Stacking interactions in stabilizing supramolecular assembly of M[9C]₂M complexes: dynamic stability with remarkable nonlinear optical features

Abstract

Continual attempts have been made to discover excellent nonlinear optical (NLO) materials. Here, we investigate the role of stacking interactions and van der Waals forces in the designed parallel stacked complexes M[9C]₂M (where M = Li, Na, K, Be, Mg, and Ca) using various quantum chemical and molecular dynamics methods. The thermodynamic stability of the present complexes is also revealed by the computed interaction energy, enthalpy of formation, and Gibbs free energy of formation (ΔG_f). Molecular dynamics simulations were performed at room temperature to determine the stability of the dimer formation and their complexes. Alkali metals act as a more prominent source of excess electrons at long-range interaction distances. Charge decomposition analysis (CDA) and natural bonding orbital (NBO) analyses suggest excellent charge transfer in the alkalide complexes. In this series, Li[9C]₂Li exhibits an excellent hyperpolarizability response up to 2.3 × 10⁶ a.u., while Ca[9C]₂Ca performs well in alkaline-earth metal complexes. The NLO response is mostly influenced by the alkalide and earthide characteristics. Dynamic NLO features were computed at externally applied frequencies. Scattering first hyperpolarizability (β_HRS) and its associated components were also measured. The effect of solvents on hyperpolarizability is also considered. The quantum theory of atoms in molecules (QTAIM) and NCI are employed to investigate the bonding nature and vdW forces in addition to stacking interactions. TD-DFT and vibrational studies are also performed. We aim for this research to pave the way for the innovative strategies in designing supramolecular assemblies tailored for NLO applications.

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1. Introduction

Over the past few decades, there has been a significant focus on the development and creation of materials that exhibit enhanced nonlinear optical (NLO) properties.^1–3 This is due to the widespread utilization of NLO materials in optoelectronics,⁴ optical communication,⁵ color displays, memory storage,⁶ switching,⁷ and laser devices.⁸ Organic NLO compounds offer several advantages over their inorganic small counterparts, including structural diversity and flexibility, ultrafast response, dielectric constants, ease of synthesis, and high π-electron delocalization.^9–11 Numerous strategies have been proposed for creating materials that exhibit a significant non-linear optical (NLO) response. These strategies include incorporation of donor and acceptor groups,¹² using organic molecules with elongated electron systems,¹³ doping of main group and transition metals,^14–16 and alternating bond lengths.¹⁷

Introducing excess electrons into organic systems is considered an effective strategy for developing high-quality NLO materials. Li et al. were pioneers in uncovering the sensitive and reactive characteristics of excess electrons.¹⁷ Additional electrons mainly enhance the NLO behaviour by reducing the transition energies of important excited states in organic systems.¹⁸ Furthermore, the excess electrons are loosely bound, and their widely spread-out characteristics contribute to a significant NLO response in compounds.^19,20 Therefore, various categories of compounds containing excess electrons, such as electrides,^21,22 alkalides,^23,24 and alkaline earthides,²⁵ have been extensively studied for their NLO characteristics.

Electrides are a different type of ionic salt where the anionic sites are exclusively filled by electrons.²⁶ Dye's group successfully created additional electrides by incorporating a complexant with carbon-to-oxygen and carbon-to-nitrogen bonds after the synthesis of Cs⁺(18-crown-6)₂ e⁻ as the pioneering electride.²⁷ Several electrides have been developed using different complexants such as Li@3⁶Adz, cyclacene, and fluorocarbons.²⁸ Alkalides are a unique form of ionic salt where anions are replaced by alkali anions such as Na⁻, K⁻, Rb⁻, and Cs⁻.^29,30 Alkali metals have a more loosely bound nature because of their lower electron affinity, which is a result of the presence of excess electrons. Due to the small transition energies and higher oscillator strengths of excited states in alkali anions, alkalides emerge as a more favourable choice compared to electrides. The fundamental design concept of alkaline earthides involves incorporating an alkali metal within a complexant to serve as a source of excess electrons (electron donors), while the alkaline earth metal is situated on the outer side of the complexant to act as an electron acceptor.³¹ They involve the participation of the polarized p-orbital in the formation of excess electrons, in contrast to alkalides where the polarized s-orbital is involved in this process. These characteristics significantly enhance the NLO of alkaline earthides, making them more effective as NLO compounds.³²

To the best of our knowledge, examples of alkaline earthides are found in the literature. Hou et al. designed alkaline earthides by strategically positioning alkali-metal (Li) and alkaline earth metals (Be, Mg, and Ca) on the polar face of the Janus all-cis-1,2,3,4,5,6-hexafluoro-cyclohexane (C₆H₆F₆) molecule.²⁵ The designed alkaline-earthides showed first hyperpolarizabilities (β_o) up to 3.51 × 10⁶ a.u. In another study, Ayub et al. incorporated alkali metals (Li and Na) within the cavity of the 3⁶Adz complexant, while the outer region of the cage is filled with alkaline earth metals (Be, Mg, and Ca).²⁸ Based on their results, the Li⁺(3⁶Adz)Mg⁻ complex exhibits the highest hyperpolarizability (5.1 × 10⁸ a.u.). Additionally, a series of complexes, including M⁺(NH₃)₆M⁻,³³ M⁺(2⁶Adz)M⁻,³⁴ and M⁺(calix[4]pyrrole)M⁻ (where M⁺ and M⁻ represent alkali metals and alkaline earth metals, respectively),³⁵ indicate impressive NLO responses.

At the atomic and molecular levels, supramolecular interactions are fundamental forces that govern the behaviour of molecules and materials.³⁶ These interactions may cause particular molecular assemblages and configurations to occur, improving the NLO characteristics.³⁷ Through the manipulation of these interactions, researchers can customize the material's characteristics to attain particular NLO responses. Previous research has shown the potential role of excess electrons in alkalide, metalide and alkaline earth metal complexes. However, the role of non-covalent interactions and stacking interactions in these complexes is still not well explored. Particularly, the role of van der Waals forces (vdW) and dispersion interaction is very crucial in various chemical processes.^38–41

In this study, 9-crown-3 (9C) and its parallel stacked dimer [9C]₂ doped with alkali and alkaline earth metals M[9C]₂M (where M = Li, Na, K, Be, Mg, and Ca) are theoretically designed using quantum chemical and molecular dynamics. Crown ethers are non-conjugated heterocyclic compounds containing oxygen atoms that have gained significant attention in recent years due to their unique ability to selectively bind to metal cations, especially alkali metals. Our study demonstrated theoretically the existence of σ–σ stacking interactions and vdW forces in the designed dimer [9C]₂ and its M[9C]₂M complexes. The overlapping in sigma stacking interactions can lead to attractive or repulsive interactions, resulting in significant impact on electronic and optical properties. In addition, enhancing intermolecular charge transport by utilising excess electrons of metals has a promising role in electronic materials and biological systems. This work can provide a theoretical framework for the development of supramolecular assembly with enhanced nonlinear optical (NLO) properties.

2. Theoretical details

Quantum chemical simulations were performed using Gaussian 16 software,⁴² and molecular dynamics computations were conducted using ORCA 5.0.⁴³ Initially, the stacked dimer [9C]₂ was formed by positioning the [9-crown-3] ether. Subsequently, M[9C]₂M complexes (where M = Li, Na, K, Be, Mg, and Ca) were constructed. The low energy and relatively stable conformations of the complexes were determined using the artificial bee colony (ABC) algorithm, implemented in the ABCluster program.^44,45 In this program, 500 initial guess structures were generated for each complex, with the initial 100 being optimized at the m062X/def2qzvpp level. The relatively low-energy structures were considered for further simulations. The entire properties and surface topology analysis is carried out using the m062X/def2qzvpp method. This method is parameterized for both metals and non-metals and has high nonlocality with double the amount of nonlocal exchange.^46,47 In general, the thermodynamic properties and relative energies calculated using the m062X method are considered reliable and are in good agreement with experimental data.⁴⁸ The wavefunction stability is considered for entire complexes with the same method. Vibrational frequency analysis is performed using the same method to verify that the designed complexes correspond to true minima on the potential energy surface. In the thermodynamic study, we have calculated the interaction energy (ΔE), Gibbs free energy of formation (ΔG), and enthalpy (ΔH) of adsorption at 300 K. The calculated thermodynamic parameters can be expressed by the following set of equations:

ΔE = E_M[9C]2M − (E_[9C] + E_[9C] + 2E_M) (1)

ΔG = G_M[9C]2M − (G_[9C] + G_[9C] + 2G_M) (2)

ΔH = H_M[9C]2M − (H_[9C] + H_[9C] + 2H_M) (3)

In addition, we performed ab initio molecular dynamics (AIMD) simulations to analyze the adsorption and thermodynamic stability at room temperature. In the AIMD study, trajectory analysis is performed by considering the already optimized structure at the m062x/def2qzvpp method. We employed a composite B97-3c method that includes corrections for various interactions, making it suitable for studying a range of molecular systems with good accuracy.⁴⁹ The trajectories of each complex were analyzed by plotting root-mean-square deviation (RMSD) at room temperature and 1-atm pressure. Furthermore, to investigate the electronic features of the M[9C]₂M complexes, the ionization potential (IP), electron affinity (EA), frontier molecular orbitals (FMO), and natural population analysis (NPA) were determined. These parameters provide valuable insights into the electronic structure, charge distribution, and overall reactivity of the complexes. Charge decomposition analysis (CDA) is carried out to characterize the donation and back-donation interaction in the present complexes. The donor–acceptor mechanism of orbitals is revealed through stabilization energy E(2) based on the second-order Fock matrix. The stabilization energy E(2) during the electron delocalization between the donor (i) and acceptor (j) interacting orbitals was estimated through NBO analysis. The following equation was used to calculate the E² of the present complexes:Here, F_i,j is the off-diagonal NBO Fock matrix element, and q_i is the donor orbital occupancy. E_i and E_j are the diagonal elements (orbital energies).

To elucidate the nature of interactions and surface bonding patterns, the quantum theory of atoms in a molecule (QTAIM) and non-covalent interaction (NCI) studies were performed using the multiwfn and visual molecular dynamics software.⁵⁰ Through QTAIM and NCI studies, the extent of vdW and dispersion interactions was correlated with the stability and NLO features of the present complexes. The molecular electrostatic potential (MESP) maps were generated to understand the charge distribution on complexes. The electron localization function (ELF) and localized orbital locator (LOL) analyses are used to determine the presence of electrons and alkalide characteristics using the same method. For optical and non-linear optical properties, dipole moment, polarizability (α_o), first-order hyperpolarizability (β_o), and static second-order hyperpolarizability (γ_o) can be determined using the following equations:

μ_o = (μ_x² + μ_y² + μ_z²)^½ (6)

α_o = 1/3(α_xx + α_yy + α_zz) (7)

where

β_x = β_xxx + β_xyy + β_xzz, β_y = β_yyy + β_yzz + β_yxx an β_z = β_zzz + β_zxx + β_zyy (9)

〈γ〉 = 1/5(γ_xxxx + γ_yyyy + γ_zzzz + γ_xxyy + γ_xxzz + γ_yyxx + γ_yyzz + γ_zzxx) (10)

where β_xxx and γ_xxxx are the first and second hyperpolarizability tensors. The γ is a fourth rank tensor that quantifies the third-order nonlinear optical response of a material or molecule. It could be employed in describing the material's respond to the externally applied dispersion frequency. In addition, the scattering hyperpolarizability (β_HRS) was determined using the following relationship:where 〈β_zzz²〉 and 〈β_zxx²〉 are the average of the orientation tensors for hyperpolarizability. The related depolarization ratio for this super alkali clusters (DR) ratio is also given by:

DR = 〈β_zzz²〉/〈β_zxx²〉 (12)

Frequency-dependent NLO parameters were analyzed at 532 and 1064 nm wavelengths. We determined the electro-optic Pockel's effect (EOPE) β(−ω;ω,0) and electric field-induced second harmonic generation (ESHG) β(−2ω;ω,ω) as part of the frequency-dependent hyperpolarizability. For the second hyperpolarizability (γ), the 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. Furthermore, a density of states (DOS) spectral study was conducted to obtain a clearer picture of orbital energies and HOMO–LUMO gaps for clusters using the GaussSum software. The absorption parameters were obtained through time-dependent density functional theory (TD-DFT) calculations, and Fourier-transform infrared (FT-IR) spectra were generated to further analyze the vibrational characteristics of the complexes.

3. Results and discussion

3.1. Electronic structure and dynamics of M[9C]₂M

We initiated the electronic structure investigation on the M[9C]₂M (where M = Li, Na, K, Be, Mg, and Ca) complexes. Initially, [9C] ether rings are stacked to create dimer [9C]₂, which is then further doped with similar alkali and alkaline earth metals on both faces. The interaction distance between [9C]–[9C] rings in the [9C]₂ ranges from 3 to 4 Å. The upper face of the [9C]₂ displays oxygen–metal (O–M) interactions, while the lower face shows hydrogen–metal (H–M) interactions (see Fig. 1). These distinct interactions on different faces of the complex suggest directional bonding and could influence the overall stability, reactivity, and electronic properties of the system. Six complexes are modelled as follows: Li[9C]₂Li, Na[9C]₂Na, K[9C]₂K, Be[9C]₂Be, Mg[9C]₂Mg, and Ca[9C]₂Ca. Fig. S1 (ESI†) displays the optimized molecular geometries of the present complexes, and their bond distances are listed in Table 1. The entire complexes have C₁ symmetry. The distances between the metal (M) and oxygen (O) atoms in the complex range from 1.90 to 2.92 Å, indicating relatively strong metal–oxygen interactions as compared to M--H. In contrast, the average distances between the metal (M) and hydrogen (H) atoms on the lower side of the complex are greater than 3.00 Å, suggesting weaker metal–hydrogen interactions. The distance between two metals (M–M), especially when they carry opposite charges (M⁺ and M⁻), plays a critical role in predicting the important role of vdW forces, and dispersion interactions operating at long-range distances. The short interaction distance between M–M is observed in the Li[9C]₂Li complex is 9.84 Å. This distance significantly increases for Li to Ca complexes. The increased metal size could potentially lead to an increase in M–M distances in the present complexes. The molecular electrostatic potential maps (MESP) surfaces of all complexes are shown in Fig. S1 (ESI†). In M[9C]₂M (M = Li, Na, K) complexes, the negative (red) region appears around the lower-side alkali metals. For the Na and K-based complexes, a positive (blue) region can be seen around the top alkali metals (see Fig. S1, ESI†), while the lower portion displays the blue color. Hence, the MESP surfaces justify the asymmetric electron polarization between similar metals in the present complexes.

Fig. 1 (a) Representation of alkalide and alkaline earth-metal complex formation. The green colored patches represent vdW stacking interactions within the dimer, and (b) the representation of [9-crown-3], dimer formation, and formation of M[9C]₂M complexes.

Table 1 The calculated bond distances and thermodynamic properties of M[9C]₂M complexes

Complexes	Interaction distances (Å)			Energetics of the complexes
	d M–O	d M–H	d M–M	ΔE (kcal mol^−1)	ΔH (kcal mol^−1)	ΔG (kcal mol^−1)	ΔS (kcal mol^−1 K)
Li[9C]2Li	1.90	3.08	9.84	−26.98	−26.04	−8.91	−0.06
Na[9C]2Na	2.29	3.36	10.56	−14.80	−12.55	−1.25	−0.04
K[9C]2K	2.66	3.81	10.57	−7.71	−9.91	−0.43	−0.03
Be[9C]2Be	2.91	3.22	11.18	−23.84	−20.20	−4.19	−0.05
Mg[9C]2Mg	2.92	3.35	11.50	−11.99	−12.36	−1.12	−0.04
Ca[9C]2Ca	2.60	3.85	11.61	−4.26	−2.93	−0.15	−0.009

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Furthermore, to characterize the strength of the interactions and the formation of [9C]₂ and their complexes, the interaction energies (ΔE) are presented in Table 1. The ΔE values for the Li[9C]₂Li, Na[9C]₂Na, K[9C]₂K, Be[9C]₂Be, Mg[9C]₂Mg, and Ca[9C]₂Ca complexes are calculated to be −26.98, −14.80, −7.71, −23.84, −11.99, and −4.26 kcal mol⁻¹. The Li[9C]₂Li complex exhibits the highest value among the complexes, while the Ca[9C]₂Ca complex indicates the lowest ΔE. The interaction distances d_M–M and d_M–H significantly impact the ΔE of complexes. Higher interaction distances lead to weaker binding, resulting in strength and their corresponding ΔE (see Table 1). This increase in interaction distances could be attributed to larger atomic radii (metal size) and reduced charge density. A comparison of interaction energy is made with reported crown-ether-based complexes. For instance, the interaction energies of present complexes are slightly lower than M@[12-crown-5] (M = Be, Mg, and Ca)³¹ but significantly improved compared with Na@[15-crown-5] and K@[15-crown-5] alkalides.⁵¹ The calculated thermodynamic parameters such as enthalpy of formation (ΔH), Gibbs free energy of adsorption (ΔG), and entropy of adsorption (ΔS), indicate the exothermic nature of complexes (Table 1). Furthermore, the negative ΔG values demonstrate the thermodynamic feasibility of complex formation for the present complexes. The adsorption process resulted in negative entropy changes for all complexes, indicating a decrease in disorder following complex formation. In addition, the geometric and thermodynamic parameters were also calculated using various DFT methods (see Table S1, ESI†). The computed results using the B3LYP-D3, MN-15, and ωB97XD methods are comparable to the selected m062X method. This suggests that these different computational methods yield similar results, providing consistency and reliability. In addition, we presented the results of AIMD simulation based on quantum mechanics (Fig. 2). These calculations were carried out for 4000 fs with a time step of 1 fs under the thermostat Berendsen temperature of 300 K for each complex. The resulting trajectories reveal no signs of dissociation or fragmentation within the supramolecular assembly, demonstrating its thermodynamic stability. In the adsorption process, the [9C] rings exhibit favourable parallel stacking. During the movement in [9C]₂, a slightly displaced stacked dimer is occasionally observed; however, the most prevalent geometry is parallel stacking. Fig. 2 presents the RMSD values of [9C]a and [9C]₂ alongside snapshots of the geometries for [9C]₂ at various time intervals. The RMSD of the spectra for the K[9C]₂K and Ca[9C]₂Ca complexes are given in Fig. S2 (ESI†).

Fig. 2 Ab initio molecular dynamics study: (a) the root mean square deviation (RMSD) of trajectories, and (b) snapshots of the [9C]₂ dimer formation at various time intervals during the simulation at 300 K.

The alkalide and alkaline-erthide characteristics were monitored through computed NBO charges analysis and frontier molecular orbital study. The results of NBO charges are reported in Table 2. Due to non-covalent interactions in stacked [9C]₂ rings, excellent charge transfer (CT) and conductivity might be expected. The calculated NBO charges on the lower hydrogen face alkali metals Q_MH are negative for the M[9C]₂M series and range from −0.77 |e| to −0.62 |e| for the Li to K-based complexes. In contrast, the positive charges were observed on the upper metals. The NBO charges are transferred from the oxygen face alkali (M_O) to the lower side metals (on the hydrogen face). A significant anionic charge can be seen on the lower Li-atom of the Li[9C]₂Li complex (see Table 2). On the other hand, alkaline-earth metal-based complexes have positive NBO charges on both faces of the complexes. The highly polarizable outer ns¹ electrons in alkali metals contribute to long-range interactions, likely weak vdW forces that enhance the alkalide characteristics. The alkaline-earth metal complexes do not exhibit significant anionic charge, which could be due to their higher charge density and a stronger tendency to retain electrons compared to group 1 metal complexes. The alkalide nature in the current complexes is improved compared to recently reported M⁺(C₅₀N₅H₅)M′⁻ (M = Li, Na; M′ = Li, Na and K) alkalide complexes.⁵² Hence, the attractive interactions between stacked [9C]₂ is more appealing for polarizing the metals and improving the electronic structure.

Table 2 The calculated electronic properties and chemical descriptors of [9C], [9C]₂ and their M[9C]₂M (where M = Li, Na, K, Be, Mg, and Ca) complexes

Parameter	[9C]	[9C]2	Li[9C]2Li	Na[9C]2Na	K[9C3]2K	Be[9C]2Be	Mg[9C]2Mg	Ca[9C]2Ca
E HOMO (eV)	−8.57	−8.02	−2.15	−2.35	−2.15	−5.35	−4.27	−2.69
E LUMO (eV)	1.08	0.67	−1.43	−1.68	−1.53	−1.14	−0.88	−1.20
E g (eV)	9.65	8.69	0.72	0.67	0.62	4.21	3.39	1.49
Electron affinity (EA eV)	8.57	8.02	2.15	2.35	2.15	5.35	4.27	2.69
Ionization potential (IP eV)	−1.08	−0.67	1.43	1.68	1.53	1.14	0.88	1.2
Dipole moment (μo D)	3.31	8.01	20.93	17.64	17.73	11.21	12.83	18.56
Chemical potential (eV)	−3.74	−3.68	−1.79	−2.02	−1.84	−3.24	−2.58	−1.95
Hardness (eV)	4.82	4.35	0.36	0.34	0.31	2.10	1.70	0.75
Softness (eV)	0.21	0.23	2.77	2.97	3.23	0.48	0.59	1.34
Electronegativity (eV)	3.74	3.68	1.79	2.02	1.84	3.24	2.58	1.95
Electrophilicity index (eV)	1.45	1.55	4.43	6.05	5.44	2.50	1.95	2.54
Q (MO |e|)	—	—	0.82	0.70	0.68	0.01	0.02	0.02
Q (MH |e|)	—	—	−0.77	−0.65	−0.62	0.13	0.01	0.05

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Charge decomposition analysis provides in-depth insights into charge transfer and bonding interactions by quantifying the electron density contributions from donation and back donation (Table S2, ESI†). In the M[9C]₂M (M = Li, Na, K) complexes, the donation (d) is highest for the Li complex and decreases with increasing atomic size, consistent with the partial charges observed in the natural bond orbital (NBO) analysis. This trend supports the strong alkalide character of the Li[9C]₂Li complex, where notable anionic behaviour is on M_H. The results closely align with the hyperpolarizability values of the complexes, indicating that the most significant electron transfers occur in the Li[9C]₂Li system, which has a value of 0.25 |e| (see Table S2, ESI†). Additionally, repulsive polarization is negative for the Mg[9C]₂Mg and Ca[9C]₂Ca complexes, reflecting high charge density in the interaction region that balances attractive and repulsive forces. Hence, the polarization is not very effective in charge transfer for alkaline-earthides.

ELF and LOL spectra are shown in Fig. 3, while those for the [9C] and its dimer are given in Fig. S3 and S4 (ESI†). Mathematically, ELF and LOL exhibit similar chemical mapping because they both depend on the kinetic energy density. The ELF includes the Pauli kinetic energy density, and the LOL analysis does not include Pauli repulsion. Strong electron localization with lower alkali metals can be seen due to the high ELF values (red color). In the alkalide M[9C]₂M (where M = Li, Na, K) series, the valence 1s¹ electrons become strongly delocalized with the increase in size and interaction distance of metal to oxygen (M–O). For these complexes, ELF analysis indicates the strong alkalide character of the M[9C]₂M complexes. The high ELF value (red color) that appeared around alkali metals is asymmetric, while that of alkaline-earth metals indicates uniform charge distribution on both faces of the dimer. Thus, the charge transfer and polarization between the two faces of alkali metals are significant, reinforcing their alkalide characteristics. Additionally, the alkali metal on one face acts as a source of excess electrons, transferring them to the opposite face to confer an anionic charge. Furthermore, the high ELF or LOL values also reveal the presence of excess electrons and resemble the Rydberg orbitals.

Fig. 3 Electron localization function (upper panel) and localized orbital locator (lower panel) of M[9C]₂M (M = Li and Be) complexes at the m062X/def2tqzvpp level.

In frontier molecular orbital (FMO) analysis, charge transfer within or between molecules is primarily influenced by two key factors: the energy gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), and the shift in orbital density. The secondary highest occupied orbitals (HOMO−1) are also shown for the present complexes (Fig. 3). For M[9C]₂M (M = Li, Na, K) complexes, the presence of HOMO electron density around alkali metals can be observed, with the HOMO associated with lower alkali metals exhibiting a spherical shape resembling s-orbitals. This suggests that the excess electrons are positioned on metal atoms, giving these metals an anionic character. The spherical orbital shape and strong electron localization with alkali and alkaline earth metals reveal their excess electron nature. This arrangement imparts “alkalide-like” properties to the complexes, where the metal behaves similarly to an alkali metal in a negative oxidation state. Such configurations are unique because they stabilize metals in an unusual anionic form, influencing the complexes’ reactivity, stability, and electronic properties. On the other hand, the LUMO shows a shift in orbital density towards the upper alkali metals (see Fig. 3). The separation of orbitals into the HOMO and LUMO also indicates efficient intermolecular electron transfer from the HOMO to a particular LUMO during transition. Table 2 reveals a notable reduction in the HOMO–LUMO energy gaps upon the formation of dimers and M[9C]₂M complexes. For instance, the HOMO–LUMO energy gap decreases from 9.65 eV for pure [9C] ether to 8.69 eV for its dimer [9C]₂. Furthermore, the M[9C]₂M (where M = Li, Na, K), shows a progressive reduction in the energy gap with values of 0.72 eV, 0.67 eV, and 0.62 eV, respectively. The HOMO–LUMO gaps in the M[9C]₂M series are significantly smaller compared to the stacked dimer [9C]₂ and the [9C]. Specifically, the energy gap for the K[9C]₂K complex is significantly reduced, being 15 times smaller compared to [9C] and 14 times smaller than its dimer, [9C]₂. In contrast, M[9C]₂M (M = Be, Mg, Ca) complexes show significantly higher E_g values, ranging from 4.21 to 1.49 eV. These E_g values are considerably lower than those of [9C] and [9C]₂, potentially making them conductive or semiconductive. Likewise, the values of chemical potential are increasing for complexes as compared to [9C] and [9C]₂, indicating an increased electron-donating capacity (see Table 2). The chemical hardness decreases for the designed complexes, indicating an increase in their softness. Additionally, the reduced electronegativity values indicate the electropositive characteristics of the present complexes. Hence, in the supramolecular assembly, the polarized ns¹ and ns² electrons could potentially introduce alkalide and alkaline-earthide characteristics, which might impact the electronic and nonlinear optical properties of the present complexes.

3.2. Static nonlinear optical properties

The influence of excess electrons in enhancing the optoelectronic properties of alkalide, metalide, and other metal complexes has already been well-documented.^41,53–56 In addition to excess electrons, it is crucial to explore the role of vdW and stacking interactions in governing the stability of supramolecular assembly and their NLO properties.⁵⁷ The polarizability (α_o) values are provided in Table 3. Alkalide series possess a significant α_o response as compared to alkaline earth metal complexes, suggesting better optical features. Alkalides involve significant charge transfer, which could be a factor in their enhanced polarizability. The static hyperpolarizability (β_o) and mean polarizability (α_o) of every complex are computed and compiled in Table 3. The complexes have α_o values ranging from 83.1 to 2.1 × 10³ a.u. Entire complexes have β_o values ranging between 2.6 × 10² and 2.3 × 10⁶ a.u. The β_o response of alkalide complexes has increased up to 2.3 × 10⁶ a.u. and is highest among the M[9C]₂M complexes. The obtained hyperpolarizability values for the Li[9C]₂Li and Na[9C]₂Na are 2.3 × 10⁶ and 3.1 × 10⁵ a.u., where Li[9C]₂Li shows the highest response among other complexes. The contribution of the x-component to β_o is more prominent in the present complexes. Possible electronic factors correlating to the β_o response are the dipole moment HOMO–LUMO gaps and ionization potential. For instance, when the compounds’ HOMO–LUMO gaps decrease, the β_o values increase. With the lowest gap (0.72 eV) and a dipole moment of 20.93 Debye, the complex Li[9C]₂Li exhibits the strongest nonlinear optical (NLO) response (β_o = 2.3 × 10⁶ a.u.). The exceptional results of β_o can be linked to the presence of excess electrons, highlighting the importance of exploring their role in optical properties. The smaller the VIP values, the higher the β_o response of the complexes. Another parameter correlating to β_o is the vertical excitation energy (ΔE) of the complexes during the crucial transition. The decreasing trend in ΔE with increasing β_o is observed in the M[9C]₂M (M = Be, Mg, Ca) series. However, β_o in the alkalide series is dependent on the alkalide character (anionic character). Although ΔE is decreasing consistently with increased metallic size (Li to K), the Li[9C]₂Li complex has a significant response of β_o which could be attributed to the alkalide character. The hyperpolarizability response of the present complexes is higher in comparison to reported MA-2-MAE and MA-3-MAE stacked complexes and superalkalides based on stacked F₆C₆H₆ molecules.^58,59 Furthermore, we have determined the effect of implicit solvent on the hyperpolarizability of the present complexes (see Table S4, ESI†). The implicit, namely the universal solvent model based on solute electron density (SMD) is considered for n-hexane, DMSO, water, ethanol, and acetone. Overall, the polar solvents promote hyperpolarizability as compared to nonpolar. We found that molecular structures are affected by solvent polarity, which may increase the charge fluctuations.

Table 3 The calculated static polarizability (α_o a.u.), hyperpolarizability and their components (β_o a.u.), projection of hyperpolarizability on dipole moment vector (β_vec a.u.), dipole moment (μ_o Debye), and HOMO–LUMO energy gap (E_g eV) of the present complexes

Complexes	α o	β o	β x	β y	β z	β vec	μ o
[9C]	8.3 × 10^1	2.6 × 10^2	0.44	0.89	2.6 × 10^2	2.6 × 10^2	3.31
[9C]2	1.6 × 10^2	5.3 × 10^2	5.3 × 10^2	2.44	0.42	5.3 × 10^2	8.01
Li[9C]2Li	1.6 × 10^3	2.3 × 10^6	2.3 × 10^6	5.7 × 10^4	4.1 × 10^4	2.3 × 10^6	20.93
Na[9C]2Na	1.5 × 10^3	3.1 × 10^5	3.1 × 10^5	5.3 × 10^3	2.3 × 10^4	3.1 × 10^5	17.64
K[9C]2K	2.1 × 10^3	4.9 × 10^5	4.9 × 10^5	8.3 × 10^3	3.3 × 10^3	4.9 × 10^5	17.73
Be[9C]2Be	2.5 × 10^2	9.1 × 10^2	9.1 × 10^2	5.8 × 10^1	5.9 × 10^1	9.1 × 10^2	11.21
Mg[9C]2Mg	3.1 × 10^2	2.9 × 10^3	2.9 × 10^3	2.1 × 10^2	3.4 × 10^2	2.9 × 10^3	12.83
Ca[9C]2Ca	5.4 × 10^2	1.4 × 10^4	1.3 × 10^4	6.3 × 10^3	1.7 × 10^3	1.3 × 10^4	18.56

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The computed values of static second hyperpolarizability (γ_o), as well as their components are provided in Table 4. From Table 4, one can observe the remarkable values of the alkalide series M[9C]₂M (where M = Li, Na, and K) complexes, where the highest value is recorded for the Li[9C]₂Li complex. The γ_o response reveals the dependence on the alkalide character of alkali metals. For alkaline earth metal complexes, a small reduction in γ_o response can be observed, however, results are quite significant as compared to pristine [9C] and the dimer [9C]₂. Furthermore, M[9C]₂M (where M = Li, Na, K) have comparable contributions to total γ_o, where γ_x is more responsive in comparison to γ_y and γ_z components. The γ_o values have a trend similar to β_o which is also most prominent for the Li[9C]₂Li complex.

Table 4 Static second hyperpolarizabilities and its components (γ_o a.u.), scattering hyperpolarizability (β_HRS) average dipolar hyperpolarizability (〈β_J=3〉), average octupolar hyperpolarizabilities (β_J=3〉 a.u.), %dipolar contribution to hyperpolarizability (Φ_J=1), and %octupolar contribution (Φ_J=3), depolarization ratio (DR), and anisotropy (ρ) of complexes

	Alkali-metals				Alkaline earth-metals
	[9C]	[9C]2	Li[9C]2Li	Na[9C]2Na	K[9C]2K	Be[9C]2Be	Mg[9C]2Mg	Ca[9C]2Ca
γ o	2.1 × 10^3	4.9 × 10^3	3.7 × 10^9	1.9 × 10^3	1.2 × 10^8	2.4 × 10^4	5.8 × 10^4	1.7 × 10^6
γ x	1.4 × 10^3	3.3 × 10^3	3.7 × 10^9	1.91 × 10^8	1.2 × 10^8	2.3 × 10^4	5.3 × 10^4	1.7 × 10^6
γ y	1.4 × 10^1	2.5 × 10^3	1.1 × 10^6	8.3 × 10^5	2.0 × 10^8	3.6 × 10^3	5.3 × 10^3	9.9 × 10^4
γ z	4.7 × 10^3	2.5 × 10^3	9.6 × 10^5	2.7 × 10^8	5.0 × 10^6	4.0 × 10^3	2.1 × 10^4	1.7 × 10^5
β HRS	7.9 × 10^1	1.7 × 10^2	1.1 × 10^6	3.1 × 10^4	5.0 × 10^4	1.1 × 10^2	2.2 × 10^2	3.3 × 10^3
〈βJ=1〉	1.5 × 10^2	3.4 × 10^2	8.7 × 10^5	5.8 × 10^5	1.0 × 10^5	1.2 × 10^2	4.0 × 10^2	6.8 × 10^3
〈βJ=3〉	1.1 × 10^2	2.0 × 10^2	7.0 × 10^5	5. 5 × 10^4	3.8 × 10^4	3.2 × 10^2	3.7 × 10^2	3.1 × 10^3
DR	5.147	6.16	5.05	4.60	7.57	2.07	4.55	7.01
ϕ J=1	38%	62%	55%	52%	73%	27%	52%	68%
ϕ J=3	62%	38%	45%	48%	27%	73%	48%	32%
ρ	0.785	0.59	0.804	0.91	0.37	2.66	0.92	0.45

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Hyper-Rayleigh scattering (β_HRS) is a parametric optical effect in which two incident photons of frequency ω are annihilated to create a scattered photon of frequency 2ω. Theoretically calculated values of the β_HRS and its associated parameters are given in Table 4. The β_HRS of pristine [9C] is negligible, while a tremendous enhancement is observed for the dimer [9C]₂ and its complexes. The highest number of β_HRS corresponds to the Li[9C]₂Li complex, while alkaline-earth metal complexes are less responsive. The significant β_HRS of alkalides can be attributed to two main factors: strong charge transfer and reduced HOMO–LUMO gaps as compared to alkaline earth metal complexes. Besides, average dipolar (〈β_J=1〉) and octupolar (〈β_J=3〉) hyperpolarizabilities are also calculated. The 〈β_J=1〉 is the maximum for alkalide complexes, while a reduction is observed in alkaline earth metal complexes. Within both series, an uptrend in 〈β_J=1〉 and 〈β_J=3〉 is observed with increased sizes of metals. Likewise, the percentage dipolar Φ_J=1 and octupolar contribution Φ_J=3 to total hyperpolarizability are calculated. Due to the significant Φ_J=3 contribution of 62% in [9C], it suggests its octupolar nature, while its dimer [9C]₂ indicates a significant contribution from Φ_J=1. The change in nature after dimerization may be attributed to the presence of vdW forces or dispersion interactions (discussed in the next non-covalent section). In Table 4, one can also notice an increasing percentage of dipolar nature for alkalide complexes except for the Na[9C]₂Na. Similarly, the alkaline earth metal series exhibits a greater octupolar nature at the beginning of the series, with a shift toward increasing dipolar nature as you move from beryllium (Be) to calcium (Ca)-based complexes. The depolarization ratio (DR) is directly related to the dipolar nature of alkalides and alkaline earth metal complexes. Alkaline earth metal-based complexes show an increasing DR ratio, indicating a gradual enhancement in dipolar character from Be to Mg complexes.

3.3. Dynamic first and second hyperpolarizability

To make the theoretical calculations more relevant to experimentalists, we take into account the effect of dispersion frequency on NLO properties. The frequency-dependent polarizabilities α(ω), first hyperpolarizability β(ω), and second hyperpolarizability λ(ω) are computed at the 0.085 and 0.0438 a.u. of applied external frequencies (see Table 5). The 1064 nm wavelength is the fundamental output of the Nd laser and lies in the near-infrared (NIR) region of the electromagnetic spectrum. For frequency-dependent first hyperpolarizability β(ω), electro-optical Pockel's effect and electric field-induced second harmonic generation (ESHG) are calculated. The Pockel's effect β(−ω;ω,0) and ESHG β(−2ω;ω,ω) are more prominent on smaller dispersion frequencies. The alkalide series is more responsive to externally applied electric fields as compared to alkaline earth metal complexes. One can conclude that alkalide complexes are more interesting candidates for obtaining frequency-dependent NLO responses. The high performance of alkalides may be attributed to strong charge transport facilitated by stacking interactions.

Table 5 Frequency-dependent polarizability β(ω), first hyperpolarizability with EOPE (β(−ω;ω,0) in a.u.), and frequency-dependent first hyperpolarizability with ESHG (β(−2ω;ω,ω) in a.u.) at ω = 532 and 1064 nm

Complexes	ω = 0.085 (532 nm)			ω = 0.0428 (1064 nm)
	α(ω)	β(−ω;ω,0)	β(−2ω;ω,ω)	β(−ω;ω,0)	β(−2ω;ω,ω)	α(ω)
[9C]	7.0 × 10^1	2.0 × 10^2	2.4 × 10^2	1.9 × 10^2	2.0 × 10^2	6.9 × 10^1
[9C]2	1.4 × 10^2	4.7 × 10^2	5.7 × 10^2	4.5 × 10^2	4.6 × 10^2	1.3 × 10^2
Li[9C]2Li	4.9 × 10^2	1.6 × 10^5	3.7 × 10^4	1.2 × 10^4	5.8 × 10^4	4.6 × 10^2
Na[9C]2Na	9.4 × 10^2	7.3 × 10^5	1.5 × 10^6	8.2 × 10^4	8.2 × 10^4	6.7 × 10^2
K[9C]2K	1.5 × 10^2	1.7 × 10^4	7.8 × 10^4	7.8 × 10^4	2.5 × 10^5	1.5 × 10^2
Be[9C]2Be	2.3 × 10^2	1.4 × 10^2	1.7 × 10^3	1.5 × 10^2	1.4 × 10^2	2.1 × 10^2
Mg[9C]2Mg	3.3 × 10^2	2.1 × 10^4	7.6 × 10^4	6.3 × 10^2	1.9 × 10^3	2.8 × 10^2
Ca[9C]2Ca	3.1 × 10^2	6.3 × 10^4	2.9 × 10^4	1.6 × 10^4	3.9 × 10^4	5.8 × 10^2

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Frequency-dependent second hyperpolarizability γ(ω) is also determined in terms of the dc-Kerr effect and electric field second harmonic generation (ESHG) at externally applied frequencies of 532 and 1064 nm. The corresponding results are indicated in Table 6. The Kerr effect γ(−ω;ω,0,0) values are most prominent at higher frequency (0.085 a.u.), where the alkali metal-based series M[9C]M series possess remarkable results of γ(−ω;ω,0,0). The γ(−ω;ω,0,0) values are slightly reduced for M[9C]₂M complexes. At the applied frequency of 532 nm, the highest value of dc-Kerr constant (4.2 × 10⁸ a.u.) is recorded for the Li[9C]₂Li, while at 1064 nm, the maximum response is for the K[9C]₂K complex. It can also be seen that alkaline-earth metal-based complexes are less responsive to the Kerr effect and ESHG at both frequencies. Remarkable frequency-dependent NLO responses of M[9C]₂M series may be attributed to their strong alkalide character and significant dipole moment. The excellent λ(ω) values of alkali metal complexes can also be correlated to the presence of loosely bound excess electrons. The role of non-covalent interactions is also crucial, which will be discussed in the next section.

Table 6 The calculated dynamic second hyperpolarizability γ(ω) at different frequencies (in a.u.) in terms of the dc-Kerr effect γ(−2ω;ω,ω,0) and second harmonic generation γ(−2ω;ω,ω,0) effect of present complexes

Complexes	Frequency-dependent second hyperpolarizability γ(ω)
	ω = 0.085 (532 nm)		ω = 0.0428 (1064 nm)
	γ(−ω;ω,0,0)	γ(−2ω;ω,ω,0)	γ(−ω;ω,0,0)	γ(−2ω;ω,ω,0)
[9C]	2.4 × 10^4	3.1 × 10^4	2.2 × 10^4	2.4 × 10^3
[9C]2	5.6 × 10^3	7.4 × 10^3	5.1 × 10^3	5.4 × 10^3
Li[9C]2Li	4.2 × 10^8	9.9 × 10^6	1.3 × 10^6	5.4 × 10^7
Na[9C]2Na	2.5 × 10^7	1.1 × 10^7	6.6 × 10^6	1.6 × 10^7
K[9C]2K	4.4 × 10^6	4.4 × 10^6	1.2 × 10^8	1.5 × 10^8
Be[9C]2Be	4.7 × 10^4	2.3 × 10^5	2.6 × 10^4	3.6 × 10^4
Mg[9C]2Mg	5.9 × 10^4	1.8 × 10^6	7.0 × 10^4	4.2 × 10^4
Ca[9C]2Ca	2.2 × 10^6	9.4 × 10^7	4.7 × 10^5	9.4 × 10^5

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3.4. Role of noncovalent interactions and dispersion forces

Noncovalent interactions are crucial in numerous chemical processes. The dispersion interaction and vdW interactions are identified as critical in promoting the hyperpolarizability of molecules in previous studies.^41,57 QTAIM based on Bader's theory was used to calculate the bond critical points (BCPs) and their topological parameters to understand the nature of bonds and interactions between [9C]₂ and M[9C]₂M. A covalent bond is identified by a bond critical point that has large values of ρ(r) > 10⁻¹ a.u. and negative values of ∇²ρ(r). The computed values of these parameters, derived from the AIM analysis, are shown in Table S5 (ESI†), while BCPs generated on the molecular structure are shown in Fig. 4. We can confidently conclude that both vdW and dispersion forces contribute to the stability of complexes and predominately regulate the NLO response of the complexes. In the designed complexes, all values for ∇²ρ(r) are positive, except for the [9C], and the ρ(r) values are always less than 0.1. Moreover, the nature of an interaction can be distinguished at the BCP using the parameter G(r)/|V(r)|. Except for the O–C interaction in the [9C], the G(r)/|V(r)| ratios are greater than 0.5. These findings suggest that non-covalent interactions are present in the designed complexes. From the computed interaction energies for BCPs, the M–H interactions are relatively weak compared with M–O interactions. The QTAIM parameters are evidence for weak interactions. A schematic representation of the bond critical sites between M (Li, K, Na, Be, Mg, Ca) and [9C]₂ is shown in Fig. 4. The CO–H and CH–O bonds can be considered stronger based on the calculated significant interaction energies.

Fig. 4 The representation of molecular orbitals HOMO, LUMO, and LUMO−1 (a) and the visualization in (b) highlights the bond critical points (BCPs) during the QTAIM study of M[9C]₂M complexes at the m062X/def2qzvpp level (isovalue = 0.250).

To understand the effect of dispersion forces and vdW interactions in triggering hyperpolarizability, non-covalent interaction NCI analysis was performed. This analysis includes the reduced density gradient (RDG) and sign(λ₂)ρ (a.u.). The isosurface and their 2D-reduced RDG for the [9C]₂, Li[9C]₂Li, and Be[9C]₂Be complexes are depicted in Fig. 5, while the remaining complexes are provided in Fig. S5 (ESI†). One can notice the sigma stacking attractive interaction between the dimer [9C]₂. Furthermore, the interactions between [9C]₂ and M can be adequately described based on the NCI study. Multiple spots are observed in the sign(λ₂)ρ (a.u.) < 0 regions, indicating the existence of highly robust non-covalent interactions between [9C] and M, notably the extremely strong vdW interactions in Li[9C]₂Li. The sign(λ₂)ρ (a.u.) < 0 region indicates areas with strong interactions, while regions where sign(λ₂)ρ (a.u.) ≈ 0 typically exhibit weak vdW interactions. Furthermore, areas in the region denoted by sign(λ₂)ρ (a.u.) > 0 suggest dispersion interactions (see Fig. 5). It is revealed that the sign(λ₂) and the electron density matrix ρ (a.u.) are highly correlated with bond strength. Hence, the vdW forces in the alkalide series (Li to K) are more prominent and monotonically reduced with the increased size of metals (Li to K). In alkaline earth metal complexes, vdW forces are further decreasing from the Be to Mg-based complexes. The β_o response is also smaller for the alkaline earth complexes. For the Li[9C]₂Li complex, both dispersion and vdW are higher, resulting in significant static and dynamic hyperpolarizability responses. On the other hand, the alkaline earth metal complexes have significant vdW interactions, while the dispersion effect is negligible in alkaline-earth M[9C]₂M (M = Be, Mg, Ca) complexes. The long interaction distance between the alkaline earth metal (M) and [9C]₂ could be the reason for the reduced dispersion effect. The attractive sigma stacking interaction in the dimer, in addition to dispersion interactions, cannot be overlooked when analyzing the role of non-covalent interactions in enhancing NLO features.

Fig. 5 The molecular isosurface from noncovalent interaction analysis and reduced density gradient (RDG) 2D spectra of [9C]₂ and M[9C]₂M (M = Li and Be) complexes at the m062X/def2qzvpp level.

To understand the donor–acceptor interactions between Lewis-type donor–acceptor orbitals, the second-order perturbation theory-based Fock matrix is considered. These interactions are important for describing the charge transfer through stabilization energy (E²). The higher the E² value, the higher the charge transfer interaction, resulting in stabilizing the complex formation. The results of the calculated NBO, featuring the orbital transition, the occupancy of the donor–acceptor orbital, and the stabilization energy, are presented in Table S3 (ESI†). Based on the results, the primary molecular interactions in the present complexes are σ–σ, lone pair (LP) to σ*, and σ–σ* electronic transitions between M and [9C]₂ atoms. The significant E² value is associated with the transition of electrons from LP (O1/O2/O3) to σ* M (M = Li, Na, K), and from σ (M) to σ* of the C23–H25, C31–H32, and C37–H38 combinations.

3.5. Excited state and vibrational study

The main use of materials with high initial hyperpolarizability values in second harmonic generation (SHG) is in frequency doubling. Therefore, effective nonlinear optical (NLO) materials should possess a significant NLO response and also maintain transparency when exposed to the laser light being used. The UV-VIS absorption study is also carried out for the designed complexes. The absorption plots of the present complexes are provided in Fig. 6 and Fig. S6 (ESI†), and the corresponding values of maximum wavelength (λ_max), transition energy (ΔE), and oscillator strength (f_o) are given in Table 7. The pristine [9C] exhibits absorption in deep UV regions, whereas [9C]₂ displays absorption at λ_max = 123 nm. However, for M[9C]₂M complexes, the absorbance maxima (λ_max) are shifted to a longer wavelength (bathochromic shift). Excitation energy (ΔE) also decreases in the same fashion and has an inverse relationship with metallic radii. The vertical excitation energies are very important and have an inverse relationship with hyperpolarizability response. For the alkalide series, K[9C]₂K, the λ_max increased up to 621 nm, where significant orbital contribution is from HOMO → L+5 (100%). The highest redshift (776 nm) among the series is observed in the Ca[9C]₂Ca complex, exhibiting the greatest orbital contribution from the HOMO to the L+5 state, which amounts to 101%. As the atomic number of doped alkali metals increases, the UV-Vis absorption wavelength shows a consistent rise. The UV-Vis spectra results suggest that the designed complexes exhibit high efficiency, making them suitable as highly efficient NLO materials.

Fig. 6 The obtained absorbance spectra from the excited-state study (upper panel) and plotted FT-IR spectra of M[9C]₂M (where M = Li, Be) complexes (lower panel) at the m062X/def2qzvpp level.

Table 7 The calculated vertical excitation energies (ΔE), absorbance maxima (λ_max), and oscillator strength (f_o) of the studied complexes

Complex	ΔE (eV)	f o	λ max (nm)	Major orbital contribution
[9C]2	10.1	0.12	123	H−3 → LUMO (10%)
				HOMO → L+5 (26%)
Li[9C]2Li	4.9	0.72	254	HOMO → L+3 (83%)
				HOMO → L+4 (22%)
Na[9C]2Na	3.2	0.84	388	H−1 → LUMO (100%)
K[9C]2K	2.0	0.38	621	HOMO → L+5 (100%)
Be[9C]2Be	4.5	1.32	276	H−1 → LUMO (92%)
				HOMO → L+5 (9%)
Mg[9C]2Mg	2.1	0.50	592	HOMO → L+3 (90%)
				HOMO → L+6 (9%)
Ca[9C]2Ca	1.6	0.44	776	HOMO → L+5 (100%)

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Theoretical calculations of FT-IR vibrational modes were conducted to observe changes in intensity following complex formation. Therefore, the structural features of [9C]₂ and its metal-doped complexes have been extensively studied via IR spectra, with the results presented in Table S6 (ESI†) and shown in Fig. 6. The remaining FT-IR spectra are shown in Fig. S7 (ESI†). [9C]₂ exhibited prominent peaks, including the stretching vibrations of C–O and C–H bonds at 1369 and 3076 cm⁻¹, respectively. The designed complexes exhibited C–O stretching vibrations at lower wavenumbers, indicating a weakening or lengthening of the C–O bond, likely due to coordination effects or electron donation in the complex. Conversely, the C–H stretching vibrations appeared at higher frequencies, suggesting that the C–H bonds experienced increased electron density or bond strengthening. Additionally, new M–O (M = Li, Na, K, Be, Mg, and Ca) stretching vibrations were observed between 193 and 525 cm⁻¹. Further electronic characteristics of the designed complexes were validated through an examination of the partial density of states (PDOS) as illustrated in Fig. S8 (ESI†). Partial atomic charges on metals and vdW forces are important in creating new energy levels across the present complexes. Therefore, the newly generated HOMO orbitals are high in energy, resulting in a significant reduction in HOMO–LUMO gaps. The weak vdW forces also regulate the charge transfer in [9C]₂. The additional interactions between the metals and the dimer have notably modified the HOMO energies. Moreover, the PDOS spectra demonstrate how metals contribute to the HOMO of the complex. Specifically, the Be[9C]₂Be, Mg[9C]₂Mg, and Ca[9C]₂Ca complexes show significant contributions in the HOMO, whereas the Li[9C]₂Li, Na[9C]₂Na, and K[9C]₂K complexes exhibit lower contributions.

4. Conclusions

In summary, we present theoretically designed M[9C]₂M complexes using the m062x/def2qzvpp method. The dispersion forces and weak vdW forces actively participate in stabilizing the dimer [9C]₂ and their M[9C]₂M complexes. Supramolecular assembly possesses significant thermodynamic and electronic stability as revealed by a computed interaction energy up to −26.98 kcal mol⁻¹ for the alkalide series. AIMD simulations also reveal the adsorption and thermodynamic stability of complexes at room temperature. In the M[9C]₂M (M = Li, Na, K) complexes, alkalide nature and charge transfer are more prominent as compared to alkaline earth metal pairing. Electronic structure calculations and NBO analysis indicate the long interaction distances between metals on both sides of the [9C]₂. The performed charge decomposition analysis reveals the prominent role of donation as compared to back donation in electron transfer. Alkalide complexes have fascinating NLO responses, where β_o is recorded up to 2.3 × 10⁶ a.u. for the Li[9C]₂Li complex. An alkalide nature dependence hyperpolarizability trend was observed for the alkali metal complexes, with the metals’ size being the main influencing factor in improving NLO features of alkaline-earth metal complexes. The effect of solvents on the hyperpolarizability response was considered. In addition to excess electrons, the critical role of vdW forces is observed in regulating NLO properties. Frequency-dependent NLO features were also computed at the 532 and 1064 nm frequencies. Theoretically, second harmonic generation (SHG) exhibits a strong dependence on the size of metals. In addition, vertical excitation energy and absorbance are determined by employing the excited-state calculations. Stretching and binding modes of vibrations become more intense when determined after supramolecular complex formation.

Data availability

The data supporting this article are included in the ESI.† The other data to support the findings of this study are available from the corresponding authors upon reasonable request.

Conflicts of interest

The authors declare no conflict of interest.

Acknowledgements

This work was supported by the National Natural Science Foundation of China and the Beijing National Laboratory for Molecular Sciences. Author A. A. also acknowledges the ANSO for the support. In addition, the present research has been supported by the Kazan Federal University Strategic Academic Leadership Program (PRIORITY-2030).

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4cp04052j

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