Spin-Orbit Coupling Effect on the Electrophilicity Index, Chemical Potential, Hardness and Softness of Neutral Gold Clusters: A Relativistic Ab-initio Study

Electrophilicity index ( ) is related to the energy lowering associated with a maximum amount of electron flow between a donor and an acceptor and possesses adequate information regarding structure, stability, reactivity and interactions. Chemical potential (μ) measures charge transfer from a system to another having a lower value of μ, while chemical hardness (η) is a measure of characterizing relative stability of clusters. The main purpose of the present research work is to examine the Spin-Orbit Coupling (SOC) effect on the behavior of the electrophilicity index, chemical potential, hardness and softness of neutral gold clusters Aun (n=2-6). Using the second-order Douglas-Kroll-Hess Hamiltonian, geometries are optimized at the DKH2-B3P86/DZP-DKH level of theory. Spin-orbit coupling energies are computed using the fourth-order Douglas-Kroll-Hess Hamiltonian, generalized Hartree-Fock method and all-electron relativistic double-ζ level basis set. Then, spin-orbit coupling (SOC) corrections to the electrophilicity index, chemical potential, hardness and softness are calculated. It is revealed that spin-orbit correction to the vertical chemical hardness has important effect on Au3 and Au6, i.e. SOC decreases (increases) the hardness of gold trimer (hexamer). Due to the relationship between hardness and softness, σ = 1 ⁄ , inclusion of spin-orbit coupling increases (decreases) the softness of Au3 (Au6) and thus destabilizes (stabilizes) it. Spin-orbit coupling (SOC) also has more important effect on the chemical potential of Au3 by decreasing its value. It is found that spin-orbit coupling has considerable effect on the electrophilicity index of gold trimer and greatly increases its value. Furthermore, SOC increases the maximal charge acceptance of Au3 more and thus destabilizes it more. As a result, spin-orbit coupling effect appears to be important in calculating the electrophilicity index of the gold trimer.


Introduction
As a kind of promising nanomaterials, metal nanoclusters (NCs) have sparked wide-spread attention. In recent years, gold nanoparticles (AuNPs) have been applied to biomedicine and biological sensing [1][2][3][4][5] due to their biocompatibility and unique physical properties. AuNPs are one of the most promising catalysts, in spite of bulk Au as an inactive material [6][7][8]. Most of the computations on small neutral gold clusters have been performed using spinfree (scalar-relativistic) methods [9][10][11][12]. To obtain reliable theoretical results for gold clusters, the scalar relativistic correction is substantial. However, the spin-orbit coupling is expected to be important. There exist theoretical studies regarding spin-orbit coupling effect using effective core potentials or plane-wave basis sets [13][14][15][16][17][18][19]. These studies mainly focused on the spin-orbit coupling effect on the highest-occupied lowest-unoccupied (HOMO-LUMO) energy gaps, geometries, binding energies per atom and optical absorption of gold clusters. Xiao and Wang [13] using

Computational Methods
All calculations are performed using the Gaussian 09 suite of program [22] and the plots of molecular configurations and contour maps are generated with the GaussView software [23]. The B3P86 functional is used in conjunction with the valence double-ζ quality plus polarization functions (DZP-DKH [8s7p4d2f]) basis set [24]. The B3P86 functional has already proven to perform well for ionization potential computations of small neutral gold clusters [10]. The second-order Douglas-Kroll-Hess Hamiltonian [25][26][27][28][29] is used instead of the Schrödinger operator. For geometry optimizations, the coordinates are chosen according to the experimentally determined structures by Gruene et al. [30]. Using the second-order Douglas-Kroll-Hess Hamiltonian, all geometries are fully optimized at the DKH2-B3P86/DZP-DKH level of theory followed by harmonic vibrational frequency analysis. Then, from these optimized geometries, the spin-orbit coupling (SOC) energies are calculated using the fourth-order Douglas-Kroll-Hess Hamiltonian and the generalized Hartree-Fock method [31]. The electronic energy including spin-orbit coupling, , is calculated using the following definition, Where, is the electronic energy obtained from spin-orbit free (scalar-relativistic) optimizations at the DKH2-B3P86/DZP-DKH level, and is the spin-orbit coupling energy calculated by the DKH2-B3P86/DZP-DKH //DKHSO-GHF/DZP-DKH level of theory. In order to determine the spin-orbit coupling (SOC) effect on the electronic properties of neutral gold clusters, the spin-orbit corrections to the chemical hardness, softness, chemical potential and electrophilicity index [20,32] of the gold clusters are computed. In the following, the spin-orbit correction to a particular property, , is defined using, Where, and are the calculated properties with considering the spin-orbit coupling energy, , and without considering the spin-orbit coupling energy, , respectively. Within the valence state parabola model [20], the chemical potential, chemical hardness, softness and electrophilicity index are introduced by, Where N, v, IP and EA are the number of electrons, the potential due to the nuclei plus any external potential, ionization potential and electron affinity, respectively. The spin-orbit corrections to these properties are calculated from, The aforementioned properties are calculated using the both vertical and adiabatic ionization potential and electron affinities. Figure 1 illustrates the steps in calculating the spin-orbit coupling (SOC) effect on the electrophilicity index, chemical potential, hardness and softness of neutral gold clusters. Calculation of the spin-orbit corrections to these properties: Calculation of the electronic energy including spin-orbit coupling: = +

Results and Discussion
The optimized structures of gold clusters at the DKH2-B3P86/DZP-DKH level of theory are displayed in Figure 2. Table 1 compares the apex angle, symmetry and electronic state of the two Jahn-Teller components of gold trimer. The isomer with the lowest energy is an acute angle triangular structure with C2V symmetry, a 58.455° apex angle, and a 2.537 A° bond length. The result is in agreement with the coupled cluster calculations of Schwerdtfeger et al. [33]. In fact, because of the Jahn-Teller effect, the D3h symmetry of Au3 distorts to an acute triangular C2V structure. The lowest-energy structures of Au4 and Au5 are a trapezoid with C2V symmetry and a W-shaped geometry, respectively. The D3h planar triangular structure of Au6 is obtained by adding one gold atom to W-shaped planar Au5.

B2
The acute gold trimer has lower energy than the obtuse one at the DKH2-B3P86/DZP-DKH level of theory.
The calculated vertical and adiabatic ionization potential, electron affinity, chemical potential, hardness, softness and electrophilicity index values are given in Tables 2 to 5. Table 6 presents the spin-orbit coupling (SOC) corrections to the chemical potential, hardness, softness and electrophilicity index. Chemical hardness has been used to characterize the relative stability of clusters. The principle of maximum hardness (PMH) states that systems at equilibrium present the highest value of hardness [34]. The computed hardness values with and without considering spin-orbit coupling (SOC) effect are plotted in Figure 3a. According to this Figure, the chemical hardness computed using adiabatic electron affinities and ionization potentials exhibits an odd-even oscillation behavior, whether spinorbit coupling is considered or not. The even-sized clusters with closed-shell electronic configurations have higher values of chemical hardness compared to their immediate open-shell neighbors, indicating their higher stability. This is in agreement with the previous study of Singh and Sarkar [11]. They performed spin-free (scalar-relativistic) calculations using B3LYP/LANL2DZ method. The spin-orbit correction to the chemical hardness with the increasing cluster size is presented in Figure 4a. As this Figure shows, the spin-orbit correction to the vertical chemical hardness has important effect on Au3 and Au6. The ηv value for acute Au3 and Au6 is negative (-0.038 eV) and positive (0.045 eV), respectively, i.e. spin-orbit coupling decreases (increases) the hardness of gold trimer (hexamer).
The softness, σ, is simply the inverse of the hardness. Figure 3b depicts the variation of softness as a function of cluster size, with and without considering spin-orbit coupling (SOC) effect. As this Figure illustrates, the oscillation behavior of softness is opposite to that of chemical hardness, due to its relationship with hardness, σ = 1/η. Moreover, the obtuse Au3 has the highest adiabatic softness (σa = 0.454 eV -1 and σa,so = 0.456 eV -1 ), indicating its high reactivity. On the other hand, hard molecules have a large HOMO-LUMO gap, whereas soft molecules have a small energy gap [32]. The HOMO-LUMO energy gap of Au2, obtuse (acute) Au3, Au4, Au5 and Au6 at the DKH2-B3P86/DZP-DKH level of theory is 2.760, 1.170 (1.454), 1.982, 1.529 and 2.739 eV, respectively. Therefore, obtuse Au3 with the highest adiabatic softness has the lowest HOMO-LUMO gap value of 1.170 eV. The variation of the spin-orbit correction to the softness of neutral gold clusters as a function of cluster size is plotted in Figure 4b. The spin-orbit correction to the softness shows an even-odd alternation behavior. Furthermore, the inclusion of spin-orbit coupling increases (decreases) the softness of Au3 (Au6) and thus destabilizes (stabilizes) it, σv = + 0.005 eV ( σv = -0.006 eV).
Chemical potential, μ, is related to the charge transfer from a system to another having a lower value of μ. Hence, it is anticipated that the odd-numbered Aun clusters present higher μ values because they have an open shell and that after transferring one electron, they will close their electronic shell and will be more stable than their original openshell clusters. The variation of chemical potential with and without considering spin-orbit coupling (SOC) effect is depicted in Figure 3c. According to this Figure, the vertical chemical potential of obtuse Au3 has the highest value, with or without considering spin-orbit coupling (μv = -4.712 eV, μv,so = -4.750 eV ), indicating its high spontaneous response and chemical reactivity. Figure 4c shows the spin-orbit correction to the chemical potential of neutral gold clusters versus the cluster size. Spin-orbit coupling increases the vertical chemical potential of Au2 by 0.021 eV, while decreases that of Au3 and Au6 by 0.038 and 0.037 eV, respectively. Hence, spin-orbit coupling has more important effect on the chemical potential of Au3.
Electrophilicity has been a measure of the energy stabilization of a cluster due to acquiring additional electronic charge from its surroundings. It is expected that the electrophilicity index (ω) should be related to the electron affinity (EA), because both ω and EA measure the capability of an agent to accept electrons. However, EA reflects the capability of accepting only one electron from the environment, whereas the electrophilicity index (ω) measures the energy lowering of a cluster due to maximal electron flow between donor and acceptor. The electron flows may be either less or more than one. Meanwhile, the electrophilicity index depends not only on EA, but also on IP. Electron affinity and electrophilicity index are related; yet they are not equal [20].      The variation of electrophilicity index as a function of cluster size, with and without considering spin-orbit coupling (SOC) effect is plotted in Figure 3d. As can be seen in this Figure, the electrophilicity index computed using adiabatic electron affinities and ionization potentials shows an odd-even oscillation behavior. The odd-numbered gold clusters present local maxima, and due to their open-shells, have more tendencies to accept electronic charge. On the other hand, the even-numbered and closed shell gold clusters are more stable and are less likely to acquire additional electronic charge. Hence, low electrophilicity index values are expected for them. However, using vertical IPs and EAs, the electrophilicity index behaves similar to vertical electron affinity (Table 2), and due to the linear geometry of the stable anionic gold trimer [33], acute Au3 has a very low vertical electrophilicity index (ωv= 3.894 and ωv,so= 4.010 eV) and the even-odd alternation is not observed (Figure 3d). To illustrate the spin-orbit coupling effect on the electrophilicity index of small neutral gold clusters, the variation of spin-orbit correction to the ω as a function of cluster size is presented in Figure 4d. An odd-even alternation behavior is obvious. Moreover, the spin-orbit correction to the vertical and adiabatic electrophilicity values of acute Au3 (∆ = 0.116 eV and ∆ =0.056 eV) and Au5 ( ∆ = 0.011 eV and ∆ = 0.013 eV) are positive, indicating that spin-orbit coupling increases the electrophilicity index of these clusters. It is evident that spin-orbit coupling has significant effect on the vertical electrophilicity index of the gold trimer.
Maximal charge acceptance, ∆ = − / , measures the maximal flow of electrons between a donor and an acceptor [20]. Figure 5a depicts the variation of the maximal charge acceptance of neutral gold clusters as a function of cluster size, with and without considering spin-orbit coupling. Similar to the electrophilicity index (Figure 3d), the adiabatic maximal charge acceptance exhibits odd-even alternation behavior, whether spin-orbit coupling is considered or not. The spin-orbit coupling corrected adiabatic maximal charge acceptance per atom of neutral gold clusters are 0.805, 0.814 (0.613), 0.517, 0.493 and 0.314 atom -1 for Au2, obtuse (acute) Au3, Au4, Au5 and Au6, respectively. Obtuse Au3 has the highest adiabatic maximal charge acceptance per atom value of 0.814 and 0.810 atom -1 with and without considering spin-orbit coupling, respectively; indicating high reactivity of this cluster due to its active sites. On the other hand, Au6 has the lowest adiabatic maximal charge acceptance per atom value of 0.314 and 0.316 atom -1 with and without considering spin-orbit coupling, respectively, indicating its high stability. The results of Nguyen et al. [12] based on the second difference of energy and fragmentation energy calculations, also predicted high reactivity (stability) for Au3 (Au6). As can be seen in Figure 5b, the spin-orbit correction to the maximal charge acceptance of neutral gold clusters exhibits an odd-even alternation behavior. Moreover, when the maximal charge acceptance is computed using vertical ionization potentials and electron affinities, inclusion of spin-orbit coupling increases the maximal charge acceptance of obtuse Au3 more (0.035 eV) and thus destabilizes it more. The natural population analysis (NPA) shows that the natural charge is positive (negative) for the apex atom of acute (obtuse) gold trimer, i.e. the natural charge changes its sign in going from acute ( 2 A1) to the obtuse ( 2 B1) Au3 [35,36]. Figure 6 represents the natural charges and electrostatic potential (ESP) contour maps of the lowest-energy gold clusters Aun (n-3-6), at the DKH2-B3P86/DZP-DKH level. It is obvious that the apex atom of acute gold trimer has the highest natural charge value of +0.172, indicating its capability to accept additional electronic charge. Red lines in the electrostatic potential contour maps correspond to the negative charges, so these regions are responsible for the nucleophilic interactions.

Conclusion
Gold clusters are attracting interest as the building blocks of novel nano-structural materials due to their biocompatibility and unique physical properties. Most of the computations on small neutral gold clusters have been performed using scalar-relativistic methods, however, the Spin-Orbit Coupling (SOC) effect is appeared to be important. Previous studies focused on spin-orbit coupling effects on HOMO-LUMO energy gaps, geometries, cohesive energies and optical absorption of gold clusters. Electrophilicity index is related to the energy lowering associated with a maximum amount of electron flow between a donor and an acceptor and provides information about structure, stability, reactivity and interactions. In the present research work, the behaviour of the electrophilicity index along with chemical potential, hardness and softness is investigated. The main purpose is to examine spin-orbit coupling (SOC) effects on these properties. We use generalized Hartree-Fock method in conjunction with the fourthorder Douglas-Kroll-Hess (DKH) Hamiltonian and all-electron relativistic double-ζ-level basis set to compute the spin orbit coupling energies and corrections to the electrophilicity index, chemical potential, hardness and softness. It is revealed that spin-orbit correction to the vertical chemical hardness has important effect on Au3 and Au6 by decreasing (increasing) the hardness of gold trimer (hexamer). Moreover, spin-orbit correction to the softness exhibits an evenodd oscillation behaviour and the inclusion of SOC increases (decreases) the softness of Au3 (Au6) and thus destabilizes (stabilizes) it. Spin-orbit coupling has more important effect on the chemical potential of gold trimer by decreasing its value. The adiabatic electrophilicity index shows an odd-even alternation behaviour and odd-numbered gold clusters present local maxima due to their open-shells. The spin-orbit corrections to the electrophilicity index of acute Au3 and Au5 are positive, i.e. SO coupling increases the electrophilicity index of these clusters. Furthermore, the open-shell and odd-numbered gold trimer has the highest adiabatic maximal charge acceptance per atom, with and without considering spin-orbit coupling, indicating high reactivity of this cluster. In addition, spin-orbit coupling increases the maximal charge acceptance of Au3 more and thus destabilizes it more. The natural population analysis reveals that the apex atom of the acute gold trimer has the highest natural charge, indicating its capability to accept additional electronic charges.

Supplementary Material
Supplementary material (Appendix I) for this article contains XYZ coordinates of the lowest-energy gold clusters optimized at the DKH2-B3P86/DZP-DKH level of theory.