Ab Initio Theoretical Investigation on the Geometrical and Electronic Structures of Gallium Aurides: $$ and $$  (n = 1–4)

3.1. Geometric and Electronic Structures of GaAu_n and $$

$$ clusters with n–Au terminals possess geometric structures and bonding patterns similar to those of the corresponding gallium hydrides $$ [38]. As shown in Figure 1, low-spin electronic states are consistently favored in $$. At all levels of theory, the ground structure GaAu⁻ anion (1, ²Σ⁺) has a bond length of r_Ga–Au = 2.52 Å and is 1.76 eV more stable than its quartet isomer (2, ⁴Σ⁺) at the CCSD(T) level. The most stable GaAu neutral structure (3, ¹Σ⁺) possesses a bond length of r_Ga–Au  = 2.45 Å. For GaAu₂, the V-shaped C_2v  $$ (5, ¹A₁) with a bond length of r_Ga–Au = 2.55 Å is the ground state and is 0.80 eV more stable than the linear C_∞v  $$ (6, ¹Σ⁺) at the CCSD(T) level. V-shaped C_2v GaAu₂ (7, ²B₂) is the most stable geometry on the potential surface of neutral GaAu₂. A large geometric change may be observed upon electron detachment from the anion $$ 4 to the neutral C_2v GaAu₂ 7, although these molecules have the same symmetry: the Ga–Au bond length increases by 0.08 Å, the Au–Au distance decreases by 1.19 Å, and the Au–Ga–Au bond angle considerably decreases by 38° in the anion relative to the neutral molecule.

The perfect planar triangular $$ structure has D_3h symmetry (9,²A₁′) with a bond length of r_Ga–Au = 2.49 Å and an Au–Ga–Au bond angle of AuGaAu = 120°. This structure is the ground state form and is 0.18 eV more stable than the off-plane C_s  $$  (10, ²A′) at the CCSD(T) level. Neutral GaAu₃ (11,¹A₁′) with an sp² hybridized Ga at the center of the molecule is a closed-shell singlet with D_3h symmetry. Compared with the anion, the neutral molecule only exhibits slight shortening of the Ga–Au bond length (0.1 Å).

On $$, we calculated several isomers and found that the perfect tetrahedral $$ (13, ¹A₁) has an sp³ hybridized Ga. This structure is the ground geometry; here, the four –Au terminals are singly σ-bound to the central Ga with a bond length of r_Ga–Au = 2.45 Å and a Wiberg bond index of WBI_Ga–Au = 0.93. T_d  $$ (13) is separated by at least 0.14 eV from other 2D and 3D isomers at the CCSD(T) level, which suggests that an Ga^– tetrahedral center is strongly favored in the $$ anion. Interestingly, T_d  $$ (13) has the shortest Ga–Au bond length and the largest HOMO-LUMO energy gap of ΔE_gap = 2.89 eV in the GaAu⁻ series. Detaching one electron from the perfect tetrahedral T_d  $$ (13) involves a John-Teller process to produce the severely distorted global minimum of C_s GaAu₄ (16, ²A′), which lies at least 0.22 eV higher than those of other low-lying isomers at the CCSD(T) level.

3.2. Bonding Consideration of GaAu_n and $$

NRT was used to calculate the bond orders and bond polarities of the molecules under study. As shown in Table 1, covalent contributions to the Ga–Au interactions continuously increase in the $$ series from n = 1 to n = 4. The Ga–Au bonds in $$ (13) have the highest percentage of covalence (78%). The Ga–Au bonds in $$ (9), $$ (5), and C_∞v  GaAu⁻ (1) show covalent contributions of 54%, 44%, and 40%, respectively. This result indicates that Ga–Au interactions in the $$ series render the characteristics of ionic structures, especially in $$ and GaAu⁻. The characteristic of the Ga–Au bond in the $$ series is also illustrated clearly by ELF analysis [27], which reflects the probability of finding an electron or a pair of pairs in specific basins (Figure 2). Contour line maps of C_∞v  GaAu⁻ (1), $$ (5), $$ (9), and $$ (13) reveal the presence of a weak electronic interaction between Ga and Au. This interaction is a highly polar covalent bond. NBO quantitatively reveals the Ga–Au bonding properties: the Ga–Au bond length of r_Ga–Au = 2.45–2.55 Å and the corresponding bond order of WBI_Ga–Au = 0.73–0.90 in the $$ series. These properties further indicate that the interaction is covalent but with ionic characteristics.

In the $$ series, the perfect tetrahedral $$ (13) is unique. Figure 3 shows the four valence molecular orbitals of the molecule, including a triply degenerate HOMO (t₂) and a singlet HOMO-1 (a₁). $$ has a bonding pattern similar to that of $$, with an sp³ hybridized Ga center surrounded by four Au atoms to form four equivalent σ single bonds. The B–Au and B–H σ bonds in $$ and $$ show subtle differences in orbital composition because of obvious relative effects in Au. $$ possesses the orbital combination of $$ and the corresponding atomic contribution of 40%Ga + 60%Au, with Au 6s contributing 95.2% and Au 5d contributing 4.7% to the Au-based orbital. In $$, Au 5d contributes 8.5%–4.7% to the Au-based orbital, which is less than that in monoboron aurides.

3.3. Geometric and Electronic Structures of Ga₂Au_n and $$

All low-lying neutral and anion clusters of $$ are summarized in Figure 4. $$ clusters with bridged Au atoms possess geometric structures similar to those of the corresponding gallium hydrides $$ [38]. As shown in Figures 4(a) and 4(b), the smallest digallium auride Ga₂Au^0/− contains a bridging Au atom. Anionic Ga₂Au⁻ exhibits three possible structures: triplet Au-bridged V-shaped (C_2v, 18, ³B₁), singlet Au-bridged V-shaped (C_2v, 19, ¹A₁), and triplet linear (C_∞v, 20, ³Σg⁻). At the CCSD(T) level, the triplet Au-bridged structure 18 with bond lengths of r_Ga–Au = 2.62 Å and r_Ga–Ga = 2.67 Å, respectively, lies 0.22 and 0.47 eV lower than the singlet Au-bridged 19 and triplet linear 20 structures. This result suggests that the triplet Au-bridged C_2v  Ga₂Au⁻ (³B₁, 18) is the ground state of Ga₂Au⁻. Similar to the V-shaped Ga₂H (Ga(μ-H)Ga), the doublet Au-bridged C_2v Ga₂Au (21, ²B₁) is a global minimum lying 0.72 eV lower than the linear C_∞v  Ga₂Au (22) at the CCSD(T) level. Adding one Au atom to bridge two Ga atoms in C_2v  Ga₂Au⁻ (18) and C_2v  Ga₂Au (21), respectively, produces the ground states of the off-plane di-Au-bridged C_2v  Ga₂Au⁻ ([Ga (μ-Au)₂Ga]⁻) (23, ²A₁) and C_2v  Ga₂Au₂ ([Ga (μ-Au)₂Ga]) (26, ¹A₁), which are at least 0.43 and 0.67 eV, respectively, more stable than other isomers. Di-Au-bridged C_2v  Ga₂Au₂ (26) possesses the same geometry as di-H-bridged C_2v  Ga₂H₂. For $$ (X=B, Al, Ga) systems, the global minima of $$ and $$ show similar V-shaped geometries. Both molecules differ from $$, which favors a linear structure containing a multiple-bonded BB core terminated by two Au atoms. This finding further demonstrates the presence of a strong chemical interaction between two B atoms in diboron aurides.

The interaction between two Ga atoms is so weak that two Au atoms prefer to bond with digallium auride isomers, such as D_∞h  Ga₂Au₂ (28). The anion $$ prefers a tri-Au-bridged [Ga(μ-Au)₃Ga]⁻ form with a singlet electronic structure. The most stable geometry of the tri-Au-bridged $$ (29, ¹A₁′) with a bond length of r_Ga–Au = 2.69 Å is at least 0.37 and 0.02 eV more stable than di-Au-bridged $$ (30, ¹A′) at the MP2 and CCSD(T) levels, respectively. Similar to Ga₂H₃ favoring a tri-H-bridged [Ga(μ-H)₃Ga] structure, the global minimum of Ga₂Au₃ is the tri-Au-bridged D_3h  Ga₂Au₃ (²A₁′, 32), which lies 0.54 and 1.17 eV lower than di-Au-bridged $$ (33, ²A′) and planar $$ (34, ²A₁), respectively. The Ga–Ga distance decreases by 0.48 Å and the Ga–Au bond length decreases by 0.09 Å in anion 29 relative to the neutral molecule 32. This result indicates a large geometry change upon electron detachment from the anion to the neutral molecule, although they have the same symmetry.

Adding one Au atom terminally to a Ga in $$ [Ga(μ-Au)₃Ga]⁻ (29, ¹A₁′) produces the ground state of tri-Au-bridged C_3v   $$ [Ga(μ-Au)₃Ga]Ga⁻ (35, ²A₁), which is 0.44, 1.33, and 0.92 eV more stable than di-Au-bridged $$ (36, ²A₁), distorted Y-shaped $$ (37, ²A′), and mono-Au-bridged $$ (38, ²A) at the CCSD(T) level, respectively. Ga₂Au₄ has the same high-symmetry ground state of tri-Au-bridged C_3v  Ga₂Au₄ Au⁺[Ga(μ-Au)₃Ga]⁻ (39, ¹A₁), which lies 1.61, 1.22, and 1.57 eV lower than Y-shaped C_s  Ga₂Au₄ (40, ¹A′), mono-Au-bridged Ga₂Au₄ (41, ¹A′), and nonbridged perpendicular D_2d  Ga₂Au₄ (42, ¹A₁), respectively. Similar to the tri-H-bridged C_3v  Ga₂H₄ [39], tri-Au-bridged C_3v  Ga₂Au₄ (39) is composed of strong interactions between a Ga⁺ cation and the face of a tetrahedral $$ anion. The global minima of Ga₂Au₄ and Al₂Au₄ have the same ionic conformer; both molecules differ from B₂Au₄, which has a di-Au-bridged covalent structure.

3.4. Bonding Consideration of Ga₂Au_n and $$

AdNDP analysis [27] is an effective tool for analyzing the bonding patterns of various organic and inorganic molecules. As shown in Figure 5, other bonds besides the lone pairs of the Au atom may be analyzed as follows. C_2v  Ga₂Au (21) contains one localized Ga–Ga 2c–2e σ-bond with an occupation number of ON = 1.96 |e|, one localized Ga–Ga 2c–2e σ-anti-bond with an occupation number of ON = 1.96 |e|, and one delocalized Ga–Au–Ga 3c–2e bond with an occupation number of ON = 2.00 |e|. C_2v  Ga₂Au₂ (26) contains one localized Ga–Ga 2c–2e σ-bond with an occupation number of ON = 1.90 |e|, one localized Ga–Ga 2c–2e σ-anti-bond with an occupation number of ON = 1.90 |e|, and two delocalized Ga–Au–Ga 3c–2e bonds with an occupation number of ON = 1.97 |e|. D_3h  Ga₂Au₃ (32) contains three delocalized Ga–Au–Ga 3c–2e bonds with an occupation number of ON = 1.94 |e|. C_3v  Ga₂Au₄ (39) contains one localized Ga₍₂₎–Au_(t) 2c–2e σ-bond with an occupation number of ON = 1.95 |e| and three delocalized Ga–Au–Ga 3c–2e bonds with an occupation number of ON = 1.93 |e|. The same number of electrons occupies the bonding and antibonding orbitals in C_2v  Ga₂Au (21) and C_2v  Ga₂Au₂ (26). Thus, no electronic effect is produced between two Ga atoms and the delocalized Ga–Au–Ga 3c–2e bond is the main interaction in high symmetric Ga₂Au_n  (n = 1–4).

Detailed NLMO analyses quantitatively reveal the existence of bridging Ga–Au–Ga 3c–2e bonds in C_2v  Ga₂Au (21), C_2v  Ga₂Au₂ (26), D_3h  Ga₂Au₃ (32), and C_3v  Ga₂Au₄ (39), as clearly shown in an image of their 3c–2e orbital and orbital combination (Figure 6). In C_2v Ga₂Au (21), the 3c–2e bond possesses the orbital combination of $$ and the corresponding atomic contribution of 15%Ga + 70%Au + 15%Ga. In the Ga–Au–Ga 3c–2e bond, Au 6s contributes 97.8% and Au 5d contributes 1.64% to the Au-based orbital, whereas Ga 4p contributes 98.2% and Ga 4s contributes 0.76% to the Ga-based orbital. Obviously, Au 6s and Ga 4p provide the largest contributions to the Ga–Au–Ga 3c–2e bond, and it can be practically approximated as τ_Ga–Au–Ga = 0.38(p)_Ga + $$ + 0.38(p)_Ga. The orbital combinations of Ga–Au–Ga 3c–2e bond in C_2v Ga₂Au₂ (26) [τ_Ga–Au–Ga = 0.39(p)_Ga + $$ + 0.39(p)_Ga] and D_3h  Ga₂Au₃ (32) [τ_Ga–Au–Ga = 0.42(p)_Ga + $$ + 0.42(p)_Ga] are surprisingly similar to that of C_2v  Ga₂Au (21). However, the composition of the 3c–2e orbital in C_3v  Ga₂Au₄ (39) is obviously different from that in Ga₂Au_n  (n = 1–3). Each 3c–2e bond has an orbital combination of $$ and the corresponding atomic contribution of 16%Ga₍₁₎ + 50%Au + 35%Ga₍₂₎.

In the bridging Ga₍₁₎–Au–Ga₍₂₎ 3c–2e bond, bridged Au 6s and 5d, respectively, contribute 80.7% and 1.4% to the Au-based orbital, Ga₍₁₎ 4s and 4p, respectively, contribute 0.7% and 98.6% to the Ga₍₁₎-based orbital, and Ga₍₂₎ 4s and 4p, respectively, contribute 17.3% and 81.8% to the Ga₍₂₎-based orbital. Obviously, the 17.3% contribution from Ga₍₂₎ is not negligible because of the Ga₍₂₎ atom of the $$ unit. Au 6s and Ga₍₁₎ 4p provide the largest contributions to the Ga₍₁₎–Au–Ga₍₂₎ bridging bond in C_3v Ga₂Au₄ (39). This finding agrees with the ionic characteristic of $$ [Ga₍₁₎(μ-Au)₃Ga₍₂₎]⁻ presented earlier. Thus, in contrast to the orbital combination in Ga₂Au_n    (n = 1–3), the bridging bond of C_3v  Ga₂Au₄ (39) can be approximated as $$. Similar 3c–2e orbital combinations exist in the corresponding anions. Compared with the bridging Au 3c–2e bond observed in electron-deficient systems (B₂Au_n, Al₂Au_n, and Ga₂Au_n), we found that bridging Au provides greater contributions to dialuminum and digallium aurides (70%–68%) than to diboron aurides (50%–45%). Specifically, Au 5d contributes less than 2% to the Au-based orbital in dialuminum and digallium aurides.

3.5. Electron Detachment Energies

The ADE and VDE values of the anions were calculated in PES experiments. As shown in Table 2, the B3LYP and CCSD(T)//B3LYP levels produced consistent one-electron detachment energies for $$ and $$ anions. Except for C_2v  $$ (5) and $$ (13), $$ (29), the calculated ADEs and VDEs at the CCSD(T) level lay at 0.30–1.87 eV. The small differences between ADE and VDE (0.03–0.32 eV) agree with the minor structural relaxation observed between the anion and the corresponding neutral molecule. At the same level, $$ (5) shows ADE = 2.33 eV and VDE = 2.72 eV. The difference between the ADE and VDE (0.39 eV) shows considerable structure relaxation between the C_2v anion (5) and the C_2v neutral molecule (7). A similar result was observed in $$ (29). $$ (13) anion has the calculated one-electron detachment energies of ADE = 3.07 eV and VDE = 4.17 eV at the CCSD(T)//B3LYP level. B3LYP approaches produced close ADE and VDE values with CCSD(T). The extremely high electron detachment energies of $$ indicate that GaAu₄ neutrals lie considerably higher than $$ anions in energy, while the big ADE-VDE differences (0.72–1.10 eV) agree with the considerable structural relaxation from $$ (13) and its closely related C_s  GaAu₄ (16). The electron binding energies of these anions fall within the energy range of the conventional excitation laser (266 nm, 4.661 eV) in PES measurements.