Electric Field Effects on Neutral Gold Clusters Au 2-10 : A First-Principles Theoretical Survey of the First- and Second-Order Hyperpolarizabilities

: Herein we report the density functional theory (DFT) calculations of nonlinear optical (NLO) properties of neutral gold clusters Au n (n=2-10) applying long-range corrected LC-M06L functional and Los-Alamos National Laboratory double-ζ polarized basis set. The effects of the incident frequency on the first and second-order hyperpolarizability together with the influence of the external electric field on the frontier orbitals of neutral gold clusters are investigated. It is found that the application of external electric field can increase or decrease the gap energy of neutral gold clusters depending on the direction and magnitude of the applied field. More importantly, by correctly controlling the direction and magnitude of the external electric field, reactive gold clusters having low gap energies can be achieved. Furthermore, the external electric field has more important effect on the virtual orbitals of gold hexamer and decreases the energy of these orbitals along the directions parallel to the molecular plane, resulting in low-energy excitations. The low-energy excitations are expected to play important role in the high second-order hyperpolarizability and better response to the applied field. The third-order nonlinear (NLO) properties of gold hexamer are also strongly affected by the frequency of the incident light and thus can be tuned using the incident frequency for applications. The present work may propose new strategies for enhancing the nonlinear optical response of neutral gold clusters.


Introduction
Metal clusters are formed by assembling a small number of atoms ranging from a few to several hundred [1]. Among them, gold nanoparticles have received great interest in electronics [2,3], catalysis [4,5], chemical sensing [6], optical limiting [7,8], optics [9] and biomedical fields [10][11][12][13][14][15]. The field of optics investigates the interaction between matter and light resulting in specific optical properties of individual material which are essentially determined by the response of the electrons to an external electric field [16]. Due to discrete energy levels and molecular-like HOMO-LUMO transitions, ultrasmall clusters exhibit a spectacular optical behavior which is fundamentally different from that of larger plasmonic nanocrystals [17]. It has been established that transition to the plasmonic regime sets in progressively with size, and at 144 atoms (Au 144 ) weak absorption saturation manifests in the z-scan curve [18]. The investigation of static first hyperpolarizabilities of eight model clusters [Au m (SH) n ] z (m=18-38) by means of density functional theory revealed that no correlation between cluster size and static first hyperpolarizability can be identified [19]. Instead, the symmetry of the clusters seems to dominate the nonlinear optics (NLO) properties. For instance, at a fundamental wavelength of 800 nm, the Au 38 (SCH 2 CH 2 Ph) 24 cluster is active, while Au 25 (SCH 2 CH 2 Ph) 18 does not yield significant second-harmonic generation (SHG) signal due to center of inversion in the Au 25 cluster [20]. By experimentally probing thickness-dependent changes of band structure using two-photon photoluminescence, a surprisingly 100-fold increase of the nonlinear signal was observed when the gold film thickness was reduced bellow 30 nm [21]. Furthermore, the first hyperpolarizability β for Au 15 cluster (509×10 -30 esu) as compared to Au 25 cluster (128×10 -30 esu) is larger considering the difference in the number of gold atoms [17]. The 10 30 β per atom values reported for Au 15 and Au 25 clusters are more than two-orders of magnitude larger than the values reported for AuNPs in the size range 10-50 nm. The Au 10 (SG) 10 cluster also presents a large first hyperpolarizability [22]. In addition, the firsthyperpolarizability β(-2 ; , ) of gold nanocluster Au 6 (GSH) 2 (MPA) 2 having six gold atoms is in comparison with larger AuNCs, the largest value ever reported [23]. Hence, by considering non-linear optical properties of gold quantum clusters, it has been revealed that the smaller gold clusters are better due to providing larger first hyperpolarizabilities [17,22,23].
Recent efforts have been done on the nonlinear optical (NLO) properties determination of metal clusters with the number of metal atoms higher than 10 [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33], while the only reported nonlinear optical properties of gold cluster with a number of metal atoms lower than 10 has been for the gold nanocluster Au 6 (GSH) 2 (MPA) 2 having six gold atoms [23]. We herein present density functional theory (DFT) calculations of the nonlinear optical (NLO) properties of gasphase gold clusters Au n (n=2-10) as a first approach towards the theoretical exploration of their NLO properties. The influence of the external electric field on the frontier molecular orbital (HOMO-LUMO) gap energies of gold clusters is also discussed. We assess the effects of incident frequency on the first-and second-order hyperpolarizability. Our study may propose new strategies for enhancing the nonlinear optical (NLO) responses of neutral gold nanoclusters Au n (n=2-10) by modulating the frequency of incident light.

Computational Details
For geometry optimizations, the coordinates of neutral gold clusters Au n (n=3-8) were selected based on the available experimentally-theoretically determined structures in the literature [34][35][36]. To obtain global minima of Au 9 and Au 10 clusters, a series of structures were built based on our designed or previously reported geometries [35][36][37][38][39][40], and optimized using the B3LYP functional in combination with the Los-Alamos National Laboratory double-ζ polarized basis set for gold. Hay and Wadt [41][42][43] have proposed these relativistic effective core potentials to enable valence electron calculations to be carried out with essentially the same accuracy as all electron calculations, but with much less computational effort. The LANL2DZ basis set incorporates the mass-velocity and Darwin relativistic effects into the potential and explicitly deals with 19-valence electrons per Au atom through a split valence double-ζ basis set [44]. The calculations were performed in the spin states, singlet, doublet, triplet, quartet, quintet and sextet. Harmonic vibrational analyses were performed in order to discriminate between minima and possible transition states on the potential energy surface. Then, depending on the energy distribution of the cluster isomers, the most stable one to three isomers for each cluster size were further optimized using the LC-M06L functional and Los-Alamos National Laboratory double-ζ polarized basis set for gold. A final vibrational analysis was also carried out.
In order to determine the best functional to be used for the calculations, we made a comparative study of the vibrational frequency, ionization potential and bond length of the gold dimer as well as the mean static polarizability of atomic gold. The results are compared with the available experimentally determined values in Table 1.  [36], [40] and [45], respectively.

Geometric and Electronic Characteristics
Results obtained for the neutral gold clusters Au 2-10 are summarized in Table 2, where their global minima structures, bond lengths, symmetries and electronic states are listed. As can be seen in this Table, planar structures are found to represent the putative global minima. The two-dimensional (2D) structures have been reported in previous studies as the most stable isomers, thus confirming that our method works well [34][35][36]38]. The main purpose is to examine the effect of an external electric field on the response of gold clusters to the applied field; however, before we proceed to compute the optical responses for neutral gold clusters, it is important to see the influence of electric field on the HOMO-LUMO gap energies of these clusters.

The Effect of External Electric Field on the Electronic Characteristics of Gold Dimer
To better understand, we first present comprehensive study of the influence of an external electric field on the electronic characteristics of the gold dimer. Survey of the First-and Second-Order Hyperpolarizabilities  Figure 1 illustrates the behavior of the highest occupied-lowest unoccupied (HOMO-LUMO) molecular orbitals of gold dimer in the presence of an external electric field along the axial direction of the linear Au 2 molecule. As this Figure shows, the Au-Au distance increases from 2.520 to 2.970 A°, as the electric field increases from 0 to 0.0395 au (0.001 au = 0.05142 V/A° [55]). The ionization of gold dimer occurs at the electric field value of 0.0396 au. Figure 1 also represents the effect of the static external electric field on the frontier orbitals (HOMO-LUMO) of gold dimer. It is clear that the application of external electric field along the axial direction of Au 2 molecule breaks the orbital symmetry of the frontier orbitals. The breaking of orbital symmetry of HOMO and LUMO is due to the polarization of gold dimer in the presence of external electric field. As in the Gaussian program package, the direction of electric field vector is from minus to plus [56] (Figure 2), the external electric field pushes the lobes of HOMO and LUMO from left to right ( Figure 1). This is beneficial to the enhancement of charge transfer along the direction of electric field. A natural population analysis (NPA) reveals that the natural charge increases from 0 to 0.93339 as the external electric field increases from 0 to 0.0395 au and at the electric field value of 0.0396 au, ionization of gold dimer occurs by transferring one electron from the left to the right gold atom. As Table 3 shows, at the external electric field value of 0.0395 au, the charge has nearly transferred from orbital "6s" located on the left atom to the orbitals "6s, 5d, 6p and 7p" located on the right gold atom. Finally, at the external electric field value of 0.0396 au gold dimer dissociates and in the presence of this field, the cationic and anionic gold atoms have the natural electron configurations "[core] 5d (9.98) 6p (0.02)" and "[core] 6s (1.78) 5d (9.99) 6p (0.23)", respectively. By removing the external electric field, the cationic and anionic gold atoms reconfigure their charge distribution and relax to "[core] 5d (10.00)" and "[core] 6s (2.00) 5d (10.00)", respectively. Figure 3a shows the effect of external electric field on the highest occupied-lowest unoccupied molecular orbitals of gold dimer. According to this Figure, at the low magnitudes of external field (0 to 0.0050 au), the energies of HOMO and LUMO remain unchanged. Then, gradual lowering of energy for the both HOMO and LUMO is observed for the field of larger magnitudes (0.0050 to 0.0395 au). However, for higher magnitudes of electric field, LUMO is relatively more susceptible to variation in energy levels than HOMO. A similar behavior is observed for the Lithium dimer, i.e., LUMO is also more sensitive to the variation in electron density than HOMO [57]. The plot of band gap energy (E g = E LUMO -E HOMO ) versus the external electric field is depicted in Figure 3b. This curve shows that the minimum band gap energy of gold dimer, i.e., E g = 6.624 eV, can be achieved for electric field value of 0.0395 au.

The Effect of External Electric Field on the Gap Energies of Au 3-10
The gold clusters Au 8 and Au 9 are transition state structures in the presence of weak electric fields perpendicular to the molecular plane; hence, the geometries of Au 3-10 are optimized in the presence of relatively weak external electric fields along the directions perpendicular ( 2 ) and parallel to the molecular plane ( → and ↓ ) and the frontier orbital (HOMO-LUMO) gap energies of these clusters in the absence (F=0) and presence of electric fields ( 2 = → = ↓ =0.0050 and 0.0100 au; 0.001 au = 0.05142 V/A° [55]) are summarized in Table 4. The HOMO-LUMO gap energy of a molecule is considered to be important electronic property which can represent its ability to participate in a chemical reaction. As Table 4 represents, the application of external electric field can increase or decrease the gap energy of neutral gold clusters depending on the direction and magnitude of the applied field. When the external electric field decreases the HOMO-LUMO gap energy of a cluster along a particular direction, a more reactive gold cluster can be obtained. However, according to Table 4, the HOMO-LUMO gap energy of the clusters Au 3 , Au 4 , Au 6 , Au 9 and Au 10 has increased along some directions of the applied electric field. In the case of Au 10 , the gap energy shows gradual increase along the direction perpendicular to the molecular plane ( 2 =0.0050 au, E g = 5.678 eV) and then decreases ( 2 =0.0100 au, E g = 5.665 eV) that is only 0.0110 eV lower in energy than its zero field gap energy value of 5.676 eV.
In order to know the behavior of the increasing HOMO-LUMO gap energy of Au 3 , Au 4 , Au 6 and Au 9 , we compute the gap energies of these clusters as a function of electric field. For Au 4 and Au 9 clusters, the increase in external electric field along the direction parallel to the molecular plane ( → ) results in threshold gap energy values of 5.801 eV ( → = 0.0250 au) and 4.252 eV ( → = 0.0150 au) that are not very different from their zero field gap energy values of 5.843 eV and 4.306 eV, respectively. However, the calculations reveal that in spite of gradual increase in the HOMO-LUMO gap energy in the presence of lower electric fields, the gold trimer and hexamer can result in more reactive clusters as the external electric field increases. Figure 4a depicts the gap energy of gold trimer versus the applied electric field along the direction parallel to the molecular plane ( → ). As this Figure shows, the external electric field increases the gap energy of Au 3 up to 5.092 eV ( → = 0.0150 au) and then suddenly decreases to its minimum threshold band gap energy value of 3.574 eV ( → = 0.0300 au), that is 1.300 eV lower in energy than its zero field gap energy. The gold trimer dissociates at the electric field value of 0.0350 au. A similar behavior is observed for the gold hexamer in Figure 4b. According to this Figure, as the external electric field increases along the direction perpendicular to the molecular plane, the band gap energy of gold hexamer shows gradual increase from 7.510 eV (F = 0) to 7.571 eV ( 2 = 0.0250 au) and then abruptly decreases to its minimum threshold band gap energy value of 4.850 eV ( 2 = 0.0400 au), that is 2.660 eV lower in energy than its zero electric field value. The gold hexamer dissociates at the external electric field value of 0.0450 au.  The results represented in Figure 4 state an important consequence, i.e., by correctly controlling the direction and magnitude of the applied external electric field, highly reactive gold clusters with low energy gaps can be achieved. To better understand the effect of external electric field on the charge distribution of the gold trimer and hexamer, we perform natural population analysis in the presence of their threshold electric fields, i.e., → =0.0300 au and 2 = 0.0400 au, respectively; and compare the obtained results with their zero field charges. Figure 5 represents the natural charges and total electron density contour maps of gold trimer and hexamer in the absence and presence of the applied field along the directions parallel and perpendicular to the molecular plane, respectively. It is clear that the external electric field changes the geometry of the clusters and pushes their electronic cloud along the direction of the applied field. In the presence of electric field along the direction perpendicular to the molecular plane, the planar gold hexamer transforms to a three-dimensional structure. A non-planar cluster has more spatial opportunities to interact with other nucleophilic and electrophilic species. Furthermore, in the presence of external electric field, more charge separations and stronger nucleophilic and electrophilic sites are achieved. As a result, the reactivity of a cluster in the presence of external electric field can increase not only due to decrease in its gap energy, but also because of the change in its geometry, electron distribution and charge separations.
As we will see in the next sections, the application of an external electric field along the directions parallel to the molecular plane may have considerable effect on the secondorder hyperpolarizability of gold hexamer. Au 6 is a totally symmetric D 3h triangle in the absence of the applied field; however, in the presence of external electric field along the directions parallel to the molecular plane, reduces its symmetry from a totally symmetric D 3h equilibrium geometry to a lower symmetry C 2v structure. Figures 6 and 7 present the frontier molecular orbitals of gold hexamer together with the corresponding contour maps in the absence ( = 0) and presence of electric field parallel to the molecular plane ( → = ↓ = 0.0100 au).  As these Figures illustrate, the application of external electric field parallel to the molecular plane breaks the symmetry of frontier orbitals due to polarization of the gold hexamer in the presence of the applied field and pushes the lobes of the highest occupied-lowest unoccupied molecular orbitals (HOMO-LUMO) along the direction of the field.

Frequency-Dependent First-order Hyperpolarizability for Neutral Gold Clusters
Materials with non-linear optical (NLO) activity are those in which a non-linear polarization of light occurs during the application of an intense electric field, which is caused by the exposure of the material to a laser beam [48]. The first important controlling factor for the static first-order hyperpolarizability is the symmetry of the cluster, i.e., the first-order hyperpolarizability tensor vanishes for the centrosymmetric molecules [48]. The geometry, size and dimensionality of a cluster may be another important determining factor due to the extensivity and flexibility of the electronic cloud. Wang et al. [58] calculated dipole moments and polarizabilities of Ge n (n=2-25) clusters under B3LYP/LANL2DZ scheme and concluded that the polarizability of the cluster is not only dependent on HOMO-LUMO gap but also closely related to geometrical characteristics. For instance, the HOMO-LUMO gap of Ge 18 is larger than that of Ge 19 , while the polarizability of the former is larger than that of the latter due to the structural difference. Furthermore, in the base of sum-over-states (SOS) approach, the following expression can be used for the static case [59,60], where ∆µ gm is the change in dipole moment between the ground and mth excited state, f gm is the oscillator strength of the transition from ground state to the mth excited state and E gm is transition energy. According to this relation, for any non-centrosymmetric system, small value of transition energy with large oscillator strength would be favorable. Table 5 contains the calculated static first-order hyperpolarizability β(0;0,0) and second harmonic generation β(-2 ; , ) coefficients for the neutral gold clusters and Figure 8a presents the variation in the static first-order hyperpolarizability β(0;0,0) versus the cluster size.  Here, we perform a time-dependent density functional calculation (TD-DFT) and present the maximum and lowestenergy excitation wavelengths along with their corresponding energies, oscillator strengths, dominant molecular orbital transitions and ground to excited state transition electric dipole moments of gold clusters Au 2-10 in Table 6. As this Table illustrates, the highest occupied-lowest unoccupied molecular orbital (HOMO-LUMO) transitions have no effect on the excitations of Au 4 , Au 8 , and Au 10 clusters. Moreover, according to Figure 8a, due to centrosymmetric geometries of Au 2 , Au 4 and Au 10 clusters, the static first-order hyperpolarizability coefficient for these structures is zero. Furthermore, β(0;0,0) value for Au 8 with C s symmetry is too low (0.00222×10 -30 esu), while its value for centrosymmetric gold octamer with D 2h symmetry is zero. As Table 6 shows, HOMO-LUMO transitions have more important effect on the lowest-energy excitations of Au 2 , Au 3 , Au 5 , Au 6 , Au 7 and Au 9 clusters and the non-centrosymmetric Au 3 , Au 5 , Au 6 , Au 7 and Au 9 clusters have nonzero static first-order hyperpolarizability coefficients (Table 5). As can be seen in Table 6, the small triangular gold trimer has the lowest excitation energy value of 0.3938 eV; however, its static first-order hyperpolarizability is only 7.26128×10 -30 esu, that may be due to its small oscillator strength value of 0.0036. For the open shell structures, "a" and "b" indicate "α spin→α spin" and "β spin→β spin" molecular orbital transitions, respectively.
The highly asymmetric gold heptamer Au 7 has the highest static first-order hyperpolarizability value of 138.002×10 -30 esu. According to Table 6, the lowest excitation energy of Au 7 is relatively small (1.7859 eV) that can enhance its firstorder hyperpolarizability coefficient, however, it appears that the asymmetric geometry and extensivity of its electronic cloud would play considerable role in its high first-order hyperpolarizability coefficient. In the case of Au 9 , the two lowest excitations with energies 1.1196 and 1.4375 eV are responsible for the "β spin→β spin" and "α spin→α spin" HOMO-LUMO transitions, respectively. The low excitation energy values of 1.1196 and 1.4375 eV, together with the distributed electronic cloud may result in the first-order hyperpolarizability value of 20.6197×10 -30 esu. For the gold hexamer, the maximum absorption is also the lowest-energy excitation and in spite of its highest HOMO-LUMO gap energy, this non-centrosymmetric cluster shows relatively high static first-order hyperpolarizability value of 44.498×10 -30 esu. According to Table 6, the HOMO-LUMO transition contribution to the lowest-energy excitation of Au 6 is about 70% and regarding its relatively high excitation energy value of 3.5861 eV, it has also high oscillator strength and transition electric dipole moment values of 0.3912 and 2.1100 au., respectively; that can enhance its static first-order hyperpolarizability.
The second-harmonic generation (SHG) is a special case of sum-frequency generation. In the second-harmonic generation (SHG), two photons at frequency combine to form a photon at frequency 2 . In other words, second harmonic generation conserves energy [48]. Figure 8b displays the variation in the second-harmonic generation (SHG) coefficient of gold clusters with increasing cluster size at the wavelength of 800 nm. By comparing Figures 8a and 8b, we find similar trend for the variation of static first-order hyperpolarizability and second-harmonic generation with the increasing cluster size. Furthermore, the calculated secondharmonic generation of Au 9 (60.2401 × 10 -30 esu) is comparable with that of protected Au 10 (SG) 10 cluster (85×10 -30 esu) [22]; however, the calculated β(-2 ; , ) value of gold heptamer Au 7 (184.283×10 -30 esu) is even higher than second-harmonic generation of protected Au 10 (SG) 10 at the wavelength of 800 nm [22], indicating the fact that the smaller clusters should exhibit higher hyperpolarizabilities than larger particles [17]. The second harmonic generation coefficient β(-2 ; , ) for Au 2-10 with varying from 0.00 to 1.63 eV are represented in Table 7. As can be seen in this Table, moving from =0 to 1.63 eV, the variation in secondharmonic generation (SHG) is more important for Au 5 , Au 7 and Au 9 clusters; hence, the frequency dispersion in the coefficients of these nanoclusters are depicted in Figure 9a. The maximum absorptions of Au 5 and Au 9 are around 3.6281 and 3.2595 eV, respectively; and the sharp increase in SHG around 1.2-1.6 eV is an indication of two-photon resonance and so our calculated values of second-harmonic generation are in good agreement. The dispersion plot in Figure 9a for Au 7 exhibits a maximum around 1.36 eV and a modest normal increase in SHG values up to this point compared to Au 5 and Au 9 in the same energy region meaning no resonance effect for Au 7 .   Table 8 lists the calculated static second-order hyperpolarizability γ(0;0,0,0), Optical-Kerr γ(-; ,0,0), electric-field induced second-harmonic generation γ(-2 ; , ,0) and degenerate four-wave mixing γ(-; ,-, ) coefficients of neutral gold clusters. The calculated values of the second-order hyperpolarizability coefficients γ(0;0,0,0,), γ(-, ,0,0), γ(-2 ; , ,0) and γ(-; ,-, ) as a function of cluster size are represented in Figure 9b. It is clear that the gold hexamer exhibits the highest second-order hyperpolarizability coefficients at the wavelength of 800 nm. Furthermore, the second-order hyperpolarizability for Au 6 displays the trend of increasing magnitude from static γ(0;0,0,0) to Kerr effect γ(-; ,0,0) to degenerate fourwave mixing γ(-; ,-, ) to electric field induced second-harmonic generation γ(-2 ; , ,0). For the gold hexamer, the maximum absorption is also the lowest-energy excitation and this cluster with the largest HOMO-LUMO gap energy has a highly symmetric D 3h structure in the absence of external electric field; however, the applied field breaks its symmetry and as Figures 5, 6 and 7 show, results in more flexible electronic cloud. To better know the reason for the observed high second-order hyperpolarizability of gold hexamer, the energy levels of neutral gold clusters Au 2-10 in the absence and presence of external electric field along the directions parallel and perpendicular to the molecular plane are also depicted in Figure 10.   As this Figure exhibits, the external electric field has negligible effect on the energy levels of Au 2 , Au 3 , Au 4 , Au 5 and Au 9 ; whereas, its influence on the energy levels of Au 8 is relatively small, resulting in γ(0;0,0,0) value of 1428. 31 10 -36 esu. In the case of Au 7 , the decrease in the energies of HOMO and LUMO is nearly the same, so its gap energy remains nearly unaltered. However, it is obvious that the applied field has considerable effect on the virtual orbitals of gold hexamer and decreases the energy of these orbitals in the directions parallel to the molecular plane ( → and ↓ ). To better understand, the density of states (DOS) plots for Au 4 , Au 5 and Au 6 in the absence and presence of external electric field along the direction parallel to the molecular plane are compared in Figure 11. By comparison, a red-shift for the virtual orbitals of Au 6 can be observed, i.e., the applied field has more important effect on the virtual orbitals of gold hexamer and decreases the energies of these orbitals more, resulting in smaller HOMO-LUMO gap energy in the presence of external electric field parallel to the molecular plane. In order to know the consequences of the decrease in the energy of virtual orbitals of gold hexamer due to the applied field, a time-dependent density functional (TD-DFT) calculation is also performed on Au 6 , in the absence and presence of external electric field and the corresponding UV-Vis spectra are presented in Figure 12. As can be seen in this Figure, the applied field along the directions parallel to the molecular plane results in a red-shift for the allowed excitations and appearing additional spectroscopic line in the low-energy region of the UV-Vis spectra of gold hexamer.  The effect of external electric field on the UV-Vis spectrum of gold hexamer is also presented in Figure 13. As this Figure shows, by increasing the external electric field, the energy of allowed excitations decreases, resulting in more flexible electronic cloud and better response to the applied field. The allowed molecular orbital transitions of the additional spectroscopic line together with the corresponding excitation energies, wavelengths and oscillator strengths are presented in Figure 14. It appears that these low-energy excitations of gold hexamer in the presence of external electric field play important role in the high second-order hyperpolarizability of this cluster. In summary, in the case of gold hexamer, the maximum absorption is also the lowest-energy excitation and regarding its high HOMO-LUMO gap energy in the absence of external electric field, the applied field has considerable effect on the non-linear response of this cluster due to changing its geometry, breaking its D 3h symmetry, a red-shift for the allowed excitations and appearing additional low-energy excitations that would result in more flexible electronic distribution and better response to the applied field.  The incident-frequency effect on the second-order hyperpolarizability of neutral gold clusters is also examined. The electro-optical-Kerr effect (EOKE) γ(-; ,0,0) and electric-field-induced second-harmonic generation (EFISHG) γ(-2 ; , ,0) coefficients for Au 2-10 with varying from 0.00 to 1.63 eV are represented in Table 9 and the frequency dispersion of these coefficients are depicted in Figure 15. The dispersion plot in this Figure is more important for gold hexamer, so that the electro-optical Kerr effect (EOKE) γ(-; ,0,0) and electric-field-induced second-harmonic generation (EFISHG) γ(-2 ; , ,0) coefficients of Au 6 disperse and increase to a different extent. In particular, moving from =0.00 to 1.63 eV, the value of second-order hyperpolarizability increases by 2.31 times for EOKE and 9.32 times for EFISHG. In fact, the electro-optical Kerr effect (electric-field-induced second-harmonic generation) begins to disperse and exhibits a large value owing to the one-photon (two-photon) resonance that occurs when (2 ) is close to the strong allowed transition energy. For the Au 6 cluster, the calculated first-strong allowed transition energy is about 3.5861 eV. As a result, at =1.793 eV the near two-photon resonance results in a higher dispersion in the electric-field-induced second-harmonic generation value. In the case of Si 10 H 16 cluster, the calculated first-strong allowed transition energy is around 6.53 eV, i.e., at =3.2 eV the near two-photon resonance would result in a large dispersion in the γ(-2 ; , ,0) values [61]. In summary, these results show that the third-order non-linear (NLO) properties of gold hexamer are strongly affected by the frequency of incident light, and thus can be tuned using the incident frequency for applications.

Conclusions
In the present work, density functional theory computations of the nonlinear optical properties of gas-phase gold clusters as a first approach towards the theoretical exploration of their NLO properties are presented. The effects of the incident frequency on the first and second-order hyperpolarizability together with the influence of external electric field on the frontier molecular orbitals of neutral gold clusters are also examined. It is revealed that the application of external electric field may increase or decrease the HOMO-LUMO gap energy of neutral gold clusters depending on the direction and magnitude of the applied field. Moreover, reactive trimer and hexamer gold clusters with small HOMO-LUMO gap energies can be obtained when the external electric field increases in the directions parallel and perpendicular to the molecular plane, respectively. Furthermore, it is found that the reactivity of a cluster in the presence of external electric field can increase not only due to the decrease in its gap energy, but also because of the change in its geometry, electronic distribution and charge separations. It is predicted that the external electric field has considerable effect on the virtual orbitals of gold hexamer and decreases the energy of these orbitals along the directions parallel to the molecular plane, resulting in the appearance of low-energy excitations that are expected to have important role on the high second-order hyperpolarizability and better response of this cluster to the applied field. The wavelength dispersion plot is also more important for gold hexamer, so that the third-order nonlinear properties of this cluster are strongly affected by the frequency of the incident light and thus can be tuned using the incident frequency for applications.