Superalkali nature of the Si₉M₅ (M = Li, Na, and K) Zintl clusters: a theoretical study on electronic structure and dynamic nonlinear optical properties

Abstract

Zintl clusters have attracted widespread attention because of their intriguing bonding and unusual physical properties. We explore the Si₉ and Si₉M₅ (where M = Li, Na, and K) Zintl clusters using the density functional theory combined with other methods. The exothermic nature of the Si₉M₅ cluster formation is disclosed, and the interactions of alkali metals with pristine Si₉ are shown to be noncovalent. The reduced density gradient analysis is performed, in which increased van der Waals interactions are observed with the enlargement of the size of alkali metals. The influence of the implicit solvent model is considered, where the hyperpolarizability (β_o) in the solvent is found to be about 83 times larger than that in the gas phase for Si₉K₅. The frequency-dependent nonlinear optical (NLO) response for the dc-Kerr effect is observed up to 1.3 × 10¹¹ au, indicating an excellent change in refractive index by an externally applied electric field. In addition, natural bonding orbitals obtained from the second-order perturbation analysis show the charge transfer with the donor–acceptor orbitals. Electron localization function and localized orbital locator analyses are also performed to better understand the bonding electrons in designed clusters. The studied Zintl clusters demonstrate the superalkali character in addition to their remarkable optical and nonlinear optical properties.

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1 Introduction

Designing novel materials with excellent nonlinear optical (NLO) properties has gained significant interest in the last three decades because of their potential applications in various optical and electrical devices.^1–4 As a prerequisite for nonlinear media, materials must be in noncentrosymmetric structures in order to generate effective non-equilibrium spontaneous electron polarization.⁵ Literature reveals that not only the hyperpolarizability but also third-rank NLO responses are equally contributing to achieving modern-day optical technologies such as optical data storage, optical communication, biological imaging, and signal processing.^6–8 For obtaining exceptional NLO features in molecule systems, numerous techniques have been proposed in the literature, such as utilizing the push–pull mechanism,⁹ creating the metal–organic framework,¹⁰ introducing diradical character in the conjugated system,¹¹ designing multidecker sandwich complexes,¹² and induction of excess electrons in molecules.¹³

Since Dye et al. carried out an innovative study of excess electron compounds based on cryptand and crown ether,¹⁴ the significant impact of excess electrons on the hyperpolarizability of molecules and clusters gives rise to new prospects for designing and exploring high-performance NLO materials.¹⁵ Almost two decades ago, Li et al. first reported the NLO properties of systems with excess electrons and uncovered the pivotal role of loosely bound electrons in amplifying the first hyperpolarizability (β_o).¹⁶ Up until now, several molecular systems that exhibited the nature of excess electrons have been designed for optical and optoelectronic applications.^17–19 The role of excess electrons in triggering the hyperpolarizability response has been revealed, and these electrons remarkably reduce excitation energies (transition energies), which would result in rapid excitation of electrons to virtual orbitals.¹³ The theory is evident that excitation energy (ΔE) has an inverse relation with β_o of molecules from the conventional two-level model.²⁰

In the development of excess electron molecules, alkali-like superatom clusters are frontline and emerging as building blocks for constructing the three-dimensional periodic table.^21,22 As an intriguing subset of the superatom family, superalkalis has characteristics similar to the alkali metals.²³ Generally, the superalkali nature of clusters is usually described by their reduced ionization potential (IPs) as compared to alkali metals (3.9–5.4 eV).²⁴ A series of experiments on molecular species as superalkali clusters were reported such as XLi₂ (X = F, Cl, Br, and I),²⁵ OM₃ (M = Li, Na, and K),²⁶ NLi₄,²⁷ BLi_n (n = 1–7)²⁸ and a variety of superalkali cluster were designed theoretically, such as bimetallic, uninuclear, and polynuclear ones.^29–31 Very recently, Ayub et al. reported the superalkali (Li₃O) and alkali metals doped crown ether complexes Li₃O[12-crown-4] for optoelectronic properties,³² where dynamic third-order Kerr effect increased up to 2.75 × 10¹² au. The alkali anion (superalkalide nature) and reduced excitation energy were observed for these complexes. Likewise, a class of bimetallic superalkali clusters was investigated for dynamic NLO properties, where a substantial hyperpolarizability value was assigned to the presence of excess electrons. Likewise, M₂OCN and oxacarbon (C₃X₃Y₃) superalkali clusters show tremendously enhanced NLO properties owing to the nature of excess electrons.^33–35 The exceptional charge transfer from superalkali clusters to complexant and their superior reducing properties make them ideal candidates for constructing high-performance molecular NLO materials. The prospective applications can be seen in the synthesis of unusual charge-transfer salts (supersalts), cluster-assembled nanomaterial in the reduction of carbon dioxide as storage materials, and noble-gas-trapping hosts.^23,36

Since the discovery by Eduard Zintl in 1930, groups 14, 15, and Zintl superatom clusters have drawn much curiosity and have been considered as potential antecedents for material design.³⁷ It is also suggested that Zintl clusters can be connected via transition metal, alkali metal, non-metal atoms, or organic linkers to construct nanonetworks or materials with novel and tunable electronic properties. As far as we know, the study of the Zintl clusters based on Si-atom is very limited. Very recently, Zintl clusters [Si₉R₂]²⁻ and [Si₉R₃]⁻ [R=SiH(tBu)₂, SnCy₃, Si (TMS)₃] have been synthesized experimentally.³⁸ Later, Sinha et al.³⁹ investigated the superhalogens nature of Si₉R₃ clusters theoretically, in which Zintl anion (Si₉²⁻) is functionalized with electronegative ligands such as –CF₃, –CN, –BO, and –NO₂.

The present study mainly focuses on exploring superalkali and excess electron nature through noncovalent interactions of 1st group metals with pure Si₉ cluster. In the designed clusters Si₉M₅ (M = Li, Na, and K), loosely bound alkali electrons are actively transferring charge and are crucial for activating the NLO of responses. In addition, van der Waals (vdW) interaction is very important in numerous chemical processes,^40–44 and its role in the formation and properties of clusters is of great interest. The present work also uncovers the nature and magnitude of vdW forces in tuning the optoelectronic properties in the designed clusters. This paper is organized as follows. The computational details are given in Section 2. The results are presented and discussed in Section 3. Finally, a brief conclusion is given in Section 4.

2 Methods and computational details

For the designed Si₉M₅ (M = Li, Na, and K) clusters, ab initio molecular dynamics (AIMD) simulations were performed using the ORCA 5.0 package,⁴⁵ and other quantum chemical simulations were performed by Gaussian 16.⁴⁶ To make the results reliable, the method and basis set should be chosen carefully.^47–51 The initial-guess structure for the pure Si₉ cluster is obtained from literature, and the present geometry of Si₉ is in good consistent with those reported in literature,⁵² which justifies that it is a global minimum. As for Si₉M₅, considering that the geometry of Si₉ is not altered or distorted greatly during the adsorption of alkali metals by vdW interactions, we performed initial optimization in dozens of designed initial-guess structures, and then find the structure that has lowest potential energy. Our AIMD calculations justify that the present structures are the global minima, and it should be noted that a global optimization procedure^53–56 would have better efficiency in searching global minimum on a potential energy surface (PES). Initial optimization is carried out using the B3LYP/6-31G level of theory. The stable molecular geometries were reoptimized, and the final optimization and the following entire electronic and NLO properties were computed using the ωb97xd/def2-qzvp level. Infrared and Raman frequencies are also computed using the same method. The thermodynamic stability of studied clusters was validated via binding energy (cohesive energy) per atom, which can be expressed as:

E_B = [E_Si9M5 − E(X)]/n (1)

where E_Si9M5 is the total electronic energy of the cluster, E(X) is the sum of the energy of isolated atoms (silicon and alkali metals) in a cluster, and n is the total number of atoms. To investigate geometry fluctuation and thermal stability of the pristine Si₉ and Si₉M₅, AIMD simulations at the B3LYP-D3/def2-SVP were performed, with 3500 geometries calculated at 300 K with a time step of 1.0 femtoseconds.⁵⁷

Superalkali nature and reactivity were identified and illustrated through global reactivity parameters. Frontier molecular orbital (FMO) and natural population analysis (NPA) were chosen to get the orbital energies and charge transfer in Si₉M₅. A second-order Fock matrix study using the NBO tool was considered for the donor–acceptor atoms and type of electronic transition. The reactivity and topological properties were further examined using molecular electrostatic potential (MESP). The bonding nature of alkali metals to Si-atoms and vdW interactions were outlined through the quantum theory of atom in molecules (QTAIM) and noncovalent interaction (NCI) study. Additionally, the electron localization function (ELF) and localized orbital locator (LOL) analyses were performed to determine the nature of bonding (localized) and delocalized (lone pair) electrons within clusters. The chemical potential (μ) and electronegativity can be calculated using the equation:

μ = −(IP + EA)/2 (2)

where IP and EA are ionization potential and electron affinity, respectively, which are obtained from Koopman's theorem. The negative of chemical potential is considered as electronegativity of clusters.

For the optical and NLO characteristics, dipole moment (μ_o), polarizability (α_o), first-order hyperpolarizability (β_o), and static second hyperpolarizability (γ_o) were calculated with the help of the following equations:

μ_o = (μ_x² + μ_y² + μ_z^2)1/2 (3)

α_o = 1/3(α_xx + α_yy + α_zz) (4)

where

β_x = β_xxx + β_xyy + β_xzz, β_y = β_yyy + β_yzz + β_yxx and β_z = β_zzz + β_zxx + β_zyy, (6)

〈γ〉 = 1/5(γ_xxxx + γ_yyyy + γ_zzzz + γ_xxyy + γ_xxzz + γ_yyxx + γ_yyzz + γ_zzxx) (7)

here, β_ijk (i,j,k = {x,y,z}) is the tensor component of β_o, γ_ijkl (i,j,k,l = {x,y,z}) is the tensor component of second hyperpolarizability (γ_o). The second hyperpolarizability (γ) is fourth rank tensor of 3 × 3 × 3 × 3 form which exhibit both static and dynamic nature.

Furthermore, the density of states (DOS) spectral study was carried out to get a comprehensible picture of orbital energies and HOMO–LUMO gaps for clusters using the GaussSum software.⁵⁸ TD-DFT study was also carried out by considering forty excited states for singlet and triplet. In this analysis, we calculated various excited state parameters and absorbance of clusters. Scattering hyperpolarizability (β_HRS) was calculated by using the following relation:

where 〈β_zzz²〉 and 〈β_zxx²〉 are the average of orientational (β) tensor. While the related depolarization ratio for these superalkali clusters (DR) ratio is also given by:

DR = 〈β_zzz²〉/〈β_zxx²〉 (9)

Moreover, the frequency-dependent (dynamic) nonlinear optical parameters were calculated at the 532 and 1064 nm wavelength. The frequency-dependent β(−ω;ω,0) electro-optic Pockel's effect (EOPE) and β(−2ω;ω,ω) electric field-induced second harmonic generation (ESHG) were calculated. The second hyperpolarizability γ(ω), dc-Kerr γ_dc-Kerr (ω) = γ(−ω;ω,0,0) and second harmonic generation γ_ESHG(ω) = γ(−2ω;ω,ω,0) were also obtained.

In the NBO analysis, donor (i)–acceptor (j) interactions between filled Lewis orbitals and empty non-Lewis orbitals are explored to further enrich the structural-reactivity relationship. For this purpose, a second-order Fock matrix is considered, where only high stabilization energy (E₍₂₎) interactions are considered:⁵⁹

where F(i,j) is the NBO Fock matrix element for non-diagonal orbits, q_i denotes the donor orbit occupancy rate, i denotes the diagonal element, and j represents the non-diagonal element of the matrix.

3 Results and discussion

3.1 Electronic structure and thermodynamic stability

We begin the neutral Zintl cluster (Si₉) geometry as core, which is decorated with alkali metals (M = Li, Na, and K) for creating Zintl-superatom clusters with superalkali nature. The pure Si₉ cluster shown in Fig. 1 has a bicapped pentagonal pyramid geometry, whereas the Si₉²⁻ and Si₉⁴⁻ clusters have different geometries (see Fig. S1†). The geometries of Si₉M₅ are consistent with previously reported clusters⁶⁰ which are also trigonal prismatic, and the obtained bond distances are given in Fig. 1 and Table S2 of the ESI.† In Si₉Li₅, the interaction distance between Si and alkali metals ranged from 2.77 to 2.92 Å, while for Si₉Na₅ increased up to 3.02 Å. The highest interaction distance (3.79 Å) is observed between the Si–K bonds in the Si₉K₅ cluster. The calculated results at the ωb97xd/def2-qzvp level are given in Table 1. As seen, the binding energy per atom (E_B) is higher for pure Si₉ cluster as compared to Si₉M₅. A monotonic reduction E_B for Si₉M₅ with the increased alkali metal size and interaction distance (d_Si–M) are observed. The binding energies at the ωb97xd/def2-qzvp level are −4.42, −3.66, −4.25, and −3.53 eV for Si₉, Si₉Li₅, Si₉Na₅, and Si₉K₅, respectively. These values are higher than those reported for the Ge₉AM₅ and P₇M₂ (M = Li, Na, and K) clusters,^60,61 indicating that the present Si-based Zintl clusters have better thermodynamic stabilities. The computed mean dipole moment (μ_o) increase with the enlargement of the size of alkali metals. The μ_o value of Si₉K₅ is 20.42 D, which is 27 times higher than that of the pristine Si₉ cluster (0.73 D). In addition, we can also see from Table 1 that, the chemical hardness of Si₉ is 2.92 eV, higher than those of the Si₉M₅ clusters, and the chemical softness of Si₉M₅ increases with the increased metal size.

Fig. 1 The optimized structure of the Si₉, and Si₉M₅ clusters at the ωb97xd/def2-qzvp level of theory. The Si atom is in green and the alkali metal atoms are in pink. The important interaction distances (in Å) of alkali metals with silicon are given below the geometries. The corresponding Cartesian coordinates are given in Table S1.†

Table 1 The calculated global reactivity descriptors of the Si₉ and Si₉M₅ (M = Li, Na, and K) clusters

Properties	Si9	Si9Li5	Si9Na5	Si9K5
μo (D)	0.73	2.61	4.46	20.42
NBO on Si (e)	−0.17/0.11	−0.60	−0.52	−0.48
NBO charges (e)	—	0.87	0.89	0.95
Ionization potential (eV)	7.58	4.35	4.10	2.41
Electron affinity (eV)	1.75	0.18	0.06	0.03
EHOMO (eV)	−7.58	−4.35	−4.10	−2.41
ELUMO (eV)	−1.75	−0.18	−0.06	−0.03
Eg (eV)	5.83	4.17	4.06	2.37
Chemical potential (eV)	−4.66	−2.27	−2.08	−1.22
Hardness (eV)	2.92	2.09	2.02	1.19
Softness (eV)	0.34	0.48	0.49	0.84
Electronegativity (χ)	4.66	2.27	2.08	1.22
Electrophilicity index (eV)	3.73	1.23	1.07	0.63
Eb (eV)	−4.42	−3.66	4.25	−3.54

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The superalkali nature of examined clusters can be monitored by their minimized ionization potential (IP) and electron affinity (EA) values. Pristine Si₉ has an IP value of 7.58 eV, higher than the designed Si₉M₅, while its EA value is 1.75 eV. Table 1 shows that the interaction of alkali metals with the Si₉ cluster has significantly lowered the IP and EA values. For instance, the obtained IP values are 4.35, 4.10, and 2.41 eV for Si₉Li₅, Si₉Na₅, and Si₉K₅, respectively. A similar trend of decreasing IP by doping alkali metals can also be found in the literature.⁶² EA value also decreased to 0.03 eV, which is much smaller than previously reported P₇M₂ clusters. The smaller EA values declare the electropositive nature and excellent reducing properties of the Si₉M₅ clusters. Furthermore, the superalkali character of these clusters can be observed by their reduced IP values compared to alkali metals. The IPs of Si₉Li₅ and Si₉Na₅ are lower than that of the Li atom, and the IP of Si₉K₅ has even lower than that of the Cs atom (3.89 eV), demonstrating that these clusters manifest superalkali characteristics.²⁷ Due electropositive nature, alkali metals become electron deficient after sharing their valence electrons ns¹ to np of Si-atoms within clusters. On the other hand, the Si₉ cluster acquires electronic charge from alkali due to its comparatively higher electronegativity. The electronic stability of Si₉M₅ can also be correlated to the famous Jellium model. The core Zintl cluster Si₉ contains 36 electrons, and after interactions with alkali metals, there are 41 electrons in total. Hence, after losing one electron, these clusters may become stable by attaining the electronic shell closure configuration of 40 electrons. The presence of loosely bound (excitable) electrons in these clusters mimics the chemistry of the first group metals.

To investigate the kinetic and thermodynamic stability of the studied clusters, a total of 3500 different geometries for each cluster are obtained during the AIMD simulations at the B3LYP-D3/def2-SVP level of theory. The snapshots are taken at every 500 fs to ensure their structural integrity. Fig. 2a displays the fluctuation curves of total energy, indicating that the designed clusters are kinetically stable with steady fluctuations in energy. In addition, the changes in root-mean-square deviation (RMSD) of the geometries within the simulations are shown in Fig. 2b. As seen, the fluctuations are more pronounced for the Si₉K₅ cluster than the other three clusters, and the RMSD value for Si₉K₅ increases up to 1.5 Å near the 3500 fs. Although RMSD curves for both Na and K-based clusters are increasing, no isomerization or dissociation is observed in the geometries of clusters (see Fig. 2c and S2†). The larger deviations in the RMSD of Si₉Na₅ and Si₉K₅ may be due to their big size of metals and interaction distances. Hence, the present designed Zintl clusters are kinetically stable at room temperature, and can be further probed for optoelectronic applications.

Fig. 2 Time evolution of the (a) total energy, (b) root-mean-square deviation (RMSD), and (c) snapshots of the Si₉K₅ geometries during ab initio molecular dynamics simulations at 300 K using the B3LYP-D3/def2-SVP method.

Manipulating the FMO analysis, it is attainable to uncover complex interactions between two components in a system based on the HOMO–LUMO band gap (E_g). For the pure Zintl Si_9, the E_g value is 5.83 eV, according to our FMO analysis. The orbital energies are presented in Table 1. As can be seen, due to the doping/interactions of alkali metals on Si₉, the energy gaps (between HOMO and LUMO orbitals) of the present Si₉M₅ clusters are reduced. This is reasonable since the delocalized electrons of alkali (loosely held) would influence the valence electrons in the Si₉ Zintl cluster, and then a diffuse excess electron system is generated, resulting in a reduced E_g gap. The HOMO–LUMO gaps of Si₉Li₅, Si₉Na₅, and Si₉K₅ are 4.17, 4.06 and 2.37 eV, respectively. The decreasing values have a periodic trend with the enlargement of the size of metals. In addition, the doping of alkali metals led to an increase in the energy of HOMO orbitals because of the residing excess electrons which lead to the generation of new HOMOs. The HOMO–LUMO gaps of Si₉M₅ are slightly higher than those of Ge₉AM₅ (ref. 60) and much lower than those of the P₇M₂ Zintl clusters.⁶¹ Fig. 3 shows the FMOs of the present clusters. As seen, the shape of LUMO of Si₉Na₅ and Si₉K₅ is spherical (similar to s-orbital) in shape. In the case of Si₉Li₅, the shape is more diffused, and the electronic density of the orbital is still higher around alkali metals than other places.

Fig. 3 Representation of frontier molecular orbitals (FMOs) at the ωb97xd/def2-qzvp level (isovalue = 0.025).

To visualize intra-and intermolecular charge transfer, we simulated the natural bonding orbital (NBO) charges on Si₉M₅, and the values are also given in Table 1. NBO charges on Si-atoms (Q_Si) have both positive and negative in magnitude. After alkali metals interaction, the Si-atoms get significant negative NBO charges, whereas partial charges on alkali metal (Q_M) atoms become positive as compared to silicon atoms. The highest magnitude (0.95 |e|) of NBO charge (positive) is displayed by the K-atom in the Si₉K₅ cluster, indicating excellent charge transfer. Such charge transfer from alkali metals to Si-atoms would result in asymmetric charge distribution within designed clusters.

Furthermore, charge transfer and delocalization can be explained using the interactions between bonds and different properties (such as stability, reactivity, and the relationship between donor and acceptor), as well as the relationship between donors (i) and acceptors (j). Table S3† lists the values from our NBO analysis, from which a number of electronic transitions are observed and the most significant transition with a major stabilization (E₍₂₎) contribution is reported. Evidently, molecular interactions between the investigated system are mainly by σ–σ* and π–π* electronic transitions between Si–Si atoms. In Table S3,† we can also observe that, some bonds are very strong in Si₉ due to strong interaction between Si–Si atoms. The most important maximum energy associated with the present clusters is to donate electrons from σ (Si4–Si6) to σ (Si4–Si9) in Si₉, from σ* (Si9–Si11) to σ* (Si9–Si13) in Si₉Li₅, from π (Si7–Si9) to π* (Si1–Si2) in Si₉Na₅ and from σ* (Si4–Si8) to σ* (Si3–Si8) in Si₉K₅, with stable energies being 80.41, 44.86, 45.66 and 50.04 kcal mol⁻¹, respectively.

3.2 Noncovalent interactions

The noncovalent interaction plays a very important role in many chemical processes.^63–65 To investigate the nature of interactions between Si₉ and alkali metals, we carried out a reduced density gradient (RDG) analysis based on the noncovalent interaction (NCI) method. Fig. 4 displays the RDG scatter graph, where the λ₂ sign is exploited to differentiate between the bonded (λ₂ < 0) and non-bonded (λ₂ > 0) interactions. As seen in Fig. 4, for the present clusters, λ₂ sign ρ function ranges from −0.05 to 0.05 au. The red peaks in the range of λ₂ > 0 correspond to the effect of steric repulsion, green spikes appearing in the region of λ₂ = 0 represent the dipole–dipole or London dispersion forces,¹³ blue-colored spikes in the regions ρ > 0 and λ₂ < 0 represent the electrostatic interactions. In addition, Fig. 4 shows a significant increase in weak vdW attractions after the doping of alkali metals to the pure Si₉ cluster. The strong, attractive interaction appears between −0.035 and −0.025 au. Hence, the isosurface plot indicates the presence of vdW interactions between the alkali metals and the pristine Si₉ cluster, in which the dispersion interaction may play an important role. A significant increase in the magnitude of noncovalent can be seen with the increased atomic number of alkali metals, which may cause a strong impact on the NLO properties of the designed Si₉M₅ Zintl cluster.

Fig. 4 Representation of reduced density gradient (RGD) spectra and three-dimensional surfaces for non-covalent interactions analysis (NCI) at the ωb97xd/def2-qzvp level.

We also performed the QTAIM analysis to identify the nature of bonding within the present clusters, and values are given in Table 2. The generated bond critical points (BCP) at (3, −1) appear in Fig. S3.† The interaction energies between alkali and silicon atoms (M–Si) are relatively weaker than the interaction between Si–Si in pure Si₉, which demonstrates the presence of non-bonding interactions in present clusters. The Laplacian of electronic density (∇²ρ(r)) and total energy density (H(r)) values at BCP (3, −1) between Si–Si bond are negative, while positive for Si–M (where M = Li, Na, and K), which further suggests the non-covalent nature of bonds. Alkali metal interactions with silicon (M–Si) have a smaller value of total electronic density (ρ(r) < 0.1) and a positive magnitude of ∇²ρ(r), indicating the weak interactions are operational in the present Si₉M₅ clusters. Similarly, G(r)/|V(r)| ratios are close to or larger than unity for M–Si interactions, which also demonstrates the presence of vdW forces between Si and alkali metals in the present clusters.

Table 2 Calculated QTAIM parameters for bond critical points (BCPs) at (3, −1) electron density. The values are in au

Cluster	Interaction	ρ(r)	Δ^2ρ(r)	G(r)	V(r)	G(r)/|V(r)|	H(r)
Si9	Si1–Si4	0.047	0.0040	0.0144	−0.0280	0.5124	−0.0136
	Si4–Si5	0.051	−0.0041	0.0155	−0.0323	0.4798	−0.0168
	Si4–Si9	0.047	0.0042	0.0143	−0.0279	0.5125	−0.0136
	Si5–Si8	0.062	−0.0292	0.0168	−0.0417	0.4028	−0.0248
Si9Li5	Li1–Si8	0.016	0.0723	0.0153	−0.0126	1.2142	0.0027
	Li2–Si12	0.016	0.0745	0.0159	−0.0131	1.2137	0.0024
	Li3–Si8	−0.016	0.0725	0.0154	−0.0127	0.4494	0.0027
	Li4–Si2	0.055	−0.0121	0.0160	−0.0356	1.2126	0.0027
	Li5–Si13	0.016	0.0729	0.0154	−0.0127	0.8717	0.0027
Si9Na5	Na10–Si3	0.012	0.0489	0.0034	−0.0039	0.0903	−0.0004
	Na11–Si5	0.011	0.0419	0.0027	−0.0299	0.8571	−0.0003
	Na12–Si4	0.012	0.0400	0.0030	−0.0035	0.9285	−0.0004
	Na13–Si6	0.010	0.0407	0.0026	−0.0028	0.8571	−0.0001
	Na14–Si2	0.012	0.0418	0.0030	−0.0035	1.2258	−0.0005
Si9K5	K10–Si7	0.006	0.0179	0.0038	−0.0031	1.2258	0.0007
	K11–Si4	0.061	−0.0282	0.0169	−0.0418	0.4043	0.0007
	K12–Si7	0.011	0.0332	0.0076	−0.0069	1.1014	0.0006
	K13–Si2	0.011	0.0345	0.0079	−0.0071	1.1126	0.0007
	K14–Si8	0.011	0.0332	0.0076	−0.0069	1.1014	0.0006

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3.3 Static nonlinear optical properties

It is rather beneficial for the NLO response of the superatom clusters that there is adequate electronic communication across their various moieties. As a new addition to the family of loosely bound excess electron compounds, the Si₉M₅ clusters designed in this work have superalkali characteristics and contain excess electrons, further prompting us to investigate their optoelectronic properties. Literature is also evident for crucial role of excess electrons and vdW interactions in triggering NLO response of molecules and clusters.^66–68 The optical and NLO responses can be characterized through polarizability, hyperpolarizability, and static second hyperpolarizability. The obtained values are given in Table 3. The determined values of polarizability (α_o) and μ_o are higher as compared to that of the urea, revealing excellent polarizability and charge separation. Overall, μ_o and α_o response decreases in the order of Si₉K₅ > Si₉Na₅ > Si₉Li₅ > Si₉. The highest value of α_o can be seen for the Si₉K₅ cluster, while the lowest is observed for the pristine Si₉ cluster. The substantial value of polarizability indicates the polarizable nature, which may be due to the presence of soft-nature alkali metals.

Table 3 Static polarizability (α_o), hyperpolarizability (β_o), projection of hyperpolarizability on dipole moment vector (β_vec), static second hyperpolarizability (γ_o), average dipolar hyperpolarizability (〈β_J=1〉), average octupolar hyperpolarizability (〈β_J=3〉), percentage contribution to dipolar nature (ϕ〈β_J=1〉), and depolarization ratio (DR) of studied clusters

Properties	Si9	Si9Li5	Si9Na5	Si9K5
αo (au)	268.82	403.913	578.84	1017.67
βo (au)	1.58 × 10^2	2.22 × 10^3	1.41 × 10^4	6.66 × 10^4
βvec (au)	1.44 × 10^2	1.01 × 10^3	1.19 × 10^4	4.72 × 10^4
βHRS (au)	1.44 × 10^2	1.01 × 10^3	1.19 × 10^4	4.72 × 10^4
γo (au)	5.94 × 10^4	6.01 × 10^5	2.37 × 10^6	7.67 × 10^7
〈βJ=1〉	122.92	1721.66	21 246.21	56 345.58
〈βJ=3〉	430.277	1898.84	21 106.19	126 518.78
ϕ〈βJ=1〉	22%	47%	50%	30%
DR	1.84	3.93	4.28	0.69

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Hyperpolarizability (β_o) is an approach for identifying nonlinear optical features of present clusters. The obtained β_o values for Si₉, Si₉Li₅, Si₉Na₅, and Si₉K₅ are 1.58 × 10², 2.22 × 10³, 1.41 × 10⁴, and 6.66 × 10⁴ au, respectively. The β_o value of the designed Si₉M₅ clusters is significantly increased compared to that of the pristine Si₉, indicating the crucial role of excess electrons and non-covalent interactions. For example, the β_o values of Si₉Li₅ and Si₉K₅ are 140 and 421 times larger than that of Si₉, and the highest β_o value of 6.66 × 10⁴ au can be found in the Si₉K₅ cluster. Formulating excess electrons after the interaction of alkali metals might cause a significant reduction in excitation energies (ΔE), and excess electrons have a dominant role in triggering hyperpolarizability values. Fig. 5 displays ΔE of the crucial state and obtained β_o of Si₉ and Si₉M₅ clusters. As seen, there is an inverse relation of ΔE and β_o. The calculated values of α_o and β_o of the present cluster are higher than those of urea and p-nitroaniline.⁶⁹ The γ_o values range from 5.94 × 10⁴ to 7.67 × 10⁷ au, and a monotonic increase is observed from Li to K. Noncovalent interactions are also crucial in promoting the optical and NLO properties of designed clusters. The uptrend in values of hyperpolarizabilities is observed with the increased vdW interactions, as revealed by NCI analysis.

Fig. 5 Plotted excitation energy (ΔE) of crucial state and obtained hyperpolarizability (β_o) of Si₉ and Si₉M₅ clusters at the ωb97xd/def2-qzvp level.

In addition, the effect of solvent on α_o and β_o is also considered at the ωb97xd/def2-qzvp level, by employing the implicit, namely the universal solvent model based on solute electron density (SMD). Calculated values are presented in Table S4.† Obtained second-order NLO parameters in solvent media are better than those of vacuum-based results. It is worthy of note that computed values of β_o using four different solvents are broadly consistent and in the same tendency. Using water as a solvent leads to slightly larger β_o values. For instance, the calculated β value (5.5 × 10⁶ au) for Si₉K₅ is found to be 83 times higher as compared to β in the gas phase. The altered electronic density and change in electronic properties due to the presence of solvent may influence the electric field, which would result in increased β_o. The presence of solvent has introduced a strong polarization effect to the applied electric field. Thus, controlling the polarizable environment around studied clusters can be seen as an effective way to adjust the NLO response.

To get deep insight into nonlinear optical features, we disclose the second-order electric susceptibility response of the present clusters theoretically by Hyper-Rayleigh scattering (HRS) simulation, and the results are listed in Table 3. As seen, the obtained β_HRS values of Si₉, Si₉Li₅, Si₉Na₅, and Si₉K₅ are 1.44 × 10², 1.01 × 10³, 1.19 × 10⁴, and 4.72 × 10⁴ au, respectively. From Si₉ to Si₉K₅, β_HRS values have the same trend as β_o values, indicating the excellent NLO characteristics of the present clusters. In particular, the β_HRS value of Si₉K₅ is 328 times larger than that of Si₉. Furthermore, the average dipolar hyperpolarizability 〈β_J=1〉 and octupolar 〈β_J=3〉 values of the Si₉M₅ clusters are higher than those of Si₉. The maximum dipolar contribution is 50% for the Si₉K_5, while the remaining clusters exhibit an octupolar-like nature. Compared with the previously reported clusters,⁴⁷ the present clusters exhibit greater γ_o values.

3.4 Dynamic nonlinear optical properties

We also examined the frequency-dependent hyperpolarizability β(ω) theoretically considering the externally applied frequency of 532 and 1900 nm, and the obtained results are summarized in Table 4. β(ω) consists of two parts, i.e., electro-optical Pockel's effect (EOPE) with β(−ω,ω,0) and second harmonic generation phenomena (SHG) with β(−2ω,ω,ω). The choice of wavelengths also relevant to Nd:YAG laser functioning, which frequently emits invisible light in at longer wavelength and serves in laser-based devices.⁷⁰ The electro-optical effect (Pockel's effect) is an essential nonlinear effect used in many applications. As shown in Table 4, for all four clusters, the values of β(−ω;ω,0) and β(−2ω,ω,ω) at ω = 0.0240 are higher than the corresponding values at ω = 0.0856. Hence, the frequency-dependent first hyperpolarizability β(ω) response dominates at smaller dispersion frequency (at longer wavelength). From Si₉ to Si₉K₅, both β(−ω;ω,0) and β(−2ω,ω,ω) show increasing trends. Obviously, the dynamic first hyperpolarizability β(ω) values of Si₉M₅ are immensely better than the pristine Si₉ cluster, indicating their excellent dynamic nonlinear optical features.

Table 4 Static polarizability frequency-dependent first hyperpolarizability β(ω) and second hyperpolarizability γ(ω) using the definition of EOPE β(−ω,ω,0), ESHG β(−2ω,ω,ω), dc-Kerr effect γ(−ω,ω,0,0), and SHG γ(−2ω,ω,ω,0) at 532 and 1900 nm

Frequency-dependent first hyperpolarizability β(ω)
	ω = 0.0856 (532 nm)		ω = 0.0240 (1900 nm)
	β(−ω;ω,0)	β(−2ω;ω,ω)	β(−ω;ω,0)	β(−2ω;ω,ω)
Si9	2.79 × 10^2	2.03 × 10^3	1.61 × 10^2	1.25 × 10^2
Si9Li5	3.39 × 10^4	1.11 × 10^5	2.41 × 10^3	2.80 × 10^3
Si9Na5	7.62 × 10^6	1.32 × 10^6	3.83 × 10^4	4.45 × 10^5
Si9K5	2.65 × 10^6	3.05 × 10^5	2.65 × 10^8	1.05 × 10^8

---

Frequency-dependent second hyperpolarizability γ(ω)
	γ(−ω;ω,0,0)	γ(−2ω;ω,ω,0)	γ(−ω;ω,0,0)	γ(−2ω;ω,ω,0)
Si9	8.53 × 10^4	1.86 × 10^5	6.10 × 10^4	6.42 × 10^4
Si9Li5	1.10 × 10^10	3.30 × 10^9	6.81 × 10^5	1.83 × 10^6
Si9Na5	3.67 × 10^9	3.90 × 10^8	3.48 × 10^6	2.44 × 10^8
Si9K5	1.18 × 10^9	1.24 × 10^9	5.86 × 10^11	1.30 × 10^11

---

Table 4 also lists the dynamic second hyperpolarizability γ(ω) values. As seen, the obtained the dc-Kerr effect γ(−ω,ω,0,0) and second harmonic generation SHG γ(−2ω,ω,ω,ω) values are much better than those of static γ_o. The calculated highest response is indicated regarding the Kerr effect with a value up to 5.86 × 10¹¹ au. Additionally, the frequency-dependent SHG and Kerr effect are much more prominent at 1900 nm (small dispersion frequency) than those of at 532 nm. Within the present designed clusters, an uptrend can be seen in both γ(−2ω;ω,ω,0) and γ(−ω,ω,0,0) values with increased metal size at 1900 nm, while at 532 nm Si₉Li₅ has the highest value of γ(−2ω;ω,ω,0) and γ(−ω,ω,0,0). The Kerr effect is caused by the instantaneous change in its refractive index in response to the applied electric field. Also, the change in the refractive index is directly related to the square of electric field strength. Hence, the applied smaller frequency (ω = 0.024) has significantly enhanced values of the Kerr effect as compared to a higher frequency. Fig. S4† reveals the variational trend frequency-dependent β(ω) and γ(ω) at 532 and 1900 nm, respectively, where the SHG and dc-Kerr effects exhibit the same increasing trend from Li to K at 1900 nm.

3.5 Spectroscopic study and bonding analysis

Fig. S5† depicts the obtained absorption spectra of the present Si₉M₅ clusters by our TD-DFT calculations. As seen, the absorbance maxima are shifted to longer wavelength (bathochromic shift), where the highest absorbance wavelength (866 nm) is seen for the Si₉K₅ cluster. The atomic number of alkali metals is the main influencing factor in the absorbance properties where increased-sized metals decrease the excitation energy. The values of excitation energies (ΔE), absorbance maxima (λ_max), and oscillator strength (O.S.) are given in Table 5. The excitation (transition) energy (ΔE) of the Si₉ cluster is higher than those with alkali metal decorated, and a gradual decrease in ΔE is observed from Si₉Li₅ to Si₉K₅ for designed clusters. The above results demonstrate that the higher size metals in Si₉ clusters enable them to harvest light at comparatively longer wavelengths.

Table 5 The obtained excitation energies (ΔE), absorption maxima (λ_max), and oscillator strengths (f_os) of the present clusters

Clusters	ΔE (eV)	λmax (nm)	fos (au)
Si9	3.47	356	0.0091
Si9Li5	2.63	469	0.0203
Si9Na5	1.80	685	0.1620
Si9K5	1.43	866	0.4689

---

In addition, we examined the total density of states (TDOS), and the obtained spectra are given in Fig. S6.† It is known that the different kinds of states inhabited by electrons at a discrete energy level are expressed by electronic states per unit of energy. For the pristine Si₉, the wide energy gap can be seen where HOMO is located at −8 eV. After interacting with alkali metals, the HOMO–LUMO gap became narrow, where HOMO has a further increase in energy. Furthermore, the vibrational frequencies of Si₉ and Si₉M₅ are listed in Table S5,† and the spectra of FT-IR and Raman are shown in Fig. S7.† The vibrational and Raman frequencies are simulated at the ωb97Xd/def2-qzvp level. In both Si₉ and Si₉M₅ cases, the Si–Si stretching vibrations agree well with experimental values. The most dominant peaks of stretching vibrations of silicon in Si₉ are ranging from 533 to 317 cm⁻¹ (Table S5†). The increased vibrational frequency of Si₉M₅ reveals the structural change after introducing alkali metals.

Fig. 6 displays the results of ELF and LOL analysis, which further justifies the nature of boning and the presence of loosely bound electrons in present clusters. The color scale of the ELF and LOL maps varies from blue to red in the range of 0 to 1.0, respectively. The area colored in red and blue corresponds to the maximum and minimum Pauli repulsion with the unity and zero values, respectively. However, the highest repulsive interaction (most localized electron state) is shown in red. The strong localization of electrons (ELF > 0.5) corresponds to the covalent bond, lone pair, or inner shells. From the ELF analysis, the Si₉Li₅ cluster shows strong electron localization after interactions with alkali metals, where a concentrated red color can be seen with alkali metals. The blue spots (ELF < 0.5) show charge delocalization regions, indicating the existence of noncovalent interactions. The electrons are highly localized when ELF is one and delocalized when ELF is zero. The high level of ELF and LOL values are indicated by the red color near alkali metals. A delocalized electron cloud can be seen as a blue color on the outer surface of the Si₉ cluster and also in the Si₉M₅ cluster. In addition, the green regions on the ELF surface and LOL colour-filled maps point out the presence of vdW interactions.

Fig. 6 Electron localization function (ELF) (upper panel) and 2D localized orbital locator (LOL) spectra (lower panel) of the present clusters. The planes that are used to plot the ELF and LOL patterns are shown in Fig. S8.†

4 Conclusions

In the concluding summary, we investigate the Zintl-based superalkali-excess electron clusters theoretically using various calculations. The designed Si₉M₅ (M = Li, Na, and K) clusters are thermodynamically stable, as indicated by the binding energies and AIMD simulations. Also, Si–M interactions are shown to be weak and dominated by vdW interactions, which contribute to optical and NLO properties. Excess electron nature, bonding structures, and partial atomic charges are calculated through FMO and NBO studies. IR and Raman frequencies are calculated to examine the amplitude of vibrations after doping. The HOMO–LUMO gap for Si₉K₅ is much reduced as compared to the pristine Si₉ cluster. Remarkable static polarizability (α_o) and hyperpolarizability (β_o) values are obtained up to 1.02 × 10⁴ au and 6.66 × 10⁴ au for the Si₉K₅ cluster. The influence of the implicit solvent model has been considered, where β_o increased 83 times as compared to the corresponding value in the gas phase for the Si₉K₅ cluster. The frequency-dependent NLO response for the dc-Kerr effect is observed up to 1.3 × 10¹¹ au, indicating the excellent change in refractive index by an externally applied electric field. Electron localization (ELF) and localized orbital locator (LOL) further reveal the bonding nature in these clusters.

Conflicts of interest

The authors declare no competing financial interest.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 22320102004, 22133003) and the Beijing National Laboratory for Molecular Sciences. Furthermore, author A. A. also acknowledges support from Alliance of National and International Science Organization (ANSO).

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Footnotes

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4ra02396j

‡ The authors contributed equally to this work.

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