For comparison, the lowest-energy structures of pure Au_{n + 1} (n = 6–15) clusters are first predicated from our evolutionary simulations and presented in Fig. 1. Their geometries are consistent with previously reported results.^[33] The most stable isomers of Au_nGd (n = 6–15) found in our predications are shown in Fig. 2. For Au₆Gd, the guest impurity atom Gd is capped on the center of the Au₆ ring, contriving a regular hexagonal pyramid with C_6v symmetry. Note that the equilibrium geometry of Au₆Gd is different from transition-metal-doped Au₆ clusters, Au₆TM (TM = Sc, Ti, V, Cr, Mn, Fe, Co, Ni), in which the TM atom is located in the center of the hexagonal Au₆ ring, forming a perfectly planar two-dimensional (2D) structure with a higher symmetry (D_6h).^[4,34] In the case of Au₇Gd, a perfectly planar structure with D_7h symmetry is its ground state isomer, where Gd is just in the center of the Au₇ ring. Moreover, our predication for the Au₇Gd cluster is in good agreement with a previous report on the structure of the Ac₇Gd cluster.^[35] It is noticeable that the most stable structure of Au₈Gd possesses C_s symmetry and can be explained as an evolution from Au₆Gd capped by two Au atoms symmetrically. Obviously, it can be found from Fig. 2 that for Au_nGd the transition from 2D planar structure to three-dimensional (3D) structure occurs at Au₉Gd. Meanwhile, our predicated results have clearly indicated that the 2D–3D crossover for Au_nGd occurs in advance in comparison with pure Au_{n + 1} clusters presented in Fig. 1. On the one hand, the 5d–6s hybridization between Au atoms has a tendency to form σ-like directional bonds,^[36] causing Au_n clusters to conserve planarity up to larger sizes. On the other hand, a rare earth Gd atom tends to have a high coordination number (CN) by bonding with the nearest neighbor atom due to its various valence orbitals. As indicated in our predication, however, the maximum of CN of Gd atom in Au_nGd stable isomers with planar structure seems to be 7, reaching this limit in Au₇Gd. With further increasing the size of Au_nGd cluster, 2D planar isomers become unstable and 3D structures like Au₉Gd (C_2v), Au₁₀Gd (C_s), and Au₁₁Gd (C_s), respectively with the CN of 9, 10, and 9, are energetically favorable. For Au₁₂Gd, the lowest energy structure with D_6d symmetry is made up of two rotated Au₆ rings linked by the central Gd atom. The cage-like feature gradually appears in Au₁₃Gd (C_s) and Au₁₄Gd (C₂) clusters. It must be pointed out that the cage-like structure of Au₁₄Gd is also analogous to that of Au₁₄Th previously reported by Gao and Wang.^[37] When n = 15, Au₁₅Gd, a nearly perfect Gd-encapsulated Au₁₅ cage is formed, which agrees well with the previous prediction.^[23] The optimized Gd–Au and Au–Au bond lengths in Au_nGd clusters are listed in Table 1. For comparison, the bond lengths of pure Au_{n + 1} clusters are given in Table 1 and are in good agreement with previously calculated values.^[33] In Au_nGd, the Au–Au bond lengths are obviously shorter than the Au–Gd bond lengths. The bond lengths gradually increase as the size of cluster increases, accompanied by an odd–even oscillation. Moreover, it can be found that the change trend in the Au–Gd bond length is almost contrary to that in the Au–Au bond length.

Fig. 1.

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	Fig. 1. (color online) Geometrical structures of the lowest-energy Aun + 1 (n = 6–15) clusters determined from the structural evolutionary simulations.

Fig. 2.

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	Fig. 2. (color online) Geometrical structures of the lowest-energy AunGd (n = 6–15) clusters determined from the structural evolutionary simulations.

Table 1.

Table 1.

Table 1. Calculated values of interatomic distance (dAu-Au, dAu-Gd), substitution energy (Es), binding energy (Eb), magnetic moment (M), and HOMO–LUMO energy gap (G) of AunGd clusters. The calculated values of pure Aun + 1 clusters are listed for comparison. . Size AunGd Aun + 1 (n) dAu-Au/Å dAu-Gd/Å Es/eV Eb/eV M/μB G/eV dAu–Au/Å Eb/eV G/eV 6 2.738 2.845 −3.944 2.41 7.421 0.42 2.7 1.846 0.87 7 2.624 3.023 −4.187 2.487 6.971 0.60 2.676 1.964 1.41 8 2.739 2.875 −4.227 2.422 7.135 0.08 2.717 1.952 0.61 9 2.728 3.021 −4.441 2.480 6.999 0.94 2.722 2.036 1.29 10 2.743 2.965 −4.503 2.448 7.132 0.04 2.721 2.038 0.61 11 2.766 3.054 −4.398 2.472 6.997 0.82 2.724 2.106 0.98 12 2.817 3.017 −4.877 2.473 7.226 0.94 2.725 2.097 0.54 13 2.805 3.101 −5.184 2.51 6.994 0.76 2.849 2.14 1.56 14 2.822 3.07 −5.65 2.515 7.04 0.01 2.816 2.138 0.52 15 2.832 3.051 −6.151 2.562 7 1.47 2.761 2.177 1.35

Table 1. Calculated values of interatomic distance (dAu-Au, dAu-Gd), substitution energy (Es), binding energy (Eb), magnetic moment (M), and HOMO–LUMO energy gap (G) of AunGd clusters. The calculated values of pure Aun + 1 clusters are listed for comparison. .

To verify whether it is favorable to dope/substitute a Gd atom in Au_n clusters, we calculate the substitution energy (E_s) which is defined as

Fig. 3.

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	Fig. 3. Calculated values of substitution energy (Es) and binding energy (Eb) of the ground state AunGd (n = 6–15) clusters. For comparison, the calculated Eb of pure Aun + 1 clusters is presented.

In order to further investigate the size-dependent thermodynamic stability of Au_nGd clusters, the average binding energy (E_b) changing with cluster size is calculated. The values of E_b of Au_nGd and pure Au_{n + 1} clusters are defined as follows:

The total magnetic moment of a cluster is a combination of the spin and orbital magnetic moments. However, the magnetic moment of a cluster is mainly dominated by the spin magnetic moment since the contribution from the orbital magnetic moment of an electron can be neglected. The total magnetic moments of Au_nGd clusters obtained from the spin-polarized calculations are presented in Table 1 and displayed in Fig. 4.

Fig. 4.

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	Fig. 4. Calculated magnetic moment (M) versus cluster size of the ground state AunGd (n = 6–15) clusters.

It can be seen that the calculated magnetic moment shows an odd–even oscillation with the increase of cluster size. The calculated magnetic moment of Au₁₅Gd is 7 μ_B, and consistent with a previously reported value.^[23] When n = 7, 9, 11, 13, the total magnetic moments of Au_nGd clusters are also almost close to 7 μ_B. However, the calculated magnetic moments of Au_nGd clusters with even n are larger than 7 μ_B. Especially, Au₆Gd atomic cluster possesses the largest magnetic moment of 7.421 μ_B. It is not surprising if one considers the fact that atomic clusters with the reduced coordination number and higher symmetry would exhibit enhanced magnetization.^[34] In all of Au_nGd atomic clusters considered here, as discussed above, Au₆Gd with C_6v symmetry has the lowest coordination number of 6. Although Au₁₂Gd has a relatively high coordination number, its symmetry with D_6d is the highest in Au_nGd clusters. Hence, the calculated magnetic moment of Au₁₂Gd with 7.226 μ_B is also significantly larger than those of other Au_nGd clusters except Au₆Gd. Additionally, our calculations indicate that the additional magnetic moments beyond 7 μ_B for Au_nGd clusters with even n are fully from Au atoms. For Au₆Gd, the local magnetic moment on each Au atom is about 0.07 μ_B. In fact, Au bulk solid is known to be weakly diamagnetic, but a great variety of magnetic phenomena have been found in polymer-capped Au nanoparticles due to the strong chemical affinity of Au atoms to the capping molecules.^[38] Moreover, recent experiments have confirmed the existence of spontaneous magnetic moments in bare Au nanoparticles.^[39]

To gain an insight into the origin of magnetism in Au_nGd clusters, the electronic density of states (DOS) is further investigated. For comparison, the calculated total DOS and partial DOS (PDOS) on Au and Gd atoms of Au₆Gd, Au₁₂Gd, and Au₁₅Gd clusters are presented in Fig. 5. It can be found that the high magnetic moments of Au_nGd clusters are mainly ascribed to the PDOS of Gd-4f electrons centered in the two energy regions between −4.8 eV and −3.4 eV, and between 0.2 eV and 1.2 eV. For Au₁₅Gd, the spin-up and spin-down PDOSs of Au-6s and Au-5d electrons are almost identical, indicating that the spin polarized effects from the Au-6s and Au-5d electrons can be ignored. In the cases of Au₆Gd and Au₁₂Gd, however, the two spin channels of Au-6s and Au-5d electrons are not strictly symmetrical and have a slight shift toward higher or lower energy levels, which generates the exchange splitting and makes a few contributions to the magnetism of Au₆Gd and Au₁₂Gd clusters, respectively.

Fig. 5.

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	Fig. 5. (color online) Calculated curves of total and partial density of states versus energy of ground state Au6Gd, Au12Gd, and Au15Gd clusters.

The calculated charge density difference between Au₆Gd and Au₇Gd is displayed in Fig. 6. It can be clearly found that most electrons withdraw to the intermediate zone between the circumambient Au atoms and the central Gd atom and transfer to Au atoms. The delimitation of delocalized electrons occurs in a finite margin around Au atoms and is cancelled out in a certain small scope. Whereas the quantity of overall electron transfer is still small, showing the covalent metallic bonding features in Au₆Gd and Au₇Gd clusters. The Coulomb repulsion between bonding basins inevitably interfere with the structure models, triggering the deformation of the electron cloud.

Fig. 6.

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	Fig. 6. (color online) Contours of charge density difference of Au6Gd and Au7Gd clusters across Au and Gd atoms. Insets show corresponding isosurfaces of charge density difference.

The calculated energy gaps between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) of Au_nGd and Au_{n + 1} (n = 6–15) clusters are presented in Fig. 7 and listed in Table 1. Generally, the HOMO–LUMO energy gap can be used to measure the chemical stability of small clusters. A big energy gap is usually related to a high chemical stability. It can be found that pure Au_{n + 1} small clusters show an obvious odd–even oscillation in their energy gaps. This phenomenon can be explained by the electron pairing effect and also be found in pure Cu_n clusters reported by Die et al.^[40] After one Au atom in each of the Au_{n + 1} clusters is replaced by a Gd atom, the energy gaps of the newly formed Au_nGd clusters are narrower than those of Au_{n + 1} clusters except Au₁₂Gd and Au₁₅Gd that may be related to their peculiar structures in comparison with Au₁₃ and Au₁₆ clusters. Especially, the energy gaps of Au₈Gd, Au₁₀Gd, and Au₁₄Gd are close to zero. The decrease of HOMO–LUMO energy gap implies the increase of metallic bonding and decrease of chemical stability of the Au_nGd cluster. This means that Au_nGd small clusters promise to be used as new catalysts with more reactive and even metallic properties.

Fig. 7.

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	Fig. 7. Calculated HOMO–LUMO energy gaps of the ground state AunGd and pure Aun + 1 clusters.