Geometrical Structures and Electronic Properties of Ga₆ and Ga₅X (X = B, C, N, O, F, Al, Si, P, S, Cl) Clusters.

Based on the unbiased CALYPSO (Crystal structure Analysis by Particle Swarm Optimization) structure searching method in combination with density functional theory (DFT), the geometrical structures and electronic properties are investigated theoretically for Ga₆ and Ga₅X (X = B, C, N, O, F, Al, Si, P, S, Cl) clusters. The PBE0 exchange-correlation functional and the 6-311G(d) basis set is carried out to determine global minima on potential energy surfaces. The relative stabilities of the clusters are examined by the binding energies and substitution reaction. Following the predictions of the Jellium model, the Ga₅B cluster with the 18 valence electrons is the most stable structure. At last, with the obtained lowest energy structures, some physical properties such as electrons transfer, molecular orbitals, and total and partial densities of states are discussed, respectively.

1. Introduction

Due to the unusual structure and bonding characteristics, al">gallium has attracted a <al">span class="Chemical">significant amount of attention for its extensive application in the thin-layer growth and the thin-film deposition in the deposition of multilayer structures [1,2,3]. In the last decade, many theoretical and experimental studies of gallium clusters have been reported [4,5,6,7,8,9,10,11]. For example, based on the density functional theory (DFT), the lowest-energy geometrical and electronic structures of Gan clusters containing up to 26 atoms have been calculated with generalized gradient approximation for the exchange-correlation potential [4]. By a simulated annealing technique with ab initio molecular dynamics (AIMD) method, Gong and Tosatti determined the lowest energy structures of small Gan (n ≤ 8) clusters [6]. The atomic structure, structural transformation, and reactivity of Ga13 clusters were investigated by the first principles’ pseudopotential calculations [7]. With a similar approach, Jones obtained the most stable structures of Gan cluster up to n = 10 [9]. Based on eight different density functional theory and hybrid DFT/Hartree–Fock methods, the structures, dissociation energies and electron affinities of the anionic and neutral gallium clusters were calculated by Zhao et al. [10]. For the experimental research, photoelectron spectra of small anionic gallium clusters were presented in the size range of n = 1–15 [12]. Results show that a change in the pattern of the spectra near n = 6 can be predicted as a transition from planar to compact three-dimensional structures. A study of the electronic spectrum of the gallium dimer was presented in the range of 33,600–36,800 cm−1 [13]. In addition, Himmel et al. have characterized isolated Ga2 dimers in an argon matrix with the aid of resonance Raman and UV/Vis spectroscopy [14]. Then, Balducci et al. characterized the dissociation energies of Ga2 cluster, which derived from the Knudsen cell-mass spectrometric [15].

In order to modify or tune phy<span class="Chemicn>an class="Chemical">al">sical and chemical properties of clusters, doping, which is the intentional incorporation of impurities into materials, is a widely used strategy. When X (X = B, C, N, O, F, Al, <n>an class="Chemical">al">span class="Chemical">Si, P, S, Cl) atom was doped into Ga6 clusters, the introduction of a doped atom in Ga6 clusters can change undoubtedly its geometrical structures and electronic properties, and further affect its chemical and physical properties significantly. A number of gallium clusters doped with a single X (X = N, O, F, Al, Si, P) atom have been observed and studied [16,17,18,19,20,21,22,23,24,25,26]. Based on the density functional theory (DFT), Costales et al. have studied the structures of GanNn (n = 1–6) clusters [16,17]. Combined with molecular dynamics and simulated annealing techniques, Song calculated the structures of GanNn (n = 3, 5, 6) clusters using a full-potential linear-muffintin-orbital method [18,19,20]. The Gan−1Al+ clusters were generated by pulsed laser ablation of a liquid aluminum/gallium alloy [21]. By an all-electron linear combination of atomic orbital approach, low-lying isomers of GanAl (n = 1–15) clusters were reported [22]. The ground state structures, stability and electronic properties were systemically investigated for neutral, cationic and anionic GanOn (n = 4–7) clusters [23]. The geometrical structures, electronic states and energies of GanPm (n = 1–7; n = 1, 2) clusters were investigated using the density functional theory [24]. The equilibrium geometries and electronic states of Ga3Sn, GaSn3 and their ions were investigated using the complete active space self consistent field (CASSCF) levels and DFT [25]. Using ab initio quantum chemistry, the feasibility of laser cooling GaF was performed. The X1∑+, 3∏ and 1∏ states of GaF was calculated using the multireference configuration interaction (MRCI) level of theory [26]. However, a systematic study on X-doped gallium (X = B, C, S, Cl) clusters has not been reported, and some problems on the Ga5X (X = B, C, S, Cl) clusters also have not been solved either. For example, how does the structure of the gallium clusters change with the X atom doped, and how can the pure gallium cluster properties be changed after doping a single X atom?

In cluster phyal">sics, the Jellium model predicts that small clusters with a certain valence electronic configuration (8, 18, 20, 34, 40, 58…), which were known as magic numbers, exhibit increased stability compared with their neighboring configuration. However, some works have shown that the number of electrons not correal">sponding to the magic numbers also produces extra stability [27,28]. The <al">span class="Chemical">nitrogen-doped (GanN, n = 1–9) clusters were investigated at the PBE/DNP level of theory. It is found that GanN (n = 3, 7, 15) clusters were particularly stable despite of the fact that these clusters do not conform to the predictions of the Jellium model because they have 14, 26 and 50 valence electrons. In addition, Rebere et al. reported that Ag13−, Ag14, and Ag15+ clusters with 14 valence electrons were resistant to reactivity with O due to their large highest occupied-lowest unoccupied molecular orbital (HOMO-LUMO) energy gaps, despite not having fully filled electronic shell configurations. Inspired by the above studies, when there are X-doped (X = B, C, N, O, F, Al, Si, P, S, Cl) gallium clusters, do the stable X-doped structures possess the magic number? Hence, it is of interest to carry on more detailed studies on the magic numbers.

With this purpose in mind, we systematically studied the geometrical structures and electronic properties of Ga6 and al">Ga5X (X = B, C, N, O, F, Al, <al">span class="Chemical">Si, P, S, Cl, Ga) clusters based on a particle swarm optimization algorithm combined with density function theory. For all of these clusters, relative stability was measured via analysis of binding energies and substitution reaction. Based on the HOMO-LUMO gaps and chemical hardness, the stable structures were determined for Ga5X (X = B, C, N, O, F, Al, Si, P, S, Cl, Ga) clusters. At last, the electron transfer, molecular orbitals, and density of states are also computed and discussed.

2. Computational Detail

It is well known that the number of posal">sible geometric configurations of clusters increases exponentially with the <al">span class="Chemical">size of the clusters, it will be more complex for binary clusters. Thus, it is necessary for us to search for an effective method in structure prediction. In the present paper, we performed through the intelligent methodology [29,30], as implemented in the CALYPSO (Crystal structure Analysis by Particle Swarm Optimization) code [30]. The algorithm can predict stable structures depending only on the chemical composition. It has been successful in correctly predicting structures for various systems [31,32,33,34]. The optimized structures obtained from particle swarm optimization (PSO) are taken as the initial structures for further simulation by the DFT method. Within the CALYPSO structure search, each generation contains 20 structures, 70% of which are generated by a particle swarm optimization algorithm. The others will be generated randomly. We followed 50 generations for each cluster to achieve convergence of the potential energy surface sampling. Then, 1000 structures of variable structures can be obtained for every Ga5X (X= B, C, N, O, F, Al, Si, P, S, Cl) and Ga6 clusters. The further geometry optimizations are performed with the PBE0 functional using the 6-311G(d) basis set, as implemented in the Gaussian09 package [35]. PBE0 has been confirmed to be suitable for describing the energy difference of isomers of X-doped gallium clusters [36]. In the process of calculation, the effect of the spin multiplicity is performed in the geometric optimization procedure. Harmonic vibrational frequencies are calculated to ensure that the obtained optimized structures are real minima with no imaginary frequencies.

The accuracy of the present DFT method was assessed by benchmark calculations for available experimental and theoretical results. For Ga2, the calculated dissociation energy and vibrational frequency are 1.34 eV and 214 cm−1, real">spectively, in excellent agreement with experimental values 1.50 eV and 180 cm−1 [14,37]. To our knowledge, there is no bond length in the experiment for the <al">span class="Gene">Ga2 dimer. The present Ga–Ga bond length (2.45 Å) is broadly consistent with the previous theoretical studies [4,6,7,9]. For GaAl and GaSi, we obtained a bond length (2.456 Å, 2.43 Å) that fits well with the theoretical values of 2.585 Å and 2.45 Å by the GGA using B3LYP function, respectively [23,38]. In addition, for Ga6 cluster, the calculated vertical electron affinity (VEA) (2.12 eV) and vertical ionization potential (VIP) (6.69 eV) are close to the experimental values (2.60 eV) and theoretical vertical ionization potential (6.52 eV), respectively [12]. In short, PBE0/6-311G(d) level is able to describe the structural and electronic properties of Ga6 and Ga5X (X = B, C, N, O, F, Al, Si, P, S, Cl) clusters in a satisfactory manner.

3. Results and Discussion

3.1. Lowest Energy Structures and Growth Pattern

The lowest energy structures of Ga6 and <span class="Chemical">Ga5X clusters determined from CALYPSO and DFT calculation are dipan class="Chemical">al">splayed in Figure 1. The correpan class="Chemical">al">sponding electronic states, symmetries and harmonic vibrational frequencies are tabulated in Table 1, and the atomic coordinates of each cluster are provided in the Supporting Information (<al">span class="Chemical">SI) for the lowest energy structures.

Figure 1

Optimized geometries of the lowest energy Ga5X (X = B, C, N, O, F, Al, Si, P, S, Cl, Ga) clusters.

Table 1

State, symmetry, vertical ionization potential (VIP) (eV), vertical electron affinity (VEA) (eV), η (eV), Egaps (eV), and harmonic vibrational frequencies (cm−1) of Ga5X (X = B, C, N, O, F, Al, Si, P, S, Cl and Ga) clusters.

Isomer	State	Sym	VIP	VEA	η	Egaps	Frequencies
							Lowest Frequency	Intensity Vibration
Ga5B	1A1	C2v	6.76	1.16	5.60	2.879	36(a2)	543(a1)
Ga5C	2A″	Cs	6.67	1.95	4.72	1.934	15(a″)	547(a′)
Ga5N	1A1	C2v	6.44	1.56	4.88	2.241	27(b1)	631(a1)
Ga5O	2A′	Cs	6.38	1.79	4.59	1.812	46(a′)	449(a′)
Ga5F	1A′	Cs	6.64	1.49	5.15	2.360	60(a′)	424(a′)
Ga5Al	3A″	Cs	6.70	2.13	4.57	1.874	28(a″)	170(a′)
Ga5Si	2A′	Cs	6.79	2.16	4.63	1.836	18(a″)	207(a″)
Ga5P	1A′	Cs	6.88	1.68	5.2	2.387	45(a″)	336(a′)
Ga5S	2A′	Cs	6.65	2.05	4.6	1.837	30(a″)	289(a′)
Ga5Cl	1A′	Cs	6.74	1.45	5.29	2.536	44(a′)	274(a′)
Ga6	3A1′	D3h	6.70	2.12	4.58	1.889	29(a1′)	115(e′)

As for the Ga6 cluster, the 3A1′ state of the trigonal prism (D3h) is almost degenerate with the 1A1 state of the <span class="Chemical">similar trigonal prism, and the latter is computed to be 0.003 eV lower in energy, which is a certainly in<al">span class="Chemical">significant amount. Our result agrees with Drebov and Jones et al.’s results [9,39]. However, Gong and Tosatti applied an ab initio molecular-dynamics method and showed the most stable structure to be a distorted prism structure with C2v point symmetry [6].

For a al">Ga5Al cluster, u<al">span class="Chemical">sing the B3LYP-DFT calculations, Guo et al. predicted that the ground state is Cs structure and analogous to Ga6 structure, which is identical to our results [23]. Simultaneously, the calculated the vertical ionization potential (VIP) of 6.70 eV and the vertical electron affinity (VEA) of 2.13 eV in our work agree with previous DFT estimates (6.44 and 2.11 eV) calculated by Guo et al. [23].

Li et al. [25] calculated geometrics of <span class="Gene">GanP cluster and predicted that the ground state is the trigonal bipyramidal state (Cs). From Figure 1, we can find that our result is identical to Li et al. In addition, our calculation predicted the VEA of the ground state (1.68 eV) agree with Li’s results (1.93 eV).

For the <span class="Chemical">Ga5N cluster, in Li’s calculation, the ground state structure with C2v point symmetry is a planar structure. In our prediction, the result is <al">span class="Chemical">similar to Li and Song’s results [20,40]. Moreover, the calculated intensity vibration and lowest frequency (631 and 27 cm−1) are close to the Li et al.’s results (672.3 and 32 cm−1) [40].

Although the ground state structures of the al">Ga5X (X = B, C, S, Cl) clusters have not been reported, the above structures are obtained and listed in Table 1. Comparing X-doped (X = B, C, N, O, F, <al">span class="Chemical">Si, P, S, and Cl) clusters, we can find that the doped atom simply replaces one Ga atom of pure Ga6 cluster, leading to a reduction of the molecular symmetry for the lowest stable configurations of Ga5B(1A1), Ga5C(2A″), Ga5O(2A′), Ga5F(1A′), Ga5Al(3A″), Ga5Si(2A′), Ga5S(2A′) and Ga5Cl(1A′). On the contrary, completely new structures are formed for Ga5N(1A1) and Ga5P(1A′) clusters. In conclusion, except for Ga5N and Ga5P clusters, the dopants of lighter elements (B, C, O, F) greatly influence the structure of the pure gallium clusters; however, those of heavier elements (Al, Si, S, Cl) generally generate a minor effect.

3.2. Relative Stability

The relative stability of clusters can be compared with the computing the binding energy (BE). Binding energies are calculated by the enthalpy change for the following reaction:5Ga + X→Ga

In addition, the stability can also be measured by the enthalpy of substitution reaction (∆Hsubst), which can be expressed as:Ga

In Figure 2, the values of the BE and ∆Hsubst, corrected for zero point vibrational energy, are shown with the cluster <span class="Chemical">size increa<al">span class="Chemical">sing.

Figure 2

(a) Binding energies and (b) ∆Hsubst of substituted gallium clusters (Ga5X) (eV).

As seen in Figure 2, the Ga6 cluster with 18 valence electrons possesses the maximum value of the binding energy. The isoelectronic al">Ga5Al and <al">span class="Chemical">Ga5B clusters are also possessed of considerably larger binding energies, indicating greater overall stability. The Ga5Al cluster has larger binding energy than that of the Ga5B cluster. Moreover, this result can be obtained for all isoelectronic X-doped systems. In general, the tendency of ∆Hsubst is similar to that of the binding energies, and the Ga5C cluster leads to the largest stabilizing effect (∆Hsubst = −3.38 eV). Theoretically, the dopants of B and Al should arouse a minor influence on the stability of the doped clusters because they are equivalent to the valence electron of the Ga atom. In fact, we find that the Al atom actually leads to an increase in stability (0.16 eV) while the B atom leads to an even greater increase in stability (2.16 eV). Doping with O and S elements leads to a 21 valence electrons system and would not be expected to lead to a significant increase in stability. However, it is clear from Figure 2 that X-doped (X = O, S) clusters increase the stability of the cluster. Generally speaking, based on the BE and ∆Hsubst, the stability of host Ga6 clusters will indeed be enhanced by the dopants of X atoms.

3.3. The HOMO-LUMO Gaps

In cluster science, the HOMO-LUMO gaps (Egap) can be usually used for estimating the chemical stability of the clusters. Larger values of the gaps indicate stronger chemical stability. In Table 1, we list the calculated energy gaps of the mentioned clusters. It is interesting to notice that the <al">span class="Chemical">Ga5B, Ga5N, Ga5F, Ga5P and Ga5Cl clusters with an even number of electrons (18, 20, 22) have higher stabilities than other Ga5C, Ga5O, Ga5Si and Ga5S clusters with an odd number of electrons (19, 21). For Ga5B, Ga5N, Ga5F, Ga5P and Ga5Cl clusters, all paired electrons form a closed-shell electronic structure caused the larger energy gap of 1.87–2.88 eV. However, the Ga5C, Ga5O, Ga5Si and Ga5S clusters show a small energy gap due to the LUMO occupied by a single electron. In addition, the local maxima of Egap are found at Ga5B, suggesting that the Ga5B cluster possesses enhanced relative stability, and the clusters with 18 valence electrons are expected to be especially stable by the Jellium model.

Chemical <span class="Disease">hardness, proposed by Pearson [41,42], can also be viewed as a parameter to measure the relative stability of the clusters. The chemical <al">span class="Disease">hardness (η) can be expressed as

VIP and VEA are the vertical <span class="Disease">ionization potential and vertical electron affinity, real">spectively. From Table 1, one can see that the tendency and the extremum (<al">span class="Chemical">Ga5B) of η completely accord with the above analysis based on the HOMO-LUMO gaps.

3.4. Charge Transfer in the Ga5X Cluster

To probe into the internal charge transfer of these X-doped al">gallium clusters, we extended the charge populations on Ga and X atoms inferred from the Hirshfeld, Mulliken, natural population analy<al">span class="Chemical">sis (NPA) and Bader approaches based on the Multiwfn program (Tian Lu and Feiwu Chen, Beijing, China) [43]. As is shown in Table 2, there is less positive charge on Ga with less negative charge on X, expect for Ga5Al clusters. The results show that the values for the X (X = B, C, N, O, F, Si, P, S, Cl) atoms are almost negative. This means that the charge transfers from the Ga host to X (X = B, C, N, O, F, Si, P, S, Cl) atoms in the corresponding clusters because of the higher electronegativity of the X (X = B, C, N, O, F, Si, P, S, Cl) atom than that of the Ga atom (Ga:1.81 eV, B:2.04 eV, C:2.55 eV, N:3.04 eV, O:3.44 eV, F:3.98 eV, Si:1.90 eV, P:2.19 eV, S:2.58 eV, Cl:3.16 eV) [44]. However, the electronegativity of the Ga (1.81 eV) is more than that of the Al (1.61 eV) [44]. One can see that the charge transfers from Al atoms to Ga frames in Ga5Al clusters. This fact has also been observed in a number of gallium-containing clusters in earlier reports [31].

Table 2

Theoretical atomic charges on X atoms (X = B, C, N, O, F, Al, Si, P, S, Cl).

Methods	B	C	N	O	F	Al	Si	P	S	Cl
Hirshfeld	−0.399	−0.436	−0.405	−0.366	−0.274	0.001	−0.175	−0.224	−0.248	−0.177
Mulliken	−0.861	−1.350	−1.150	−0.822	−0.471	0.254	0.023	−0.061	−0.156	−0.262
NPA	−2.865	−1.395	−2.179	−0.789	−0.755	0.189	−0.332	−0.921	−0.503	−0.485
Bader	−1.282	−1.816	−1.673	−1.343	−0.773	0.350	−0.371	−0.857	−0.974	−0.591

3.5. The Molecular Orbitals

In Figure 3, an examination of the molecular orbitals (MOs) is presented for understanding the shell model of the most stable al">Ga5X (X = B, C, N, O, F, Al, <al">span class="Chemical">Si, P, S, and Cl) clusters. In an effort to obtain the specific orbital composition of every MO, the corresponding orbital compositions are also plotted in Figure S1 (see SI).

Figure 3

The molecular orbitals for the lowest energy structure of Ga5X (X = B, C, N, O, F, Al, Si, P, S, Cl) clusters.

From the Figure 3, it can be seen that the HOMO and LOMO are localized around the Ga atoms. For <al">span class="Chemical">Ga5X (X = B, C, O) clusters, the HOMO involves the s, px, py and pz from all of Ga atoms, the s, p of B atoms, the p of C atoms; the contribution of py is very small from oxygen atoms. The LUMO of Ga5X (X = B, C) cluster is composed of (s, py, pz), (px, py, pz) of Ga and py of B atoms; the effect of p of C is very small. Moreover, the LUMO involves Ga-s, px, py and pz atomic orbitals as well as px and py from oxygen atom in Ga5O clusters. For Ga5F and Ga5Cl clusters, the HOMO mainly comes from the (s, pz) and (s, px, pz) orbitals of Ga atoms along with small mixture of px and py orbitals in X (X = F, Cl) atoms. The LUMO is almost identical in Ga5F and Ga5Cl clusters. It results from the identical orbital. Gallium atoms involve s, px, py, and pz component orbitals; F and Cl atoms possess the same pz atomic orbitals. The HOMO of the Ga5Al and Ga5Si clusters is very similar. However, the HOMO includes the px, py, pz states of the Ga atoms and the s, px, py states of the Al atom in the Ga5Al cluster, whereas the HOMO mostly involves the px of gallium and silicon atoms as well as a small mixture of s orbitals of the Ga and Si atoms in the Ga5Si cluster. For Ga5X (X = N, P, S) clusters, the HOMO mainly comes from the same p, and p orbitals of Ga atoms in Ga5N and Ga5P clusters. In LUMO, all of the atoms are composed of the p, p, and p atomic orbitals in Ga5N, Ga5P, and Ga5S clusters, respectively. The contribution of N, P, and S atoms are almost zero in HOMO and LUMO. These molecular orbitals indicate the presence of sp hybridization of the X atom and Ga atoms.

3.6. The Density of States

To understand the nature of the chemical bonding of these clusters, the denal">sity of states (DOS) of <al">span class="Chemical">Ga5X (X = B, C, N, O, F, Al, Si, P, S, and Cl) and Ca6 clusters are shown in Figure 4. It is clear that there is finite electronic density, mainly coming from Ga atoms, at the Fermi level for all clusters, which means that they exhibit metallic behavior. At the Fermi level, the total density of states (TDOS) of Ga5X clusters is lower than that of the Ga6 cluster except for Ga5N and Ga5Si clusters. The fact indicates that the impurity atoms reduce the metallicity of the Ga5X (X = B, C, O, F, Al, P, S, and Cl) clusters. The contribution of the impurity atoms primarily centralizes in the valence bands for Ga5X (X = B, C, N, O, F, S, and Cl) clusters, and they are comparative in the valence bands and conduction band for Ga5X (X = Al, Si, and P) clusters. In addition, the HOMO energy levels shift to higher energy levels when X (X = C, N, O, F, and S) are doped gallium clusters, whereas they shift to lower energy levels when other atoms are doped gallium clusters.

Figure 4

Calculated total densities of states (TDOS) and partial densities of states (PDOS) of Ga5X (X = B, C, N, O, F, Al, Si, P, S, Cl) clusters.

4. Conclusions

In the present study, we have performed a global minimum search for the ground state structures of Ga6 and al">Ga5X (X = B, C, N, O, F, Al, <al">span class="Chemical">Si, P, S, Cl) clusters by using the CALYPSO method in combination with DFT geometry optimization. The optimized geometries reveal that the dopant atoms (B, C, O, F, Al, Si, S, and Cl) simply replace one Ga atom of pure Ga6, leading to a reduction of the molecular symmetry. However, completely new structures are formed for Ga5N and Ga5P. Trends of the binding energies and substitution reaction showed that the dopant atoms can lead to increased stability relative to Ga6 clusters. Based on the HOMO-LUMO gaps and chemical harnesses, the Ga5B clusters with 18 valence electrons exhibit superior stability and can be viewed as the magic number clusters. In addition, analysis of molecular orbitals and DOS show that the HOMO and LOMO are localized around the Ga atoms, and the contribution of X atoms is almost zero. These molecular orbitals indicate the presence of sp hybridization of the X atom and Ga atoms. At last, the electrons transfer from the Ga frames to X (X = B, C, N, O, F, Si, P, S, Cl) atoms. On the contrary, the direction of charge transfer reverses and the amount of charges transfer from Al atoms to Ga atoms in Ga5Al clusters.