Inside a Superatom: The M7q (M=Cu, Ag, q=1+, 0, 1−) Case

DOI: 10.1002/cphc.200900714

Inside a Superatom: The M7q (M = Cu, Ag, q = 1 + , 0, 1) Case Alvaro MuÇoz-Castro, Desmond Mac-Leod Carey, and Ramiro Arratia-Prez*[a] All-electron relativistic density functional calculations are performed to obtain the electronic structure and nucleus-independent chemical shifts (NICS) of D5h pentagonal-bipyramidal (PBP) Cu7q and Ag7q (q = 1 + ,0,1) clusters. Scalar and spin– orbit relativistic effects are taken into account at two levels: the two-component zero-order regular approximation (ZORA) Hamiltonian and fully relativistic four-component calculations via the Dirac equation. These clusters are treated by including

the spin–orbit effect in the jellium model, within the doublevalued point group (D5h*) establishing the symmetry correlations between the molecular and the atomic spinors given by the full rotation group. These clusters show highly spherical aromaticity, which is suggested to increase the hardness of the superatom. Thus, the calculations suggest that the paramagnetic Cu7 and Ag7 clusters can be regarded as pseudohalogens.

1. Introduction Clusters continue to attract considerable interest due to their rich structural chemistry. An exciting feature of certain stable clusters is that they can mimic the chemical behavior of individual atoms.[1–8] Thus, some clusters can be regarded as superatoms, which provide a new dimension in the design of novel materials[9, 10] and serve as elemental building blocks for new nanostructured materials constructed according to the bottom-up approach.[11, 12] These nanoscale atoms must retain their atomlike behavior and identity, even in conjunction with other clusters or ligands. A few well-know examples are Al13,[13, 14] X3O (X = Li, Na, K),[15–17] AlPb10 + , and AlPb12 + [18] among others.[19, 20] In particular, the paramagnetism of the Cu7 and Ag7 clusters is desirable for designing and obtaining tunable magnetic properties that could lead to novel nanoscale optoelectronics as well as miniature spintronics and storage devices.[21] These clusters have been characterized by EPR measurements[22] and theoretical calculations[24, 25] as belonging to the D5h point group of symmetry, with pentagonal-bipyramidal (PBP) threedimensional structure (Scheme 1). The seven Cu and Ag atoms respectively have a singly occupied outer ns shell, and together they provide seven free electrons to the total cluster. These seven electrons in the compact three-dimensional PBP geometry can be regarded according to the superatom sugges-

tion,[9, 10] and thus must occupy a new set of orbitals defined in the center of the cluster in accordance with the jellium model,[26, 27] by placing the seven electrons in an s2p5 configuration,[24] similar to a halogen. In addition, the paramagnetic properties of the Cu7 and Ag7 clusters allow them to be regarded as magnetic superatoms.[21] Relativistic effects (scalar and spin–orbit) certainly affect both electronic and geometrical structure, and are needed in order to obtain a reliable electronic structure and for calculating spin-dependent properties.[24, 25, 28–33] The two-dimensional (planar) structure observed for the Au7 cluster, as opposed to the three-dimensional PBP geometries of Cu7 and Ag7, is attributed to increased relativistic[25] and electronic correlation effects that allow significant “sd” hybridization in the molecular orbitals.[28] Here we combine fully relativistic four-component calculations via the Dirac equation and the two-component approximation via the zero-order regular approximation (ZORA) Hamiltonian, by employing the DSW[30–33] and ADF[34] packages, respectively. Thus, we obtained the relativistic electronic structure including both scalar and spin–orbit effects. In the spin–orbit coupling regime,[35] the double-valued point group must be used according to the inclusion of the spin (s) into the orbital momentum (‘) of each electron to give the total angular momentum (j = ‘  s), which couple both spin and spatial degrees of freedom into the double-valued representations. In this context, to achieve proper application of the jellium model approximation,[26, 27] we must find the correlation between the full rotation group (Dj  ) and the D5h* [a] Dr. A. MuÇoz-Castro, Dr. D. Mac-Leod Carey, Prof. Dr. R. Arratia-Prez Departamento de Ciecias Quimicas Universidad Andres Bello, Av. Republica 275, Santiago (Chile) Fax: (+ 56) 2 6551396 E-mail: [email protected]

Scheme 1. Pentagonal-bipyramidal (D5h) M7 (M = Cu, Ag).

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Supporting information for this article is available on the WWW under http://dx.doi.org/10.1002/cphc.200900714.

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Inside a Superatom double-valued point group for PBP geometry, which are derived conveniently for j = 1/2, 3/2, and 5/2 to obtain the extra irreducible representations of the s1/2, p1/2, p3/2, d3/2, and d5/2 atomic shells. An interesting stabilizing factor of these hollow polyhedral clusters is the spherical aromaticity inside the cage, which can be estimated by the magnitude of the diatropic currents at the center of the hollow cluster,[36–47] quantified by calculating nucleus-independent chemical shifts (NICS).[48–49] Hence, we also report here the calculated NICS values at the center of the PBP structure to obtain a picture of the electronic current inside the cage by incorporating spin–orbit effects due to its direct and indirect effect over the atomic shells.[50–52] For the paramagnetic clusters, the spin magnetization density based on the singly occupied molecular spinor (SOMS) was plotted.

Computational Details Fully relativistic four-component calculations via the Dirac equation were done with the Born–Oppenheimer self-consistent-field Diracscattered-wave (DSW) molecular orbital method, developed by Yang, Case, and Arratia-Prez,[28–33] by using the symmetry basis constructed by Alvarez-Thon et al. for the D5h* point group.[24] Total valence population analyses were done by using the numerical relativistic extension of the Case and Karplus algorithm[53–54] implemented into the DSW code. The two-component relativistic approximation to the Dirac equation was carried out by using the Amsterdam density functional (ADF) package[34] including scalar and spin–orbit (SO) effects via the ZORA Hamiltonian.[55, 56] Geometry optimizations were done by the analytical energy gradient method implemented by Verluis and Ziegler by employing the local density approximation (LDA) with the Vosko, Wilk, and Nusair parameterization (VWN) for the local correlation energy,[57, 58] and the generalized gradient approximation (GGA) for the nonlocal exchange correlation (XC) according to the Perdew–Burke–Ernzerhof (PBE) functional.[59] We employed allelectron Slater-type orbital (STO) basis sets with triple-z plus polarization functions (TZP) for all atoms in the ADF package. The openshell systems were treated by using the unrestricted noncollinear approximation. The NICS[48–49] calculations were carried out at the optimized geometries by using the nonlocal XC correction within the generalized gradient approximation (GGA) OPBE functional according to Zhang et al.[60]

2. Results and Discussion 2.1. Symmetry Considerations The D5h* double-valued point group employed for the treatment of the Cu7 and Ag7 clusters is obtained as the direct product of the irreducible representations (irrep) of the singlevalued D5h and the double-valued spin irrep g1/2 for the D5h* point group. Thus the compatibility relationship between D5h and D5h* is given as shown Table 1. 2.2. Molecular Structure The geometries, relative binding energies (RBE), and the gap between the highest occupied and the lowest unoccupied moChemPhysChem 2010, 11, 646 – 650

Table 1. Relationship between D5h and D5h*. D5h

g1/2

D5h*

a1’ a2’ e1’ e2’ a1’’ a2’’ e1’’ e2’’

! ! ! ! ! ! ! !

g1/2 g1/2 g7/2g9/2 g3/2g5/2 g9/2 g9/2 g1/2g3/2 g5/2g7/2

lecular spinors (HOMS–LUMS gap) are summarized in Table 2. For Cu7q and Ag7q (q = 1 + , 0, 1) the D5h PBP geometry is the global energy minimum. These results are in accord with the

Table 2. Optimized distances [], HOMS–LUMS (H–L) gap [eV], relative binding energy (RBE) [eV], and NICS values [ppm] at the center of M7q (M = Cu, Ag; q = 1 + , 0, 1) clusters. Cluster D5h

eq–eq

eq–ax

ax–ax

H–L gap

RBE

NICS[a]

Cu7 + Cu7 Cu7 Ag7 + Ag7 Ag7

2.483 2.414 2.346 2.819 2.745 2.665

2.417 2.420 2.456 2.750 2.759 2.812

2.353 2.559 2.865 2.690 2.939 3.328

1.075 1.229 1.182 1.002 1.086 1.100

8.198 2.077 0.000 7.810 2.067 0.000

46.7 254.0 54.2 23.5 249.7 42.6

[a] NICS values for a point located at the center of the hollow cluster. Calculated at the OPBE/ZORA + SO level.

reported ESR measurements[22, 23] (on neutral Cu7 and Ag7) and theoretical calculations on Cu7q and Ag7q (q = 0, 1),[24, 25] which show that these clusters have a pentagonal bipyramidal (PBP) geometry which belongs to the D5h symmetry point group (Scheme 1). The distance between the equatorial atoms (eq– eq) decreases from M7 + to M7 (M = Cu, Ag) clusters, while the equatorial–axial (eq–ax) and axial–axial distances (ax–ax) increase. The RBE values (Table 1), which indicate the relative stability of these clusters, show that M7 is 2.0 eV more stable than M70 and about 8.0 eV more stable than M7 + . The neutral cluster with seven Cu0 or Ag0 atoms, each with an electronic configuration of (n1)d10ns1 with a half-filled ns shell, can be considered as seven free electrons of the valence shell of a cluster with a quite spherical density and three-dimensional PBP geometry. Thus, the electronic structure and stability can be rationalized according to the simple jellium model which implies approximation of an atom or a cluster by a homogenous sphere, in which the seven electrons should occupy several shells according to spherical harmonics patterns.[26, 27] In this sense, the seven free electrons, according to the jellium sphere, must be arranged as an s2p5 configuration towards a magic number of eight electrons that suggests an increase in the stability of the title clusters. Moreover, on going from the s2p4 configuration of Cu7 + and Ag7 + to the closedshell s2p6 configuration of Cu7 and Ag7 , the geometry changes from oval-shaped to more spherical with radii of

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R. Arratia-Prez et al. 1.995 and 2.267 , respectively, as is schematically shown in Figure 1. The more spherical geometry observed for the anionic clusters suggests that the closed-shell configuration of the superatom favors the jellium-sphere approximation for a threedimensional cluster.

between the single and double point groups of symmetry. The SOMS (g9/2) in both clusters is constituted mainly by s and p orbitals of the two axial atoms, while the HOMS1, LUMS + 1, and LUMS + 2 principally have contributions from s and p orbitals of the equatorial atoms (Supporting Information). Downwards from HOMS2, the molecular spinors are mainly composed of the corresponding d shell. Note that the shape of the spinors from the HOMS1 to LUMS + 2 clearly resembles the atomic p and d shells (Figure 3); the s shell is located in the region of low-lying spinors and is not described here.

Figure 1. Side view of the geometrical structure of M7q (M = Cu, Ag, q = 1 + ,0,1) approaching to a sphere.

2.3. Electronic Structure The electronic structures of the neutral clusters are depicted in Figure 2. The gap between the highest occupied and lowest unoccupied molecular spinors (HOMS–LUMS gap) of these systems retains its value of about 1 eV (Table 1), which suggests

Figure 2. Energy-level diagram of Cu7 and Ag7.

similar stability along the series. Both exhibit a 2A2’’ ground state or, according to the relativistic case, a stable Kramer doublet ground state spanning the G9/2 irrep.[24] This half-filled spinor accounts for the magnetic properties of these clusters, which can be regarded as magnetic superatoms.[21] The total valence population, calculated by using the relativistic extension of the Case and Karplus algorithm[53, 54] implemented in the DSW code, suggests that the formal atomic charges for these neutral clusters could be assigned as Cu(ax)0.012Cu(eq)0.005 and Ag(ax)0.048Ag(eq)0.020 (Supporting Information), that is, electron transfer has occurred from the equatorial to the axial atoms in the Cu7 cluster, but from the axial to the equatorial atoms in Ag7. In the closed-shell M7 cluster, the majority of the incoming electron is located mainly in the ns and np shell of the equatorial atoms M(eq), with 88 % for Cu7 and 90 % for Ag7 (Supporting Information). The percentage compositions of several molecular spinors, depicted in the Supporting Information, denote the correlation

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Figure 3. Plot of the HOMS1 to LUMS + 2 of Cu7 and Ag7 at 0.024 e¯ 3 with indication of the atomic electronic shells (p1/2, p3/2, d3/2 and d5/2).

The inclusion of spin–orbit coupling affects the electronic structure of the clusters by splitting the nonrelativistic HOMO1 (e1’) into two double-valued irreps,[24, 35] namely, g7/2 and g9/2, with a spin–orbit splitting xso of 723.2 and 827.9 cm1 for Cu7 and Ag7, respectively. The nonrelativistic LUMO (e2’) and LUMO + 2 (e1’’) exhibit xso = 170.8 (Cu7) and 340.8 cm1 (Ag7), and xso = 24.8 (Cu7) and 176.2 cm1 (Ag7), which indicate increasing strength of the spin–orbit coupling going down the group. This confirms that the relativistic effects are active even for elements than lighter gold.[29] Within the relativistic theory, the atomic p and d shell split into p1/2p3/2 and d3/2d5/2 (designated ‘j), respectively, and the s shell is considered as s1/2.[35] Thus, the proper treatment of our fully relativistic calculations according to the jellium-sphere model requires derivation of the relationship between the full rotation group (Dj  ) and the extra irreducible representation of the double-valued D5h*, in order to achieve the fully relativistic representation of the jellium sphere model. To achieve this model, we propose that the decomposition of the Dj  into the irreps of the D5h* doublevalued point group for j = 1/2, 3/2 and 5/2 must be derived. Here we show the correlation between the D5h* irreps and the relativistic atomic shell (denoted in parentheses according to the ‘j nomenclature; Table 3). Hence, the g7/2g9/2 spinors of HOMS1 can be regarded as the p3/2 shell, the SOMS (g9/2) as p1/2, and the LUMS (g3/2g5/2), LUMS + 1 (g1/2), and LUMS + 2 (g1/2g3/2) as the d3/2 and d5/2 shell (Figure 3) of a superatom. Note that the energy levels on Figure 2 follow the relativistic jellium-sphere model. Moreover, the s2p5 clusters (neutral Cu7 and Ag7) exhibit a high electron affinity (EA) ranging from 2.077 to 2.067 eV, as calculated from

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Inside a Superatom

Table 3. Correlation between the D5h* irreps and the relativistic atomic shell. Dj

Dj + D1/2 + (s1/2) D3/2 + (d3/2) D5/2 + (d5/2)

g1/2 g1/2g3/2 g1/2g3/2g5/2

D1/2(p1/2) D3/2(p3/2) D5/2

g9/2 g9/2g7/2 g9/2g7/2g5/2

the difference in energy between the neutral and anionic clusters (Table 1). The EA of these neutral clusters are lower than those of the halogens Cl and Br (EA = 3.61 and 3.46 eV respectively) and pseudohalogen[61] clusters such as Al13 (EA = 3.40 eV).[13, 14] Thus, M7 (M = Cu and Ag) can be regarded as pseudohalogens.

spontaneous spin current density J(0) [64, 65] in addition to the external first-order induced current density J(1),[65, 66] which act as an external uniform-field (Figure 4) into the system,[64–66] increasing the diatropic currents at the center of the cluster, as can be extracted from the calculated NICS values (ca. 250.0 ppm). Note that for Cu7 the spin-density magnetization at the center of the cluster is higher than in Ag7. This fact is accounted for the NICS index, which shows higher ring currents for the former. Recently, a four-component relativistic formalism was implemented to estimate aromaticity via the induced current density in closed-shell molecular systems by Saue et al.[67] These results suggest that the highly spherical aromaticity inside the cluster induces an increment in the hardness of the superatom, which accounts for the small EA compared to isolated halogen atoms (see above).

2.4. Spherical Aromaticity

3. Conclusions

An important stabilizing factor in some hollow clusters are highly diatropic electronic currents inside the cage, which can be quantified by the negative values of the nucleus independent chemical shifts (NICS).[48, 49] These diatropic currents are interpreted as spherical aromaticity (negative values). In Table 1, the calculated NICS values at the center of the hollow cluster suggest highly spherical aromatic behavior (3D aromaticity).[36–47] The diatropic currents increase on going from s2p4 to s2p6 electronic configuration, and this accounts for the aromatic stabilization of the clusters towards an eight-electron magic number. These results are in accordance with expansion of the two-dimensional (4 N + 2)pe Hckel rule for aromaticity to the three-dimensional 2 (N+1)2e Hirsh rule, proposed initially to explain the aromaticity of icosahedral fullerenes.[62] In fact, the s2p6 electrons (i.e., 8 e for N = 1) exhibit high aromaticity. The relativistic direct effect in the ns1/2 shell of Cu and Ag, accounts for the lower aromaticity of the Ag7 cluster, because the SOMS and HOMS1 are constituted mainly by the more contracted ns1/2 shell (5s1/2 vs 4s1/2 in Cu7[35]). It is noteworthy that in the title paramagnetic clusters or magnetic superatoms,[21] the unpaired electron of the 2g9/2 spinor leads to a substantial increase of the diatropic currents at the center due to a large magnetization of the clusters, as can be seen from the plot of the spin magnetization density m for the SOMS 2g9/2 (m + 9/2, Figure 4).[63] This magnetization comes from the fact that the unpaired electron introduces a

We have extended the jellium-sphere model to the fully relativistic regime by including spin–orbit coupling and proved its validity in the quasispherical Cu7 and Ag7 clusters. The calculations suggest that the noble gas configuration (s2p6) is favored by about 2 eV over those of the neutral clusters (s2p5) and by about 8 eV over those of the cationic clusters (s2p4). The cluster with s2p6 closed-shell configuration has a quasispherical structure that favors the jellium-sphere approximation. It is noteworthy that the diatropic currents inside the cluster arising from the interaction between the metallic members affect the chemical behavior of the superatom. The spherical aromaticity inside the superatom induces a harder character in the clusters. Hence, the paramagnetic Cu7 and Ag7 clusters can be regarded as pseudohalogens.

Figure 4. Spin magnetization density (m + 9/2 = f + 9/2†sf + 9/2) for the SOMS 2g9/2.

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Received: September 10, 2009 Revised: November 5, 2009 Published online on January 13, 2010

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