Revisiting The Trapping of Noble Gases (He – Kr) By The Triatomic H3+ and Li3+ Species: A Density Functional Reactivity Theory Study

Small atomic clusters with exotic stability, bonding, aromaticity and reactivity properties can be made use of for various purposes. In this work, we revisit the trapping of noble gas atoms (He – Kr) by the triatomic H 3+ and Li 3+ species by using some analytical tools from density functional theory, conceptual density functional theory, and the information - theoretic approach. Our results showcase that though similar in geometry, H 3+ and Li 3+ exhibit markedly different behaviour in bonding, aromaticity, and reactivity properties after the addition of noble gas atoms. Moreover, the exchange - correlation interaction and steric effect are key energy components in stablizing the clusters. This study also finds that the origin of the molecular stability of these species is due to the spatial delocalization of the electron density distribution. Our work provides an additional arsenal towards better understanding of small atomic clusters capturing noble gases.


Introduction
During the past few decades, atomic clusters have been the subject of intensive research due to their indispensable roles in biomedicine, optics and synthetic materials. [1][2][3] Since Bartlett first synthesized Xe + [PtF6] -, [4] the underexplored chemistry of noble gases has witnessed rapid developments, motivating both synthetic and theoretical chemists jumping into this field. [5] It has found that noble gas insertion compounds play an important role because of their unique properties in geometrical structure, molecular stability, bonding, chemical reactivity, and other physicochemical properties.
In continuation with our previous work, [6] we intend to revisit the trapping of noble gases by H3 + and Li3 + species from the perspective of stability, bonding, aromaticity, and reactivity properties using the framework of conceptual density functional theory (CDFT) augmented by the information-theoretic approach (ITA).
Also employed are well-established analytical tools, such as aromaticity descriptors NICS (nucleus independent chemical shift) and GIMIC (gauge-including magnetically induced currents), "spike" region for noncovalent interactions and our newly developed analytical tools in analyzing strong covalent interactions and reactivity patterns in terms of local temperatures, that are intrinsically different from their thermodynamic definitions. Combining these effective methods allows us to gain insights into these species and to obtain many unexpected properties of these small atomic clusters.

Methodology
In Kohn-Shan DFT, we can decompose the total energy difference (Δ ) into its components as [7,8] and where Ts, Ee, and Exc are the noninteracting kinetic, electrostatic, and exchange−correlation energies, respectively. The electrostatic energy Ee includes three independent components: the nuclear−electron attraction, Vne; the classical interelectron Coulombic repulsion, J; and the nuclear−nuclear repulsion, Vnn. The last term Exc consists of exchange (Ex) and correlation (Ec) components. Es stands for the energetic contribution from the steric effect, and Eq signifies the contribution originating from Fermionic quantum effect (due to the exchange−correlation interaction). The steric effect Es has been shown to be simply the Weizsäcker kinetic energy as will be shown later in Eq. (8). The definition of Eq is by simply combining Eqs. (1), (2) and (8), which reads This new formulation has its own distinct physical meaning with a corresponding physical state. [8] It has been applied to a number of molecular systems and phenomena, such as conformational changes, [8][9][10][11][12][13] anomeric effect, [14][15][16] the cis-effect, [17] , etc. The results are consistent with our chemical intuition and conventional wisdom.
In CDFT, [18][19][20] where τ(r), ρ(r), and kB are the KED, electron density, and Boltzmann constant, respectively. Among various KEDs, two well-appreciated definitions are the Hamiltonian KED with φi(r) as the occupied Kohn−Sham orbitals, and the Lagrangian KED where ∇ 2 ( ) is the Laplacian of the electron density. In addition, in the literature [24][25][26][27][28][29] there exist two famous approximate forms of KEDs: the Thomas−Fermi formula (derived for the homogeneous electron gas) and the Weizsäcker KED (exact for one-electron and two-electron Hartree-Fock systems), as defined by Eqs. (7) and (8), respectively: and for electrophilic attack, we define where TN, TN+1, and TN-1 are local temperatures for the system with N, N + 1, and N -1 electrons, respectively, in the N electron (fixed) geometry. Condensed-to-atom local temperatures were also defined. An in-depth discussion and applications of local temperatures can be found in our recent publication.
[23] The uniqueness of the descriptor of local temperature is manifold, compared to its predecessor Fukui function. The scope of applications is greatly expanded due to its connection to the information theory. Moreover, the local temperature can be used to explore bonding can be used to determine multiple bond orders (up to quintuple bond). Suffice to note that CDFT is still an active field with lots of progresses keep emerging.
Next, we will give a brief introduction to ELF and SCI. Earlier, we assumed that with the Weizsäcker kinetic energy TW as defined in Eq. (12), TS as the total noninteracting kinetic energy, and tS(r) and tP(r) the corresponding local energy density. To convert the Pauli energy to a dimensionless quantity, we define a local function (r), [34] ( ) ≡ P ( ) and then the SCI index is defined as the reciprocal of (r),[34] This index is similar to the ELF (electron localization function) index,[32,33] We have to mention that the SCI index has its own unique features compared to that

Results and Discussion
We The observation is in line with our previous work of "bare" H3 + and Li3 + . Furthermore, in Figures 1c and 1f, no signature of SCI isosurface for double or triple or other bonds is observed, indicating that there is no obvious strong covalent bond.
Next, we will delve into the aromaticity of the H3 + and Li3 + clusters and the corresponding trapped noble gas clusters. It is well-documented in the literature that a negative NICS value indicates the existence of aromaticity. We clearly know that H3 + and Li3 + structures are aromatic, though Li3 + is not σ-aromatic in nature. [56,57]. Does the addition of noble gases change its aromaticity? Collected in Table 1  for H3 + , He3H3 + , Ne3H3 + , Ar3H3 + , and Kr3H3 + , respectively. The similar trend is observed for Li3 + and its trapped noble gas clusters. In addition, we employ the GIMIC diagrams as shown in Figure 2 to determine aromaticity/anti-aromaticity. It is lucidly shown that all currents run in a counterclockwise manner, which is indicative of aromaticity.
It is worthwhile to note that for H3 + and its trapped noble gas clusters, sparsity of current distributions around the H3 + local motif is in line with the decreased NICS values as previously mentioned. However, this is not the case with Li3 + . One possible reason is that H3 + is assembled together because of electron delocalization but for Li3 + it is weak van der Waals interactions that counts.
[56] In a nutshell, addition of noble gas atoms to H3 + and Li3 + leads to different aromaticity properties.
We employ the local CDFT descriptors, the Fukui function and the local temperature, to unveil the possible electrophilic and nucleophilic sites of a molecular system. Shown in Figure 3 are the Fukui function and local temperature results for HeH3 + and HeLi3 + . One can see that the overall trend for the Fukui function and the local temperature is similar. For HeHe3 + (see Figure 3A), the electrophilic (Θ -/f -) and nucleophilic (Θ + /f + ) sites are located at the He nucleus and the H3 motif, respectively.
However, the trend is reversed for electrophilic (Θ -/f -) attacks of HeLi3 + . Moreover, the electrophilic local temperature is more localized than the Fukui function for HeH3 + as shown in Figure 3B (upper panel). This indicates that the local temperature is a more definitive quantity in determining the reactivity sites of a molecular system.
To summarize, though similar in geometry structures, HeH3 + and HeLi3 + can have very different chemical reactivity behavior.
We further dissect the molecular stability of all the noble gas trapping clusters by H3 + and Li3 + . A total of 12 possible chemical reactions for each of them are proposed as shown in Table 2 by gradually adding a noble gas atom. Collected in From Table 2, we find that the exchange-correlation (ΔExc) part is negative, indicative of its positive role in stabilizing the complex. The negative steric hindrance (ΔEs) is largely compensated by the positive quantum effect (ΔEq). What is the origin and nature of molecular stability? In other words, is there a single energetic component that is mostly responsible for the molecular stability of these systems?
Shown in Figure 4 are the strong correlations between the total energy difference and the exchange-correlation (ΔExc) and the steric hindrance (ΔEs), as well as ITA quantities, Shannon entropy (ΔSS)[58] and Fisher information (ΔIF) [59], whose values are given in Table 3. In a nutshell, from the viewpoint of energetics, it is the exchange-correlation that is essentially responsible for the total energy difference, which is a good supplement to our previous publications where the electrostatic potential (ΔEe) dominated. Moreover, from the perspective of information theory, spatial delocalization of the electron density gauged by the Shannon entropy is the origin of the molecular stability of these trapping clusters. This result provides us a novel insight into this complicated phenomenon of molecular stability associated with intermolecular interactions. Similar results have been reported by us for isomeric (intramolecular) stability of fullerene buckyballs. [60] Finally, we also tabulate some ITA quantities (GBP entropy SGBP[30] and information gain ΔIG [61,62]) in Table 3 and the correlation coefficient (R) between the total energy and its components with ITA quantities as shown in Table 4. It is known that ITA and energetic quantities can be strongly intercorrelated. [63][64][65] These different but strongly correlated relationships provide effective measurements about the electron density distribution of the systems; thus, they attribute useful and novel insights into the nature and origin of various physicochemical phenomena, including isomeric stability of fullerenes and trapping noble gas clusters as in this work.

Conclusions
To summarize, in this work, we present the results of bonding, aromaticity and reactivity properties for the trapping of noble gas atoms (He-Kr) by the triatomic H3 + and Li3 + species from certain theoretical and analytical tools. Though similar in geometrical structure, the trapping noble gas clusters of H3 + and Li3 + differ much in bonding, aromaticity, and reactivity properties. The core reason lies in the structural motifs of H3 + and Li3 + . We further analyzed the origin and nature of the molecular stability in terms of energetics and information theory. We have shown that it is the exchange-correlation interaction and steric effect that are responsible for the total molecular stability. Shannon entropy results corroborate that electron delocalization is the origin of the molecular stability. We mention in passing that more future studies along this direction will help us better appreciate atomic clusters with many unconventional properties and unveil many novel and potential applications.

Declarations
Funding: SJZ and CYR acknowledge support from the science and technology Availability of data and material: All data in this work are in the text.