# An Investigation of Electronic Structure and Aromaticity in Medium-Sized Nanoclusters of Gold-Doped Germanium

Authors: Xiao-Jun Li; Hong-Jiang Ren; Li-Ming Yang
Journal: Journal of Nanomaterials
 (2012)

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## Abstract

The electronic property and aromaticity of endohedrally dopedAuGe10- and AuGe12- clusters are investigated using the density-functional theory (DFT) within the hybrid B3LYP method. The calculated results reveal that the two clusters have high thermodynamic stability reflected by reaction energy. At the same time, it could be hoped that their high stability may arise from the closed-shell spherical aromaticity with eight π-electrons satisfying the 2(Nn+1)2 counting rule with Nπ=1. A popular nucleus-independent chemical shifts (NICSs) calculation on basis of magnetic shieldings is also performed to confirm the aromaticity of the three-dimensional nanoclusters with largely negative NICS values. In addition, the electronic features and chemical bonding of the two clusters are analyzed with the help of the density of states (DOS) and electron localization function (ELF), and the majority of Ge–Ge bonds on the cage show more covalent characters.


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## Body

## 1. Introduction

Semiconductor clusters, especially germanium, have been extensively investigated both experimentally and theoretically due to their potential applications in the microelectronics industry [1–6] and are of considerable scientific interest to explore the chemical structure and bonding as well as their thermodynamic stability and photoelectric effect. Unlike silicon, the effective masses of electron and hole in germanium are light; it can generate high carrier mobility and makes germanium attractive for high-performance field-effect logic transistors [7]. However, pure germanium clusters with medium or large size are chemically unstable and quite reactive because of the existence of abundant dangling bonds [5]. Interestingly, the encapsulation of different transition-metal (TM) atoms [8–10] in the medium-sized clusters can saturate the dangling bonds on the germanium cage surface and form sp3 hybridization, and thereafter stabilize the germanium cages. Consequently, the metal-encapsulated germanium clusters exhibit many fascinating features such as nanostructures, for example, thermodynamic stability [11, 12], magnetic and superconducting behaviors [8, 13, 14], photoelectric effect [15, 16], and reduced energy gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) [9, 17, 18]. Some metal-encapsulated Ge clusters were also proposed as novel cluster-assembled materials due to the small HOMO-LUMO gap within about 2 eV, for example, AuGen (n=2–13) [18] and TM@Ge12 (TM = Zn, W, and Os) [19, 20]. In an experimental study, the bimetallic CoGe10- cluster in the mass spectrum was measured using laser vaporization technique [21], and its molecular structure was determined by the density-functional theory (DFT) calculations [22].To our knowledge, there have been a number of interesting findings on the calculation of metal-encapsulated Ge clusters, for example, NiGen (n=1–13) [23], CrGen (n=1–13) [24], WGen (n=1–17) [25], TMGen (TM = Ti, Zr, Hf and Cu; n=1–20) [26, 27], and so on. A few outstanding predictions are medium-sized fullerene-like with 10 or 12 atoms. Kumar et al. [19, 28] reported a series of systematic investigations of TM@Xn (X = Si, Ge, Sn, and Pb; TM = Ni, Pd, Pt, Zn, and Cd; n=10 and 12) clusters using an ab initio ultrasoft pseudopotential method and generalized gradient approximation, and these clusters can be further stabilized by doping of an Ni or Pt atom, leading to some of the smallest metal-encapsulated clusters. King and coworkers [29, 30] explored the possible structures of M@Ge10zsystems (M = Co and Be; z=-5 to +2) at the DFT-B3LYP/6-31G(d) level of theory and found that a singlet D5h pentagonal prismatic structure is the global minimum for the trianionic Co@Ge103- cluster, which is in accordance with the experimental result. Tang et al. [31] investigated the structure, stability, and optical property of TM@Ge12 (TM = Sc−Ni) using the relativistic all-electron density functional theory. It was found that all the TM@Ge12 clusters are perhaps partial metallic, and the ground-state geometry of them is probably pseudoicosahedron. Furthermore, eight π-electron natures make closed-shell spherical systems aromatic according to the 2(Nπ+1)2[Nπ=1] rule proposed by Hirsch et al. [32]. In 2005, Chen and coworkers [33] also reported the nucleus-independent chemical shifts (NICSs) approach as an aromaticity criterion based on magnetic properties, which have been applied to characterize the all-metal aromatic/antiaromatic clusters [34].In this paper, we report an investigation of electronic structure and aromaticity of theAuGe10- and AuGe12- nanoclusters using the hybrid DFT-B3LYP functional in conjunction with LanL2DZ basis set. The choice of endohedral gold atom is due to the fact that the bimetallic gold clusters have some fascinating physical and chemical properties and were experimentally observed using photoelectron spectroscopy [16, 35–37].


## 2. Theoretical Methods

The molecular structure, electronic property, and spherical aromaticity ofAuGe10- and AuGe12- clusters were carried out using the hybrid DFT-B3LYP functional [38, 39] implemented in the Gaussian 09 package [40]. The standard double-ξ LanL2DZ basis set [41–43], which provides effective core potentials (ECPs) to consider the relativistic effects of transition metal atoms, was utilized here. Our previous calculations [18] also confirmed the credibility of the theoretical level, and the deviations within 1−6% are acceptable. To search for the most stable structures, a great number of structural isomers were considered for each size, on basis of a global search published previously [18, 44], and the local minima of all the stationary point geometries were determined by vibrational frequency analysis. The density of states (DOS) of the most stable AuGe10- and AuGe12- clusters was constructed using GaussSum 2.2 program [45] with the full width at half maximum (FWHM) of 0.3 eV, and the molecular orbitals were plotted with the isodensity surfaces (0.02 e/a.u.3). In addition, the nucleus-independent chemical shifts (NICS) were calculated by gauge-independent atomic orbital (GIAO) method [46, 47] at the B3LYP/LanL2DZ level of theory. In order to obtain more accurate NICS values, the correlation-consistent cc-pVTZ basis set [48] was also used for Ge atom, whereas the energy-adjusted small-core (19 valence electrons) pseudopotentials (PPs) [49] were used for Au atom, labeled as cc-pVTZ-PP, and the single-point GIAO calculations at the B3LYP/cc-pVTZ-PP level of theory were performed based on the B3LYP/LanL2DZ-optimized geometries.


## 3. Results and Discussion

### 3.1. Geometry and Stability

Using the computation scheme described above, we have extensively explored a number of initial isomers formed from taking the previously reported structures of different TMGe10 systems, or doping one Au dopant at all possible positions of the lower-lying Ge10 isomers, or replacing one Ge atom of the lower-lying Ge11 isomers with an Au atom to form new isomers. As a result, the most stable AuGe10- cluster is found to be an endohedrally Au-doped pentagonal prism (see Figure 1(a)). It is worth mentioning that, when an electron is added to the lowest-energy neutral AuGe10 cluster in C2v symmetry [18], the high D5h-symmetric anionic AuGe10- cluster with a A1′1 electronic state forms due to the equal electron distribution for each Ge atom [44], indicating that the neutral AuGe10 cluster is an electron-deficient base. The geometry of the lowest-energy AuGe10- cluster in D5h symmetry is also consistent with the neutral FeGe10 cluster [9] as well as the trianionic Co@Ge103- cluster [29]. In the structure, there are 10 equivalent Ge–Ge bond lengths on two five-numbered germanium rings, and 5 equivalent Ge–Ge bond lengths on side-jointed prism. At the B3LYP/LanL2DZ level of theory, the equilibrium Ge–Ge bond lengths of them are predicted to be 2.722, and 2.858 Å, respectively; and the equilibrium Au–Ge bond lengths in the pentagonal prism are calculated to be 2.721 Å, as shown in Table 1.Table 1
Optimized bond distances (in Å), NICS values (in ppm) at different positionsa, and electron configurations of the lowest-energy AuGe10- (D5h) and AuGe12- (D2d) clusters.
Cluster
d
Ge
-
Ge
d
Ge
-
Au
NICSb
Electron configuration
0.0 Å
0.5 Å
1.0 Å
1.5 Å
2.0 Å
3.0 Å
Au
Ge
10
-
2.7222.858
2.721
−259.6(−236.1)
−183.7(−177.4)
−50.7(−47.3)
−25.2(−24.4)
−14.1(−15.7)
−8.5(−10.6)
(1a1′)2(1e1′)4(1a2′′)2(2a1′)2
(2e1′)4(2a2′′)2(1e1′′)4(1e2′)4
(3a1′)2(2e1′′)4(2e2′)4(1e2′′)4
(4a1′)2(5a1′)2(3e2′)4(3e1′)4
(3a2′′)2(4e1′)4(2e2′′)4
Au
Ge
12
-
2.6112.6442.6472.855
2.8142.942
−296.4(−277.1)
−199.0(−196.4)
−61.3(−60.8)
−36.3(−37.7)
−25.8(−28.0)
−18.0(−19.4)
(1a1)2(1b2)2(1e)4(2a1)2
(2b2)2(2e)4(1b1)2(3e)4
(3a1)2(3b2)2(4a1)2(4e)4
(1a2)2(2b1)2(4b2)2(5e)4
(5a1)2(5b2)2(6a1)2(6e)4
(6b2)2(7e)4(3b1)2(7a1)2
(8e)4(2a2)2
aNICS values were computed at the GIAO-B3LYP/LanL2DZ//B3LYP/LanL2DZ level, based on the positions located on the radial Lab connecting the center site (see Figure 1).
bNICS values in parentheses were obtained from the GIAO-B3LYP/cc-pVTZ-PP//B3LYP/LanL2DZ calculations.Equilibrium structures of the lowest-energyD5h-symmetric AuGe10- (a) and D2d-symmetric AuGe12- (b) clusters in [44]. The different NICS sites (purple balls) are indicated.
(a)
(b)Analogous to the formation ofAuGe10- cluster, all the possible isomers of AuGe12- cluster are considered. An Au-doped bicapped pentagonal prism is optimized to be the lowest-energy AuGe12- structure, which displays a A11 electronic state in D2d symmetry (see Figure 1(b)). By comparison, the icosahedral structure of ZnGe12 [50] cluster is confirmed to be the global minima in the potential energy surface (PES), but the icosahedral AuGe12- structure is only a local minimum which is higher in energy than the lowest-energy structure by 0.32 eV [44]. In the bicapped pentagonal prism, the D2d structure has two C2 axes and two σd planes. As a result, four different Ge–Ge bond lengths are predicted to be 2.611, 2.644, 2.647 and 2.855 Å, respectively, whereas the two equilibrium Au–Ge bond lengths are calculated to be 2.814 and 2.942 Å.As discussed above, the endohedrally dopedAuGe10- and AuGe12- clusters are the most stable. Obviously, they do not obey the 18-electron counting rule (also known as the octet rule) [51] which may give rise to the high stability of cluster. In 2003, Sen and Mitas [52] also reported the limitation of the 18-electron rule and found that the filling of electron shell according to 18-electron rule is not the only factor in determining the structural stability, and the stability should depend on structural geometry, choice of metal atom, neutral or charged species, and so forth. In addition, one needs to calculate reaction energy to explain the thermodynamic stability of the cluster, and the reaction energy (Er) can be computed according to the following equation:
(1)Er(AuGen-)=[E(AuGen-)-nE(Ge)-E(Au-)](n+1),n=10or12,
where E(AuGen-),  E(Ge), and E(Au-) represent the total energies of AuGen-, Ge, and Au- atoms, respectively. All the total energies include the correction of zero-point vibrational energy (ZPVE). According to our knowledge, the formations of AuGe10- and AuGe12- clusters are exothermic with the large negative Er values, being −2.44 and −2.50 eV. Interestingly, their high thermodynamic stabilities are mainly associated with the geometry and the number of dangling bonds in the two clusters, as mentioned in previous study [18]. Our results also reveal that both of them can appear as stable species, and the two clusters could be synthesized through the exothermic reaction pathway.


### 3.2. Electronic Structure and Chemical Bonding

In order to explore the electronic features related to the stability of the clusters, we have performed a detailed analysis of the molecular orbitals by examining the partial density of states (PDOS) from the contribution of different orbital components (Au:s, p, d; Ge: s, p) and the electron density of the HOMO and LUMO states. The plots of PDOS of the AuGe10- and AuGe12- clusters are shown in Figures 2(a) and 2(b), respectively.Density of states (DOS) of the lowest-energyAuGe10- (a) and AuGe12- (b) clusters computed at the B3LYP/LanL2DZ level of theory.
(a)
(b)As for a typical case, each Ge atom is expected to contribute its four valence electrons to the electron shell configuration of the molecular systems. TheAuGe10- cluster with D5h symmetry thus contains 42 valence electrons which are distributed in the following orbital configuration: (1e1′′)2(1e2′)4(3a1′)2(2e1′′)4(2e2′)4(1e2′′)4(4a1′)2(5a1′)2(3e2′)4(3e1′)4(3a2′′)2(4e1′)4(2e2′′)4, as listed in Table 1. Due to the high D5h symmetry, additionally, the molecular orbitals strongly resemble the spherical features. From the Figure 2(a), we clearly see that the electronic states at high-energy region of around −3.9 eV above come mainly from p-Ge state; and the contribution from s-Ge state is very little. The difference is that the valence molecular orbitals at around −6.0 eV below mainly ascribe to the contribution of d-Au ands-Ge states; in particular, the E2′-type valence orbital at −6.77 eV, holding a doubly degenerate state, is composed of d-Au state mixed withs-Ge state. Similar behavior is observed for the D2d-symmetric AuGe12- cluster which contains 50 valence electrons with an orbital configuration of [(3e)4(3a1)2(3b2)2(4a1)2(4e)4(1a2)2(2b1)2(4b2)2(5e)4(5a1)2(5b2)2(6a1)2(6e)4(6b2)2(7e)4(3b1)2(7a1)2(8e)4(2a2)2]. The electron density (0.02 e/a.u.3) of HOMO and LUMO states of the two clusters is represented in Figure 2. One can see that both the HOMO and LUMO states are mainly localized around germanium cage, while some electronic distributions around Au atom are also found in the LUMO. It is remarkable that the encapsulation of the Au dopant is responsible for the geometric rearrangement of pure germanium clusters and the reduction of the HOMO-LUMO gaps of pure Ge10 and Ge12 clusters. Especially for the AuGe12- cluster the HOMO and LUMO are nondegenerate and are slightly separated from the adjacent occupied and unoccupied orbitals, resulting in its quite small HOMO-LUMO gap of 1.63 eV. Thus, although the germanium is semiconductor element, the AuGe10- and AuGe12- clusters may be considered as novel cluster-assembled materials with partially metallic features [19, 44].In 1994, Silvi and Savin [53] reported that the electron localization function (ELF) is currently used to describe the nature of bonding. According to their descriptions, the molecular space is divided into regions or basins of localized electron pairs or attractors. Typically, the existence of a high ELF isovalue (around 0.7 and above) in the bonding region between two atomic basins signifies a localized chemical bond in the region [44, 54]. For the D5h-symmetric AuGe10- cluster, the ELF basins along Ge–Ge bond upon two five-number germanium rings merge at an isovalue of 0.71 (Figure 3(a)), revealing that these Ge–Ge bonds are covalent in nature, while the five Ge–Ge prismy bonds have more metallic or less covalent character reflected by the slightly low ELF isovalue at 0.60 (Figure 3(b)). The ELF contour at a low isovalue of 0.32 (Figure 3(c)) shows two polarized basins between Au and Ge atoms, so all the Au–Ge bonds are far from being covalent in nature but show an ionic character. The most stable AuGe12- cluster has a high (almost spherical) D2d symmetry. The covalent bonding pattern between germanium atoms becomes more preferable because all atomic basins along the surface bonds in ELF contours almost merge at an isovalue of 0.72 (Figure 3(d)), whereas the Au–Ge bonds display a slightly increased ionic character confirmed by lower ELF isovalue (0.29, Figure 3(e)) and elongated Au–Ge bond lengths (0.093−0.221 Å), as discussed above.Figure 3
The ELF isosurfaces of theAuGe10- and AuGe12- clusters. Some different isovalues are indicated: (a) 0.71, (b) 0.60, (c) 0.32, (d) 0.72, and (e) 0.29.


### 3.3. The2(N+1)2 Rule

Aromaticity is well known as one of the important measures of compounds. Compared to nonaromatic compounds, analogous aromatic compounds typically give enhanced chemical stability. In general, aromaticity of planar structure can be qualitatively explored using the4N+2 Hückel rule [34]. Another electron counting rule, namely, 2(N+1)2 rule proposed by Hirsch et al. [32], is proven as an effective aromaticity criterion for spherical species (Ih symmetry), as well as the extension of the treatment to inorganic cage molecules [55]. In the latter electron rule, the π-electron system of the species can be approximately considered as a spherical electron gas, which surrounds the surface of a sphere [56]. According to the Pauli principle, if the number of π electrons in a spherical structure satisfies the 2(N+1)2 rule, the structure can show a spherical charge distribution and is therefore expected to be aromatic.The molecular orbitals (MOs) of theAuGe10- and AuGe12- clusters are depicted in Figures 2(a) and 2(b), respectively. We see that the valence electron orbitals of them are divided into two different subsets occupied by σ or π electrons. In Figure 2(a), the D5h-symmetric AuGe10- cluster contains eight valence π-electrons in four MOs, for example, −5.52(A1′, 1), −2.98(E1′, 2), and −2.60(A2′′, 1), and these π-electrons satisfy the 2(Nπ+1)2[Nπ=1] counting rule. Similarly, eight valence π-electrons are also found for the D2d-symmetric AuGe12- cluster (Figure 2(b)) and occupy the −5.89(A1, 1), −3.31(B2, 1), and −2.92(E, 2) orbitals. As a consequence, the two π-electron systems make the AuGe10- and AuGe12- clusters spherically aromatic. This aromatic feature can be regarded as one of the main reasons in the structural stabilization of endohedrally doped AuGe10- and AuGe12- clusters. However, it is noteworthy that the 2(N+1)2 electron counting rule cannot be solely used to explain the aromaticity of compounds. For instance, the bianionic Si122- cluster contains eight π electrons, but gives an antiaromatic character [57]. Therefore, the aromaticity of both the AuGe10- and AuGe12- clusters needs to be further confirmed by NICS values given in next section.


### 3.4. NICS Values of Clusters

The aromaticity of a chemical compound can be usually discussed in terms of various criteria, but the criteria used to validate the aromaticity are sometimes controversial, likelySi122- cluster. Herein, we performed a calculation of NICS proposed by Chen and coworkers [33], based on magnetic shieldings. Aromaticity is expected to be evaluated by a negative NICS value, and antiaromaticity by a positive NICS value. In general, a ghost atom is placed at the center of a spherical geometry to obtain NICS value. In order to get a reasonable value, however, we considered many different sites along central axis inside or outside the cage to test the variation of NICS value, as depicted in Figure 1. A radius (Lab) from the center site of the cage to each ghost atom is also defined with 0≤Lab≤3.0.Using the GIAO-B3LYP/LanL2DZ level of theory, we have calculated the NICS values of theAuGe10- and AuGe12- clusters. In order to give a proper description of magnetic and electronic properties, a large basis set (cc-pVTZ-PP) in our NICS calculations is also necessary with the same method. The NICS values of the two clusters are given in Table 1. We see that the NICS value at the center site is the largest one among all considered sites, being −259.6 and −296.4 ppm for AuGe10- and AuGe12- clusters, respectively. From Figure 4 we see that the NICS values of the AuGe10- cluster rapidly decreases to Lab=1.5 (−25.2 ppm) and then slowly decreases to Lab=3.0 (−8.5 ppm). Obviously, the NICS values close to the outer surface are relatively smaller than those inside the cage. Similar behavior is found for the AuGe12- cluster, and it gives a slightly larger NICS value than AuGe10- cluster. We also find that the NICS values with LanL2DZ basis set are in good agreement with the results with cc-pVTZ-PP basis set, except for the value at the center site that is relatively large with the deviation of about 20.0 ppm. Thus, the aromaticity of both the AuGe10- and AuGe12- clusters, characterized by the 2(Nπ+1)2[Nπ=1] electron counting rule, is confirmed by largely negative NICS values. Moreover, the two clusters have very strong aromatic character at the center site, and the aromaticity will weaken outside the cage and vanish beyond.Figure 4
Comparison of NICS values of anionicAuGe10-, and AuGe12- clusters using two different LanL2DZ and cc-pVTZ-PP basis sets with the same B3LYP functional, based on the B3LYP/LanL2DZ geometries.


## 3.1. Geometry and Stability

Using the computation scheme described above, we have extensively explored a number of initial isomers formed from taking the previously reported structures of different TMGe10 systems, or doping one Au dopant at all possible positions of the lower-lying Ge10 isomers, or replacing one Ge atom of the lower-lying Ge11 isomers with an Au atom to form new isomers. As a result, the most stable AuGe10- cluster is found to be an endohedrally Au-doped pentagonal prism (see Figure 1(a)). It is worth mentioning that, when an electron is added to the lowest-energy neutral AuGe10 cluster in C2v symmetry [18], the high D5h-symmetric anionic AuGe10- cluster with a A1′1 electronic state forms due to the equal electron distribution for each Ge atom [44], indicating that the neutral AuGe10 cluster is an electron-deficient base. The geometry of the lowest-energy AuGe10- cluster in D5h symmetry is also consistent with the neutral FeGe10 cluster [9] as well as the trianionic Co@Ge103- cluster [29]. In the structure, there are 10 equivalent Ge–Ge bond lengths on two five-numbered germanium rings, and 5 equivalent Ge–Ge bond lengths on side-jointed prism. At the B3LYP/LanL2DZ level of theory, the equilibrium Ge–Ge bond lengths of them are predicted to be 2.722, and 2.858 Å, respectively; and the equilibrium Au–Ge bond lengths in the pentagonal prism are calculated to be 2.721 Å, as shown in Table 1.Table 1
Optimized bond distances (in Å), NICS values (in ppm) at different positionsa, and electron configurations of the lowest-energy AuGe10- (D5h) and AuGe12- (D2d) clusters.
Cluster
d
Ge
-
Ge
d
Ge
-
Au
NICSb
Electron configuration
0.0 Å
0.5 Å
1.0 Å
1.5 Å
2.0 Å
3.0 Å
Au
Ge
10
-
2.7222.858
2.721
−259.6(−236.1)
−183.7(−177.4)
−50.7(−47.3)
−25.2(−24.4)
−14.1(−15.7)
−8.5(−10.6)
(1a1′)2(1e1′)4(1a2′′)2(2a1′)2
(2e1′)4(2a2′′)2(1e1′′)4(1e2′)4
(3a1′)2(2e1′′)4(2e2′)4(1e2′′)4
(4a1′)2(5a1′)2(3e2′)4(3e1′)4
(3a2′′)2(4e1′)4(2e2′′)4
Au
Ge
12
-
2.6112.6442.6472.855
2.8142.942
−296.4(−277.1)
−199.0(−196.4)
−61.3(−60.8)
−36.3(−37.7)
−25.8(−28.0)
−18.0(−19.4)
(1a1)2(1b2)2(1e)4(2a1)2
(2b2)2(2e)4(1b1)2(3e)4
(3a1)2(3b2)2(4a1)2(4e)4
(1a2)2(2b1)2(4b2)2(5e)4
(5a1)2(5b2)2(6a1)2(6e)4
(6b2)2(7e)4(3b1)2(7a1)2
(8e)4(2a2)2
aNICS values were computed at the GIAO-B3LYP/LanL2DZ//B3LYP/LanL2DZ level, based on the positions located on the radial Lab connecting the center site (see Figure 1).
bNICS values in parentheses were obtained from the GIAO-B3LYP/cc-pVTZ-PP//B3LYP/LanL2DZ calculations.Equilibrium structures of the lowest-energyD5h-symmetric AuGe10- (a) and D2d-symmetric AuGe12- (b) clusters in [44]. The different NICS sites (purple balls) are indicated.
(a)
(b)Analogous to the formation ofAuGe10- cluster, all the possible isomers of AuGe12- cluster are considered. An Au-doped bicapped pentagonal prism is optimized to be the lowest-energy AuGe12- structure, which displays a A11 electronic state in D2d symmetry (see Figure 1(b)). By comparison, the icosahedral structure of ZnGe12 [50] cluster is confirmed to be the global minima in the potential energy surface (PES), but the icosahedral AuGe12- structure is only a local minimum which is higher in energy than the lowest-energy structure by 0.32 eV [44]. In the bicapped pentagonal prism, the D2d structure has two C2 axes and two σd planes. As a result, four different Ge–Ge bond lengths are predicted to be 2.611, 2.644, 2.647 and 2.855 Å, respectively, whereas the two equilibrium Au–Ge bond lengths are calculated to be 2.814 and 2.942 Å.As discussed above, the endohedrally dopedAuGe10- and AuGe12- clusters are the most stable. Obviously, they do not obey the 18-electron counting rule (also known as the octet rule) [51] which may give rise to the high stability of cluster. In 2003, Sen and Mitas [52] also reported the limitation of the 18-electron rule and found that the filling of electron shell according to 18-electron rule is not the only factor in determining the structural stability, and the stability should depend on structural geometry, choice of metal atom, neutral or charged species, and so forth. In addition, one needs to calculate reaction energy to explain the thermodynamic stability of the cluster, and the reaction energy (Er) can be computed according to the following equation:
(1)Er(AuGen-)=[E(AuGen-)-nE(Ge)-E(Au-)](n+1),n=10or12,
where E(AuGen-),  E(Ge), and E(Au-) represent the total energies of AuGen-, Ge, and Au- atoms, respectively. All the total energies include the correction of zero-point vibrational energy (ZPVE). According to our knowledge, the formations of AuGe10- and AuGe12- clusters are exothermic with the large negative Er values, being −2.44 and −2.50 eV. Interestingly, their high thermodynamic stabilities are mainly associated with the geometry and the number of dangling bonds in the two clusters, as mentioned in previous study [18]. Our results also reveal that both of them can appear as stable species, and the two clusters could be synthesized through the exothermic reaction pathway.


## 3.2. Electronic Structure and Chemical Bonding

In order to explore the electronic features related to the stability of the clusters, we have performed a detailed analysis of the molecular orbitals by examining the partial density of states (PDOS) from the contribution of different orbital components (Au:s, p, d; Ge: s, p) and the electron density of the HOMO and LUMO states. The plots of PDOS of the AuGe10- and AuGe12- clusters are shown in Figures 2(a) and 2(b), respectively.Density of states (DOS) of the lowest-energyAuGe10- (a) and AuGe12- (b) clusters computed at the B3LYP/LanL2DZ level of theory.
(a)
(b)As for a typical case, each Ge atom is expected to contribute its four valence electrons to the electron shell configuration of the molecular systems. TheAuGe10- cluster with D5h symmetry thus contains 42 valence electrons which are distributed in the following orbital configuration: (1e1′′)2(1e2′)4(3a1′)2(2e1′′)4(2e2′)4(1e2′′)4(4a1′)2(5a1′)2(3e2′)4(3e1′)4(3a2′′)2(4e1′)4(2e2′′)4, as listed in Table 1. Due to the high D5h symmetry, additionally, the molecular orbitals strongly resemble the spherical features. From the Figure 2(a), we clearly see that the electronic states at high-energy region of around −3.9 eV above come mainly from p-Ge state; and the contribution from s-Ge state is very little. The difference is that the valence molecular orbitals at around −6.0 eV below mainly ascribe to the contribution of d-Au ands-Ge states; in particular, the E2′-type valence orbital at −6.77 eV, holding a doubly degenerate state, is composed of d-Au state mixed withs-Ge state. Similar behavior is observed for the D2d-symmetric AuGe12- cluster which contains 50 valence electrons with an orbital configuration of [(3e)4(3a1)2(3b2)2(4a1)2(4e)4(1a2)2(2b1)2(4b2)2(5e)4(5a1)2(5b2)2(6a1)2(6e)4(6b2)2(7e)4(3b1)2(7a1)2(8e)4(2a2)2]. The electron density (0.02 e/a.u.3) of HOMO and LUMO states of the two clusters is represented in Figure 2. One can see that both the HOMO and LUMO states are mainly localized around germanium cage, while some electronic distributions around Au atom are also found in the LUMO. It is remarkable that the encapsulation of the Au dopant is responsible for the geometric rearrangement of pure germanium clusters and the reduction of the HOMO-LUMO gaps of pure Ge10 and Ge12 clusters. Especially for the AuGe12- cluster the HOMO and LUMO are nondegenerate and are slightly separated from the adjacent occupied and unoccupied orbitals, resulting in its quite small HOMO-LUMO gap of 1.63 eV. Thus, although the germanium is semiconductor element, the AuGe10- and AuGe12- clusters may be considered as novel cluster-assembled materials with partially metallic features [19, 44].In 1994, Silvi and Savin [53] reported that the electron localization function (ELF) is currently used to describe the nature of bonding. According to their descriptions, the molecular space is divided into regions or basins of localized electron pairs or attractors. Typically, the existence of a high ELF isovalue (around 0.7 and above) in the bonding region between two atomic basins signifies a localized chemical bond in the region [44, 54]. For the D5h-symmetric AuGe10- cluster, the ELF basins along Ge–Ge bond upon two five-number germanium rings merge at an isovalue of 0.71 (Figure 3(a)), revealing that these Ge–Ge bonds are covalent in nature, while the five Ge–Ge prismy bonds have more metallic or less covalent character reflected by the slightly low ELF isovalue at 0.60 (Figure 3(b)). The ELF contour at a low isovalue of 0.32 (Figure 3(c)) shows two polarized basins between Au and Ge atoms, so all the Au–Ge bonds are far from being covalent in nature but show an ionic character. The most stable AuGe12- cluster has a high (almost spherical) D2d symmetry. The covalent bonding pattern between germanium atoms becomes more preferable because all atomic basins along the surface bonds in ELF contours almost merge at an isovalue of 0.72 (Figure 3(d)), whereas the Au–Ge bonds display a slightly increased ionic character confirmed by lower ELF isovalue (0.29, Figure 3(e)) and elongated Au–Ge bond lengths (0.093−0.221 Å), as discussed above.Figure 3
The ELF isosurfaces of theAuGe10- and AuGe12- clusters. Some different isovalues are indicated: (a) 0.71, (b) 0.60, (c) 0.32, (d) 0.72, and (e) 0.29.


## 3.3. The2(N+1)2 Rule

Aromaticity is well known as one of the important measures of compounds. Compared to nonaromatic compounds, analogous aromatic compounds typically give enhanced chemical stability. In general, aromaticity of planar structure can be qualitatively explored using the4N+2 Hückel rule [34]. Another electron counting rule, namely, 2(N+1)2 rule proposed by Hirsch et al. [32], is proven as an effective aromaticity criterion for spherical species (Ih symmetry), as well as the extension of the treatment to inorganic cage molecules [55]. In the latter electron rule, the π-electron system of the species can be approximately considered as a spherical electron gas, which surrounds the surface of a sphere [56]. According to the Pauli principle, if the number of π electrons in a spherical structure satisfies the 2(N+1)2 rule, the structure can show a spherical charge distribution and is therefore expected to be aromatic.The molecular orbitals (MOs) of theAuGe10- and AuGe12- clusters are depicted in Figures 2(a) and 2(b), respectively. We see that the valence electron orbitals of them are divided into two different subsets occupied by σ or π electrons. In Figure 2(a), the D5h-symmetric AuGe10- cluster contains eight valence π-electrons in four MOs, for example, −5.52(A1′, 1), −2.98(E1′, 2), and −2.60(A2′′, 1), and these π-electrons satisfy the 2(Nπ+1)2[Nπ=1] counting rule. Similarly, eight valence π-electrons are also found for the D2d-symmetric AuGe12- cluster (Figure 2(b)) and occupy the −5.89(A1, 1), −3.31(B2, 1), and −2.92(E, 2) orbitals. As a consequence, the two π-electron systems make the AuGe10- and AuGe12- clusters spherically aromatic. This aromatic feature can be regarded as one of the main reasons in the structural stabilization of endohedrally doped AuGe10- and AuGe12- clusters. However, it is noteworthy that the 2(N+1)2 electron counting rule cannot be solely used to explain the aromaticity of compounds. For instance, the bianionic Si122- cluster contains eight π electrons, but gives an antiaromatic character [57]. Therefore, the aromaticity of both the AuGe10- and AuGe12- clusters needs to be further confirmed by NICS values given in next section.


## 3.4. NICS Values of Clusters

The aromaticity of a chemical compound can be usually discussed in terms of various criteria, but the criteria used to validate the aromaticity are sometimes controversial, likelySi122- cluster. Herein, we performed a calculation of NICS proposed by Chen and coworkers [33], based on magnetic shieldings. Aromaticity is expected to be evaluated by a negative NICS value, and antiaromaticity by a positive NICS value. In general, a ghost atom is placed at the center of a spherical geometry to obtain NICS value. In order to get a reasonable value, however, we considered many different sites along central axis inside or outside the cage to test the variation of NICS value, as depicted in Figure 1. A radius (Lab) from the center site of the cage to each ghost atom is also defined with 0≤Lab≤3.0.Using the GIAO-B3LYP/LanL2DZ level of theory, we have calculated the NICS values of theAuGe10- and AuGe12- clusters. In order to give a proper description of magnetic and electronic properties, a large basis set (cc-pVTZ-PP) in our NICS calculations is also necessary with the same method. The NICS values of the two clusters are given in Table 1. We see that the NICS value at the center site is the largest one among all considered sites, being −259.6 and −296.4 ppm for AuGe10- and AuGe12- clusters, respectively. From Figure 4 we see that the NICS values of the AuGe10- cluster rapidly decreases to Lab=1.5 (−25.2 ppm) and then slowly decreases to Lab=3.0 (−8.5 ppm). Obviously, the NICS values close to the outer surface are relatively smaller than those inside the cage. Similar behavior is found for the AuGe12- cluster, and it gives a slightly larger NICS value than AuGe10- cluster. We also find that the NICS values with LanL2DZ basis set are in good agreement with the results with cc-pVTZ-PP basis set, except for the value at the center site that is relatively large with the deviation of about 20.0 ppm. Thus, the aromaticity of both the AuGe10- and AuGe12- clusters, characterized by the 2(Nπ+1)2[Nπ=1] electron counting rule, is confirmed by largely negative NICS values. Moreover, the two clusters have very strong aromatic character at the center site, and the aromaticity will weaken outside the cage and vanish beyond.Figure 4
Comparison of NICS values of anionicAuGe10-, and AuGe12- clusters using two different LanL2DZ and cc-pVTZ-PP basis sets with the same B3LYP functional, based on the B3LYP/LanL2DZ geometries.


## 4. Conclusions

Using the hybrid DFT-B3LYP functional, we have studied a large number of structural isomers for medium-sizedAuGe10- and AuGe12- clusters. The endohedrally Au-doped pentagonal prism and bicapped pentagonal prism are found to be the most stable AuGe10- (D5h) and AuGe12- (D2d) structures, respectively. Analysis of reaction energy indicates that the endohedrally doped AuGe10- and AuGe12- clusters have the high thermodynamic stability. We see that the high-energy regions of molecular occupied orbitals of the two compounds come mainly from the contribution of p-Ge state by means of the partial density of states (PDOS). The electron localization functions (ELFs) are also considered to explore the chemical bonding of the two clusters, and in some regions more metallic (or less covalent) character between germanium atoms becomes more preferable, for example, pentagonal prismy bonds. In the AuGe10- and AuGe12- structures, eight valence π-electrons satisfy the 2(Nπ+1)2[Nπ=1] counting rule and make the AuGe10- and AuGe12- clusters spherically aromatic. An NICS calculation on basis of magnetic shieldings is carried out to confirm the aromaticity of the three-dimensional clusters reflected by largely negative NICS values. Additionally, NICS is also relatively insensitive to the level of theory used in the study, for example, basis set.



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*Source: 518593-2012-12-06.xml*