Alkali and alkaline-earth metal hybrid doping enhances NLO response: from a weak alkaline-earthide of (Be^δ−@C₅₀N₅H₅)^{(δ′−δ)−}‒M^δ′+ to dual-alkali-metal salts of (Mg^δ+@C₅₀N₅H₅)^{(δ+δ′)−}–M^δ'+ and (Ca²⁺@C₅₀N₅H₅)^(2+δ)−–M^δ+

Abstract

Aiming to obtain enhanced NLO performance for C₅₀N₅H₅ cages through hybrid alkali/alkaline-earth metal doping in comparison with dual alkali-metal doping schemes, (M′@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca) were theoretically constructed by placing an alkali and alkaline earth metal atom outside and inside the C₅₀N₅H₅ cage, respectively. Binding characteristic analysis indicates that it changes from a weak alkaline-earthide of (Be^δ−@C₅₀N₅H₅)^{(δ′-δ)−}–M^δ+ to dual-alkali-metal salts of (Mg^δ+@C₅₀N₅H₅)^{(δ+δ′)−}–M^δ′+ and (Ca²⁺@C₅₀N₅H₅)^(2+δ)−–M^δ+ (M = Li, Na and K) as the atomic number of the alkali and alkaline earth metal increases. Notably, our species possess better stability and NLO performance than their alkali analogues, (M@C₅₀N₅H₅)M′ (M, M′ = Li, Na and K, J. Mol. Liq., 2023, 385, 122400). The static first hyperpolarizability exhibits an initial decrease followed by a subsequent increase in the (Be@C₅₀N₅H₅)–M and (Ca@C₅₀N₅H₅)–M series but increases monotonically in the (Mg@C₅₀N₅H₅)–M (M = Li, Na and K) series. The alkalide of (Ca²⁺@C₅₀N₅H₅^(2+δ)+)–K^δ+ holds prominently high β₀ up to 4.05 × 10⁵ a.u. A similar trend has emerged for the dynamic first hyperpolarizability.

Hybrid alkali/alkaline-earth metal doping is shown to be a better strategy in comparison with dual alkali-metal doping schemes.

Introduction

High-performance nonlinear optical (NLO) materials have long been a research hotspot owing to their applications in optical communication, terahertz generation, photonic computing and photonic crystals.^1,2 Substantial efforts have focused on modulating the static first hyperpolarizability (β₀). For instance, alkali metal fluoroxiborates (AB₄O₄X; X = K, Rb, and Cs) have been reported to display prominent second-harmonic generation (SHG);³ Ghosal et al. further demonstrated that the electronic-structure modulation of tetragonal silicene and germanene quantum dots can boost NLO responses.⁴ Doping with (super)alkali or alkaline-earth metal atoms offers an effective strategy for NLO modulation, attracting extensive research attention.^5–27 Incorporating s-electron-rich metal atoms into complexants can lead to the emergence of two distinct effects: the salt effect and the alkalide effect. For the former, Xu et al. confirmed that lithiation-induced salt effects can enhance β₀ values of acenes and heteronanotubes,^5,6 and our previous work also revealed considerable β₀ on Li@AR (AR = benzene and naphthalene)⁷ and Ca@SP-Helix (SP-Helix = aza-single-strand pyridine helix).⁸ Alkalides are a unique class of species where alkali metals serve as the anion. Experimental and computational evidence indicates typical alkalides feature superalkali–alkalide interactions,²⁸ and they act as potent, regioselective reducing agents.²⁹ Numerous alkalide and superalkalide systems have been proposed as NLO candidates,^9–19 including crown-ether-based alkalides M@[15-crown-5]¹² and Li₃O@[12-crown-4]M.¹³ We have reported a series of alkalide-based NLO switches of Li(HF)₃Li and Li(HCN)₃Li.^9,10

Nevertheless, conventional alkali-metal-based alkalides suffer from high chemical reactivity, imposing strict synthetic and storage constraints under ambient air and temperature. To address this issue, alkaline-earthides (alkaline-earth-based analogues) with rigid complexants have been developed.^20–27 However, published alkaline-earth salt and alkaline-earthide systems generally deliver lower β₀ than their alkali analogues, which is rarely discussed in previous studies. For example, M⁺(calix[4]pyrrole)M′⁻ (3.80 × 10³–1.98 × 10⁴ au)²⁰ has a narrower β₀ range than Li⁺(calix[4]pyrrole)M⁻ (M = Li, Na and K; M′ = Be, Mg and Ca) (1.10 × 10⁴–3.59 × 10⁴ au).¹⁴ This raises two open questions: whether hybrid doping outperforms single-type metal doping in NLO performance and whether stable alkaline-earthide/salt can achieve superior NLO responses to alkali analogues.

Motivated by the above issues, we designed a series of (M′@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca). The C₅₀N₅H₅ cage is derived from the synthetically available C₅₀H₁₀ cage.³⁰ Distinct from the reported all-alkali (⁺M@C₅₀N₅H₅)M′⁻ (M, M′ = Li, Na and K),¹⁹ our systems feature an alkaline-earth metal encapsulated inside the cage and an alkali metal anchored to the exterior. The calculated results show that these hybrid clusters have improved stability compared with all-alkali analogues and comparable stability to most reported alkalides, as well as appreciable NLO responses. This work provides a theoretical strategy for designing stable, high-efficiency NLO materials and guidance for related experimental syntheses.

Computational details

The geometric structures and relevant property calculations for (M@C₅₀N₅H₅)M′ (M, M′ = Li, Na and K)¹⁹ were obtained with the M06-2X³¹ functional and the 6-31G(d) basis set. To compare, the geometric structures of (M′@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca) with all real frequencies were also realized at the M06-2X/6-31G(d) level. The natural population analysis (NPA),³² vertical ionization energies (VIE), and interaction energies (E_int) values were obtained at the M06-2X/6-311++G(d,p) level. The VIE was estimated as:

VIE = E[(M′@C₅₀N₅H₅)–M]⁺ − E[(M′@C₅₀N₅H₅)–M] 1

The thermodynamic stabilities of (M@C₅₀N₅H₅)–M′ are in terms of E_int, which is calculated as:

E_int = E_{(M′@C50N5H5)–M} − E_M − E_M′ − E_C50N5H5 + BSSE 2

where E_int, E_{(M@C50N5H5)M′}, E_M, E_M′ and E_C50N5H5 are, respectively, the interaction energy and the energies of (M@C₅₀N₅H₅)–M′, the bare alkali (M), the alkaline earth metal atom (M′), and the complexant C₅₀N₅H₅. The counterpoise-corrected method was used to remove the basis set superposition error (BSSE).³³ Electron excitations were investigated by time-dependent DFT (TD-DFT) with the 6-311++G(d,p) method. To investigate which excitation path primarily contributes to the total β^e₀, 250 excited states were considered.

In previous alkalide and related systems, the M06-2X method has been widely used to calculate β^e₀. Recently, among CAM-B3LYP, LC-BLYP, M06-2X, MP2, MP3, MP4(SDQ), and QCISD methods, the M06-2X results are close to those obtained by the more sophisticated QCISD methods for the alkalide of M(HF)₃M (M = Na or Li).⁹ Table S1 shows that, at the M06-2X/6-311++G(d,p) level, the static β^e₀ values of (Be@C₅₀N₅H₅)–Na and (Be@C₅₀N₅H₅)–K are 3.74 × 10³ and 2.75 × 10⁵ a.u., respectively. Despite similar geometries, the β^e₀ values vary by over two orders of magnitude, suggesting potential drawbacks in the chosen computational approach. For the alkali and alkaline earth metal doped open-cage fullerenes, this may involve strong static electron-correlation effects. It is known that the static β components are obtained from the second derivative of the dipole moment with respect to an electric field, and errors in the dipole moment are nonlinearly amplified. Moreover, functionals with high HF exchange are unsuitable for multireference systems.³⁴ So, it is necessary to identify whether the multi-reference characteristics are significant before calculating β₀ values. The fractional orbital density (FOD) analysis³⁴ was done at the TPSS/def2-TZVP (T = 5000 K) level with a convergence threshold of “tightSCF” and a truncation threshold of “normalPNO” by the ORCA program.^35,36 The FOD analysis plot at σ = 0.003 e Bohr⁻³ (TPSS/def2-TZVP (T = 5000 K) level) for (M′@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca) is shown in Fig. S1. From Fig. S1, the FOD values are in the range of 2.162 to 3.396. So, it may be a significant yet relatively localized or rather delocalized FOD visible for these (M′@C₅₀N₅H₅)–M from M/M′ = Li/Be to K/Ca. Then, the frequency-dependent NLO properties of (M′@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca) were obtained with the TPSSh method with 10% HF exchange–correlation.

The frequency-dependent β(ω) can be obtained from:

β(ω) = (β_x² + β_y² + β_z²)^1/2 3

where β_i(ω) = [β_ijj(−2ω;ω,ω) + 2β_jji(−2ω;ω,ω)]/5 (i, j = x, y, and z) for the second harmonic generation (SHG).³⁷

Considering two measurable properties, the projection of β onto the dipole moment vector (β_prj) and the hyper Rayleigh scattering³⁸ response (β_HRS),³⁹β(ω) was also obtained at the TPSSh/def2-SVP level.

To further investigate the thermodynamic stability of our species, taking (Ca@C₅₀N₅H₅)–M (M = Li, Na, and K) as examples, we performed AIMD simulations at the B97-3c level of theory. For each molecule under consideration, AIMD simulations were carried out for 4000 fs with a time step of 1 fs and a fixed temperature of 298.15 K and 1 atm pressure with the Canonical Sampling through Velocity Rescaling (CSVR) thermostat⁴⁰ in the ORCA program. This method includes corrections for diverse interactions, which makes it suitable for studying a series of molecular systems with good accuracy.⁴¹ This trajectory analysis has been suggested to study the dynamic stability of parallel stacked complexes M[9C]2M (M = Li, Na, K, Be, Mg, and Ca).⁴²

β _HRS and β_prj were obtained from:

4

5

For β₀, ω = 0 au. Using the Born–Oppenheimer approximation, β₀ can be mainly written as an electronic contribution (β^e_ijk) and a nuclear relaxation contribution (β^nr_ijk, i, j, k = x, y, z). β^nr_ijk is given in terms of vibronic energies and dipole moment matrix elements over the vibronic wave functions.^43,44 Details of β^nr_ijk and β^e_ijk can be found in the SI.

All quantum chemistry calculations were performed using the Gaussian 16 package.⁴⁵ After calculations, the data analysis, including maps of electron density difference (EDD),⁴⁶ charge-transfer spectra (CTS),⁴⁷ and the hyperpolarizability values, were obtained using the Multiwfn 3.8(dev) code.^48,49

Results and discussion

Optimized geometries and stabilities

After trapping an alkaline earth atom into the C₅₀N₅H₅ cage and doping an alkali earth atom over the N₅H₅ subunit, structures of (M′@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca) with all real frequencies were obtained. From Fig. 1, the M–M′ distance (L₁) increases as the atomic number of M increases in each series (Be@C₅₀N₅H₅)–M, (Mg@C₅₀N₅H₅)–M and (Ca@C₅₀N₅H₅)–M (M = Li, Na and K). L₂ represents the height at which the alkaline earth metal atoms leave the bottom of the cage. Note that L₂ is higher for (Be@C₅₀N₅H₅)–M, (Mg@C₅₀N₅H₅)–Li and (Mg@C₅₀N₅H₅)–Na than for (Mg@C₅₀N₅H₅)–K and (Ca@C₅₀N₅H₅)–M (M = Li, Na and K). L₂ values are, respectively, in the ranges of 3.481–3.496 Å and 2.325–2.487 Å for the former and the latter. Unfortunately, we do not find any isomer with a different L₂ value. So, the difference in L₂ values arises from the balance of different forces.

Fig. 1. Optimized geometries of (M′@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca). Selected bond lengths (L₁ and L₂, Å), NPA charges of M and M′ (Q_M and Q_M′, |e|), interaction energies (E_int, kcal mol⁻¹), vertical ionization energies (VIE, eV), and HOMO–LUMO gaps (ε_gap, eV).

As an NLO molecular material, we are also concerned about its chemical and electronic stability, which reflect the tendency of a molecule to transform into another molecule or to lose an electron, respectively. Molecular electronic, chemical, and thermodynamic stability can be characterized by VIE, ε_gap, and E_int, respectively. For the open-shell systems of (Ca@C₅₀N₅H₅)–M (M = Li, Na and K), ε_gap gives the difference in energies between the HOMO(α) and LUMO(α). The data in Fig. 1 show that, for each series of (Be@C₅₀N₅H₅)–M, (Mg@C₅₀N₅H₅)–M, and (Ca@C₅₀N₅H₅)–M (M = Li, Na and K), an almost monotonic decrease is observed for both VIE and ε_gap values as the atomic number of the alkali metal atom (M) increases. So, both electronic and chemical stability monotonically decrease for these series as the atomic number of M increases. The VIE and ε_gap values are, respectively, in the ranges of 5.650–6.624 eV and 2.962–3.756 eV for (Be@C₅₀N₅H₅)–M, (Mg@C₅₀N₅H₅)–Li, and (Mg@C₅₀N₅H₅)–Na with higher L₂, while they are in the ranges of 4.755–6.149 eV and 1.154–2.463 eV for (Mg@C₅₀N₅H₅)–K and (Ca@C₅₀N₅H₅)–M (M = Li, Na and K) with lower L₂, respectively. Thus, the higher the height of the alkaline-earth metal from the bottom, the higher the electronic and chemical stability of the structure.

As illustrated in Fig. 1, two distinct structural categories can be classified according to the variation in the L₂ geometric parameter. The first category features large L₂ values ranging from 3.481 to 3.496 Å, covering all (Be@C₅₀N₅H₅)–M, (Be@C₅₀N₅H₅)–M, (Mg@C₅₀N₅H₅)–Li, and (Mg@C₅₀N₅H₅)–Na. The second category exhibits obviously reduced L₂ values of 2.325–2.487 Å, including (Mg@C₅₀N₅H₅)–K and all (Ca@C₅₀N₅H₅)–M (M = Li, Na and K). Excluding the special case of (Mg@C₅₀N₅H₅)–K, a clear structure–property correlation is observed: species with larger L₂ distances exhibit smaller interaction energies (|E_int| of 22.09–29.90 kcal mol⁻¹) and wider band gaps (ε_gap of 2.962–3.756 eV), whereas those with shorter L₂ distances possess stronger binding (|E_int| of 50.40–58.08 kcal mol⁻¹) and narrower band gaps (ε_gap of 1.048–2.463 eV). Accordingly, the former species with larger L₂ distances show inferior thermodynamic stability but improved chemical stability, while the latter species with smaller L₂ values feature superior thermodynamic stability accompanied by reduced chemical stability.

Among all nine studied species, (Mg@C₅₀N₅H₅)–K presents the minimum L₂ distance (2.325 Å) and the maximum L₁ distance (7.474 Å). Consequently, it possesses the narrowest ε_gap value (1.154 eV) and the smallest |E_int| value (13.45 kcal mol⁻¹), demonstrating that (Mg@C₅₀N₅H₅)–K features the poorest thermodynamic and chemical stability among the nine species of (M′@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca). The comparison between (Mg@C₅₀N₅H₅)–Na (longer L₂) and (Mg@C₅₀N₅H₅)–K (shorter L₂) shows that the latter displays a narrower ε_gap value, which is consistent with the above-mentioned correlation trend. However, its |E_int| behavior deviates from the above-predicted rule: despite its shorter L₂ distance, which favors stronger binding, (Mg@C₅₀N₅H₅)–K delivers a weaker E_int, suggesting the presence of unique underlying mechanisms.

For all (Ca@C₅₀N₅H₅)–M (M = Li, Na and K), the encadpsulated Ca atom carries highly positive NPA charges exceeding 1.0|e|. In contrast, the Mg atom in (Mg@C₅₀N₅H₅)–K only presents a moderate positive NPA charge of 0.744|e| (see Fig. 1). This significant charge difference demonstrates that Ca nearly donates both 4s valence electrons to the C₅₀N₅H₅ cage, corresponding to an effective oxidation state of +2, while the Mg atom only transfers one 3s valence electron, yielding a lower oxidation state of +1. Given that electrostatic interaction strength is proportional to the square of the transferred charge, the Ca-cage electrostatic interaction in (Ca@C₅₀N₅H₅)–M (M = Li, Na and K) is far stronger than the Mg-cage electrostatic interaction in (Mg@C₅₀N₅H₅)–K. Notably, although the short L₂ distance (2.325 Å) of (Mg@C₅₀N₅H₅)–K is geometrically favourable for electrostatic stabilization compared with the longer L₂ distances of all (Ca@C₅₀N₅H₅)–M (2.434–2.487 Å, M = Li, Na and K), this structural advantage may be largely offset. The shorter L₂ distance in (Mg@C₅₀N₅H₅)–K can bring a stronger repulsion between the Mg atom and the bottom of the cage than that between the Ca atom and the bottom of the cage in (Ca@C₅₀N₅H₅)–M, which obviously undermines structural stability. Collectively, the weaker electrostatic stabilization and the additional destabilizing repulsive component result in a much smaller |E_int| value (13.45 kcal mol⁻¹vs. 50.40–58.08 kcal mol⁻¹) and inferior thermodynamic stability of (Mg@C₅₀N₅H₅)–K relative to all (Ca@C₅₀N₅H₅)–M (M = Li, Na and K).

In comparison with all (Ca@C₅₀N₅H₅)–M (M = Li, Na and K), the species of (Be@C₅₀N₅H₅)–M (M = Li, Na and K), (Mg@C₅₀N₅H₅)–Li and (Mg@C₅₀N₅H₅)–Na exhibit evidently longer L₂ distance (3.481–3.496 Å vs. 2.434–2.487 Å) and negligible alkaline-earth metal NPA charges (−0.019–0.021|e| vs. 1.298–1.639|e|). These two factors jointly weaken the electrostatic interactions in these Be- and Mg-based systems, leading to lower |E_int| values relative to their Ca-based counterparts (22.09–29.90 kcal mol⁻¹vs. 50.40–58.08 kcal mol⁻¹). As a result, (Ca@C₅₀N₅H₅)–M possesses distinctly higher thermodynamic stability than (Be@C₅₀N₅H₅)–M (M = Li, Na and K), (Mg@C₅₀N₅H₅)–Li and (Mg@C₅₀N₅H₅)–Na.

Comparing (Mg@C₅₀N₅H₅)–K and (Mg@C₅₀N₅H₅)–Na, the former has a higher NPA charge on the Mg atom (0.744 vs. 0.021|e|) and a shorter L₂ distance (2.325 vs. 3.496 Å) than the latter, which is theoretically beneficial for electrostatic stabilization. Nevertheless, two unfavorable factors counteract this advantage in (Mg@C₅₀N₅H₅)–K. First, its considerably long Mg–K distance (L₁, 7.474 Å) may significantly weaken both the alkali metal-cage interaction and the alkali metal-alkaline earth metal interaction, in sharp contrast to the shorter Mg–Na distance (L₁, 5.841 Å) in (Mg@C₅₀N₅H₅)–Na. Second, the compressed L₂ distance can induce stronger repulsive interaction between the Mg atom and the C₅₀N₅H₅ cage for (Mg@C₅₀N₅H₅)–K than for (Mg@C₅₀N₅H₅)–Na. These destabilizing factors collectively result in an even smaller |E_int| value for (Mg@C₅₀N₅H₅)–K compared with (Mg@C₅₀N₅H₅)–Na, rendering the former thermodynamically less stable than the latter. Overall, (Mg@C₅₀N₅H₅)–K exhibits the smallest |E_int| magnitude and narrowest ε_gap value among all nine species of (M′@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca). Subsequently, it has the poorest thermodynamic and chemical stability throughout the studied systems.

The VIEs of (⁺M@C₅₀N₅H₅)M′⁻ (M, M′ = Li, Na and K)¹⁹ range from 3.46 to 4.13 eV. So, our (M′@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca) with VIEs of 4.755–6.396 eV exhibit considerably higher electronic stability than their alkali analogues. For relevant alkaline-earth-based compounds, the VIEs are, respectively, in the ranges of 2.412–3.374 eV, 3.5–4.7 eV, and 2.77–3.26 eV for reported (M–HMHC)–M′,^11,27 M⁺(calix[4]pyrrole)M′⁻,²⁰ and M⁺(NH₃)₆M′⁻ (M = Li, Na and K; M′ = Be, Mg and Ca).²¹ Then, (M′@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca) also have considerably higher electronic stabilities than them.

The ε_gap values are in the range of 2.750–3.758 eV for (M–HMHC)–M′ and within the ranges of 3.22–5.63 eV, 3.55–5.03 eV and 2.66–3.18 eV for M(HF)₃M (AM = Li and Na),¹⁰ M⁺(calix[4]pyrrole)M′⁻ and M⁺(NH₃)₆M′⁻ (M = Li, Na and K; M′ = Be, Mg and Ca),^20,21 respectively. So, the chemical stability of our (M′@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca) with ε_gap values of 1.048–3.481 eV is comparable.

The E_int values range from −27.35 to −29.90 kcal mol⁻¹ for the series of (Be@C₅₀N₅H₅)–M, from −13.45 to −22.17 kcal mol⁻¹ for the series of (Mg@C₅₀N₅H₅)–M, and from −50.40 to −58.08 kcal mol⁻¹ for the series of (Mg@C₅₀N₅H₅)–M. The latter are comparable to those of reported (M–HMHC)–M′(−38.86–56.38 kcal mol⁻¹).^11,27 The E_int values are in the range of −26.60–38.02 kcal mol⁻¹ for their alkali analogues of (⁺M@C₅₀N₅H₅)M′⁻ (M, M′ = Li, Na and K).¹⁹ Then, our (M′@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca) holds good thermodynamic stability.

Comparison of the series of (Be@C₅₀N₅H₅)–M, (Mg@C₅₀N₅H₅)–M, and (Ca@C₅₀N₅H₅)–M shows that, along with increasing atomic number of the alkaline earth metal atom, the E_int value first decreases and then significantly increases, which indicates that the thermodynamic stability first drops and then rises. This may be related to the changes in the NPA charges.

Therefore, the thermodynamic stability of our (M′@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca) is not only better than the alkalide analogues of (M@C₅₀N₅H₅)–M′ (M, M′ = Li, Na and K),¹⁹ but also has good comparability with many similar alkalides.^{9,11,20,21,27}

Fig. 2 provides the results of AIMD simulations based on quantum mechanics. To qualitatively describe the molecular deformation due to thermal fluctuations, for each trajectory, we aligned each structure (at each time t) to the reference structure (at time 0) to eliminate translation and rotation, and then averaged deviations of all atoms in the molecule. Thus, the root mean square deviation (RMSD) at each time was obtained. As shown in Fig. 2, the RMSD associated with each trajectory was bounded within about or less than 0.3 Å and revealed no signs of dissociation or fragmentation of such molecules, demonstrating their thermodynamic stability.

Fig. 2. Ab initio molecular dynamics study for (Ca@C₅₀N₅H₅)–M (M = Li, Na, and K): the root-mean-square deviation (RMSD) of trajectories.

Binding characteristics

The NPA charges of M (Q_M) and M′ (Q_M′) in (M′@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca) are listed in Fig. 1. Both Q_M′ and Q_M exhibit gradual trends. From the series of (Be@C₅₀N₅H₅)–M to (Mg@C₅₀N₅H₅)–M and then to (Ca@C₅₀N₅H₅)–M, Q_M′ gradually changes from small negative values (>−0.1|e|) to small positive values (<0.1|e|) and then to larger positive values (>0.7|e|), and Q_M changes from small positive values (<0.8|e| for M′ = Be and Mg) to larger positive values (>1.0|e| for M′ = Ca).

The Q_M and Q_M′ for the series of (Be@C₅₀N₅H₅)–M are very small positive and negative values, respectively. So, they are weakly electropositive and electronegative, respectively. The NPA charge values of Be in them are in the range of −0.019–0.036|e|. The NPA charges range from −0.28 to −0.47|e| for (M⁺@C₅₀N₅H₅)M′⁻ (M, M′ = Li, Na and K).¹⁹ The reported weak superalkalide is a C_2v OLi₃–Li–Li₃O with Q_Na = −0.12|e| to the best of our knowledge.¹⁷ So, (Be@C₅₀N₅H₅)–M (M = Li, Na and K) exhibit considerably weak alkaline-earthide behavior. Their salt-like structures may be (Be^δ−@C₅₀N₅H₅)^{(δ′−δ)−}M^δ′+.

The electron density difference (EDD) maps, suggested by Lu et al.,⁴⁶ are shown in Fig. 3. These blue and green isosurfaces indicate the decrease and increase in electron density in the regions, respectively. A small blue lobe is observed over the alkali atom (M = Li, Na, and K) and a small green lobe over the Be atom for the series of (Be@C₅₀N₅H₅)–M. This electron density may provide support for the above NPA charges of M and Be in the series of (Be^δ−@C₅₀N₅H₅)^{(δ′−δ)−}–M^δ′+. Also, a very small blue isosurface is observed on the Mg atom for both (Mg@C₅₀N₅H₅)–Li and (Mg@C₅₀N₅H₅)–Na, which indicates a small positive NPA charge on the Mg atom. For (Mg@C₅₀N₅H₅)–K and (Ca@C₅₀N₅H₅)–M (M = Li, Na and K), a large blue isosurface exists on the alkaline earth metal atom (Mg or Ca), which provides support for the large positive NPA charge of the Mg atom in them. The salt-like structures of (Mg@C₅₀N₅H₅)–M (M = Li and Na) with fractional oxidation number or fractional valence may be (Mg^δ+@C₅₀N₅H₅)^{(δ′+δ)−}–M^δ′+, while those of (Mg@C₅₀N₅H₅)–K and (Ca@C₅₀N₅H₅)–M are, respectively, (Mg⁺@C₅₀N₅H₅²⁻)–K⁺ and (Ca²⁺@C₅₀N₅H₅^(2+δ)+)–M^δ+.

Fig. 3. Electron density difference (EDD) maps for (M′@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca).

In summary, the results of the NPA charges for the alkaline earth metal atoms show that, as the atomic number increases, there is an evolution from a very small negative charge to a very large positive charge for the alkaline earth metal atom in (M′@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca), and the structure changes from the weak alkaline-earthide of (Be^δ−@C₅₀N₅H₅)^{(δ′−)−}–M^δ+ to the dual-alkali-metal salts of (Mg^δ+@C₅₀N₅H₅)^{(δ′+δ)−}–M^δ′+ and (Ca²⁺@C₅₀N₅H₅^(2+δ)+)–M^δ+ (M = Li, Na and K) as the alkali and alkaline earth metal increases. More importantly, the NPA charge and EDD maps indicate that there is a charge transfer between the metal atoms and the C₅₀N₅H₅ cage and that the interaction between them should be primarily electrostatic.

Static first hyperpolarizabilities

In Table 1, the alkali and alkaline earth metal atomic number dependences are shown for (M′@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca). As the atomic number of M increases, the µ₀ value increases monotonically in each series of (Be@C₅₀N₅H₅)–M, (Mg@C₅₀N₅H₅)–M, and (Ca@C₅₀N₅H₅)–M (M = Li, Na and K) but decreases monotonically in each series of (M′@C₅₀N₅H₅)–Li, (M′@C₅₀N₅H₅)–Na, and (M′@C₅₀N₅H₅)–K along with the increase in the atomic number of M′ (M′ = Be, Mg and Ca). Then, (Li@C₅₀N₅H₅)–K has the largest µ₀ value (16.02 D) among the nine species. In addition, all (M′@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca) have downward directions of µ₀ or µ_z.

Table 1. Dipole moment (µ₀), static first hyperpolarizability (β^e₀) and its z-components (β^e_zzz), the projection of β onto the dipole moment vector (β_prj) hyper Rayleigh scattering response (β_HRS), the sum-over-states estimation values (β^SOS₀), dipole moment between the ground state and the crucial excited state (Δµ), oscillator strength (f₀), and transition energy (ΔE) of the crucial excited state for the chair forms of (M′–@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca).

	µ 0 (D)	β HRS (a.u.)	β prj (a.u.)	β e (a.u.)	β e (a.u.)	β SOS (au)	Δµz (au)	Δµ0 (au)	f 0 z	f 0	ΔE (eV)	Major contributiona
(Li@C 50 N 5 H 5 )–Li
				(9.23 × 103)b		2.18 × 104	3.587	3.637	0.182	0.182	1.230	H → L + 3
							1.123	2.025	0.000	0.145	2.440	H − 1 → L + 1
							−2.423	3.460	0.304	0.307	2.471	H → L + 7
							−0.277	1.558	0.000	0.200	3.841	H − 2 → L + 3
(Be@C 50 N 5 H 5 )–M (M′ = Be)
M = Li	13.02	2.33 × 104	−3.51 × 104	5.84 × 104	−5.43 × 104	1.34 × 105	−1.067	1.815	0.364	0.364	2.040	H(α) → L(α) + 1
							−2.038	4.629	0.000	0.061	2.313	H(α) → L(α) + 4
							−0.865	4.691	0.000	0.112	3.826	H(β) → L(β) + 4
M = Na	13.13	1.53 × 104	−2.41 × 104	4.02 × 104	−3.33 × 104	1.23 × 105	2.009	1.718	0.349	0.350	2.203	H(α) → L(α) + 1
							−0.348	5.525	0.095	0.095	2.437	H(β) → L(β) + 4
							−1.153	4.339	0.000	0.062	3.794	H(α) → L(α) + 3
							−1.591	4.342	0.000	0.066	3.838	H(α) − 2 → L(α) + 3
M = K	16.02	2.42 × 104	−3.94 × 104	6.56 × 104	−4.55 × 104	5.73 × 104	0.603	2.524	0.383	0.383	1.514	H(α) → L(α) + 1
							3.854	4.462	0.081	0.081	2.022	H(α) → L(α) + 4
							−0.767	5.828	0.000	0.104	3.815	H(α) − 2 → L(α) + 4
(Mg@C 50 N 5 H 5 )–M (M′ = Mg)
M = Li	11.52	2.69 × 104	−6.64 × 104	6.64 × 104	−6.35 × 104	6.29 × 104	−0.612	5.274	0.045	0.045	1.796	H(α) → L(α) + 1
							−1.694	1.767	0.317	0.317	2.078	H(α) − 1 → L(α) + 1
							−3.750	4.356	0.000	0.045	2.318	H(α) − 1 → L(α) + 4
							−1.211	4.396	0.000	0.106	3.813	H(β) − 2 → L(β)
M = Na	11.77	2.58 × 104	−6.65 × 104	6.65 × 104	−5.80 × 104	6.48 × 104	0.396	4.909	0.035	0.035	1.846	H(α) → L(α) + 1
							−1.851	6.538	0.349	0.349	2.257	H(α) − 1 → L(α) + 1
							−2.058	4.184	0.000	0.087	2.752	H(α) → L(α) + 4
							0.828	4.204	0.000	0.139	3.822	H(β) − 3 → L(β) + 5
M = K	15.98	8.59 × 104	−2.20 × 105	2.20 × 105	−1.93 × 105	3.54 × 105	1.961	5.803	0.083	0.083	0.669	H(β) → L(β) + 1
							8.985	8.071	0.000	0.064	2.128	H(β) → L(β) + 4
							8.137	9.771	0.165	0.166	2.304	H(β) → L(β) + 5
							8.036	8.105	0.000	0.064	3.809	H(β) → L(β) + 11
(Ca@C 50 N 5 H 5 )–M (M′ = Ca)
M = Li	5.09	2.96 × 105	−7.01 × 105	7.10 × 105	−6.82 × 105	1.03 × 105	−0.732	4.167	0.031	0.031	0.958	H(β) → L(β) + 3
							3.479	5.016	0.176	0.176	1.794	H(α) − 1 → L(α) + 4
							−0.882	4.515	0.128	0.127	2.312	H(β) → L(β) + 5
							1.537	2.777	0.019	0.130	3.758	H(β) − 2 → L(β) + 4
M = Na	5.17	7.40 × 104	−1.29 × 105	1.46 × 105	−2.44 × 104	8.53 × 104	−0.791	4.059	0.040	0.040	0.981	H(β) → L(β) + 3
							6.311	4.409	0.123	0.123	1.875	H(α) − 1 → L(α) + 1
							−1.401	4.760	0.279	0.279	2.882	H(α) − 1 → L(α) + 5
							1.207	2.688	0.009	0.121	3.762	H(β) → L(β) + 4
M = K	11.58	1.63 × 105	−4.05 × 105	4.05 × 105	−3.83 × 105	7.43 × 105	4.460	5.772	0.073	0.073	0.861	H(α) − 1 → L(α)
							−4.340	7.713	0.000	0.091	1.987	H(α) − 1 → L(α) + 4
							−3.663	6.088	0.142	0.142	2.437	H(α) − 1 → L(α) + 6
							−2.199	7.865	0.002	0.054	3.634	H(β) → L(β) + 6

^aH, L, H(α), H(β), L(α), and L(β) are HOMO, LUMO, HOMO(α), HOMO(β), LUMO(α), and LUMO(β), respectively.

^bThe value in parentheses is from ref. 17.

The β_HRS and β_prj, as well as β^e₀, are also shown in Table 1. The main components of the polarizability (α, au) and hyperpolarizability (β, au) tensors are shown in Table S2. Overall, the alkali and alkaline earth metal atomic number dependences are also shown for β_HRS and β_prj as well as β^e₀. As the atomic number of M (M = Li, Na, K) increases, the β_HRS, β_prj, and β^e₀ values undergo an initial decrease and a subsequent increase in each series of (Be@C₅₀N₅H₅)–M and (Ca@C₅₀N₅H₅)–M (M = Li, Na and K). In the (Mg@C₅₀N₅H₅)–M series, the β_HRS value presents the aforementioned variation tendency, while the β_prj and β^e₀ values rise in a monotonic manner. It can be found that, for the (Be@C₅₀N₅H₅)–M series, which feature long L₂ distances and weak alkaline-earthide characteristics, the variation orders of β_HRS, β_prj and β^e₀ are consistent with the trend of the NPA charge of alkaline earth metal atom (Q_M′) as the atomic number of M (M = Li, Na, K) increases, while the NLO ordering follows the variation trend of the L₂ distances with increasing atomic number of M (M = Li, Na and K) for the (Ca@C₅₀N₅H₅)–M species, characterized by short L₂ distances and dual-alkali-metal salt-like features. In contrast, for the (Mg@C₅₀N₅H₅)–M series with hybrid salt-like features and both long and short L₂ distances, the orders of β_prj and β^e₀ coincide with the trend of Q_M′, whereas the β_HRS order follows the NPA charge variation of the alkali metal atom (Q_M) as the atomic number of M (M = Li, Na and K) increases. For each series of (M′@C₅₀N₅H₅)–Li, (M′@C₅₀N₅H₅)–Na and (M′@C₅₀N₅H₅)–K (M′ = Be, Mg and Ca), β_HRS, β_prj, and β^e₀ values all increase monotonically with the increase in atomic number of M′. These β orders exhibit an inverse correlation with the sequences of Q_M′ and band gap values (ε_gap) within their respective series. Then, (Ca@C₅₀N₅H₅)–K has the largest absolute values of β_HRS, β_prj, and β^e₀ among the nine species. These values are 1.63 × 10⁵, −4.05 × 10⁵, and 4.05 × 10⁵ au for β_HRS, β_prj, and β^e₀, respectively. All (M′@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca) have downward directions of β_prj or β_zzz. The main components of the polarizability (α, au) and hyperpolarizability (β, au) tensors are shown in Table S2.

The β^e₀ values of our species are in the range of 4.02 × 10⁴–4.05 × 10⁵ au. For the reported (M⁺@C₅₀N₅H₅)–M′⁻ (M, M′ = Li, Na and K),¹⁹ the β^e₀ values are in the range of 5.86 × 10³–1.91 × 10⁵ au. So, our (M′@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca) have better NLO responses than their alkali analogues. For the relative alkalides, the β₀ values of M⁺(calix[4]pyrrole)M′⁻ and M⁺(NH₃)₆M′⁻ (M = Li, Na and K; M′ = Be, Mg and Ca)^20,21 are, respectively, in the range of 3.80 × 10³–1.90 × 10⁵ a.u. and 2.1 × 10³–6.4 × 10⁴ a.u. So, our (M′@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca) also have better NLO responses than those of M⁺(calix[4]pyrrole)M′⁻.

The β^e₀ value can be estimated by using the sum-over-states estimation:

β^e₀ ∝ (Δµ·f₀)/ΔE³ 6

where ΔE, f₀, and Δµ are, respectively, the transition energy, oscillator strength, and the difference in transition dipole moment between the ground state and the crucial excited state. The estimated first hyperpolarizabilities (β^SOS₀) are listed in Table 1. Obviously, the difference between β^SOS₀ and β^e₀ is significant. So, given that the β^SOS₀ results cannot be directly used to explain the β^e₀ results, we turned our attention to the ΔE and the z-components of f₀ and Δµ. The charge-transfer spectra of (M′@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca) are shown in Fig. 4–6.

Fig. 4. Charge-transfer spectra of (Be@C₅₀N₅H₅)–M (M = Li, Na and K) and (Li@C₅₀N₅H₅)–Li. The molecular orbitals at an isovalue of 0.02 a.u.

Fig. 5. Charge-transfer spectra of (Mg@C₅₀N₅H₅)–M (M = Li, Na and K). The molecular orbitals at an isovalue of 0.02 a.u.

Fig. 6. Charge-transfer spectra of (Ca@C₅₀N₅H₅)–M (M = Li, Na and K). The molecular orbitals at an isovalue of 0.02 a.u.

From Fig. 4–6, each spectrum exhibits a broad peak in the low-energy region and one or two sharp peaks in the high-energy region. For the sharp peak(s) in the high-energy region, the wavelengths are 323–325 nm. From Table 1, it can be found that the z-component of f₀ is 0.000 (f₀^z) for the high-energy sharp peaks. So, it has nearly no contribution to β^e_zzz. Thus, it is the low-energy broad peak that makes the key contribution to β^e_zzz. For these low-energy broad peaks, it can be found that the wavelength first decreases and then increases in each series of (Be@C₅₀N₅H₅)–M, (Mg@C₅₀N₅H₅)–M, and (Ca@C₅₀N₅H₅)–M (M = Li, Na and K) as the atomic number of M increases. The longer the wavelength, the smaller the ΔE. Considering that β^e₀ is proportional to the reciprocal of ΔE's cube, the order of the wavelength of the broad peak may explain the order of the β^e₀ for each series of (Be@C₅₀N₅H₅)–M, (Mg@C₅₀N₅H₅)–M, and (Ca@C₅₀N₅H₅)–M (M = Li, Na and K).

In addition, for the (Be@C₅₀N₅H₅)–M (M = Li, Na, and K) series, both Δµ₀ and f₀^z of the first excited state initially decrease and then increase with the rising atomic number of M, which follows the variation trends of the β_HRS, β_prj, and β^e₀ as well as that of the above Q_M′. For the (Mg@C₅₀N₅H₅)–M series, the |Δµ₀^z| of the excited state with the maximum oscillator strength (f₀) increases monotonically as the atomic number of M increases, showing the sequence of 1.694 (M = Li, f₀ = 0.317) < 1.851 (M = Na, f₀ = 0.349) < 8.137 (M = K, f₀ = 0.165); this trend is consistent with the variations of β_prj, β^e₀ and above Q_M′. For the (Ca@C₅₀N₅H₅)–M series, the |Δµ₀^z| of the maximum-f₀ excite state exhibits a trend of first decreasing and then increasing with the increase of M atomic number, with a |Δµ₀^z| order of 3.479 (M = Li, f₀ = 0.176) < 1.401 (M = Na, f₀ = 0.279) < 3.663 (M = K, f₀ = 0.142), which well matches the variation patterns of β_prj, β^e₀ and above L₂ distance.

Considering the charge transfer, Fig. 4–6 show that the low-energy broad peak mainly belongs to redistribution in M′–M and transfer between M′–M and C₅₀N₅H₅ for each case of (M′@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca). The orbital transition mainly responds to s → p or p → p transition in the z-axis (HOMO(α) → LUMO(α) + 1 ((Be@C₅₀N₅H₅)–M)), or HOMO(α) − 1 → LUMO(α) + 1 or LUMO(α) + 5 or LUMO(α) + 6 ((Mg@C₅₀N₅H₅)–M (M = Li or Na) and (Ca@C₅₀N₅H₅)–M (M = Li or Na or K)), and HOMO(β) → LUMO(β) + 5 (Mg@C₅₀N₅H₅)–K) for all the nine species. For M′@C₅₀N₅H₅)–K (M′ = Be, Mg and Ca) in their respective series, i.e., (Be@C₅₀N₅H₅)–M, (Mg@C₅₀N₅H₅)–M, and (Ca@C₅₀N₅H₅)–M (M = Li, Na and K), there exist some shoulder absorptions with large f₀. In addition, there is a low-energy absorption with wavelength > 1000 nm for (Mg@C₅₀N₅H₅)–K and (Ca@C₅₀N₅H₅)–M (M = Li, Na and K), with a wavelength that first decreases and then increases in series (Ca@C₅₀N₅H₅)–M (M = Li, Na and K) as the atomic number of M increases. This may be the reason why M′@C₅₀N₅H₅)–K (M′ = Be, Mg and Ca) has the largest β^e₀ value in their respective series.

Comparing the CTS maps of (Be@C₅₀N₅H₅)–Li and (Li@C₅₀N₅H₅)–Li (see Fig. 4), it can be observed that it mainly involves the s → p transition from Be–Li to the C₅₀N₅H₅ cage, which corresponds to the characteristic absorption peak (λ_max) for (Be@C₅₀N₅H₅)–Li, but mainly involves the s → p transition around the outside Li atom, which corresponds to λ_max for (Li@C₅₀N₅H₅)–Li. More importantly, the λ_max is 107 nm larger in wavelength for (Be@C₅₀N₅H₅)–Li than for (Li@C₅₀N₅H₅)–Li, which means the ΔE of the former is smaller than that of the latter. This may be the most important reason why the β^e₀ value of (Be@C₅₀N₅H₅)–Li with co-doping of alkali and alkaline earth metal elements is slightly larger than that of (Li@C₅₀N₅H₅)–Li with single-element doping, although the latter holds a significant low-energy absorption at 1008 nm that cannot be ignored.

It has been⁴³ suggested that the vibrational first hyperpolarizability tensor (β^nr_zzz) is also important for the total hyperpolarizability. The Raman frequency is required for the calculation of β^nr_zzz. So, the β^nr_zzz of (M′–@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca) were obtained at the M06-2X/6-31G(d) level. Table 3 also gives the β^e_zzz at the same level.

Table 3. β _HRS and β_prj of the chair forms at different incident laser frequencies (ω).

	ω = 1907 nm		ω = 1460 nm		ω = 1340 nm		ω= 1064 nm
	β HRS (a.u.)	β prj (a.u.)	β HRS (a.u.)	β prj (a.u.)	β HRS (a.u.)	β prj (a.u.)	β HRS (a.u.)	β prj (a.u.)
(Be@C 50 N 5 H 5 )–M (M′ = Be)
M = Li	2.92 × 104	7.31 × 104	3.76 × 105	9.15 × 105	9.16 × 104	−2.13 × 105	2.05 × 105	−5.87 × 105
M = Na	5.68 × 103	1.36 × 104	2.66 × 104	7.22 × 104	4.91 × 105	1.19 × 106	1.76 × 105	4.75 × 105
M = K	6.50 × 104	1.74 × 105	1.15 × 105	3.17 × 105	3.47 × 105	9.37 × 105	1.26 × 105	−3.39 × 105
(Mg@C 50 N 5 H 5 )–M (M′ = Mg)
M = Li	9.84 × 105	−2.37 × 106	2.51 × 104	6.30 × 104	3.78 × 105	9.20 × 105	1.64 × 104	−4.31 × 104
M = Na	2.60 × 107	−6.28 × 107	9.54 × 103	2.88 × 103	4.67 × 104	−8.17 × 104	3.87 × 106	9.33 × 106
M = K	8.64 × 104	2.25 × 105	5.44 × 104	1.52 × 105	2.73 × 105	7.59 × 105	2.22 × 105	−5.76 × 105
(Ca@C 50 N 5 H 5 )–M (M′ = Ca)
M = Li	8.29 × 105	−1.99 × 106	3.06 × 104	−7.36 × 104	4.23 × 104	1.06 × 105	1.03 × 105	2.66 × 105
M = Na	6.37 × 104	−1.52 × 105	3.07 × 104	7.87 × 104	4.98 × 104	1.22 × 105	4.72 × 105	−1.10 × 106
M = K	2.03 × 106	4.64 × 106	1.46 × 105	3.85 × 105	6.07 × 105	5.93 × 105	2.01 × 105	−5.40 × 105

Comparing Tables 1 and 2, the β^e_zzz obtained at the M06-2X/6-31G(d) and M06-2X/6-311++G(d,p) levels are, respectively, −5.93 × 10³ and −2.62 × 10⁵ a.u., which indicates that the M06-2X/6-31G(d) method significantly underestimated the β^e_zzz value. Still, from Table 3, it can be found that, numerically, β^nr_zzz is an order of magnitude smaller than β^e_zzz at the same level. All the η values are very small (<0.20) except that of (Ca@C₅₀N₅H₅)–M. So, it is not β^nr_zzz but β^e_zzz that makes the pivotal contribution to the total β₀ of each of (M′–@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca).

Table 2. Electronic and vibrational first hyperpolarizability tensors (β^e_zzz and β^nr_zzz) and ratio η of β^nr_zzz/β^e_zzz for (M′–@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca) at the M06-2X/6-31G(d) level.

	(M′–@C50N5H5)–M
	β e (a.u.)	β nr (a.u.)	H
(Be@C 50 N 5 H 5 )–M (M′ = Be)
M = Li	−1.36 × 104	−1.33 × 103	0.10
M = Na	−1.69 × 104	−1.90 × 103	0.11
M = K	−1.07 × 104	−1.42 × 103	0.01
(Mg@C 50 N 5 H 5 )–M (M′ = Mg)
M = Li	−2.66 × 104	−2.64 × 103	0.10
M = Na	−3.26 × 104	−3.01 × 103	0.09
M = K	−1.29 × 105	−6.91 × 103	0.05
(Ca@C 50 N 5 H 5 )–M (M′ = Ca)
M = Li	−2.70 × 104	−4.69 × 103	0.17
M = Na	−3.34 × 104	−4.50 × 103	0.13
M = K	−5.93 × 103	−1.01× 104	1.80

Dynamic first hyperpolarizabilities

For (M′–@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca), the β_HRS and β_prj with the frequency dispersion at wavelengths (ω) of 1907, 1460, 1340, and 1064 nm are listed in Table 3. Their tensors are shown in Table S2.

With increasing atomic number of M (M = Li, Na, K), the β_HRS and |β_prj| of (Be@C₅₀N₅H₅)–M exhibit three distinct variation trends. At ω = 1907 nm and 1460 nm, they first decrease and then rise markedly. At ω = 1340 nm, they increase significantly at first and then decrease, while they display a monotonic decreasing trend at ω = 1064 nm. Similarly, the β_HRS and |β_prj| of (Ca@C₅₀N₅H₅)–M also present three distinct variation trends, whereas only two trends are observed for (Mg@C₅₀N₅H₅)–M. For the (Mg@C₅₀N₅H₅)–M series, β_HRS and |β_prj| increase significantly first and then decrease at ω = 1907 nm and 1604 nm, while they exhibit the opposite tendency of initial decrease followed by a subsequent increase at ω = 1460 nm and 1340 nm. In terms of (Ca@C₅₀N₅H₅)–M, its β_HRS and |β_prj| values decrease significantly at first and then rise at ω = 1907 nm and present an initial increase followed by a decrease at ω = 1640 nm. In contrast, a monotonic increasing behavior is observed at ω = 1460 nm and 1340 nm.

With the rise in atomic number of M, β_HRS and |β_prj| display two variation trends for each (M′@C₅₀N₅H₅)–Li and (M′@C₅₀N₅H₅)–Na series, while three distinct trends are observed for (M′@C₅₀N₅H₅)–K (M′ = Be, Mg and Ca). For the (M′@C₅₀N₅H₅)–Li and (M′@C₅₀N₅H₅)–Na series, β_HRS and |β_prj| first increase significantly and then decrease at ω = 1907 nm, whereas they exhibit the opposite tendency of initial decrease followed by a subsequent increase at ω = 1460 nm. (M′@C₅₀N₅H₅)–Li at ω = 1340 nm and (M′@C₅₀N₅H₅)–Na at ω = 1064 nm, these parameters show an initial increase followed by a significant decrease. Conversely, they first decrease significantly and then increase for (M′@C₅₀N₅H₅)–Na at ω = 1340 nm and (M′@C₅₀N₅H₅)–Li at ω = 1064 nm. For the (M′@C₅₀N₅H₅)–K series, β_HRS and |β_prj| values decrease significantly first and then rise at ω = 1460 nm and 1340 nm, while they present a slight decrease after an initial increase at ω = 1064 nm. In addition, a monotonic increasing behavior is observed at ω = 1907 nm. As a result, all (M′@C₅₀N₅H₅)–Li, (M′@C₅₀N₅H₅)–Na, and (M′@C₅₀N₅H₅)–K (M′ = Be, Mg and Ca) exhibit a consistent trend of initial decrease followed by a subsequent increase at ω = 1460 nm.

Conclusions

To compare the NLO performances of hybrid alkali–alkaline-earth metal doping and dual-alkali-metal doping, we theoretically constructed a series of (M′@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca), in which an alkaline-earth metal atom is encapsulated inside the C₅₀N₅H₅ cage and an alkali metal atom is anchored on the cage exterior.

Along with the increase in atomic number of the doped metal atoms, the structure changes from a weak alkaline-earthide of (Be^δ−@C₅₀N₅H₅)^{(δ′−δ)−}–M^δ′+ to a dual-alkali-metal salt of (Mg^δ+@C₅₀N₅H₅)^{(δ′+δ)−}–M^δ′+ and (Ca²⁺@C₅₀N₅H₅^(2+δ)+)–M^δ+ (M = Li, Na and K). Analysis of VIE and ε_gap results shows that our (M′@C₅₀N₅H₅)–M possesses improved stability relative to their all-alkali alkalide analogues, (M@C₅₀N₅H₅)M, with stability performance comparable to that of most reported alkalide systems.

It is noteworthy that our species also exhibit higher NLO responses than their alkali analogues. With increasing atomic number of M (M = Li, Na, K), the β^e₀ presents an initial decrease followed by a subsequent increase for the (Be@C₅₀N₅H₅)–M and (Ca@C₅₀N₅H₅)–M series, while a monotonic increasing trend is observed for the (Mg@C₅₀N₅H₅)–M series. A similar trend is exhibited for the dynamic first hyperpolarizability.

This work reveals the potential of rationally designed cage-confined hybrid alkali–alkaline-earth metal-doped systems for NLO applications. We hope this study can motivate experimental efforts toward the synthesis and characterization of such emerging alkaline-earthide and dual-alkali-metal salt candidates for advanced NLO materials.

Conflicts of interest

There are no conflicts to declare.

Data availability

The data will be available on request.

Supplementary information (SI): data of optimized geometries, computational details on β^nr_ijk and β^e_ijk of (M′@C₅₀N₅H₅)–M (M = Li, Na and K; M′ = Be, Mg and Ca). See DOI: https://doi.org/10.1039/d6ra04746g.