Theoretically designed M@diaza[2.2.2]cryptand complexes: the role of non-covalent interactions in promoting NLO properties of organic electrides

ABSTRACT Organic excess electron compounds with significant nonlinear optical (NLO) properties are widely employed in optoelectronic applications. Herein, single-alkali metals with diaza[2.2.2] cryptand (M@crypt,M=Li, Na, and K) are investigated for optoelectronic and NLO properties by using the density functional theory. Thermodynamic and kinetic stabilities of present complexes are computed through interaction energy (Eint) and ab-initio molecular dynamic (AIMD) simulations. M@crypt complexes carry excess electrons and mimic molecular electrides. Quantum theory of atoms in molecules (QTAIM) analysis and reduced density gradient (RDG) spectra demonstrate the roles of the weak van der Waals (vdW) interactions between metal and complexant. The remarkable hyperpolarizability (βo) value up to 1.41 × 106 au may be credited to the presence of loosely bound excess electrons. The hyper Rayleigh scattering hyperpolarizability (βHRS) is recorded up to 1.31 × 106 au for the K@crypt. Furthermore, frequency-dependent first-order and second-order hyperpolarizability is more prominent at the applied frequency of ω = 0.042823 au. The electron localizing function (ELF) and localized orbital locator (LOL) analysis further disclose the nature of interaction between alkali metal and complexant. The TD-DFT method is adopted to get excited state parameters and absorbance properties. An electron density difference map (EDDM) is exploited to evaluate the orbital contributions in excited states. Hence, the studied electride may become a promising candidate for NLO materials. We anticipate that the present work will provide insight into further development of molecular electride for optoelectronic applications.

Asymmetric charge distribution [11], significant dipole moment (µ o ), and low excitation (ΔE) energy, are considered crucial factors in enhancing the NLO response of molecular materials [12,13].Up till now, various kinds of measures have been considered to improve the molecular structure for promoting the optoelectronic at the molecular level.Among the adopted strategies, constructing donor-conjugatedbridge-acceptor (D-A) models [14], introducing pushpull effects [15], making sp 2 -hybridized carbon-based materials [16], designing multidecker sandwiches complexes [17], creating octupolar molecules [18], bond length alternation (BLA) theory [19], and introduction of loosely bound electrons (excess electrons) are very renowned for constructing NLO materials [20,21].Among the above-stated techniques, investigating excess electron molecules is perhaps the most fascinating way to enhance the NLO properties.The doping of transition and main group metals [on host molecules] is an emerging approach for introducing excess electrons into materials [22,23].Research has shown that the existence of loosely bound electrons greatly reduces the excitation energy (ΔE), facilitates electron promotion to unoccupied orbitals, and is required for regulating the optical and NLO characteristics [24][25][26].
Alkalides (alkali anion) [27], alkaline earth metals earthides [28], and electrides [23] are the main classes of excess electron compounds and were studied for various applications.In alkalides, the alkali metals act as anions (Li − , Na − and K − ) for charge transfer, while the 2nd group (earth metals) contains anionic nature in earthides.Theoretically designed alkalides were reported using the alkali metals and superalkali clusters with various complexant [27,29].In such complexes, the role of alkali metal is to transfer charge, and strong polarization was seen.For instance, Ayub et al. [29] have designed an alkalides Li 3 O@[12-crown-4]M (where M=Li, Na, and K) as excess electrons complexes by simultaneous doping of Li 3 O and alkali metals on the crown ether.Similarly, a variety of diffuse excess electron compounds were designed for NLO applications based on cages and crown ether complexes [30,31].Electrides containing excess electrons (diffuse excess) in lattice voids as anions have grabbed considerable attention in both fundamental research and application development [32].The very first evidence about the synthesis of electride was received in 1983 by Dye and co-workers.Since then, numerous organic and inorganic electrides have been reported [33,34].Johnson and colleagues investigated eight organic electrides through a DFT study and the presence of localized interstitial electrons justifies their electrides properties [35].They also claimed the strong adsorption affinity Na + through complexation with cryptand (crypt), containing eight tertiary amine nitrogen [36].Owing to the excess electron nature of electride, they are continuously considered to construct remarkable NLO materials.Very recently, Wajid et al. theoretically designed alkali metalsdoped C 6 O 6 Li 6 organometallic electrides for optical and NLO applications [37].For the K@C 6 O 6 Li 6 complex, the hyperpolarizability (β o ) value was recorded up to 2.9 × 10 5 au.Likewise, M 3 O@C 6 S 6 Li 6 (M = K, Na, Li) electrides were also reported for geometric, electronic, and optical properties [38].
Cryptands belong to the family of synthetic biand polycyclic organic compounds and have received great attention due to their excellent structural feature.Cryptand-based hosts are very famous for unique three-dimensional structures, which can be exploited in constructing novel complexes [39,40].In 1987, Lehn and their co-workers [41] successfully synthesized cryptand for the first time.The interior cavity can be operational for the sitting of guest molecules and atoms in modifying its properties.Using the cavity of benzocryptand, Nimra et al. have reported alkali metals doped benzocryptand for NLO application, where hyperpolarizability response was recorded up to 9.1 × 10 5 au [42].The evidence of the strong interaction of alkali metal cations with cryptand and their trapped electrons (electride) has already been emphasized [43].The dye group has experimentally reported thermally stable K + (cryptand 2.2.2)e − electride, where ionic solids with cavity-trapped electrons serve as the anions [44,45].
Inspired by earlier developments in molecular electrides and their NLO properties, we theoretically designed single alkali metal-based electrides based on diazacryptand [2.2.2].We aim to uncover the electride nature, and role of noncovalent interactions in promoting optoelectronic properties of M@crypt (where M=Li, Na, and K) complexes (Figure 1).It should be noted that the van der Waals (vdW) interaction is a kind of important noncovalent interactions, which plays very important roles in many chemical processes [46,47], and includes dispersion interaction as well as others.Its role in the formation and properties of molecular complexes is of great interest.

Computational methodology
The ground state structures of the designed M@crypt were fully optimized with the ωB97×d/def2-tzvp level of theory.The entire simulations were performed using the Gaussian 16 and ORCA 5.0 package [48,49].The long-range corrected method (ωB97×d) yields satisfactory results for thermochemistry, adsorption study, and non-covalent interactions [50].Interaction energies are calculated at the same method to justify the thermodynamic stability of present complexes.Additionally, we have performed ab-initio molecular simulation (AIMD) at 300 K to assess the kinetic and thermodynamic stability using the ORCA 5.0 program.Results of AIMD comprise 1500 distinct geometries in each trajectory of the complex, which are adopted for further analysis.Furthermore, AIMD is executed for 750 femtoseconds (fs) with a step size of 0.5 using the B3LYP-D3/def2-SVP method [44].The adsorption nature of complexes is determined through calculated interaction energy (E int ), which is given by the following relation; The charge transfer study, electride character, and orbital energies are also calculated through NBO and frontier molecular orbital (FMO) theory.Vibrational frequencies are computed using the same method, where the mode of vibrations is compared with the experimental study.Vertical ionization potential (IP), electron affinity (EA), and hardness (η) were computed for the present complexes.These parameters are given by the following equations [51,52]; The optical and nonlinear optical (NLO) properties were characterized through calculated values of polarizability (α o ), hyperpolarizability (β o ), and projection of hyperpolarizability on dipole moment (β vec ) using the ωB97×d/def2-tzvp level of theory.A comparison of static NLO properties is made using the CAM-B3LYP and M062X, and ωB97×d method with def2tzvp basis set.Scattering hyperpolarizability values are calculated for hyper Rayleigh scattering measurement at the ωB97×d/def2-tzvp level of theory.The effect of solvent on NLO response is also considered using the same method.Frequency-dependent (dynamic) firstorder hyperpolarizability β(ω) and second-order hyperpolarizability γ(ω) were calculated at the applied frequencies of 532 and 1064 nm using ωB97×d/def2tzvp at which frequencies experimental studies were carried out.
Perturbation theory may be used to calculate the energy and nonlinear optical parameters.The Taylor expansion describes the energy of the perturbed system.
The magnitude of hyperpolarizability can be defined as; Hyper-Rayleigh scattering (β HRS ) response is described as [53]; The β HRS is composed of dipolar φ(β J = 1 ) and octupolar φ(β J = 3 ) contribution to total hyperpolarizability and given by [54]; To unfold the nature of the interaction between metal and crypt, topology analysis was exploited using Bader's quantum theory of atoms in molecules (QTAIM) and reduced density gradient (RDG) spectra [55].In this analysis, we considered bond critical points (BCP) at (3,-1) electron density.Electron density difference maps are also obtained with the same method.An excited state analysis was carried out using TD-ωB97×d/def2-tzvp to get the absorbance pattern, excitation energy, and orbital contributions during the crucial transition.The projected density of states (PDOS) spectra were plotted using GaussSum and multiwfn software [56].In addition, the electron localizing function (ELF) and localized orbital locator (LOL) are plotted to understand the bonding properties in present complexes.The kinetic energy density plays a vital role in the description of a chemical bond, since the driving force in forming a covalent bond.The ELF function can nicely unveil the location of atomic shells, core binding electrons, and lone pairs electrons in atomic and molecular orbitals.Mathematically, the ELF can be expressed based on kinetic-energy density [57]; Where D σ 0 is uniform electron gas with a spin density equal to the local value of ρ σ (r) χ σ ratio is a dimensionless quantity, indicating localization index and calibrated by considering uniform-density electron gas as reference.
The possible ELF values range with the upper limit ELF = 1, indicating perfect localization and the value of ELF = 0.5 represents the electron gas-like pair probability.
On the other hand, the LOL is similar to the ELF, where just the orbital kinetic energy density and the local spin density approximation (LSDA) kinetic energy density are compared, without considering the Pauli repulsion [58]:

Electronic structure and energetic analysis of complexes
The optimized geometries of M@crypt (M=Li, Na, K, and crypt = diazacryptand[2.2.2]) complexes using ωB97×d/def2tzvp are shown in Figure 1.The adsorbed alkali metals have unequal interaction distances from bridgehead nitrogen atoms (d M-N ) and with the oxygen of the cyclic ring (d M-O ).The interaction of alkali metals with nitrogen and oxygen is crucial to characterize the physico-chemical properties of complexes.Only influential bond distances were computed and values are given in supplementary information (Table S1).In a reported study on the same system, the geometries of K@crypt and Na@crypt complexes were slightly distorted from the ideal D 3 -group [59].In the present results, a similar scenario is seen except for a slight distortion in Li@crypt, where Li is asymmetrically adsorbed in the cavity.In Li@crypt, the calculated bond distances between Li63-N7 and Li63-N8 are 3.95 and 2.46 Å.The Li-atom has a higher distance with N7 while strongly interacting with lower bridgehead nitrogen (N8).Similarly, average optimized distance of Li-atom with oxygen (d M-O ) is 2.34 to 2.70 Å.For Na@crypt, the interaction distance between the Na1-N8/N9 is 3.19 Å (Table S1).Similarly, Na1 is equidistant from O4/O5 calculated up to 2.48 Å.The interaction distance of K63-N7 and K63-N8 is 3.07 Å (see Table S1).K-metal is adsorbed exactly in the middle of the cavity and is equidistant from nitrogen (N7, and N8) and oxygen of the ring.The obtained structural results were also compared with the reported experimental synthesis of (2.2.2-cryptand)potassium nitrate hydrate complex [60], and values are given in supplementary information (Table S1).The K-O interaction distance reported by Boldyrev et al. [59] has a comparable value to the present results (see Table S1).There is no imaginary frequency associated with complexes, denoting their stability on potential energy surfaces.
The obtained significant interaction energies (E int ) quantify their thermodynamic stability (see Table 1).The interaction energies range from -11.10 to -21.91 kcal/mol, where the highest value is obtained for Li@crypt, while the lowest is calculated for Na@crypt.Overall, the E int of present electrides are higher in comparison with the reported electride, evidencing the strong interaction and formation of stable complexes.The significant E int of Li@crypt is in accordance with the least interaction distances of Li with N7 (2.46 Å) and oxygen atoms (2.70 Å).The E int (-21.91 kcal/mol) of present Na@crypt is higher than alkalidoped benzocryptand [42], superalkalis doped all-cis -1,2,3,4,5,6-hexachlorocyclohexane [61], and M 3 O@C 6 S 6 Li 6 (M = K, Na, Li) electrides.
A comparison of the vibrational study of the pure complexant (diazacryptand) and alkali metalencapsulated complexes is made with reported FT-IR analysis to monitor changes in vibrational modes.The vibrational frequency and their comparison with the experimental values are summarized in supplementary information (Table S2).For the crypt, the intense peaks of C-H stretching vibrations were observed in the range of 2812-2981 cm −1 .C-H bending peaks range from 1280 to 1523 cm −1 and are second-strong peaks.The experimental value of C-H stretching ranges from 2790 cm −1 to 2877 cm −1 (see Table S2) [62].Further details of characteristics IR peaks is given in supplementary information (see section S2).
Ab-initio molecular dynamics (AIMD) calculations of designed complexes were performed using ORCA 5.0.2, time step 0.5 (fs), and at room temperature (300 K).AIMD results show that M@crypt remained stable during simulation, with rigid structures against isomerization and decomposition.The root-mean-square deviation (RMSD) for complexes and pure crypt is plotted and given in supplementary information (see Figure S1).RMSD is a standard measure of structural distance between coordinates.The variation in kinetic energy versus time is also plotted in Figure S1b.In the beginning, kinetic energy increases for complexes up to 200 fs while equilibrium is attained after 300 fs.Additionally, the drift energy (K) versus time (fs) spectra were also given in supplementary information (Figure S2).The drift energy (K) of Li@crypt is higher in the beginning and near 800 fs.Likewise, the drift energy for Na@crypt is maximum near 200 fs and noticeably reduced at 700 fs.The snapshots of geometries after 100 fs are captured to ensure the rigidity of complexes against dissociation and are shown in the supplementary information (Figure S3).The curves of complexes have an uptrend with the increased size of metals (Li to K).Hence, the AIMD study reveals their thermodynamic and kinetic stability at room temperature.
The computed partial NBO charges are given in Table 1.The calculated charges on alkali metals Q(M) are slightly positive and range from 0.40 to -0.0001 |e|.In Li@crypt, the small Li to nitrogen distance (d M-N ) causes significant charge transfer (due to strong interaction) to nitrogen (N7), which ranges from -0.65 to -0.57|e|.Similarly, for the Na@crypt complex, slightly reduced NBO charge (positive) on Na-atom up to + 0.013 |e| while, the K-atom shows negligible positive NBO charge.Due to the interaction of metals with cryptand, ns 1 electrons are polarized by neighbour atoms (N and O) of complexant.The lowest polarization of partial charge is observed on the K-atom, attributing to symmetric adsorption of K-metal [a] Values were reported using the PBE0/6-311++G** method [61].
inside cavity.One can conclude that loosely bound electrons are trapped inside the complexant and induce the electride nature to complexes.The electride nature and charge transfer predominately enhance the optical properties, which will be shown later.Further justification of electride nature and orbital energies can be established through frontier molecular orbital (FMO) analysis (see section S3).The orbital shapes of present complexes are given in Figure 2. The HOMO is not warping a particular atom distributed in space in the complexant.After the alkali-metals encapsulation, these complexes have HOMO and LUMO in space distributed within the entire complexant, which is reminiscent of the molecular electride.The HOMO has diffused shape and indicates excess electrons are trapped inside the complex.Vertical ionization potential (VIP), electron affinity (VEA), and chemical hardness (η) are also calculated and values are given in Table 1.The VIP of the pristine crypt is 7.56 eV, while its VEA is 3.14 eV.After the interaction of alkali metals (Li, Na, and K), a significant reduction in VIP and VEA values is observed.The calculated VIP values of complexes are 1.90, 1.01, and 1.08 eV for Li@crypt, Na@crypt, and K@crypt, respectively.VIP values are endowed to be useful to manifest their capability to donate and accept electrons.The reduced VIP values with increased alkali metals size (Li to K) suggest the electropositivity and excellent reducing ability of present complexes.Smaller ionization potential than Cs-atom (3.87 eV) also suggests their superalkali-like nature [63].The VIP, VEA, HOMO-LUMO gaps, and interaction energies are compared with similar reported complexes and values are given supplementary information (Table S3).

Static nonlinear optical (NLO) properties of complexes
Owing to the electride nature of present complexes, one may envisage interesting NLO features.Various excess electron compounds were investigated as efficient NLO candidates, in which electrides are the most promising candidates [22,26,[64][65][66][67]. NLO parameters of M@crypt (electrides) are given in Table 2.The dipole moment (µ o ) plays a vital role in studying the polarization, packing, and asymmetric charge distribution.A notable value of µ o can be correlated to higher NBO charges on and asymmetric interaction between metal (M) and cryptand.
A remarkable µ o of Na@crypt complex may be credited to the strong asymmetric interaction of metal Na1 with N7, N8, and oxygen atoms of the cryptand ring.The lowest value of µ o can be seen in K@crypt, where K63 is exactly in the middle of the cryptand cavity.Static polarizability (α o ) provides information on the distribution of electrons in a molecule and is critical in deciding the polar nature and reactivity of present complexes.The obtained (α o ) value of pristine complexant is 2.54 × 10 2 au, while after the interaction of alkali metals M@crypt, the α o increased up to 9.30 × 10 2 au (Table 2).Also, the z-component shows a significant contribution to the total polarizability response.The gradual increase in α o is observed as follows: K@crypt >Li@crypt > Na@crypt.The values of α o at M06-2X/def2qzvp and CAM-B3LYP /def2tzvp method are close to ωB97×d with same basis set and are provided in supplementary information (Table S4).
On the other hand, the hyperpolarizability (β o ) values of complexes are 5.32 × 10 3 , 1.41 × 10 6 , and 9.30 × 10 5 au for Li@crypt, Na@crypt, and K@crypt, respectively, where the highest value is observed for the Na@crypt complex (Table 2).The calculated significant value of 1.41 × 10 6 is higher than that of the organic reference molecule (P-nitroaniline).Likewise, the computed β o value of 1.41 × 10 6 au for Na@crypt is quite higher than those of previously reported M@C 6 O 6 Li 6 electrides, superalkali@Al 12 P 12 electrides [37,68].The non-monotonic trend of β o for the Na@crypt complex using ωB97×d/def2tzvp might be attributed to its high dipole moment and low excitation energy (1.22 eV).Also, the asymmetric interaction has significantly altered the β o response in Na@crypt.The β o response of designed electrides can be correlated to electronic properties like VIP, EA, and chemical hardness.In the current study, lower VIP values indicate an increased β o response.For instance, the highest β o value of Na@crypt has the lowest VIP value (1.0), followed by K@crypt, where the VIP value (1.08 eV) is slightly higher using ωB97×d/def2-tzvp.A similar trend in values of projection of hyperpolarizability on dipole moment vector (β vec ) and total hyperpolarizability (β o ) can be seen which indicates excellent nonlinear optical properties of present complexes.The static second-order hyperpolarizability (γ o ) is considered to be an important index to measure nonlinear optical coefficients of designed electrides.These values are calculated using ωB7XD/def2-tzvp and given in Table 3.The γ o values are 1.05 × 10 4 , 4.30 × 10 5 , 2.96 × 10 8 , and 4.96 × 10 9 au, respectively for crypt, Li@crypt, Na@crypt, and K@crypt.A gradual increase in γ o can be seen with increased size of alkali metals (see Table 3).In contrary to hyperpolarizability, γ o responses exhibit a dependence on the size of alkali metals.The obtained significant γ o value of 4.96 × 10 9 au for K@crypt is quite higher than previously reported alkali metal doped C 6 O 6 Li 6 organometallics electride, superalkalis doped C 6 O 6 Li 6 electrides [69], superalkalis@Al 12 P 12 inorganic electrides [68], and alkaline-earth-based alkaline salt electrides M-H 3 C 4 N 2 ⋯Ca (M=H, Li, and K) [70].Observed β o and γ o values of M@crypt were significantly higher as compared to alkali and alkaline earth metal-based benzocryptand, superalkali@C 6 F 6 H 6 electride, and Li 3 @C 60 electride.Table S4 reveal the comparable β o values at M062X and ωB97×d method, while values at CAM-B3LYP are slightly declined.A comparison of NLO parameters of present complexes is also made with reported similar molecular electrides and excess electron compounds using the ωB97×d method (Table S5).
The hyper-Rayleigh scattering (HRS) first-order hyperpolarizability (β HRS ) of pure cryptand and electride complexes is also calculated using the same method.β HRS is the most fundamental nonlinear chiral optical (chiroptical) effect to characterize the nonlinearity of molecular materials even with zero dipole moment.β HRS is an important theoretical tool to measure the hyperpolarizability of centrosymmetric molecules [71].The calculated values of β HRS ,  depolarization ratio (DR), dipolar contribution to hyperpolarizability (DR), percentage of dipolar contribution Φβ(j = 1) and octupolar contribution to total hyperpolarizability Φβ(j = 3) are given in Table 3.The undoped crypt holds a very low HRS (170.21)response as compared to M@crypt.For M@crypt, obtained β HRS values are 1.33 × 10 4 , 9.0 × 10 5 , and 1.31 × 10 6 au for Li@crypt, Na@crypt, and K@crypt, respectively.K@crypt is more responsive to HRS measurement, which can be correlated to the size of metal.The higher depolarization ratio (DR) indicates the dipolar nature of complexes.Pristine (crypt) shows the lowest DR ratio justifying its octupolar nature.
The octupolar nature of the crypt is seen in its 99% contribution to hyperpolarizability at Φβ(j = 3).
Complexes with a high percentage contribution from dipolar factor are characterized as dipolar (vide supra).
The dipolar contribution to hyperpolarizability Φβ(j = 1) is almost 52% for Li@crypt, while Na@crypt and K@crypt complexes are octupolar due to significant contribution from Φβ(j = 3).

Frequency-dependent (dynamic) NLO properties
The frequency-dependent hyperpolarizability β(ω) is calculated as β(-ω,ω,0) for electro-optical Pockel's effect (EOPE) and β(-2ω, ω, ω) for second harmonic generation phenomena (SHG) parameter.Entire frequency-dependent NLO parameters are given in Table 4.To investigate the frequency-dependent NLO responses of the current complexes, we chose 532 and 1064 nm as transparent regions since the complexes display significant absorbance in between 700 nm and 1016 nm.The choice of wavelength (1064 nm) is also relevant to Nd:YAG laser functioning, which frequently emits invisible light in 1064 nm and serves in laser-based devices [72].The electro-optic effect is the modification of a medium's refractive index produced by an electric field.The electrooptical effect (Pockels effect) is an essential nonlinear effect used in many applications [73].Only non-centrosymmetric materials (mostly nonlinear crystal materials) show the linear electro-optic effect, also known as the Pockels effect, in which the change in refractive index is proportional to the strength of the electric field.Overall, EOPE values are higher than those of electric field-induced second harmonic generation (ESHG) values at both frequencies.The EOPE values are 7.99 × 10 3 , 1.34 × 10 4 , 1.11 × 10 5 au, respectively, for Li@crypt, Na@crypt, and K@crypt at larger frequency (ω = 0.085645).EOPE effect is most prominent at both frequencies.The highest value (9.55 × 10 7 au) of EOPE is calculated for Na@crypt complex.The frequency-doubling nonlinear optical process is determined through ESHG.The ESHG values are almost comparable at both frequencies, while significantly lowered than those of EOPE.Frequency-dependent second-order hyperpolarizability γ(ω) is estimated using γ(-ω, ω,0,0) for dc-Kerr effect and γ(-2ω,ω, ω,0) as second harmonic generation phenomena (SHG).The computed parameters are given in Table 4.The Kerr effect, also known as the quadratic electro-optic (QEO) effect, is a change in a material's refractive index in response to an applied electric field.The Kerr effect (electro-optic effect) reveals the nonlinear change in the refractive index of materials after externally applied fields.
The nonlinear refractive index is given as [74]; where λ, K, and E are the wavelength of light, Kerr constant, and electric field strength, respectively.One can observe the higher Kerr effect takes place at a smaller frequency dispersion (ω = 0.042823 au).At 1064 nm, the highest dc-Kerr value of 7.36 × 10 12 au is recorded for Na@crypt, while lowest is while lowest is Li@crypt (Table 4).The higher value of the Kerr effect also demonstrates a larger change in the refractive index of studied complexes.Overall, at both frequencies, the dc-Kerr γ(-ω, ω,0,0) beats SHG response.

Role of non-covalent interactions in promoting NLO responses
Quantum theory of atoms in molecules (QTAIM) has been exploited to identify the bonding nature in M@crypt complexes.The presence of van der Waals (vdW) forces and other interactions are recognized as additional influencing factors in describing NLO properties of present complexes.Additionally, noncovalent interactions have crucial roles in describing chemical events and the stability of complexes in various organometallic and biological processes [75][76][77][78][79][80][81].
For the present complexes, we considered only bond critical points (BCP) using (3, -1) especially for the interactions zone to elucidate the nature and strength of attractive interactions.In Li@crypt, the Li63 atom shows interaction with O6/O4 and N8 atoms, while Na1 and K63 interact with oxygen (O1, O2, O3, O4, O5, and O6) atoms of cryptand only.Labeled structures with critical points and atoms are given in Figure 3a, while their corresponding parameters from CP are given in supplementary information (Table S6).Generally, greater values of electronic density (ρ r ) > 0.1 and negative Laplacian of electronic density (∇ 2 ρ ) indicate strong interactions like hydrogen bonding and electrostatic interactions.In contrast, the smaller value of ρ r and positive ∇ 2 ρ represents the presence of vdW interactions.The ∇ 2 ρ has a positive value for entire interactions involving Li63 with nitrogen and oxygen atoms.Also, ρ r is reduced and becomes less than 0.1, rationalizing the non-covalent nature of the interaction of Li63 inside the cavity.BCP 146 and 107 depict the N7-C54 and O4-C21 interactions having negative values for ∇ 2 ρ , which disclose the presence of strong interaction (covalent bond).The ∇ 2 ρ and energy density (H r ) values at BCP (3, -1) for Li63 interaction with N8, O1, O4, O6, and O4 are positive, indicating the non-covalent nature of bonding between atoms.Similarly, ∇ 2 ρ and H r at BCP (3, -1) are positive for Na1 interactions with O2, O4, O5, O6, and O7 pointing to its weak bonding.Na@crypt where Na1-O6/O7/04/O2/O5 has weak noncovalent interactions with complexant (crypt).For K63-O1, a small value of total electronic density (0.0127 au) is observed.Likewise, total electronic density values are also small for K63-O2, K63-O3, K63-O4, and K63-O5 interactions, justifying the presence of weak interactions or vdW forces.The values of ρ r are less than 0.1 au, and positive values of ∇ 2 ρ rationalize the presence of non-covalent interaction.The BCP for N7-K63 and N8-K63 also unfold the weak interaction (noncovalent) from the value of their electronic density and ∇ 2 ρ .On the other hand, the BCP (98) shows the O2-C36 bond, which has negative ∇ 2 ρ , while ρ r is higher than 0.2, indicating strong interactions (covalent bond).Figure 3b displays the reduced density gradient (RDG) scatter graph, where the λ 2 sign is exploited to differentiate between the bonded (λ 2 <0) and non-bonded (λ 2 >0) interactions.RDG scatter plots with colour-graded depict the type of interactions between alkali metals and cryptand, where the strong attraction (blue), the weak interaction (green), and the strong repulsion (red) are shown in spikes ranging from -0.035 to + 0.020 au.The blue color side of Figure 3b indicates strong inductive or attractive interactions.One can observe an increase in vdW forces due to interactions between metal (M=Li, Na, and K) and cryptand.The vdW interactions are more prominent in Li@crypt and Na@crypt, which agrees with the previous analysis.Therefore, the existence of noncovalent interaction between M -O/M -N may have a significant impact on triggering the complexes' optical and NLO response.
Electron localized function (ELF) and localized orbital locator (LOL) are shown in Figure 4. ELF map is designed in the range of 0.0 to 1.0; however, the delocalized electronic region falls below 0.5.The covalent region (electron-rich) shows high electron density in space where electrons are localized.From the results of color filled ELF map, the red-colored regions of the diaza cryptand [2.2.2] are observed for the hydrogen of the group.Generally, the higher ELF and LOL values, significant localization of electrons, which may be responsible for the existence of covalent bonds, inner shells, or a lone pair of electrons.The synaptic nature of the non-bonding lone pair of electrons on oxygen atoms with alkali metals in M@crypt is displayed in blue color regions in the ELF and LOL map [82].In the Li@crypt, Li63 has a high ELF value (red region) which indicates that ns 1 is not completely ionized and still bound with its valence.However, due to the strong interaction of Li63 to the lower side of the crypt, a significant increase in ELF value (red color) can be observed for N8, C12, C18, C15, and hydrogen atoms (H28, 25, 13, and 16).For Na@crypt, ELF value is decreased for Na1, which reveals the ns 1 valence electrons are more delocalized to ring in composing a diffuse excess electrons model.Na1 is separated by a blue color region, displaying its noncovalent interaction with O2, O4, O5, O6, and O7.In the K@crypt complex, K63 shows further reduced ELF value, while the CH 2 -group hold strong localization (red region).The valence electrons of K are diffused to the entire ring composing an anionic interstitial in cryptand.The Blue color shade can be seen in the LOL spectra of complexes, indicating the delocalized nature of electrons in complexes.The existence of blue circles around the alkali metals unveils the electron deficiency, transferring to complexant (crypt).Mathematically, ELF and LOL exhibit similar chemical mapping because they depend on the kinetic energy density.The ELF function can nicely unveil the location of atomic shells, core binding electrons, and lone pairs electrons in atomic and molecular orbitals.It is a dimensionless quantity, in the range of 0-1, to provide a visual description of the chemical bond for present complexes.Furthermore, ELF includes the Pauli kinetic energy density, while the LOL analysis does not include Pauli repulsion.Mathematically, the ELF can be expressed based on kinetic energy density.In contrast, the increased delocalization of electrons can be observed from Li@crypt to K@crypt.Moreover, the neutral region on LOL (light green) can be seen on bridgehead nitrogen atoms of aza-cryptand, while methylene (CH 2 ) moieties of cryptand indicate higher delocalization of electrons.

UV-visible study
For the designed M@crypt complexes, UV-Vis spectral study is exploited using the TD-DFT/ωB97×d/ def2tzvp method.In this analysis, the absorbance wavelength (λ), excitation energy (ΔE), and oscillator strength (O.S.) are determined.The contribution of major orbitals during the excitation from HOMO-LUMO is very crucial in the NLO study.Out of 40 excited states, we considered only the crucial state (with significant oscillator strength) for obtaining the excited state parameters (see Table 5).The pure cryptand shows absorbance in the deep ultraviolet region while after complexation with alkali metals, a dramatic increase in absorbance wavelength (bathochromic shift) is seen (see Figure 5).The highest absorbance maxima (λ max ) is accounted for Na@crypt complex at 1016 nm.Due to an appreciable reduction in excitation energy (ΔE) of Na@crypt, a vital excitation is observed from HOMO→LUMO with a contribution of 43.8%.The increased absorbance maxima can be strongly correlated to the increased atomic number of metals (M).Also, the smaller the excitation energy higher the hyperpolarizability response (vide supra).The oscillator strength (O.S.) of Na@crypt and K@crypt is higher as compared to Li@complex.

Conclusion
In summary, we have presented the optical, and NLO properties of designed complexes M@crypt based upon various quantum chemical calculations as well as the non-covalent interactions between the metal center and the cryptand.The designed complexes are similar to molecular electride and could be grouped in the excess electron family.AIMD simulations further confirm their kinetic and thermal stabilities at room temperature.The electronic structure calculations unveil their nature reminiscent of excess electron compounds, where the HOMO-LUMO gap was significantly reduced to 0.38 eV or lower.The C-N and C-O stretching vibrations become more intense after metal coordinating at the cavity of the cryptand.The remarkable value (1.41 × 10 6 au) of the first-order hyperpolarizability (β o ) is recorded for Na@crypt electride using ωB97×d/def2-tzvp.The β HRS is recorded up to 2.5 × 10 7 au.The highest static second-order hyperpolarizability γ o value of 4.96 × 10 9 au is calculated for K@crypt.γ o increases  gradually with the size of alkali metals.Furthermore, the frequency-dependent NLO properties are more responsive than those of static properties.Calculated NLO responses are almost identical regardless of DFT methods.The obtained results are compared to previously reported molecular electride and excess electron compounds.Increasing hyperpolarizability can be affiliated with strong van der Waals interactions, confirmed by QTAIM and RDG analyses.Electron density difference maps are employed to predict the charge transfer and distribution of electronic density after the de-excitation of electrons.The ELF and LOL analysis further disclose the nature of chemical bonds between alkali and complexant.TD-DFT calculations reveal the excited state parameters and absorbance properties.Hence, the examined electrides (as excess electrons compounds) could be a novel addition to optical materials.