this closed electronic structure is disrupted in configuration b as 8 electrons occupy the shell orbitals, which satisfy the 4n rule. It is well - -

23

known that annulenes where the π electrons satisfy the 4n rule are antiaromatic in their singlet ground state. However, it corresponds to an aromatic behavior in the corresponding lowest triplet state.63 In general, annulenes can be regarded as aromatic in the lowest-lying singlet, quintet,… total spin state when the number of π electrons is equal to 4n + 2, and in the case of lowest-lying triplet, septet… states in the case of 4n π electrons. This latter is the case in configuration b, hence it also induces an aromatic character. This shows that Au6Mn represents an outstandingly interesting case, where configuration a satisfies the 4n + 2 rule, and configuration b satisfies the 4n rule in the same compound, and both give rise to a strong aromatic character. This peculiar situation would not have been possible without an inclusion of the magnetic dopant atom. However a legitimate question arises here. One of the most important characteristics of aromatic compounds is the presence of diatropic ring currents. So how does the presence of the magnetic dopant atom (and it localized magnetic moment) influence the ring currents and consequently the aromaticity? To some extent, this is analogous to that of the 1,3,5tridehydrobenzene (TDB). TDB has in its lowest doublet ground state 6 π electrons, while the unpaired electrons are localized in the sigma system.64 In this case the unpaired electrons have no influence on the aromaticity, which is comparable to that of benzene. We have thus seen the behavior of doped coinage metal rings, where the shell orbitals are responsible for their aromatic character. Nevertheless, this is not always the case, and the Cu4Sc+ cluster provides us with a different situation.65 The latter cation attracts interest as it has a pyramid shape with a planar Cu4 ring, while it is non-planar in the neutral counterpart. It contains six electrons in the shell orbitals, and therefore the question of its aromaticity arises. The NICS value is -22.1 ppm at the center of the Cu4 ring and -22.3 ppm at the center of the cage, which further supports for the presence of aromaticity. The corresponding shell orbitals can be assigned as described above and are depicted in Figure 9.

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24

LUMO 1Pz

HOMO, 1Px

HOMO, 1Py

6.53ppm, 6.39ppm (1px)

6.53ppm, 6.39ppm (1py)

1Px'

1Py'

-3.68ppm, -2.45ppm

-3.68ppm, -2.45ppm

1S -8.82ppm, -9.77ppm

Figure 9. Shell orbitals of Cu4Sc+, and their CMO-NICS value in the center of the ring (first number) and in the center of the cage (second number).

Figure 9 clearly shows that the cation cluster has the expected 1S2{1Px2 1Py2} configuration, while the 1Pz level is unoccupied. The CMO-NICS values demonstrate that the 1S level has an aromatic contribution, but the 1Px and 1Py levels induce an antiaromatic character. Overall the shell orbitals are non-aromatic, and the global aromaticity is due to the orbitals bearing high - -

25

contributions from the d AOs (Figure 9). As these orbitals have similar shape to that of the P shell orbitals, these are labelled by 1Px' and 1Py', respectively. They provide some support for an aromatic behavior. Thus the new finding from the CMO-NICS analysis is that Cu4Sc+ is aromatic due to the d AOs, and not due to the shell orbitals. The situation regarding the d-aromaticity of Cu4Li2,66 which according to the PSM has also six itinerant electrons, is less clear-cut. Analysis if the ring currents computed for Cu4Li267 showed a strong diatropic contribution also from the HOMO, which is in contrast to the results derived from the CMO-NICS, supports that this cluster is aromatic due to the shell orbitals. Nevertheless, irrespective of its origin, the electron count based on the shell model correctly predicts the aromatic character of these compounds. Let us now consider Al42- which was well analyzed.28 The electronic structure of this dianion can also be interpreted in terms of PSM. Each Al has three itinerant electrons (Table 1), and the excess negative charge is also expected to be itinerant, thus there are overall 14 itinerant electrons are in this cluster. As the ring is planar, one would expect that only those shell orbitals are occupied, which have no nodal surface in the ring plane. This is however not true, because due to the excess negative charges the Pz orbital is also occupied. This yields the 1S2 1Px2 1Py2 1Dx2-y22 1Dxy2 2S2 1Pz2 configuration. As Pz has a nodal surface in the ring plane, it overlaps poorly with the other shell orbitals, which minimizes the Coulomb repulsion arising the excess negative charges. Pz corresponds to a perfectly delocalized π orbital, while the other shell orbitals can also give rise to the aromatic behavior. b) Three Dimensional Clusters. We now examine the concept of spherical aromaticity with the example of Cu16Sc+.57 This cluster exhibits an exceptionally large abundance in the mass spectrum, which indicates a very high stability as compared to the neighboring clusters. According to PSM, there are 18 itinerant electrons (one from each Cu plus two from Sc, taking the positive charge into account). For a nearly spherical cluster with an electropositive central atom, it corresponds to a closed electronic structure. The Pauling electronegativity of Cu is 1.90, while it is 1.36 for Sc, which fulfills the above requirement. Quantum chemical computations showed that the most stable cluster has a truncated nearly spherical tetrahedron shape (called a Frank-Kasper tetrahedron, as seen in Figure 10), in which Sc occupies a central position. The analysis of the total and partial densities of states (cf. Figure 10) confirm that this cluster can be - -

26

described by the 1S2 1P6 1D10 configuration, which in a nearly spherical cluster corresponds to a closed electronic structure. This explains its observed high stability. The DOS plots reveal that the d atomic orbitals of the central Sc also participate to the shell orbitals. The electron donation to the formally empty d AOs of Sc is also a stabilization effect in this cluster. The large HOMOLUMO gap also indicates the stability of this cluster.

Figure 10. Geometry and total and partial (computed from the different atomic orbitals) densities of states of the Cu16Sc+ cluster in its ground state. The exceptional stability, high symmetry (consequently the balanced bond lengths and bond orders) and closed electronic structure of Cu16Sc+ naturally pose the question of its aromaticity. The central Sc atom makes it difficult to compute the NICS, while it is complicated to analyze a NICS distribution due to its 3D shape. Therefore the aromaticity was investigated using the Cu163- cluster as a model system. The trianion has the same number of valence electrons as Cu16Sc+, and consequently it exhibits the same 1S2 1P6 1D10 configuration. The NICS is calculated to be -65.5 ppm at the center of the cage, which thus supports an aromatic character. This suggests that the electronically equivalent Cu16Sc+ is similarly aromatic. Both clusters have spherical aromaticity due to their dimensional shape.

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27

The smaller cluster Cu6Sc+ has also a higher abundance than the surrounding clusters in mass spectrometry studies.57 The PSM can also rationalize this finding. Each Cu contributes one electron, while the scandium ion contributes two electrons to the shells.68 Having 8 itinerant electrons, the cluster cation has, according to PSM, a closed {1S2 1P6} electronic structure. The optimized geometries of Cu6Sc+ clusters are shown in Figure 11. The most stable cluster has a C3v symmetry, which leads to an approximately oblate shape. We analyze the electronic structure and stability of Cu6Sc+-I. Due to the reduced symmetry, the degeneracy of the P orbitals is split and the electronic structure corresponds to a 1S2{1Px2 1Py2}1Pz2 configuration, as found from the analysis of the DOS and pDOS plots (Figure 12).

Figure 11. Optimized geometries of the Cu6Sc+ isomers.

As expected from the PSM (see CNM model in Figure 2), due to the oblate shape, the Px and Py orbitals remain degenerate and lower in energy than the Pz orbital. In view of the closed electronic structure, stability and relatively high symmetry, an aromatic behavior is expected. Nevertheless the loss of the degeneracy of the P orbitals tends to decrease the degree of aromaticity. The NICS value is -28.5 ppm at the central tetrahedron, while it amounts to -25.3 ppm at the center of the capping tetrahedrons. Using the total and partial electron localizability indicator (computed from the shell orbitals), it is also confirmed that the shell orbitals are responsible for the chemical bonding in Cu6Sc+.

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28

Figure 12. Total and partial (computed from the valence s orbitals) density of states of Cu6Sc+. The shell orbitals are assigned on the figure. Cu5Sc is isoelectronic to Cu6Sc+ with respect to the number of valence electrons, hence, according to the PSM, a similar aromatic behavior can be expected for the neutral hexaatomic cluster. The optimized Cu5Sc isomers are depicted in Figure 13. The lowest-energy structure is calculated to have a C4v point group, and an approximately prolate shape as the electron density is to some extent distorted. However the distortion from an approximately spherical shape is smaller than in the case of the Cu6Sc+ cluster. In Cu5Sc, the quasi-degenerate P orbitals remain, in clear contrast to a difference of 0.99 eV between the {Px Py} and Pz orbitals in Cu6Sc+. The NICS is -49.9 ppm in the center of the cluster, which is comparable to the value of -65.5 ppm of the Cu163- cluster. In agreement with the PSM expectation, Cu5Sc has also an aromatic character.

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29

Figure 13. Low-energy Cu5Sc isomers. The superscript t indicates a triplet ground state.

In the last example, we consider the doubly charged tetramercury cluster Hg42+ in both cyclic and linear shapes.69 The computations showed that the cyclic isomer of Hg42+ has a perfect D4h symmetry (Figure 14). Each mercury atom contributes two electrons to the shell orbitals (Table 1), thus there are together six itinerant electrons in this dication. Its molecular orbitals are similar to those of benzene, except of course for the different nodal properties in the plane of the ring. This already suggests an aromatic behavior, which is indeed confirmed by the NICS value of -14.1 ppm. Similar aromatic character was observed in the PSM isoelectronic species such as M32- (M = Mg,70 Zn, Cd, Hg71 and their sodium salts), and Mg42+ 39. However, it turns out that the linear isomer of Hg42+ is calculated to be 111 kJ/mol lower in energy than the cyclic form. Thus the criterion of thermodynamic stability seems not to be satisfied in this case. The reason for such an unusual behavior is that the stabilization effect by aromaticity is overcompensated by the inherent ring strain. Also, the average bond distance is smaller in the ring than that of the linear isomer. Hence the repulsion of the positive charges is larger in the ring isomer, which is also a significant destabilization factor. The D6h cyclic isomer of Hg62+ is also found to be aromatic, but its linear isomer is again calculated to be lower in energy by 102 kJ/mol, due to similar reasons as in Hg42+. It is interesting to note that Hg42+ is PSM isoelectronic with Hg2Au2,72 hence a comparable behavior can be expected. The cyclic isomer of Hg2Au2 is also aromatic (NICS is -11.4 ppm), but it lies 48 kJ/mol higher in energy than the corresponding linear cluster. Here the energy difference is mainly due to the absence of ring strain in the linear shape. The last example on Hg clusters shows that properties of chemical compounds are affected not only by the aromaticity, but also by effects like the charge distribution or the ring strain. These need also to be taken into account to understand the stability of a cluster. In the same vein, although Li3+ is σ-aromatic in its D3h cyclic form and contains two itinerant electrons, the ring currents were not observed.73

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30

LUMO +1 (B2G)

LUMO (ΣU)

LUMO (B1G)

HOMO (ΣG)

HOMO (EU)

HOMO -1 (ΣU)

HOMO -1 (A1G) a

HOMO -2 (ΣG) b

Figure 14. Geometry and the HOMOs and LUMOs of Hg42+.

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31

5. Concluding Remarks The connection between the phenomenological shell model (PSM) and aromaticity of metal clusters is presented in this chapter. This model allows us to probe the aromaticity of metal clusters, and also organic compounds, in a general framework and context. The 4n + 2 rule of planar rings is shown to be a special case of this model. It can be applied to the cases of planar, spherical and distorted (oblate and prolate) clusters. The different criteria (stability, symmetry, magnetic properties, electronic structure) are considered with the examples of different scandium-doped copper clusters CuNSc and CuNSc+. Overall, a closed electronic structure according to the PSM often results in an aromatic behavior. Acknowledgements. The Leuven group is indebted to the KULeuven Research Council for continuing support (GOA, IDO and IUAP programs). TH thanks the INPAC for a partial doctoral scholarship. We are grateful to Dr. Ewald Janssens, Dr. Ling Lin and Vu Thi Ngan for illuminating discussion.

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