Minimization of a system’s energy functional is the most fundamental step in the determination of its ground state. Reaching the global minimum (GM) geometry, however, poses a number of challenges, the most important being the high probability of getting stuck in local minima in the PES. Swarm intelligence (SI)-based algorithms have turned out to be very effective in searching for optimal solutions in a given search space. Here we discuss three different techniques, PSO combined with DFT, PSO with CNN, and DFT-integrated FA. They do not need to implement any symmetry constraint, or consider bond characterization. DFT-PSO adjusts each particle’s trajectory at every time stamp while following the convergence criteria. We have successfully implemented these techniques to find the GM configurations for small-sized nonmetallic clusters such as Boron (B₅ and B₆) (Yuan et al., 2014), Carbon (C₅) (Jana et al., 2019), and polynitrogen clusters (N₄ ^2- and N₆ ^4-) (Mitra et al., 2021), and metallic clusters such as Al₄ ^2- (Mitra et al., 2020), Au _n (n = 2–8), and Au _n Ag _m (2 ≤ n+m ≤ 8) (Mitra et al., 2021).

It is well known from Hückel’s (4n+2) π electron theory (Hückel, 1931) and Pauling’s quantum mechanical description (Pauling and Sherman, 1933) that aromaticity of conjugated systems with cyclic and planar geometry is associated with their increased stability. Various structural, energetic, electronic, and magnetic behavior-based parameters are considered to analyze the aromaticity of the systems, among which the Nucleus-Independent Chemical Shift (NICS) (Schleyer et al., 1996) is, perhaps, the most widely used criterion for aromaticity. CDFT also plays an important role in aromaticity determination (Chattaraj et al., 2005; Chattaraj et al., 2006; Chattaraj et al., 2007), and it does so with the help of reactivity descriptors and associated electronic structure principles (Chattaraj et al., 1993; Chattaraj and Maiti, 2001; Pan et al., 2013b; Chakraborty and Chattaraj, 2021). The relative aromaticity index ΔX, where X could be energy (E), polarizability (α), electrophilicity (ω), or hardness (η), is defined as the difference between the respective indices in the cyclic and open (or localized) systems, i.e., ΔX = X _CYCLIC – X _{OPEN(/LOCALIZED)}. They are observed to show a similar performance as those of the NICS and MCI values. They also provide valuable insights into the stability, reactivity, and electronic properties of the associated cluster.

All-metal aromatic Al₄ ²⁻ (Li et al., 2001) and antiaromatic Al₄ ⁴⁻ (Kuznetsov et al., 2003) clusters are investigated at the B3LYP/6-311G(d, p) level of theory to analyze their aromaticity from a CDFT perspective (Chattaraj et al., 2005; Chattaraj et al., 2006; Chattaraj et al., 2007). While Δη > 0 represents aromaticity and Δη < 0 represents antiaromaticity, the reverse is true in the cases of E, α, and ω indices. Their values for the aluminum clusters are compared with those of benzene (C₆H₆) and cyclobutadiene (C₄H₄). While Al₄ ²⁻ and C₆H₆ show positive Δη and negative ΔE, Δα, and Δω values indicating an aromatic behavior, C₄H₄ shows an exact opposite trend to account for its antiaromaticity. For the Al₄ ⁴⁻ cluster, however, we have obtained somewhat contradictory results. ΔE and Δα values indicate the cluster’s antiaromatic nature, whereas Δη, Δω, and NICS values reflect its aromatic nature. It is in conformity with the current knowledge that this cluster exhibits conflicting aromaticity. Experimental synthesis and theoretical studies reported along with ELF analysis (Li et al., 2001; Santos et al., 2005) suggest the cluster to be antiaromatic. Its σ-aromaticity directs the overall aromaticity by dominating over its π-counterpart as studied through NICS (Chen et al., 2003) and magnetic field induced current density (Havenith et al., 2004) analyses. Such conflicting aromaticity, along with other varieties of multiple aromaticity and antiaromaticity, δ- and Φ-aromaticity, bond stretch isomerism, etc. are exhibited by several other all-metal clusters (Zubarev et al., 2009; Zubarev and Boldyrev, 2011). Be₃ ²⁻, Ca₃ ²⁻, and Mg₃ ²⁻ clusters are classified as aromatic in terms of the ΔX indices (Roy and Chattaraj, 2008; Giri et al., 2010). Other applications of aromatic clusters are studied by our group through molecular electronic transport (Khatua et al., 2008), hydrogen storage (Havenith et al., 2005; Giri et al., 2011a; Giri et al., 2011b; Das and Chattaraj, 2012; Srinivasu et al., 2012; Pan et al., 2012b), and Zn–Zn and Be–Be bond stabilization (Chattaraj et al., 2008; Roy and Chattaraj, 2008) in the domain of CDFT. The ΔX and NICS parameters show their versatility in quantifying the aromaticity of not just planar and cyclic systems, but also any other nonplanar closed structure. They are both easily computable and ΔX also has a conceptual lucidity since it originates from the electronic structure principles of CDFT.

Certain chemical entities contain loosely bound electrons not directly connected to any atom(s) within the cluster, but trapped within a hollow space (cavity of cage compounds or packing void in crystals) that act as anions. Such entities are known as electrides and they are known for their nonlinear optical (NLO) properties. Besides showing NLO properties, which is considered to be an identifiable character of an electride, it is also widely applicable in electron emission, catalysis, reversible hydrogen storage, super conductivity, etc. (Toda et al., 2007; Xu et al., 2007; Kitano et al., 2012). Organic electrides of crown ethers and cryptands (Ellaboudy et al., 1983; Ward et al., 1988; Dawes et al., 1991; Xie et al., 2000), and inorganic electrides such as [Ca₂₄Al₂₈O₆₄]⁴⁺(4e ⁻) (Matsuishi et al., 2003), Y₅Si₃ (Lu et al., 2016), [Ba₂N₂](e ⁻), [Li₂Ca₃N₆](2e ⁻) (Qu et al., 2019), [Ca₂N]⁺(e ⁻) (Lee et al., 2013), [Y₂C]^1.8+·1.8e ⁻ (Zhang et al., 2014), Sr₅P₃ (Wang et al., 2017), and Yb₅Sb₃ (Lu et al., 2018) are well reported in the literature. A different class of electrides, known as molecular electrides, are basically guest@host complexes containing a significant amount of localized electron cloud within the void of the host. Cavity-containing molecular structures such as decaborane (Muhammad et al., 2009; Muhammad et al., 2011), pyrrole (Chen et al., 2005), tetracyanoquinodimethane (TCNQ) (Li et al., 2009), fullerene cages (Das et al., 2020), C₂₀F₂₀ (Wang et al., 2012), C₆₀F₆₀, extended (3.1.3.1) porphyrin (EP) (Saha and Chattaraj, 2018), and many more are utilized for this purpose. The guest atoms are usually alkali and alkaline earth metals. For a guest@host complex to be characterized as a molecular or cluster electride, certain criteria need to be fulfilled such as the presence of a non-nuclear attractor/maximum (NNA/NNM) (a non-nuclear critical point with a local maximum of electron density), a negative Laplacian of electron density [∇² ρ(r _c )], presence of an ELF basin near the NNM, high NLO properties, and a green region in the NCI plot showing accumulation of electron density.

The Mg₂EP complex studied at the M06-2X-D3/6-311G(d,p) level of theory (Saha and Chattaraj, 2018) shows the presence of NNA in between the two Mg atoms where the value of ∇² ρ(r _c ) < 0, with an ELF basin nearby. The electron population at said NNA is 1.02 e with 46% localization. The NLO properties in terms of α ¯ , β, and γ ‖ are calculated and compared with other electride systems that show that α ¯ is higher while β and γ ‖ are lower in the studied systems. The donation of the loosely trapped electron to antibonding MOs of certain bonds in small molecules (H-H in H₂, C-O in CO₂, N-O in N₂O, and C-H in CH₄ and C₆H₆) results in the activation followed by the dissociation of the respective bonds (Saha and Chattaraj, 2018). Another study (Saha et al., 2019) performed by our group utilizes the modified form of β-diketiminate ligand (^DippNacnac) to hold four Mg atoms with two equivalent Mg(I)-Mg(I) bonds ([Mg₄( L H Dipp )₂]^2–) in its lower energy singlet state. Two NNAs are found to be present at the center of each Mg(I)-Mg(I) bond with an electron population of 1.18 |e| and 52% localization. The calculated values of α ¯ , β, and γ ‖ are 891.7, 0.0, and 8.9 × 10⁵ a.u., respectively, which clearly indicate the system to be classified as an electride. This system is further stabilized by sandwiching it between two K@crown-6-ether⁺ [K@CE]⁺ counter cations. Again, the C₆₀ cage trapping a magnesium dimer (Mg₂@C₆₀) (Das et al., 2020) and a lithium trimer (Li₃@C₆₀) (Das and Chattaraj, 2021a) act as electrides as indicated by the presence of NNA at the center of the Mg-Mg bond path and Li₃ cluster in the respective complexes. The Li₃ cluster encapsulated within a B₄₀ cage (Li₃@B₄₀) (Das and Chattaraj, 2020) shows a similar behavior with a lower electron population at the corresponding NNA owing to the electron deficiency in the B cage atoms. Binuclear sandwich complexes formed with Be and Mg dimers with C₅H₅ ⁻, N₅ ⁻, P₅ ⁻, and As₅ ⁻ ligands forming M₂(η ⁵-L)₂ complexes (Das and Chattaraj, 2020) contain NNAs at the center of the M-M bonds with population varying between 0.95 and 1.39 and percentage localization ranging within 43–62%. Discernible substituent effects on electride characterizers have also been reported (Das and Chattaraj, 2021b).

The encapsulation of noble gas (Ng) atoms in B₄₀ cavitand along with their structure and interactions of Ng with the Ng and B atoms in the host–guest complex is discussed with the help of DFT-based computations (Pan et al., 2018). Dissociation energy (∆E _diss) and Gibbs free energy change (∆G _diss) are calculated to study the stability of the encapsulated complexes. NBO, EDA, and NOCV calculations have been done for studying the nature of bonding. The optimized structure of the B₄₀ cage along with the Ng@B₄₀ and Ng₂@B₄₀ systems (at ωB97X-D/def2-TZVP level) are depicted in Figure 5. The size of the B₄₀ cage is suitable to accommodate He and Ne atoms at its center, whereas for heavier atom (Ar-Rn) encapsulation, the cavity diameter expands with the help of a certain amount of energy (preparation energy, ΔE _prep). Although these complexes are thermochemically unstable, they remain in their encapsulated form on account of their high kinetic barrier. Despite that, the lighter Ng encapsulated complexes have very low ∆E _diss (−1.8 kcal mol⁻¹ for He and −7.1 kcal mol⁻¹ for Ne) as compared with the experimentally identified He@C₂₀H₂₀ complex (−33.8 kcal mol⁻¹). For the heavier Ng atom-encapsulated Ng@B₄₀ systems (Ng = Kr-Rn), the ∆E _diss increases with the size of Ng. The possibility of releasing Ng atoms in the dissociation process is either through B₇ or B₆ holes for the lighter He-Ar atoms, whereas the heavier ones can only escape through the B₇ holes due to their larger size. Decapsulation through the B₇ hole has ΔG ^≠ values ranging within 84.7–206.3 kcal mol⁻¹. The rate constant (k) calculated at 298 K for the dissociation through either the B₇ or the B₆ hole comes out to be pretty low suggesting that all the Ng@B₄₀ systems are kinetically stable.

In the case of two Ng atoms encapsulation, the inter atomic distance decreases from that of their free state. For Ar₂@B₄₀ and Kr₂@B₄₀, larger repulsion results in the exergonic dissociation of Ng₂@B₄₀ into Ng and Ng@B₄₀. Along with this, large ∆E _prep indicates the nonviability of Ar₂B₄₀ and Kr₂B₄₀. For even heavier Ng atoms (Xe and Rn), no minimum energy structures are obtained for the corresponding dimer encapsulation. The corresponding transition states of the Ng release process of Ng₂@B₄₀ (Ng = He-Ar) suggest that one Ng atom approaches and leaves through the B₇ rings, while the other remains near the center of the cavity. The associated ΔG ^≠ values for the He and Ne dimer encapsulated complexes are high enough for them to be kinetically viable.

The B₄₀ cage exhibits a fluxional behavior due to the continuous interconversion between the B₆ and B₇ rings caused by the transfer of one B center from B₇ to B₆. The encapsulated system, Ng@B₄₀, also shows similar dynamic behavior as that of the bare B₄₀. The Ng atom inside the cage does not have any substantial influence on the fluxionality of the cage which is reflected in the free energy barrier values (16.4 kcal mol⁻¹ for the bare cage, and a range of 16–18.9 kcal mol⁻¹ for Ng@B₄₀). Topological analysis of Ng@B₄₀ explains the nature of the interactions therein. ∇² ρ(r_c) and H(r_c) values being positive at the BCPs of Ng-B bond suggest the presence of noncovalent character in most of the complexes except for Rn@B₄₀, Ar₂B₄₀, and Kr₂B₄₀, where H(r_c) < 0. For the encapsulation of Ng₂ (Ng = Ar₂ & Kr₂) the Ng-Ng bond becomes partially covalent in contrast to their noncovalent character in the free state, except in the cases of He₂@B₄₀ and Ne₂@B₄₀ where no covalency is imparted. From NBO analysis it has been shown that Ng → B₄₀ charge transfer increases with increasing the size of the Ng atoms. The electron transfer further increases in the case of Ng₂ encapsulation. Along He–Rn, an increase in WBI values indicates that the increasing size of Ng atoms increases the degree of covalency between the Ng and B centers. EDA analysis reveals a high positive ∆E _pauli which leads to positive ∆E _int suggesting the interaction to be repulsive in case of the heavier noble gas encapsulated B₄₀ systems. Also, both the attractive terms, ∆E _elstat and ∆E _orb, increase with the increasing size of Ng atoms.

Small gas molecules such as C₂H₂, C₂H₄, C₂H₆, CO, CO₂, NO₂, NO, N₂, H₂, and Ng atoms (He _n -Xe _n , n = 1,2) are selected as guest molecules encapsulated in OA cavitand (Figure 6) and analyzed via DFT approach (Chakraborty et al., 2016). The systems under study are optimized at the ωB97X-D/6-311G(d,p) level of theory (LanL2DZ basis set with ECP is used for Xe). There are two possible cavities for the accommodation of the guest atoms, the inner cavity of OA is more suitable as it increases the host–guest interaction. In the case of Ng@OA, due to encapsulation of Ng atoms, no notable distortion is observed in the OA. For Ng₂@OA systems, one guest atom can occupy the center of the host cavity, whereas the other one remains in the outer cavity. The Ng–Ng bond distances inside OA are 3.6, 3.7, 3.8, 3.9, and 4.1 Ǻ, respectively, for He₂, Ne₂, Ar₂, Kr₂, and Xe₂. In the cases of CO@OA and NO@OA, the guest molecules prefer to stay in the inner cavity of OA. Their orientation with respect to the two nearest benzene-like fragments is almost perpendicular, whereas it is parallel to the rest of the benzene fragments of OA. The presence of these π electron clouds close to the encapsulated guests is expected to have a significant impact on the stability of the complexes, the nature of interaction, and dynamical behavior as well. For N₂@OA and H₂@OA, N₂ and H₂ remain well inside the inner cavity. Thermochemical study reveals that except for He@OA, in all cases, D ₀ value is positive indicating the stability of the host–guest complexes concerning their dissociation into the corresponding individual components. Going from lighter to heavier Ng atoms (also for Ng₂@OA), D ₀ value increases, i.e., the host–guest interaction increases. This could most likely be due to the increasing polarizability of the Ng atoms down the group. The dissociation channels for all the encapsulated complexes have positive D ₀ values. Most of them, however, dissociate spontaneously at room temperature except in the cases of Kr, Kr₂, Xe, C₂H₂, C₂H₄, C₂H₆, and N₂ guest molecules. Thus, the encapsulation of the mentioned guest molecules is favorable at 298 K. In the cases of NO/NO₂@OA and CO/CO₂@OA, an increase in D ₀ value is observed from NO to NO₂ and from CO to CO₂. It can thus be deduced that the encapsulation of NO₂ and CO₂ inside OA forms more stable complexes compared with NO and CO, respectively. Having said that, it is to be noted that all the four complexes have favorable dissociation channels at room temperature. The hydrocarbons have better interaction with the OA and hence are not prone to dissociation at ambient temperatures.

The interaction between the host and guest moieties is analyzed with the help of NBO, NCI, and EDA. All the Ng atoms acquire some positive charges, i.e., transfer of electron density occurs from the Ng atoms to the host OA surface. The donation primarily occurs from the lone pair (LP) of Ng to the C-H antibonding orbital of OA for all the Ng atoms [except Ne where it takes place from LP to antibonding Rydberg state (Ry*)] and for CO, NO, N₂, and CO₂ molecules. WBI values for Ng–Ng and Ng–OA interactions indicate a purely noncovalent character therein. From this discussion it is clear that the confinement brings about an increase in the reactivity of all the guest atoms/molecules within the OA. NCI isosurfaces show the presence of green surface around the guest molecules indicating van der Waals interaction that stabilizes the host–guest complexes. EDA results show that there exists closed shell type of interaction between guest Ng/H₂ and OA. The ΔE _disp and ΔE _orb are the largest and the smallest contributors toward the total attractive interaction, respectively. The former increases in magnitude with increasing size of the encapsulated Ng atoms. For the guest hydrocarbons, the contribution from ∆E _disp increases and ΔE _elstat decreases with the increasing number of H atoms. This is because the molecules such as C₂H₂ and C₂H₄ containing labile electron cloud can accumulate enough positive charge to favorably interact with the electron-rich fragments of the OA. This makes the contribution from ∆E _elstat very important in stabilizing these complexes. The contribution from charge transfer and polarization are very less toward ∆E _tot as indicated by the very low values of ΔE _orb. A similar type of situation is observed for CO/CO₂@OA complexes. For nitrogen-containing guests, the ∆E _orb outweighs the ∆E _elstat contribution toward ΔE _tot. Since the main stabilizing factor for all the complexes is ∆E _disp, the nitrogen-containing guest molecules are prone to be affected by the polarization or charge transfer by OA, in comparison with the other guest molecules. ADMP simulation performed at 298 K shows that all the Ng atoms (except Ne) are prone to leaving the cavity, whereas at 50 K they remain within the OA. Polar molecules such as CO, CO₂, NO, and NO₂ have a higher tendency to stay within the OA than the nonpolar H₂. Again, C₂H₂, C₂H₄, and N₂ containing π electron cloud also prefer to stay inside OA. From the above thermochemical, kinetic, and dynamical analyses, it can be said that OA makes a reasonably good choice for accommodating gas molecules.

Cucurbiturils are methylene-linked macrocyclic molecules having glycoluril unit [=C₄H₂N₄O₂=] as a building block. This repeating glycoluril unit can bind with hydrogen with sufficient amount of binding energy. Thus CB[n] can be designed as an effective hydrogen storage compound. The nitrogen and the oxygen centers are found to be the most active centers to bind with hydrogen with positive binding energy. It was found that (CH₃)₂C₄H₂N₄O₂(CH₃)₂ unit can interact with total 13H₂ atoms (Pan et al., 2013a). Since hydrogen has an electric quadrupole moment, a charge–quadrupole interaction plays a pivotal role in binding the hydrogen with the host. NPA charge analysis reveals that the charge transfer occurs from the N and O centers to the σ * orbital of H 2 molecule.

Among the CB[n] family, CB[7] can accommodate five H₂ molecules endohedrally. The O centers can adsorb a total of 28 H 2 molecules (two per O atom) and 19 H₂ molecules get adsorbed at the N centers exohedrally, making it a total of 52 hydrogen molecules. The binding energy and adsorption enthalpy are positive and negative, respectively, indicating CB[7] to be a potentially promising H₂ storage material. The gravimetric wt% of hydrogen for the CB[7] adsorbing 52 H₂ molecules comes out to be 8.3 with an average binding energy of 7.8 kJ mol⁻¹. These values are very encouraging when compared with various other potential hydrogen storage materials such as α-cyclodextrin (Zhu et al., 2010), COFs (Klontzas et al., 2008; Li et al., 2010), MOFs (Bhatia and Myers, 2006; Vitillo et al., 2008), Li-doped nanotubes (Wu et al., 2008), and polyacetylenes (Li and Jena, 2008), with average binding energies ranging within 5–8 kJ mol⁻¹.

The endohedral adsorption of gas molecules such as C₂H₂, C₂H₄, C₂H₆, CH₄, CO, CO₂, NO₂, NO, N₂, H₂, F₂, Cl₂, H₂S, and SO₂ into the CB[7] cavitand is depicted in Figure 7. Geometry optimizations are performed at the ωB97X-D/6-31G(d,p) and ωB97XD/6-311+G(d,p) levels (Pan et al., 2017). Both the CB[7] cage and the encapsulated gas molecules remain unaffected by the encapsulation. For SO₂ encapsulation, the binding enthalpy shows the highest value (14.3 kcal mol⁻¹) followed by Cl₂ and C₂H₂ (11.6 and 10.4 kcal mol⁻¹, respectively). CB[7] also encapsulates C₂H₄ and C₂H₆ more favorably than CO₂, NO₂, and H₂S. The binding enthalpies for NO/F₂/N₂/CO/CH₄@CB[7] systems vary from 4.7 to 5.8 kcal mol⁻¹, the highest value corresponding to CH₄. From the enthalpy values it is clear that CB[7] can selectively adsorb SO₂ among various gas molecules, and hence can be applicable in the SO₂ separation process from gas mixtures. ∆G value suggests that C₂H₆ is less prone to be encapsulated than CO₂ inside CB[7]. SO₂, Cl₂, and C₂H₂ adsorb with negative ΔG values, whereas those of C₂H₄ and CO₂ are slightly endergonic (0.6–0.7 kcal mol⁻¹). The corresponding ΔG values for the adsorption of C₂H₆, N₂, F₂, NO₂, NO, and H₂S vary within 1.3–2.8 kcal mol⁻¹ at 298 K temperature. EDA results show that for all the discussed complexes, ∆E _disp contributes more toward the stabilization of the host–guest systems. ∆E _elstat term also plays an important role here. In the hydrocarbons, as the number of H atoms increases, ∆E _elstat decreases gradually due to the reduction in the acidic character of the H atoms. Higher ∆E _elstat and lower ∆E _pauli values in the SO₂ encapsulated complex make the interaction between the host and the guest stronger than for C₂H₄ and C₂H₆ analogues. This again validates the higher SO₂ selectivity of CB[7].

Cucurbit[6]uril, among the CB[n] family, is found to be a compatible host for the encapsulation of Ngs within its cavity (Pan et al., 2015). The optimized geometries at the ωB97X-D/6-311G(2d,p) level of theory of the Ng _n @CB[6] complexes are provided in Figure 8. CB[6] effectively accommodates three Ne atoms, but can only trap two of the large Ar and Kr atoms. No significant distortion in the cage is observed for trapping all the three Ne atoms or for the first atom of Ar and Kr, whereas inserting a second atom deforms the shape of the host. The Ng dissociation process becomes more endothermic as we move from Ne to Kr. At 298 K, the dissociations of all Ng _n @CB[6] are exergonic except Kr@CB[6]. At 77 K, apart from the second Ng (Ar and Kr) atom dissociation from Ng₂ encapsulated CB[6], all dissociations become endergonic. Kr encapsulation at 298 K and 1 atm pressure is thermochemically favorable, whereas for Ne and Ar encapsulation, high pressure and moderately low temperature are preferred.

NPA charge analysis reveals N and O to be negatively charged in Ng _n @CB[6], while C and H have positive charges. Slight charge transfer (∼0.01 e ^- ) occurs from Ng→CB[6] moiety. Small ρ(r_c) value and positive ∇² ρ(r_c) and H(r_c) values from topological analysis suggest the interaction to be of closed shell type. ELF analysis shows an absence of electron localization between the Ng-Ng and Ng-cage atoms, corroborating the result obtained from AIM. EDA analysis reveals the contribution from ∆E _disp to be the largest, followed by ∆E _elstat, and the smallest contribution is from ∆E _orb, all of which gradually increases going from Ne to Kr. The green surfaces observed between the Ng and CB[6] units in the NCI isosurface (Figure 9) are an indication of a small van der Waals interaction, which increases with the size of the Ng atoms. The dynamical study (ab initio MD) for 1 ps and at 298 K reveals that Ne and Ar remain inside the cavity, whereas Kr and all the Ng₂ in Ng₂@CB[6] move toward the open end but do not leave the cage. At 77 K, all guests stay inside the host.

The activation of small molecules by a metal-supported boron cluster is studied through DFT calculations (Saha et al., 2017). They are known to be applicable in nanomaterial building blocks, automobiles (Norbye, 1971; Jiménez‐Halla et al., 2010) etc. The global minimum energy structures calculated at PBE/def2-TZVPPD (Perdew et al., 1996; Weigend and Ahlrichs, 2005) level of MB₁₂ ^-, CO@MB₁₂ ^-, N₂@MB₁₂ ^- clusters and their corresponding TSs for the internal rotation of the B₃ ring are provided in Figure 10. The coordination of the small molecules with the MB₁₂ ^- cluster forms an umbrella-shaped structure in which the M-L bonds act like the stick of the umbrella. The coordination of CO with the metal center can take place through both the C and O ends, the former producing a more stable isomer. For both OCMB₁₂ ^_ and NNMB₁₂ ^_, the Ir-L bond has the highest strength, followed by Co and Rh, while for a particular M center, CO forms a stronger bond than N₂. ∆G values of these complexes are highly positive which suggest that the corresponding complexes are thermodynamically stable concerning the dissociation process. The O-side bound isomers, however, have low positive ∆G for Co and Ir, and become slightly negative for Rh. They can be made viable by lowering the temperature. The N−N and C–O bonds get lengthened due to complexation in the order COMB₁₂ ^- < N₂MB₁₂ ^- < OCMB₁₂ ^- causing a red shift in their bond stretching frequencies which is the highest for the Ir analogues.

NBO analysis reveals that upon complexation, the metal centers get more negatively charged, apart from COIrB₁₂ ^_ and NNIrB₁₂ ^_, where Ir still contains positive charge (less than that in IrB₁₂ ⁻). L→M and M→L back transfers take place, and in certain complexes the latter completely compensates (or overcompensates) the former which is indicated by the zero (or negative) charge on the ligand. The Wiberg bond indices suggest that the covalent character follows the order M-C in OCMB₁₂ ^- > M-N in N₂MB₁₂ ^- > M-C in COMB₁₂ ^-. From EDA-NOCV, it is seen that the bonding between the metal and the ligand is predominantly orbital and electrostatic interactions (in more or less equal contributions), indicating the L-M bonds to have both covalent and ionic characters. For OCMB₁₂ ^-, however, the contribution from ∆E _elstat is higher than ∆E _orb. Figure 11 shows the deformation densities [Δρ(r)] for the pairwise orbital interactions for the LMB₁₂ ^_ complexes, where a shift in the electron density occurs from the red to the blue region. The Δρ( σ 1 ) plot reveals that the shift of electron density occurs through the L→M→B scheme. Δρ(π₁) and Δρ(π₂) explain the π electron density shift from the d _L→M. The extent of L←M π-back-donation is greater than the L→M σ-donation, causing a red-shift in its stretching frequency. The σ-donation occurs from the HOMO_(CO) to the LUM O ( LM B 12 − ) fragment and π-back-donations occur from the degenerate HOM O ( M B 12 − ) to the degenerate π* LUMO_(CO).

An internal rotation of the inner B₃ ring with respect to the outer B₉ ring occurs within the MB₁₂ ^- cluster. The energy barrier associated with this rotation is reported (Popov et al., 2014; Liu et al., 2016) to follow the order Co > Rh > Ir. BOMD simulation at 800 K shows the L-M bonds to be intact during the rotation. This makes the complex seem like a spinning umbrella with the L-M bond as the stick.

Clathrate hydrates, a class of inclusion compounds, are known to encapsulate guest compounds within its hydrogen bonded polyhedral cage (Mao et al., 2002; Mao and Mao, 2004; Lee et al., 2010). They constitute a very effective host for hydrogen storage. Four types of clathrate hydrates and their maximum possible hydrogen-encapsulated complexes studied at B3LYP/6-31G(d) level (Chattaraj et al., 2011) are depicted in Figure 12. 5¹² represents the cavity having 12 pentagonal faces, whereas 5¹²6 ^k (k = 2,4,8) represents 12 pentagonal faces along with k hexagonal faces. Here we discuss the structure, bonding, and stability of the bare and hydrogen-encapsulated complexes from a density functional theory perspective. For nH₂@5¹² complexes it is seen that for the first H₂ encapsulation, the process is energetically favorable although the overall nH₂ encapsulation is method dependent. Owing to the small size of 5¹² cavity, it can accommodate a maximum of five H₂ molecules, after which a deformation in the cavity is observed. The GM for H₂ confinement in the 5¹² cavity occurs endohedrally. The H₂ molecules favor the inside of 5¹² more than the outside. In the case of 5¹²6² cage, it can also take up a maximum of five H₂ molecules. Slight distortion is observed in the system that becomes more noticeable during the third H₂ encapsulation which slowly decreases for the fourth and the fifth hydrogen molecule encapsulation. This is reflected in the slightly conflicting trend in the corresponding interaction energies. Now in the case of 5¹²6⁴ clathrate, obtaining the minimum energy structure was difficult. It is fascinating to note that the encapsulation of one H₂ into the cage stabilizes the structure although it could not provide with the minimum energy structure. Further incorporation of guest molecules deforms the structure of the system. For 5¹²6⁸, the interaction energy for all the six H₂ encapsulation is negative making the process favorable. The large size of the host cavity makes it feasible to accommodate all the six guest molecules efficiently. Positive ∆G value suggests that the complexes are kinetically stable. Finally, it can be concluded that the 5¹² and 5¹²6² clathrates can encapsulate up to two hydrogen molecules without undergoing any structural distortions, whereas the 5¹²6⁸ clathrate may entrap up to six H₂ molecules depending upon the level of theory used. Calculation of CDFT-based reactivity descriptors of the complexes with and without H₂ encapsulation suggests that for most of the systems, stability increases with the increase in number of trapped hydrogen molecules. This is concluded from the increasing hardness and decreasing electrophilicity values.

Li ion is popularly known to bind well with hydrogen molecule owing to its positive charge (Pan et al., 2012a). Inspired by this, a number of efforts have been made to effectively polarize the Li center of various clusters to increase its hydrogen adsorbing ability. Here we study the H₂ storage potential of the Li-doped clusters, M₅Li₇ ⁺ (M = C, Si, Ge), M₄Li₄ (M = Si, Ge) at the M06/6-311+G(d,p) level, and some super-alkali ions at the M052X/6-311+G(d) level (Figures 13, 14). The Li centers attain a net positive charge due to the high polarizability of the clusters, facilitating electrostatic interactions to bind with the H₂ molecules. The negative values of interaction energies and enthalpies indicate the efficacy of these clusters to be good H₂ storage materials. The gravimetric wt% of adsorbed H₂ are 28.0, 18.3, 9.3, 14.7, and 7.1 for C₂Li₇ ⁺, Si₅Li₇ ⁺, Ge₅Li₇ ⁺, Si₄Li₄, and Ge₄Li₄, respectively. For the super-alkali ions, the values range from 13.2 to 40.9%, with the highest being that for BLi₆ ⁺. On applying electric field, a gradual improvement is observed in the interaction energy value. Thus, in terms of gravimetric wt%, BLi₆ ⁺ is preferable whereas the interaction energy per H₂ molecule suggests B₂Li₁₁ ⁺ to be the preferred choice for hydrogen storage.

The influence of encapsulation on the hydrogen bond strength in (HF)₂ within the fullerene cages is studied using DFT and ab initio MD (Khatua et al., 2014b). The optimized geometries of (HF)₂@C _n (n = 60, 70, 80, 90) complexes wB97X-D/6-31G are depicted in Figure 15. The dissociation energy, enthalpy, and change in free energy are negative for the (HF)₂@C₆₀ system which indicates that the encapsulation process is thermodynamically unfavorable, whereas positive values for the rest of the HF encapsulated C_n cages imply them to be favorable (highest being for the C₈₀ cage). Owing to the smaller size of the C₆₀, the HF units orient themselves antiparallelly to reduce repulsion at the cost of hydrogen bond strength. Thus, the energy associated with the HF-HF interaction is observed to be highest in the C₆₀ cage (positive, and hence repulsive in nature). For all the studied cases, upon encapsulation, the hydrogen bond distance reduces from that in the free state, the least being inside the C₇₀ cage. The EDA study reveals that the contribution from ∆E _pauli increases and the ∆E _int value decreases with decreasing the C _n cage cavity except for C₈₀ cage. For the C₆₀ cage, a very large value of ∆E _pauli makes the overall ∆E _int value positive. On account of the smaller H-bond distance within the C₇₀ and C₉₀ cages compared to the same within the C₈₀ cavity, both the ∆E _elstat and ∆E _orb contribute more to the attractive interaction than those in C₈₀. AIM analysis reveals that for all these confined systems, ∇² ρ(r_c) > 0 and H(r_c) < 0 implying the partial covalent nature of the hydrogen bonds. The hydrogen bond is mostly covalent in case of (HF)₂@C₇₀ ELF analysis that also supports this observation.