Germanium-based superatom clusters as excess electron compounds with significant static and dynamic NLO response; a DFT study

Abstract

Herein, the geometric, electronic, and nonlinear optical properties of excess electron zintl clusters Ge₅AM₃, Ge₉AM₅, and Ge₁₀AM₃ (AM = Li, Na, and K) are investigated. The clusters under consideration demonstrate considerable electronic stability as well as superalkali characteristics. The NBO charge is transferred from the alkali metal to the Ge-atoms. The FMO analysis shows fabulous conductive properties with a significant reduction in SOMO–LUMO gaps (0.79–4.04 eV) as compared with undoped systems. The designed clusters are completely transparent in the deep UV-region and show absorption in the visible and near-IR region. Being excess electron compounds these clusters exhibit remarkable hyperpolarizability response up to 8.99 × 10⁻²⁶ esu, where a static second hyperpolarizability (γ_o) value of up to 2.15 × 10⁻³⁰ esu was recorded for Ge₉Na₅ superatom clusters. The excitation energy is the main controlling factor for hyperpolarizability as revealed from the two-level model study. The electro-optical Pockel's effect and the second harmonic generation phenomenon (SHG) are used to investigate dynamic nonlinear optical features. At a lower applied frequency (=532 nm), the dynamic hyperpolarizability and second hyperpolarizability values are significantly higher for the studied clusters. Furthermore, for the Ge₉K₅ cluster, the hyper Rayleigh scattering (HRS) increases to 5.03 × 10⁻²⁶ esu.

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1 Introduction

The past several decades have witnessed increasing scientific and technology-driven interest in developing nonlinear optical (NLO) materials because of their tremendous importance in photonic applications.^1,2 The developments in the field of nonlinear optics and laser-based technologies started after the discovery of the ruby laser by Maiman in 1960.³ Thus nonlinear optical materials have emerged rapidly during the last few decades, mainly due to their extensive applications in optoelectronic and photonic devices, second harmonic generation (SHG), endoscopy, and laser surgery^4–8 To date, numerous approaches for designing nonlinear optical materials with high hyperpolarizability have been used, including diradical character,⁹ designing octupolar molecules,¹⁰ the push–pull effect in conjugated chromophores,¹¹ multidecker sandwich complexes,¹² and excess electron models.¹³ Among the studied nonlinear optical materials inorganic materials exhibit prime interest because of their physicochemical stability and thermal stability.¹⁴

The excess electron system is well-known for triggering second and third-order nonlinearity.^15,16 In the family of excess electron compounds, electride,¹⁷ alkalides,¹⁸ and alkaline-earthides^19,20 are well known.

Electrides are compounds in which electron trapping into the complexant acts as an anion.²¹ Similarly, the alkalides are compounds in which alkali metals possess a negative charge and become anion (Li⁻, Na⁻, K⁻).²² Alkaline-earthides are also a fabulous excess electron system that contains a negative charge on alkaline-earth metals (Be⁻, Mg⁻, Ca⁻).²³ Excess electron compounds can be designed by doping any complexant with alkali metals,^24,25 alkaline earth metals,²⁶ transition metals^27,28 and superalkali clusters.^29,30

Superalkali clusters belong to the superatom clusters family and exhibit alkali-like characteristics with tunability in their electronic and geometric properties.³¹ The term ‘superalkalis’ was first time introduced by Gutsev and Boldyrev for Li₃O, Li₂F, NLi₄ etc. through DVMxα calculations. The superalkali clusters materials are of prime interest and show significant applications in numerous fields including, catalysis,³² reduction of CO₂ and N₂,³³ hydrogen storage materials, and nonlinear optics.³⁴ Thus, being excess electron compounds superalkali clusters can be adopted for making high-performance nonlinear optical materials. Furthermore, several studies have been proposed that reveal that superalkali being excess electron clusters can be doped with different molecules and nanocages to form excess electron compounds for triggering the NLO response. In this regard, superalkali-doped nanocages Li₃O@Al₁₂N₁₂ were theoretically designed with electride characteristics for enhanced nonlinear optical response.³⁵ Similarly, superalkali doped 2D graphdiyne M₂X@GDY (where M = Li, Na, K, and X = F, Cl, Br) were studied, and it was observed that there is a significant decrease in the HOMO–LUMO gap with a notecable increase in hyperpolarizability response.³⁶ Moreover, superalkali clusters were doped with different 2D materials and nanocages to get excess electron compounds i.e. alkalide, alkaline earthides. These excess electron compounds were reported as excellent nonlinear optical materials with significant hyperpolarizality.^24,37,38 However, a very limited number of studies were reported that reveal the nonlinear optical response of pure superalkali clusters as efficient nonlinear optical materials.

A few number of pure superalkali clusters are investigated as excess electron compounds and show remarkable hyperpolarizability. In this regard, Misra et al., investigated the electronic and nonlinear optical response of hyper lithiated superalkali clusters and reported these clusters as efficient NLO materials. The nonlinear optical response increases up to 1.2 × 10⁴ au.³⁹ Subsequently, another class of superalkali clusters CNLi_n (n = 1–10) was investigated for nonlinear optical response and second-order NLO response was much pronounced.⁴⁰ The literature reveals that only conventional types of superalkali clusters are explored for optical and nonlinear optical studies however several models can be efficiently utilized to be used as nonlinear optical responses. Literature also reveals, there are several superatom clusters (silicon-based) encapsulated by transition metals which were also studied for optical and magnetic excitation^41,42

Zintl polyanions, discovered by Eduard zintl in 1930 belong to the group (14,15) and show excellent physicochemical stability.⁴³ It is previously reported that zintl P₇³⁻ anion as core material can be used to design organo-zintl superalkali clusters which contain superb electron properties.⁴⁴ Similarly, zintl based superalkalis as a building block, when treated with superhalogens make novel supersalt compounds with significant electronic and nonlinear optical properties.⁴⁵ Moreover, superalkali clusters other than alkali metals might possess better stability and a high nonlinear optical response. We become interested in zintl-based superatoms, particularly germinum-based superatoms (which belong to group 14 elements) for electronic and optical properties. Furthermore, the stoichiometry of these clusters obeys magic number nuclei, in which their valence shells are organised as 1S2, 1P6, 1D10, 2S2, 1F14 after losing one electron, achieving electronic shell closure (according to Jellium model). When designed with alkali metals, Ge semimetallic clusters may have improved optoelectronic and NLO features. Although the Ge₅Li₃, Ge₉Li₅, and Ge₁₀Li₃ were theoretically studied by Sun et al.⁴⁶ Their investigations were limited to electronic properties whereas we adopted the alkali decorated zintl clusters Ge₅AM_3, Ge₉AM₅ and Ge₁₀AM₅ (where AM = Li, Na, K) for exploring optoelectronic and nonlinear optical properties.

In these studied clusters, we are mainly concerned with the following issues. Do these clusters belong to superalkali with better thermal and electronic stability than conventional superalkalis do these clusters possess nonlinear optical responses for declaring them as efficient NLO materials. The NLO response was confirmed by hyperpolarizability calculation and second hyperpolarizability.

2 Computational details

Initially, all the studied alkali decorated zintl polyanions Ge₅AM_3, Ge₉AM_5, and Ge₁₀AM₃ (where AM = Li, Na, K) are considered and optimized at CAM-B3LYP/6-311+G(d,p) level of theory. The geometries of Ge₅Li_3, Ge₉Li_5, and Ge₁₀Li₃ were reported in the previous literature and we adopted the similar design for the rest of the alkali metals.⁴⁶ All the calculations were performed with Gaussian 09 software.⁴⁷ The CAM-B3LYP (Coulomb attenuating method) is a hybrid exchange–correlation functional that includes the hybrid properties of B3LYP functional and long-range corrected Coulomb-Attenuating Method (CAM).⁴⁸ B3LYP is a hybrid part of the above method that contains Beckes 3-parameters for exchange functional and Lee–Yang–Parr-correlation functional. This hybrid density functional theory-based method comprises 0.19 HF plus 0.81 (B88) exchange interactions at short range and 0.65 HF plus 0.35 (B88) long-range interactions.⁴⁹ The CAM-B3LYP is a well-known approach for linear and nonlinear optical characteristics of various clustered materials, and it has already been demonstrated to give appropriate geometries and comparable hyperpolarizability values with CCDST.^50–53 Besides, the triple zeta split valence basis set 6-311+G(d,p) with diffuse and polarized function is adopted throughout the calculations.

To explore the electronic stabilities of these clusters, we calculated vertical ionization potential and electron affinity.

IE = E_X⁺ − E_X⁰ (1)

EA = E_X⁻ − E_X⁰ (2)

The chemical hardness is also calculated to understand their reactivity and soft nature and given by equation below ⁵⁴

Chemical hardness (η) = VIP − VEA (3)

To further explore the electronic properties, we performed frontier molecular orbital (FMOs) analysis which included SOMO, LUMO, and E_H–L gap. The FMOs analysis also provides evidence of the excess electron nature of studied superalkali clusters. Natural bonding orbitals (NBO) study is conducted to explore the nature and charge distribution as studied superatom clusters. The time-dependent density functional (TD-DFT) theory is adopted to calculate the excited state parameters and absorbance behavior of studied zintl superatom clusters. The time-dependent density functional (TD-DFT) is well know for obtaining excited states parameters and absorption spectra of molecules and clusters. The TD-DFT was chosen because of its performance and its correspondence to experimental results. For this purpose excited-state calculations with TD-CAM-B3LYP/6-311+G(d,p) are performed. Mathematically system under the constant field can be expressed as:

where F is an external applied electric field on the molecular system, F_i is its component of force along i direction; E⁰ is the total energy of the superalkali clusters without a static electric field, μ_i, α_ij, β_ijk, and γ_ijkl are dipole moment, polarizability, hyperpolarizability, and second-order hyperpolarizability, respectively. Thus regarding nonlinear optical properties, the following parameters are estimated including the mean dipole moment (μ_o), change in dipole moment (Δμ), static polarizability (α) and static first hyperpolarizability (β).

α_o = 1/3(α_xx + α_yy + α_zz) (5)

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.

μ_o = (μ_x² + μ_y² + μ_z²)^½ (7)

Besides, the second static hyperpolarizability (γ_o) and the projection of hyperpolarizability on the dipole moment vector (β_vec) is also calculated for our studied superalkali clusters at the same level of theory. Vector part of hyperpolarizability (β_vec) and second hyperpolarizability (γ) are expressed defined as

〈γ〉 = 1/5(γ_xxxx + γ_yyyy + γ_zzzz + γ_xxyy + γ_xxzz + γ_yyxx + γ_yyzz + γ_zzxx) (8)

Mathematically β_HRS can be expressed as;

where 〈β_zzz²〉 and 〈β_zxx²〉 are 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〉

The frequency-dependent NLO analysis was conducted at 532 and 1064 nm wavelength. Frequency-dependent hyperpolarizability involves the electro-optic Pockel's effect (EOPE) β(−ω;ω,0) and electric field induced second harmonic generation (ESHG) β(−2ω;ω,ω) respectively. While for second hyperpolarizability (γ), dc-Kerr γ^dc-Kerr (ω) = γ(−ω;ω,0,0) and second harmonic generation γ^ESHG(ω) = γ(−2ω;ω, ω,0) were considered.

3 Results and discussion

3.1 Geometries and electronic properties

All the optimized geometries of Ge₅AM₃, Ge₉AM_5, and Ge₁₀AM₃ (where AM = Li, Na, K) zintl superalkali clusters and their equilibrium bond length are depicted in Fig. 1. The geometric parameters of these clusters are identical to those reported in the earlier literature⁴⁶ and are included in the ESI† (S1).

Fig. 1 Optimized geometries with NBO charge of Ge₅AM₃, Ge₉AM₅ and Ge₁₀AM₃ clusters.

These clusters have no imaginary frequency (negative frequency) according to the frequency calculations, hence they are true minima on the potential energy surface (PES).

The computed ionization potential and electron affinity are used to analyze the electronic stability and superalkali nature of the investigated clusters. The difference in ground state energy between the cationic and neutral systems is described as the ionization potential (IP), meanwhile, the difference in ground state energy between the neutral and anionic systems is defined as the electron affinity.

The vertical ionization potential (VIP) values for Ge₉AM₃ are in the range of 5.49 to 3.91 eV, and it decreased dramatically as the alkali-metal size rose (Li to k). The VIP values for the Ge₉AM₅ series follow a similar pattern, reaching to 2.15 eV for F (Table 1). On the other hand, the VIP values for Ge₁₀AM₃ are slightly higher than for Ge₉AM₅ (Table 1). The lower VIPS values for the Ge₉AM₅ series can be attributed to the higher number of alkali metals (AM = 5), which is responsible for the cluster's electropositive nature. Additionally, their superalkali nature is demonstrated by their relatively low ionization potential values than Li atoms (5.4 eV).

Table 1 Vertical ionization potential (VIP, in eV), vertical electron affinity (VEA in eV), maximum chemical hardness (η in eV), average NBO charge upon germanium (Q_Ge in |e|), an average charge upon alkali metal (Q_AM in |e|), of Ge₅AM₃ Ge₉AM₅ and Ge₁₀AM₃ (where AM = Li, Na, K)

Superalkalis	VIP	VEA	η	QGe	QAM
Ge5AM3 (where AM = Li, Na, K)
Ge5Li3 (A)	5.49	0.98	4.51	−0.69	0.86
Ge5Na3 (B)	4.69	0.64	4.05	−0.65	0.87
Ge5K3 (C)	3.91	0.29	3.62	−0.65	0.90
Ge9AM5
Ge9Li5 (D)	4.36	0.65	3.71	−0.61	0.85
Ge9Na5 (E)	2.81	0.07	2.74	−0.52	0.88
Ge9K5 (F)	2.15	0.02	2.13	−0.50	0.90
Ge10AM3
Ge10Li3 (G)	4.98	1.15	3.83	−0.38	0.86
Ge10Na3 (H)	4.34	0.67	3.67	−0.36	0.85
Ge10K3 (I)	3.50	0.11	3.39	−0.46	0.90

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The calculated vertical electron affinity (VEA) values of these superalkali clusters range from 0.02 to 1.15 eV. In the designed series, Ge₉AM₅ has a smaller VEA value than those of Ge₃AM₃ and Ge₁₀AM₃. The electro-positive feature of the examined superalkalis is revealed by the tiny VEA values. Furthermore, the reduced VEA values indicate that these alkali-decorated zintl clusters are unable to completely grasp the valence (loosely bound) electron, which could result in interesting electrical properties. Table 1 shows that the examined clusters are polarizable and soft, based on the computed minimal values of chemical hardness. The calculated chemical hardness (η) values decrease with increased alkali metals size within these clusters. Among the series, Ge₉AM₅ clusters shows lower values of chemical hardness which may be attributed to the higher number of alkali smaller metals (soft) in these clusters.

3.2 Natural bonding orbital (NBO) analysis

The NBO analysis is a useful tool for interpreting intramolecular and intermolecular interactions and conjugative interactions in compounds and clusters.⁵⁵ For Ge₅AM₃ clusters, the obtained charges on alkali metals (QAM) vary from 0.86 to 0.90|e| (positive magnitude). The C cluster has the highest computed positive charge of 0.90|e|, while A has the lowest value of 0.86|e| in the Ge₅AM₃ family. Similarly, the Ge₉AM₅ clusters, computed average charge (QAM) ranges from 0.85 to 0.90|e| (Table 1). For the Ge₁₀AM₃ superalkali clusters, similar NBO (positive magnitude) charges are observed. Hence, the computed positive NBO charges upon alkali metals shows the significant charge transfer (from alkali metals to Ge-atom) within the superalkali clusters. Additionally, the compted NBO charges upon germanium metals (Q_Ge) for Ge₅AM₃ range from −0.13 to −0.69|e| (negative in magnitude). Therefore from the computed NBO charges (Table 1) one can observe the excellent separation of charges within the clusters. Likewise, the computed NBO charges (Q_Ge) for the Ge₉AM₃ clusters are range from −0.09 to −0.61|e| where the highest value of −0.61|e| is obtained for Ge₉Li₅ while the lowest value (−0.50) is obtained for Ge₉K₅ superalkali. However, the reported NBO charges (Q_Ge) for the third series Ge₁₀AM₃ are reduced (in negative magnitude) with corresponding increased cluster size. The overall NBO charges is an order of Ge₅AM₃ > Ge₉AM₅ > Ge₁₀AM₃.

3.3 Frontier molecular orbital (FMO) analysis and excess electron character

FMO analysis is adopted to validate the reactivity of the studied cluster system. Thus according to the frontier molecular orbital treatment of chemical reactivity, the rate, and site of reactivity of a molecule with a nucleophile is dominated by the interaction of the LUMO of the molecule in question with the HOMO of the nucleophile. The closer these orbitals are in energy, the more intensely they will interact, and the higher is the reactivity will be. Result in Table 2 show that the computed values for singly occupied molecular orbitals (SOMO), LUMO, and E_S–L gaps. For the Ge₅AM₃, the obtained values of singly occupied molecular orbitals (SOMO) vary from −4.99 to −3.51 eV and increase from A to C. For Ge₉AM₅, predicted SOMO energies are in the range of −3.97 to −1.80 eV, as well. The Ge₁₀AM₃ clusters, on the other hand, have lower SOMO energies than those of Ge₉AM₅ clusters, and these range from −4.46 to −3.14 eV. It can also be concluded that the SOMO energies of the examined superalkali clusters rise monotonically as the size of the alkali metals increases. Thus the observed trend of SOMO energies for studied superalkali clusters is Ge₉AM₅ < Ge₁₀AM₃ < Ge₅AM₃. Furthermore, the estimated LUMO energies for Ge₅AM₃ clusters vary with cluster size. However, the estimated energies of virtual orbitals for the Ge₉AM₅ series are increasing, while the Ge₁₀AM₃ clusters are increasing even high.

Table 2 Energies of SOMO and LUMOs (in eV), HOMO–LUMO gaps (E_H–L in eV), excitation energies (ΔE in eV), the wavelength of maximum absorbance (λ_max in nm), oscillator strength (f_o in au), ground-state dipole moment (μ_o in au), and excited-state dipole moment (Δμ in au) of Ge₅AM₃, Ge₉AM₅ and Ge₁₀AM₃ superalkali clusters

Superalkalis	SOMO	LUMO	EH–L	ΔE	λmax	fo	μ	Δμ
Ge5AM3 (where AM = Li, Na, K)
Ge5Li3 (A)	−4.99	−0.94	4.04	2.16	571	0.015	1.63	1.05
Ge5Na3 (B)	−4.24	−0.99	3.25	2.16	572	0.015	2.26	1.14
Ge5K3 (C)	−3.51	−0.73	2.77	2.24	553	0.041	3.12	2.34
Ge9AM5
Ge9Li5 (D)	−3.97	−0.07	3.89	2.25	548	0.007	0.27	0.62
Ge9Na5 (E)	−2.42	−1.07	1.34	1.77	688	0.076	1.69	2.49
Ge9K5 (F)	−1.80	−1.02	0.79	1.12	1101	0.219	2.21	1.86
Ge10AM3
Ge10Li3 (G)	−4.40	−0.87	3.73	2.31	534	0.007	1.45	0.62
Ge10Na3 (H)	−4.00	−0.83	3.16	2.47	500	0.008	1.55	0.65
Ge10K3 (I)	−3.14	−0.69	2.45	2.14	578	0.005	1.67	0.49

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For Ge₅AM₃ clusters, there is a considerable reduction in the SOMO–LUMO gap, with values ranging from 4.06 to 2.77 eV. The HOMO–LUMO gaps for Ge₅AM₃ clusters are around 4.06 to 2.77 eV, indicating that these compounds behave like semiconductors. Similarly, the observed reduction in Ge₉AM₅ E_H–L values from 3.89 to 0.79 eV could be attributable to their higher SOMO energies. Furthermore, the Ge₁₀AM₃ series' HOMO–LUMO gap values are slightly higher than the Ge₉AM₃ series, and range from 3.37 to 2.45 eV (Table 2). The significant reduction in HOMO–LUMO gaps reveals the soft nature (higher reactivity) and conductive properties of these clusters. Furthermore, the observed reduced SOMO–LUMO gap within clusters is attributed to the larger size of alkali metals (ease in the transition of electron from occupied to virtual orbitals).

Moreover, the excess electron character of clusters can be seen in the electronic density distribution (Fig. 2). The excess electron cloud is indicated by the electronic density of SOMO and LUMO scattered throughout the alkali metals that resemble to s-orbital. The LUMO electronic density for Ge₁₀AM₃ clusters wraps around the alkali metals, whereas the LUMO densities for Ge₉AM₅ clusters move away from the alkali metals.

Fig. 2 Representation of Frontier molecular orbital densities along with orbitals contribution of superalkali clusters (iso-value of 0.030).

3.4 TD-DFT analysis of clusters

Time-dependent density functional theory was used to examine the absorbance behavior of Ge₅AM₃, Ge₉AM₅, and Ge₁₀AM₃ superalkali clusters. For nonlinear optical applications, the cluster materials employed should be transparent in the used region. In the absorption study, we calculated the maximum absorbance, excitation energy, and oscillator strength (during the electronic transition). The computed excited state parameters for the studied superalkali clusters are given in Table 2. From the computed results one can observe that the studied clusters are completely transparent in the UV-region (<400 nm) and show broadband absorbance in the visible region. The longer absorbance wavelength for the Ge₅AM₃ (λ_mam = 572 nm) is observed for the Ge₅Na₃ superalkali cluster whereas the lowest absorbance maxima (λ_mam = 553 nm) is obtained for the Ge₅K₃ superalkali cluster (Fig. 3). However, the Ge₉AM₅ series shows higher absorbance maxima (λ_mam = 1101 nm) for Ge₉K₅ clusters. Similarly, for the second series, the highest absorbance maxima (λ_mam = 578 nm) is observed for the Ge₁₀K₃ cluster in the Ge₁₀AM₃ series. Overall, the clusters of Ge₉AM₅ series shows absorption maxima at a longer wavelength (bathochromic shift) whereas the Ge₅AM₃ and Ge₁₀AM₃ have slightly blue-shifted wavelengths. As a result of their total transparency under deep UV-region, the optoelectronic properties of the examined clusters fall under the UV-region. Furthermore, the excitation energies (ΔE) of Ge₅AM₃ (i.e. excitation of an electron from HOMO to LUMO) are also very small and ranges from 2.16 to 2.24 eV whereas the observed excitation energies (ΔE) for the Ge₉AM₅ are further reduced and lie in the range of 1.12 to 2.25 eV. Alternatively, the computed excitation energies of the Ge₁₀AM₃ series are slightly higher than those of Ge₉AM₅ and display a range of 2.14 to 2.47 eV. Moreover, the calculated oscillator strength (probability of absorbance during excitation) shows significant values for Ge₅AM₃ superalkali clusters while the Ge₉AM₅ and Ge₁₀AM₃ have slightly reduced values of oscillator strength (f_o).

Fig. 3 Absorbance spectra of Ge₅AM₃, Ge₉AM₅, and Ge₁₀AM₃.

3.5 Dipole moment and change in dipole moment

Table 2 shows the computed values of mean dipole moment (μ_o) and change in dipole moment (Δμ). The magnitude of polarisation in clusters and the asymmetric charge distribution are shown by the significant values of dipole moments. For the Ge₅AM₃ series, the observed mean dipole moments range from 1.63 to 3.13 au. As the size of alkali metals grows larger (Li to K), the dipole moment gradually increase. Likewise, the calculated values of the dipole moment of Ge₉AM₅ range from 0.27 to 2.21 au. Thus the calculated values of dipole moment for the Ge₉AM₅ are slightly smaller than those of Ge₅AM₃ clusters. On the other hand, the computed mean dipole moments are enhanced for the Ge₁₀AM₃ series and the values lie in the range of 1.42 to 1.67 au. Hence, from the computed results, one can conclude that a significant dipole moment that is associated with the Ge₅AM₃ series would result in larger polarization. Our calculations also show that the studied alkali decorated zintl polyanions clusters possess polar bonds (asymmetric electronic density) that might be an important factor for imparting optical and nonlinear optical properties. Besides, the computed changes in dipole momen between the ground state and crucial excited state for Ge₅AM₃ clusters lies in the range of 1.05 to 2.34 au where the highest value of 2.34 au is observed for Ge₅K₃ while the lowest value of 1.05 au is obtained for the Ge₅Li₃ cluster. A similar decreasing trend of change in dipole moment is observed for the Ge₉AM₅ clusters. The value of excited-state dipole moment for the Ge₉AM₅ series lie in the range of 0.62 to 1.86 au. However, the computed values of the excited-state dipole moment are increasing with the increased size of alkali metals within clusters. Finally, the Ge₁₀AM₃ series of clusters show further decreased values of change in dipole moment. As a result of the fascinating electronic features of clusters examined, increased optical and nonlinear optical properties might be expected.

3.6 Static nonlinear optical properties

The alkali-like superatom clusters Ge₅AM₃, Ge₉AM₅, and Ge₁₀AM₃ examined here have an excess electron nature. As a result, large optical and nonlinear optical (NLO) responses are reasonable predictions. Literature reveals that compounds and clusters with excess electrons characteristics are significantly adopted for triggering nonlinear optical response.^{19,36,39,56–65} Hence, the calculated polarizability (α_o), hyperpolarizability (β_o), second hyperpolarizability (γ_o), and associated electronic parameters are computed and given in Table 3. The calculated values of polarizability (linear optical response) of Ge₅AM₃ lie in the range of 3.70 × 10⁻²³ to 5.3 × 10⁻²³ esu. Similarly, the computed values of polarizability (α_o) for Ge₉AM₅ series are significantly enhanced and lie in the range of 6.5 × 10⁻²³ to 1.9 × 10⁻²¹ esu. Alternatively, the obtained values of polarizability of Ge₁₀AM₃ series are slightly smaller than those of Ge₉AM₅ and lie in range of 60 × 10⁻²⁴ to 71 × 10⁻²⁴ au. The increasing trend of polarizability may be seen as the size of alkali metals increases (Li to K). Overall, the highest value of 1.9 × 10⁻²¹ esu is obtained for F wheres the lowest value of 37 × 10⁻²⁴ esu is observed for A. Furthermore, the significant polarizability values obtained demonstrate the amount of polarity within the examined clusters. Asymmetric distribution of charges and electronic densities inside clusters also contributed to the higher linear optical response.

Table 3 Polarizability (α_o in ×10⁻²⁴ esu), first static hyperpolarizability (β_o in ×10⁻³³ esu) scattering hyperpolarizability (β_vec in ×10⁻³³ esu), static second hyperpolarizability (γ_o in ×10⁻⁴⁰ esu), HOMO–LUMO gaps (E_H–L in au), and vertical ionization potential (VIP in au) of Ge₅AM₃, Ge₉AM₅ and Ge₁₀AM₃ superalkali clusters

Superalkalis	αo	βo	βvec	γo	EH–L	VIP
Ge5AM3
Ge5Li3 (A)	3.7 × 10^−23	3.44 × 10^−29	3.39 × 10^−30	1.37 × 10^−34	4.04	5.49
Ge5Na3 (B)	4.5 × 10^−23	1.01 × 10^−27	9.94 × 10^−30	5.5 × 10^−34	3.25	4.69
Ge5K3 (C)	5.3 × 10^−23	3.41 × 10^−28	2.97 × 10^−29	2.80 × 10^−33	2.77	3.91
Ge9AM5
Ge9Li5 (D)	6.5 × 10^−23	1.80 × 10^−28	1.81 × 10^−29	4.07 × 10^−34	3.89	4.36
Ge9Na5 (E)	3.6 × 10^−22	1.57 × 10^−26	1.57 × 10^−26	2.15 × 10^−30	1.34	2.81
Ge9K5 (F)	1.9 × 10^−21	8.99 × 10^−26	8.99 × 10^−27	7.68 × 10^−34	0.79	2.15
Ge10AM3
Ge10Li3 (G)	6.0 × 10^−23	1.88 × 10^−29	1.88 × 10^−29	2.26 × 10^−34	3.73	4.98
Ge10Na3 (H)	6.8 × 10^−23	3.87 × 10^−29	3.87 × 10^−29	3.87 × 10^−34	3.16	4.34
Ge10K3 (I)	7.1 × 10^−23	4.57 × 10^−29	4.57 × 10^−29	7.68 × 10^−34	2.45	3.50

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Furthermore, the calculated values of static hyperpolarizability (β_o) lie in the range of 3.44 × 10⁻²⁹ to 8.99 × 10⁻²⁶ esu. The calculated static first and second hyperpolarizabilities values are significant. Hyperpolarizability values in the proposed superalkali clusters follow the order Ge₉AM₅ > Ge₅AM₃ > Ge₁₀AM_3. The hyperpolarizability values obtained for the Ge₅M₃ series rise monotonically with alkali metal size. The highest value of 3.41 × 10⁻²⁸ esu is observed for the C cluster while the lowest value (3.44 × 10⁻²⁹ esu) is calculated for the A cluster. Similarly, the obtained values of the β_o for the Ge₉AM₅ series range from 1.80 × 10⁻²⁸–8.99 × 10⁻²⁶ esu. With the increased metal size and the number of alkali metals in the Ge₉AM₅ clusters, there is a huge increase in β_o values. In particular, the Ge₉K₅ shows a remarkable β_o value (8.99 × 10⁻²⁶ esu) which may be attributed to the larger size of alkali metal (K). In comparison, the Ge₁₀AM₃ (AM = Li, Na, and K) clusters have range of 1.88 × 10⁻²⁹ to 4.57 × 10⁻²⁹, which is lower than the Ge₉AM₅ and Ge₅AM₃ clusters. In our designed and studied clusters the hyperpolarizability response is quite larger than previously reported Li_nF (n = 2–5) superalkali clusters,³⁹ and M₂OCN & M₂NCO (M = Li, Na, K) clusters.⁶⁴ Excess electron nature might account for considerable hyperpolarizability response found in these clusters. Furthermore, the studied fabulous electronic properties contribute to the hyperpolarizabilities values. As a result, decreased E_H–L gaps and ionization potential (IP) with increasing alkali metals (AM) size ultimately prompt the β_o response.

We used a conventional two model with the sum-over-state (SOS) method to develop a full understanding of hyperpolarizability and its governing factors. The two-level model can be written as follows: β_tl = Δμ × f_o/ΔE³

Where the Δμ, f_o, and ΔE are changes in dipole moment, oscillator strength, and excitation energy for crucial excitation (excitation with maximum oscillator strength). From the above model, one can observe that β_tl has a direct relation with change in dipole moment and oscillator strength (f_o) while it is inversely related with cubic of excitation energy (ΔE). The obtained values of β_tl range from 1.03 × 10⁻²⁹–1.90 × 10⁻²⁷ esu (Table 4). The Ge₉AM₅ has the largest β_tl values in the examined clusters series, which is comparable to β_o. For the first series (Ge₅AM₃), the β_tl is increased with an increase in Δμ and oscillator strength. In this series, Ge₅K₃ shows a significant β_tl response which may be attributed to its noticeable change in dipole moment (2.34 au) and small excitation energy (1.58 eV). As a result, excitation energy (E) is considered to be extremely vital in impacting hyperpolarizability response. A similar trend is observed for the second series (Ge₉AM₅) where the Ge₉Na₅ shows a notable value (1.90 × 10⁻²⁷ esu) of β_tl which is due to small excitation energy (transition energy) and higher change in dipole moment (2.49 au). Likewise, β_tl values for the Ge₁₀AM₃ series, there is an increase β_tl with reduced excitation energy (ΔE). Hence, the excitation energy is deciding factor in triggering the hyperpolarizability response from the two-level model where its effect is inversely related to β_tl. Furthermore, the estimated β_o and β_tl values are highly correlated, providing additional insight into the hyperpolarizability response. From the plotted graph (Fig. 4), one can observe that overall, the β_tl values increase dramatically for the Ge₉Na₅ cluster.

Table 4 Computed hyperpolarizability from the two-level model (β_tl in ×10⁻³³ esu), change in dipole moment (Δμ in au), excitation energy (ΔE in eV), oscillator strength (f_o in au) hyper Rayleigh scattering (β_HRS in ×10⁻³³ esu), and depolarization ratio (DR in au) of superalkali clusters

Superalkalis	βtl	Δμ	ΔE	fo	βHRS	DR
Ge5AM3
Ge5Li3 (A)	1.03 × 10^−29	1.05	2.16	0.05	1.31 × 10^−29	6.87
Ge5Na3 (B)	3.11 × 10^−29	1.14	1.67	0.04	4.23 × 10^−29	5.61
Ge5K3 (C)	9.79 × 10^−29	2.34	1.58	0.16	1.32 × 10^−28	6.41
Ge9AM5
Ge9Li5 (D)	2.90 × 10^−30	0.62	1.69	0.01	1.13 × 10^−29	2.44
Ge9Na5 (E)	1.90 × 10^−27	2.49	0.01	0.002	9.61 × 10^−27	2.41
Ge9K5 (F)	9.83 × 10^−28	1.86	0.75	0.06	5.03 × 10^−26	2.45
Ge10AM3
Ge10Li3 (G)	1.54 × 10^−30	0.62	1.87	0.01	1.21 × 10^−29	2.37
Ge10Na3 (H)	1.79 × 10^−30	0.65	2.02	0.21	2.08 × 10^−29	2.45
Ge10K3 (I)	9.64 × 10^−31	0.49	1.86	0.11	2.73 × 10^−29	2.57

---

Fig. 4 Representation of β_tl for Ge₅AM₃, Ge₉AM_5, and Ge₁₀AM₃ superalkali clusters.

Projection of hyperpolarizability on dipole moment vector is also evaluated through β_vec simulations. The β_vec is a vector part of hyperpolarizability and is very important to characterize the nonlinearity of molecules and clusters. The calculated β_vec values are given in Table 3. From Table 3, the observed β_vec response of designed clusters strongly correlates with total hyperpolarizability which indicates the dipole moment has the same direction projection of hyperpolarizability (β_o). Furthermore, the quite comparable values of first hyperpolarizabilities with β_vec also indicate that the charge transfer is parallel to the molecular dipole moments. The resemblance of β_o and β_vec also suggests that these clusters can be synthesized in laboratory.

We also carried out calculations for estimation of static second hyperpolarizability (γ_o) which is a third rank tensor (χ³). The obtained values of γ_o for the Ge₅AM₃ range from 1.37 × 10⁻³⁴–2.80 × 10⁻³³ esu which shows an increasing trend with the increased size of alkali metals. Similarly, for the second series (Ge₉AM₅), the γ_o response increases up to 2.15 × 10⁻³⁰ esu for E cluster. The γ_o values for the Ge₁₀AM₃ superalkali clusters, on the other hand, decrease considerably. Clusters with a higher number of alkali metals (AM) have a substantial γ_o response. Likewise, the γ_o response for the Ge₁₀AM₃ clusters lie in the range of 2.26 × 10-³⁴ to 7.68 × 10⁻³⁴ esu and a slight increase is observed with growing alkali metals size. Hence, the γ_o values vary in order of Ge₉AM₅ > Ge₅AM₃ > Ge₁₀AM_3.

3.7 Hyper Rayleigh scattering measurement (β_HRS)

The hyperpolarizability of nonlinear optical molecules can be determined using hyper-Rayleigh scattering. The β_HRS is a widely used method for determining the nonlinear optical characteristics of centrosymmetric molecules with zero dipole moment. At the same level of theory, we theoretically evaluated the β_HRS response for these clusters. Overall, the calculated highest value (5.03 × 10⁻²⁶ esu) of β_HRS is obtained for Ge₉K₅ while the lowest value 1.13 × 10⁻²⁹ esu is observed for D. For the Ge₅AM₃ series β_HRS lie in the range of 1.31 × 10⁻²⁹ esu to 1.32 × 10⁻²⁸ esu, and are increased with size (Li to K). In the Ge₉AM₅ series, the usual trend of beta-HRS value of Ge₉Li₅ may be attributed to the small size of Li-atom. Similarly, other nonlinear optical parameters for Ge₉Li₅ are also small. It has been reported in the literature previously that dominating factor changes from structure to structure (and it ultimately leads to irregular trends). However, factors affecting the β_HRS response might be the same as total hyperpolarizability values. A similar increasing trend of β_HRS with the size of metal (AM) can be seen for the second series of clusters(Ge₉AM₅). Furthermore, β_HRS values do not increase with clusters size, rather these show dependence upon the size of and the number of alkali metals. As a result, the β_HRS values for the Ge₁₀AM₃ become slightly smaller, ranging from 1.22 × 10⁻²⁹ to 2.73 × 10⁻²⁹ esu. Interestingly, the Ge₉K₅ exhibits a significant HRS response which suggests better NLO properties. Additionally, the depolarization ratio (DR) is higher for Ge₅AM₃ clusters and increases up to 6.87 au for Ge₅Li₃.

3.7.1 Dynamic nonlinear optical properties. The frequency-dependent NLO properties are the fundamental molecular parameters that are required for the description of many nonlinear optical phenomena. The theoretical understanding and accurate determination of the frequency-dependent hyperpolarizabilities β(ω) and second hyperpolarizabilities γ(ω) are therefore crucial to classify nonlinearity of materials. The frequency-dependent NLO response is estimated at dispersion frequencies of 532 and 1064 nm, and we calculated electro-optic pockel's effect (EOPE) with β(−ω;ω,0) and electric field induced second harmonic generation (EFSHG) with β(−2ω;ω,ω) at applied frequency. The obtained values of electro-optical pockel's effect (EOPE) for Ge₅AM₃ clusters at dispersion frequency of 532 nm range from. The EOPE effect is much pronounced at a small dispersion frequency of 532 nm. Similarly, the obtained dynamic hyperpolarizabilities β(−2ω;ω,ω) values noticeably at smaller frequencies. Overall, the highest values (3.54 × 10⁻²⁶ esu) of EOPE is obtained for C whereas the lowest value of 3.02 × 10⁻²⁸ esu is observed for D (Table 5). The is a gradual increase in β(ω) with an increased size of alkali metals metal. Both EOPE and SHG show significant values at smaller applied frequencies, and their values slightly decrease at higher dispersion frequency (1064 nm).

Table 5 Frequency-dependent hyperpolarizability β(ω) in form of electro-optic pockel's effect (EOPE) β(−ω;ω,0) in ×10⁻³³ esu, and electric field induced second harmonic generation (EFSHG) with β(2 − ω;ω,ω) in ×10⁻³³ esu at ω = 532 nm and ω = 1064 nm

Superalkalis	ω = 0.0856 au (532 nm)		ω = 0.0428 au (1064 nm)
	β(−ω;ω,0)	β(−2ω;ω,0)	β(−ω;ω,0)	β(−2ω;ω,0)
Ge5AM3
Ge5Li3 (A)	3.28 × 10^−27	4.42 × 10^−27	6.99 × 10^−29	1.81 × 10^−28
Ge5Na3 (B)	1.29 × 10^−27	3.63 × 10^−27	2.50 × 10^−28	3.72 × 10^−28
Ge5K3 (C)	3.54 × 10^−26	1.30 × 10^−26	2.24 × 10^−28	9.50 × 10^−28
Ge9AM5
Ge9Li5 (D)	3.02 × 10^−29	3.28 × 10^−29	7.86 × 10^−29	1.47 × 10^−30
Ge9Na5 (E)	7.68 × 10^−28	4.42 × 10^−27	1.56 × 10^−26	2.94 × 10^−27
Ge9K5 (F)	6.39 × 10^−27	1.04 × 10^−26	4.93 × 10^−29	1.39 × 10^−27
Ge10AM3
Ge10Li3 (G)	9.67 × 10^−29	1.98 × 10^−28	2.24 × 10^−29	6.99 × 10^−29
Ge10Na3 (H)	5.71 × 10^−28	3.71 × 10^−28	2.85 × 10^−29	5.62 × 10^−29
Ge10K3 (I)	1.21 × 10^−27	1.04 × 10^−27	5.87 × 10^−29	2.42 × 10^−28

---

The calculated frequency-dependent second hyperpolarizability γ(ω) that includes dc-Kerr γ^dc−Kerr (ω) = γ(−ω;ω,0,0) and second harmonic generation with γ^ESHG(ω) = γ(−2ω;ω,ω,0) are calculated at the same level of theory. The calculated values of dynamic second hyperpolarizability are given in Table 6. The calculated dc-Kerr constant and EFSHG are higher at 532 nm and their values slightly decreased at higher dispersion frequency (1064 nm). Likewise, Ge₉AM₅ and Ge₁₀AM₃ are also larger at a small dispersion frequency of 532 nm. Hence, the studied clusters offer tremendous dynamic NLO properties at the smaller external applied frequency.

Table 6 Frequency-dependent second hyperpolarizability with dc-Kerr effect γ (−ω;ω,0,0) and electric field induced second harmonic generation (ESHG) γ(−2−ω;ω,ω,0) in ×10⁻⁴⁰ esu at ω = 532 and 1064 nm

Superalkalis	ω = 0.0856 au (532 nm)		ω = 0.0428 au (1064 nm)
	γ(−ω;ω,0,0)	γ(−2ω;ω,ω,0)	γ(−ω;ω,0,0)	γ(−2ω;ω,ω,0)
Ge5AM3
Ge5Li3 (A)	7.84 × 10^−31	1.37 × 10^−30	3.39 × 10^−34	1.40 × 10^−30
Ge5Na3 (B)	1.30 × 10^−32	1.64 × 10^−31	2.45 × 10^−33	2.31 × 10^−33
Ge5K3 (C)	5.43 × 10^−36	2.52 × 10^−29	7.37 × 10^−32	2.09 × 10^−31
Ge9AM5
Ge9Li5 (D)	1.36 × 10^−33	5.74 × 10^−32	3.16 × 10^−32	1.19 × 10^−33
Ge9Na5 (E)	1.76 × 10^−29	8.80 × 10^−30	1.58 × 10^−31	1.03 × 10^−32
Ge9K5 (F)	5.23 × 10^−32	2.53 × 10^−32	2.15 × 10^−30	3.76 747 508 × 10^−32
Ge10AM3
Ge10Li3 (G)	8.45 × 10^−32	1.24 × 10^−31	3.24 × 10^−40	5.28 × 10^−34
Ge10Na3 (H)	1.22 × 10^−32	6.84 × 10^−34	1.72 × 10^−32	2.41 × 10^−33
Ge10K3 (I)	5.23 × 10^−32	2.53 × 10^−32	2.15 × 10^−30	3.76 × 10^−32

---

4 Conclusion

In the summary, we explored the zintl-based superalkali for geometric, electronic and nonlinear optical properties. The studied zintl superalkali clusters Ge₅AM₃, Ge₉AM₅, and Ge₁₀AM₃ (AM = Li, Na, K) belong to excess electron compounds. These are superalkali clusters as their calculated vertical ionization potential (VIP) values are smaller than Li atom (5.39 eV). The calculated significant VIP values suggest their electronic stability. The calculated chemical hardness (η) lie in the range of 2.13 to 4.51 eV, and Ge₉AM₅ shows the higher softness among the series. There is a significant charge (positive in magnitude) on alkali metals, and the charge is transferred from alkali to Ge-atom. The charge is transferred from alkali metals to Ge-atoms within the clusters. There is a notable reduction in E_H–L (0.79–4.04 eV) which reveals their conductive applications. These clusters are completely transparent in the deep UV region, and show absorption maxima (λ_max) at the longer wavelength. Being excess electron compounds these clusters shows remarkable hyperpolarizability response up to 8.99 × 10⁻²⁶ esu where the static second hyperpolarizability (γ_o) value recorded up to 2.15 × 10⁻³⁰ esu for Ge₉AM₅ clusters. The adopted two-level model study reveals the controlling factors of hyperpolarizability. The obtained significant β_tl value of 1.90 × 10⁻²⁷ esu may attributed to smaller excitation energy (0.01 eV) The frequency-dependent hyperpolarizabilities and second hyperpolarizabilities values are much higher at smaller dispersion frequencies (ω = 532 nm). Moreover, the hyper Rayleigh scattering (β_HRS) increases up to 5.03 × 10⁻²⁷ esu for the Ge₉K₅ cluster.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The authors acknowledge the financial and technical support from the Higher Education Commission of Pakistan and COMSATS University, Abbottabad Campus.

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58. N. Kosar, et al., Significant nonlinear optical response of alkaline earth metals doped beryllium and magnesium oxide nanocages, Mater. Chem. Phys., 2020, 242, 122507.
59. A. S. Rad and K. Ayub, Change in the electronic and nonlinear optical properties of Fullerene through its incorporation with Sc-, Fe-, Cu-, and Zn transition metals, Appl. Phys. A, 2019, 125(6), 430.
60. S. Sajjad, A. Ali, T. Mahmood and K. Ayub, Journal of Molecular Graphics and Modelling Janus alkaline earthides with excellent NLO response from sodium and potassium as source of excess electrons ; a fi rst principles study, J. Mol. Graphics Modell., 2020, 100, 107668.
61. F. Ullah, K. Ayub and T. Mahmood, Remarkable second and third order nonlinear optical properties of organometallic C6Li6− M3O electrides, New J. Chem., 2020, 44, 9822–9829.
62. K. Ayub, Are phosphide nano-cages better than nitride nano-cages? A kinetic, thermodynamic and non-linear optical properties study of alkali metal encapsulated X 12 Y 12 nano-cages, J. Mater. Chem. C, 2016, 4(46), 10919–10934.
63. S. Munsif, et al., Remarkable nonlinear optical response of alkali metal doped aluminum phosphide and boron phosphide nanoclusters, J. Mol. Liq., 2018, 271, 51–64.
64. A. Ahsin and K. Ayub, Theoretical Investigation of Superalkali Clusters M2OCN and M2NCO (where M= Li, Na, K) as Excess Electron System with Significant Static and Dynamic Nonlinear optical response, Opt., 2020, 166037.
65. A. Ahsin and K. Ayub, Superalkali-based alkalides Li3O@[12-crown-4]M (where M= Li, Na, and K) with remarkable static and dynamic NLO properties; A DFT study, Mater. Sci. Semicond. Process., 2022, 138, 106254.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d1ra08192f

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