Noble Gas—Silicon Cations: Theoretical Insights into the Nature of the Bond

The structure, stability, and bonding situation of some exemplary noble gas-silicon cations were investigated at the MP2/aVTZ level of theory. The explored species include the mono-coordinated NgSiX3+ (Ng = He-Rn; X = H, F, Cl) and NgSiF22+ (Ng = He-Rn), the di-coordinated Ar2SiX3+ (X = H, F, Cl), and the “inserted” FNgSiF2+ (Ng = Kr, Xe, Rn). The bonding analysis was accomplished by the method that we recently proposed to assay the bonding situation of noblegas compounds. The Ng-Si bonds are generally tight and feature a partial contribution of covalency. In the NgSiX3+, the degree of the Ng-Si interaction mirrors the trends of two factors, namely the polarizability of Ng that increases when going from Ng = He to Ng = Rn, and the Lewis acidity of SiX3+ that decreases in the order SiF3+ > SiH3+ > SiCl3+. For the HeSiX3+, it was also possible to catch peculiar effects referable to the small size of He. When going from the NgSiF3+ to the NgSiF22+, the increased charge on Si promotes an appreciable increase inthe Ng-Si interaction, which becomes truly covalent for the heaviest Ng. The strength of the bond also increases when going from the NgSiF3+ to the “inserted” FNgSiF2+, likely due to the cooperative effect of the adjacent F atom. On the other hand, the ligation of a second Ar atom to ArSiX3+ (X = H, F, Cl), as to form Ar2(SiX3+), produces a weakening of the bond. Our obtained data were compared with previous findings already available in the literature.


Introduction
About 130 years after the discovery of argon [1], the chemistry of the noble gases seems to be a fascinating "saga" [2] wherein combative scientists never tire of challenging, and defeating, the proverbial inertness of the elements. The field currently embraces a rich synthetic chemistry of xenon and krypton [3][4][5][6][7] and countless species of helium, neon, argon, krypton, and xenon that are obtained in the gas phase [8,9], in liquid and supercritical fluids [10], in cold matrices [11], or at high pressures [12]. The binding partners include main-group or transition elements, and the bonding motifs range from the weakest non-covalent contacts to strong covalent bonds.
The capability of the noble gases, especially krypton and xenon, to combine with carbon is well established. The chemistry of synthesized xenon compounds is already rich [13,14], and numerous neutral species having Kr-C and Xe-C bonds were detected in cold matrices [11]. Over the years, interest was also extended to the interaction of noble gas (Ng) atoms with the heaviest elements of group XIV, particularly silicon. Neutral Ng-Si compounds are still experimentally elusive, even though there are theoretical predictions of species such as FXeSiF [15], FArSiF 3 [16], FKrSiF 3 [17], H n SiNgNSi (n = 1, 3; Ng = Xe, Rn) [18], and FNgSiY (Ng = Kr, Xe, Rn; Y = N, P) [19]. Experimental progress was made instead in the study of ionic species, particularly cationic. Noble gas-silicon cations are indeed of interest for several reasons. The complexes of Ng atoms with simple silylium ions, SiX 3 + , are prototypical systems to assay the activating interactions conceivably promoted by the more complex cationic silicon Lewis acids employed in catalysis [20]. The "tagging" of silicon cluster cations with Ng atoms is an effective mode to investigate the otherwise elusive structure and stability of these ionic intermediates [21], and the study of Ng-Si interactions is of prime interest to interpret the experimental findings [22]. Information about Ng-Si ionic species is also of interest in connection with the reactive ion etching of silicon by energetic Ng ions [23].
The first evidence about noble gas-silicon cations emerged in 1995, when Cipollini and one of us [24] reported the ligand-exchange reaction between Xe and protonated SiF 4 so as to form gaseous XeSiF 3 + , a stable species with a Xe-Si bond. Working under different experimental conditions, Hopkinson, Bohme, and their coworkers [25] could subsequently obtain the three congeners ArSiF 3 + , KrSiF 3 + , and XeSiF 3 + from the direct addition of Ng to SiF 3 + . They also gained evidence for the high-energy "inserted" isomers FXeSiF 2 + and FKrSiF 2 + . Some years later, Roithová and Schröder [26] prepared gaseous NeSiF 2 2+ and ArSiF 2 2+ from the reaction between Ne and Ar and the superelectrophilic dication SiF 3 2+ . The experimentally observed NgSiF 3 + were theoretically investigated by Morino, Chattaraj, and their coworkers [27] as part of their extensive ab initio study on the structure, stability, and bonding character of NgSiX 3 + (X = H, F, Cl, Br; Ng = He-Rn), Ng 2 SiH 3 + , and Ng 2 SiF 3 + . The structure and stability of the FArSiF 2 + , FKrSiF 2 + , FXeSiF 2 + , NeSiF 2 2+ , and ArSiF 2 2+ were also assayed by density functional theory (DFT) calculations performed in conjunction with experimental studies [25,26]. The bonding situation of these species remained, however, unexplored, and the congeners FNgSiF 2 + (Ng = He, Ne, Rn) and NgSiF 2 2+ (Ng = He, Kr, Xe, Rn) are still unreported. Taking into account this only limited available information even on experimentally observed species, it was the purpose of the present study to perform a comparative theoretical analysis of noble gas-silicon cations, with emphasis on the relationships between the various experimentally observed bonding motifs and the nature of Ng-Si bonds. In order to obtain strictly comparable data, we re-examined the NgSiX 3 + (Ng = He-Rn; X = H, F, Cl) and Ar 2 SiX 3 + (X = H, F, Cl) explored in the previous study [27] and extended the investigation to the still-unexplored NgSiF 2 2+ (Ng = He-Rn) and FNgSiF 2 + (Ng = Kr, Xe, Rn). The analysis was accomplished by the method that we recently proposed [28][29][30][31] to assay the bonding situation of noble gas compounds. The obtained results were also compared with previous findings reported in the literature [24][25][26][27].
The paper is organized as follows. Sections 2 and 3 give a brief account of the method of bonding analysis and the relevant computational details. Section 4 presents the obtained results, discussed family by family so as to best highlight analogies and differences between the various bonding situations. Some concluding remarks are given in Section 5.
Step 1. Ng-X contact is ascertained by analyzing the ρ(r) and locating the corresponding bond path (BP) and bond critical point (BCP) (the classical AIM analysis).
Step 2.The topological analysis of the H(r) of the whole molecule is accomplished. This typically produces various critical points (HCPs) of rank 3 and signature −3, −1, +1, or +3. The contour lines these points belong to are collected as the HCP lines.
Step 4. The HCP/STD lines are plotted as 2D or 3D graphs, and the visual inspection of these graphs allows the assignment of the bond as type A, B, or C. As discussed in our previous studies [28][29][30][31], H(r) generally partitions the atomic space in two well recognizable regions, namely an inner one of negative values, indicated H − (r), and an outer one of positive values, indicated H + (r). The boundary of these regions falls at distance R − , which is typical of each atom; at this distance, H(r = R − ) = 0. When two atoms form a chemical bond, their H − (r) and H + (r) regions combine in modes that signal the nature of the interaction. Particularly for Ng-X bonds, it is possible to recognize three major situations. In interactions of type A, the atoms overlap all of the contour lines of their H + (r) regions and part of the contour lines of their inner H − (r) regions, the bond appearing as a continuous region of negative values of H(r)and plunging in a zone of positive values. The bond is topologically signed by a (3,+1) HCP falling on the bond axis. Typical examples are covalent bonds, or donor-acceptor interactions with some degree of electron sharing. In interactions of type B, the H − (r) region of Ng, again, overlaps with the H − (r) region of the binding partner, but (i) no HCP exists on the bond axis, and (ii) the Ng-X inter-nuclear region includes a (more or less wide) region of positive H(r). Typical examples are the complexes of Ng donors with strongly electropositive Lewis acceptors. In interactions of type C, the Ng and the binding partner overlap only part of their H + (r) regions, their H − (r) regions remaining perfectly closed and separated by a (more or less wide) region of positive H(r). The bond thus appears as two clearly distinguishable H − (r) regions, separated by a region of positive values of H(r). Typical examples are noncovalent contacts of variable nature.
Step 5. The assignment of the bond as being type A, B, or C is further refined by examining the H(r) along the Ng-X BP, particularly at around the BCP. This serves to confirm the nature of interactions of type A and to distinguish the interactions of type B and C into B-loose (B l ) or B-tight (B t ) and C-loose (C l ) or C-tight (C t ). The adopted criteria are given in Table 1. Table 1. Criteria to assign the Ng-X bonds of type A, B, or C in terms of the sign of the H(r) at around the BCP.

Bond type
Ng side X side Step 6. Once designated being type A, B l /B t , or C l /C t , the Ng-X bond is assayed in terms of the contribution of covalency. This is accomplished by integrating the ρ(r) and the H(r) over the volume Ω s enclosed by the s(r) isosurface associated with the Ng-X BCP. The value of the s(r) is chosen by examining, particularly at around the BCP, the s(r) vs. sign(λ 2 ) × ρ(r) 2D plot [λ 2 is the second eigenvalue of the Hessian matrix of ρ(r), with λ 1 < λ 2 < λ 3 ]. The selected value of s(r) is the highest one that still avoids the contribution of the tails of the atomic densities. Typical values range between 0.2 and 0.5. Relevant quantities calculated over Ω s include the average value of ρ(r), ρ s (ave) and the average, maximum, and minimum values of H(r), H s (ave,max,min). Based on the values of these quantities, and on the sign of H(r) over Ω s , H(Ω s ), the bond is designated covalent (Cov), partially covalent (pCov), or noncovalent (nCov) according to the criteria listed in Table 2. Step 7. The bond is finally assigned using the notations Cov(Type), pCov[Type/H(Ω s )] or nCov(Type), for example Cov(A), pCov(B t /H −/+ ), or nCov(C l ).
Additional indices. In developing the method, we found it convenient to introduce two additional numerical indices that allow the further assay of the degree of the various interactions. Thus, for interactions of type C, we defined [28,29,37] the degree of polarization of Ng, DoP(Ng), as the dimensionless index given by the equation is the radius of the H − (r) region of Ng along the axis formed by Ng and the Ng-X BCP, and R − Ng is the radius of the H − (r) region of the free atom. The DoP(Ng) measures, in essence, the deformation of the H − (r) region of Ng arising from the interaction with X. Its positive/negative sign signals Ng atoms polarized toward/opposite to X, and its magnitude is related to the extent of the polarization. For interactions of type A, borrowing a concept introduced so far by Espinosa et al. [38], we defined [28] the bond degree (BD) as the minus ratio between the H(r) and the ρ(r) calculated at the HCP: BD = −H(HCP)/ρ(HCP). We introduce here the average BD over Ω s , BD s (ave), defined as the average over Ω s of the ratio H(r)/ρ(r). Formulated in this way, the index is applicable to any type of interaction (A, B, or C).

Computational Details
The calculations were performed at the Møller-Plesset level of theory truncated at the second order (MP2) [39], using Dunning's correlation consistent aug-cc-pVTZ [40] basis set (denoted here as aVTZ). The Xe and Rn atoms were treated with the aVTZ-PP basis set developed in conjunction with the Stuttgart/Cologne small-core, scalar-relativistic effective core potentials (ECP-28 and ECP-60, respectively) [41]. The geometry optimizations and frequencies calculations were performed with Gaussian 09 (G09) (Revision D1) [42], and the analysis of the ρ(r), the H(r), and the s(r) was accomplished with Multiwfn (version 3.8.dev) [43], using the wfx files generated with G09. ρ(r) is defined by the equation [32]: where η i is the occupation number of the natural orbital ϕ i , in turn expanded as a linear combination of the basis functions. H(r) [28,33,34] is the sum of the kinetic energy density G(r) and the potential energy density V(r): The presently employed definition [32,44] of G(r) is given by the equation: where the sum runs over all the occupied natural orbitals, ϕ i , of occupation numbers η i . V(r) is evaluated [32] from the local form of the virial theorem: The planar (2D) plots of the HCP/STD lines of H(r) (vide supra) were produced with the Multiwfn [43]. s(r) is defined by the equation [35,36]: and the integration of ρ(r) or H(r) over the volume Ω s enclosed by a given s(r) is accomplished by producing an orthogonal grid of points that encloses the isosurface and applying the formula where P(r i ) is the value of ρ(r) or H(r) at the grid point r i , and d x , d y , and d z are the grid step sizes in the x, y, and z directions, respectively. The used values are d x = d y = d z = 0.025 a 0 , and the summation is carried out on all grid points r i having RDG < s. All the calculations were performed using s = 0.3.

The MP2/aVTZ Predicted Data
The connectivities and point groups of the presently investigated ions are shown in Figure 1. Their geometries were optimized at the MP2/aVTZ level of theory, and harmonic frequencies calculations confirmed that all the located structures were true minima on the corresponding potential energy surface. The electronic dissociation energies with the loss of Ng, D e , of the NgSiX 3 + (Ng = He-Rn; X = H, F, Cl), NgSiF 2 2+ (Ng = He-Rn), and Ar 2 SiX 3 + (X = H, F, Cl) were computed as the difference between the MP2/aVTZ electronic energy of the complex and that of Ng and of the SiX 3 + /SiF 2 2+ /ArSiX 3 + cation at the geometry it takes in the complex. The basis set superposition error (BSSE) was corrected using the counterpoise (CP) method by Boys and Bernardi [45]. The bonding analysis was as well accomplished at the MP2/aVTZ level of theory. The discussed data, given in Table 3, include the Ng-Si distances, R(Ng-Si); the D e ; and the values of Ω s , N(Ω s ), ρ s (ave), H s (ave/max/min), and BD s (ave). The 2D plots of the H(r) are given in Figures 2-6. These Figures are, indeed, of major interest in discussing the bonding situation of the investigated species. They allow, in fact, the classification of the bonds as being type A, B, or C (vide supra) and the catching of subtle features of the Ng-Si interactions by examining, in particular, the shape of the H(r) at around the Ng atom (vide infra). The full Cartesian coordinates and the results of the classical AIM analysis are also available as Data S1 and Table S1, respectively, in the Supplementary Information (SI).               The good accuracy of our predicted MP2/aVTZ data is supported by the following arguments. First, as shown in Table S2 of the SI, the T 1 diagnostics [46] (the norm of the vector t 1 of the single-excitation amplitudes from a coupled-cluster calculation with the inclusion of single and double excitations, CCSD, divided by the square root of the number of correlated electrons N, T 1 = t 1 ·t 1 N )of the investigated ions resulted invariably within the threshold of 0.02 used to establish the validity of a mono-determinantal method (such as the MP2) to describe a wave function. The good performance, in particular, of the MP2/aVTZ is suggested by the comparison with the results obtained by Morino, Chattaraj, and their coworkers [27] from the study of NgSiX 3 + (X = H, F, Cl, Br; Ng = He-Rn), Ng 2 SiH 3 + , and Ng 2 SiF 3 + . They computed the geometries and stabilities of these complexes at both the MP2 and the CCSD(T) (CCSD with an estimate of connected triples) using the def2-TZVP and def2-QZVPPD basis sets. They found that, for both basis sets, the MP2 and the CCSD(T) delivered comparable results. However, the in-principle more accurate def2-QZVPPD furnished D e and R(Ng-Si) are larger and smaller, respectively, than those predicted using the def2-TZVP. Thus, taking also into account computational costs, they performed most of the calculations, including the bonding analysis, at MP2/def2-QZVPPD. According to our experience, for a given level of theory, the aVTZ generally furnishes results comparable with those obtained with def2-QZVPPD. Consistent with this expectation, we found that our MP2/aVTZ data are in very good agreement with the previous MP2/def2-QZVPPD estimates [27]. Thus, for NgSiX 3 + (Ng = He-Rn; X = H, F, Cl) and Ar 2 SiX 3 + (X = H, F), the two sets of values of the R(Ng-Si) feature a mean unsigned deviation (MUD) of 0.031 Å, and, for NgSiX 3 + (Ng = He-Rn; X = H, F, Cl), the values of D e (arriving up to ca. 45 kcal mol −1 ) feature a MUD of only 1.2 kcal mol −1 . The AIM indices of NgSiH 3 + (Ng = He-Rn) and Ar 2 SiH 3 + also unraveled quite similarly. The good accuracy of our MP2/aVTZ bonding analysis is also expected based on the extensive test calculations performed in our previous study [30], showing that this computational level furnishes results strictly similar to those obtained at the benchmark CCSD/aVTZ. The ligation of any Ng to the Si atom of the singlet ground state SiX 3 + (X = H, F, Cl) produces NgSiX 3 + mono-coordinated structures with C 3v symmetry (see Figure 1). The values of the R(Ng-Si) of NgSiH 3 + and NgSiF 3 + progressively increase when going from Ng = He to Ng = Rn, ranging between ca. 2.12 and 2.74 Å and ca. 2.06 and 2.65 Å, respectively. On the other hand, the R(Ng-Si) of NgSiCl 3 + decreases from ca. 2.99 to ca. 2.67 Å when going from Ng = He to Ng = Kr and then increases up to ca. 2.79 Å for Ng = Rn. In any case, for any X, the values of N(Ω s ), ρ s (ave), and BD s (ave) invariably increase when going from the He to the Rn congener, and H s (ave) becomes progressively more negative in the same order. The values of the D e follow the same trend, and we ascertained, in particular, positive correlations between BD s (ave) and D e , well-fitted (r 2 > 0.99) by exponential equations of the form D e = A·exp[B·BD s (ave)], with comparable A/B values of 1.898 kcal mol −1 /6.964 e hartree −1 (X = H), 1.918 kcal mol −1 /6.571 e hartree −1 (X = F), and 1.975 kcal mol −1 /6.179 e hartree −1 (X = Cl). One also notes that, for any Ng, the values of R(Ng-Si), D e , and BD s (ave) follow invariably the same order, namely NgSiF 3 + > NgSiH 3 + > NgSiCl 3 + . These trends actually herald the different bonding situations occurring in the various complexes. All of these systems are, in fact, stabilized by donor-acceptor interactions between Ng and SiX 3 + , the nature of the ensuing Ng-Si bonds depending on the size and polarizability of Ng and on the Lewis acidity of SiX 3 + . We first examine NgSiH 3 + . In HeSiH 3 + (Figure 2a), the small He penetrates so close to Si as to undergo an appreciable deformation of its H − (r) region, measured by a DoP(He) as high as 40.1. This polarization, however, is not sufficient to promote contact with the H − (r) region of SiH 3 + , and the bond is, therefore, of type C. The H(r) at around the BCP of the He-Si bond is, however, negative at both the He and the Si side, and its values over Ω s range from negative to positive, being slightly negative on the average. The bond is, thus, designated pCov(C t /H −/+ ). In the NeSiH 3 + , the H − (r) region of Ne looks (nearly) spherical (Figure 2b), and the DoP(Ne) amounts to only 6.42. As a matter of fact, with respect to He, the bigger Ne is located further away from Si (2.330 Å vs. 2.122 Å; see Table 3), and this produces an interaction of type C with a lower degree of polarization of Ng. The polarizability (α) of Ne (0.3956 Å 3 ), is, however, sufficiently higher than that of He (0.2055 Å 3 ) to promote quantitative effects that are higher than those occurring in HeSiH 3 + . Thus, for the Ne-Si bond, not only the H(r) at around the BCP is negative at both the Ne and the Si side, but its values over Ω s are also invariably negative. The interaction is thus designated pCov(C t /H − ). One also notes from Table 3 that, when going from HeSiH 3 + to NeSiH 3 + , N(Ω s ) and ρ s (ave) increase, respectively, from 4.86 to 7.57 me, and, from 0.0156 to 0.0168 ea 0 −3 , H s (ave) decreases (becomes more negative) from −0.000075 to −0.0013 hartree a 0 −3 , and BD s (ave) increases from 0.0042 to 0.0747 hartree e −3 . The D e also increases from 2.0 to 3.1 kcal mol −1 , and this is consistent with the major stabilizing role of the polarization unraveled by the energy decomposition analysis performed by Morino, Chattaraj, and their coworkers [27].
The change in the bonding situation of NgSiH 3 + is even more dramatic when going from NeSiH 3 + to ArSiH 3 + . The α of Ar (1.6411 Å 3 ) is, in fact, sufficiently large to promote an extensive overlapping of its H − (r) region with that of SiH 3 + (see Figure 2c),represented by a (3,+1) HCP along the Ar-Si axis. H(r) is also invariably negative over the Ω s , and the interaction is, therefore, of the A/H − type. Not unexpectedly, the same character is assigned to the Ng-Si bonds occurring in the heaviest congeners KrSiH 3 + , XeSiH 3 + , and RnSiH 3 + , the α values of Kr, Xe, and Rn being, indeed, higher than that of Ar (2.4844, 4.044, and 5.3 Å 3 , respectively). The ρ s (ave) of these Ng-Si bonds, ranging between 0.0325 (Ar-Si) and 0.0421 ea 0 −3 (Rn-Si), is, however, well below the threshold of covalency (0.08 ea 0 −3 ) and all are thus designated pCov(A/H − ).
Based on the criteria proposed by Boggs et al. [47], Morino, Chattaraj, and their coworkers [27] assigned the Ng-Si bonds occurring in all of the NgSiH 3 + as possessing a covalent or partially covalent character. They noticed, however, that the geometries and AIM indices of HeSiH 3 + and NeSiH 3 + were most-suggestive of noncovalent interactions. As a matter of fact, our analysis confirms the assignment based on the Boggs criteria, all of the bonds occurring in NgSiH 3 + featuring a contribution of covalency. This highlights the importance of comparing the results of different methods when assaying bonding situations that are at the borderline of different characters.
As discussed previously [27], the Lewis acidity of SiH 3 + , SiF 3 + , and SiCl 3 + decreases in the order SiF 3 + > SiH 3 + > SiCl 3 + , and this trend is clearly recognizable in the nature of the Ng-Si bonds occurring in the various NgSiX 3 + (X = H, F, Cl). Most illustrative in this regard are the three helium complexes. Thus, the comparison between Figures 2a and 3a clearly shows that, when going from HeSiH 3 + to HeSiF 3 + , the polarization of He is enhanced to such an extent that its H − (r) region comes into contact with the H − (r) region of SiF 3 + . The type of the He-Si bond thus changes from C to A, and the BD s (ave) increases by more than ten times, passing from 0.0042 to 0.043 hartree e −3 . The H(r) over Ω s , however, is still partially positive, and the bond is overall designated pCov(A/H −/+ ). On the other hand, in HeSiCl 3 + , the He atom is appreciably less polarized than in HeSiH 3 + . As shown in Figure 4a, itsH − (r) region is nearly spherical, and the DoP(He) amounts to only 3.53. The H(r) is also positive at both sides of the BCP and is invariably positive over Ω s . The He-Si bond is thus designated nCov(C l ), with a negative BD s (ave) of −0.272 hartree e −1 . Likewise the Ne-Si bond of NeSiH 3 + , the Ne-Si bonds of both NeSiF 3 + and NeSiCl 3 + are of type C, as evinced from the graphs shown in Figures 3b and 4b. The three Ne-Si bonds feature, however, differences again related to the Lewis acidity of the cation decreasing in the order SiF 3 + > SiH 3 + > SiCl 3 + . The contacts occurring in NeSiH 3 + and NeSiF 3 + are, in fact, both designated pCov(C t /H − ), but the BD s (ave) are appreciably different and are predicted to be 0.0774 and 0.135 hartree e −1 , respectively. The contact occurring in NeSiCl 3 + is, instead, nCov(C l ), with a BD s (ave) of −0.086 hartree e −1 .
The acceptor ability of SiF 3 + /SiCl 3 + being higher/lower than that of SiH 3 + also became apparent when examining the three argon complexes. As evinced from Figure 3c, the Ar-Si bond of ArSiF 3 + is of type B, the Ar atom being polarized to such an extent that its H − (r) region comes (nearly) into contact with the H + (r) region of SiF 3 + . This is, indeed, typical of complexes of Ng donors with strong Lewis acceptors [30]. The interaction also features an appreciable contribution from covalency, which is overall designated pCov(B t /H − ). On the other hand, likewise ArSiH 3 + , the Ar-Si bond of ArSiCl 3 + is designated pCov(A/H − ), but all of the bond indices appreciably decrease (see Table 3). We note, for example, the BD s (ave) passing from 0.283 to 0.157 hartree e −1 . The Ng-Si bonds of the complexes of Kr, Xe, and Rn with SiF 3 + (Figure 3d The results obtained by Morino, Chattaraj, and their coworkers [27] clearly uncovered that the ligation of a second Ng atom to any NgSiX 3 + produces a weakening of the Ng-Si interaction. They found that, as the Ng-Si distances increase, the complexation energies per Ng atom decrease, and the indices employed within various methods of bonding analysis invariably suggested a decreased degree of covalency. To understand the information gained with our taken approach, we explored the three exemplary Ar 2 (SiX 3 + ) (X = H, F, Cl). The comparison with the corresponding mono-coordinated ArSiX 3 + confirmed the indications from the previous study [27]. The weakening of the Ar-Si bond produced by the ligation of a second Ar atom is particularly evident for the chlorine complexes. A comparison between Figures 4c and 5c shows, in fact, that when going from ArSiCl 3 + to Ar 2 SiCl 3 + , the type of the interaction changes from A to C and the overall assignment changes from pCov(A/H − ) to pCov(C t /H −/+ ) (see Table 3). In essence, the polarization exerted by SiCl 3 + when interacting with two Ar atoms is insufficient to promote the overlapping of the H − (r) regions. This reduces the role of covalency, and BD s (ave) also drastically declines from 0.157 to 0.0036 hartree e −1 . As for Ar 2 SiH 3 + and Ar 2 SiF 3 + , based on the graphs shown in Figure 5a,c and our adopted criteria of classification, their Ar-Si bonds are designated pCov(A/H − ). All the bond indices, however, indicate that they are weaker than the bonds occurring in the mono-coordinated ArSiH 3 + and ArSiF 3 + . We note, in particular, values of BD s (ave) decreasing, respectively, from 0.283 to 0.190 hartree e −1 and from 0.354 to 0.265 hartree e −1 . The complexation energies per Ar atom also decrease, respectively, from 14.1 to 8.1 kcal mol −1 and from 21.7 to 10.6 kcal mol −1 .

The NgSiF 2 2+ (Ng = He-Rn): The Ng-Si Bond in Mono-Coordinated DoublyCharged Complexes
As mentioned in the Introduction, the bonding situation of the six NgSiF 2 2+ (Ng = He-Rn) is still unexplored. The plots of their H(r) (Figure 6) clearly uncover the extensive polarization of Ng toward the strong Lewis acceptor SiF 2 2+ with the formation of peculiarly tight bonds. This is consistent with the short bond distances between 1.721 (Ng = He) and 2.518 Å (Ng = Rn) and the high complexation energies between 14.0 (Ng = He) and 137.4 kcal mol −1 (Ng = Rn). The BD s (ave) are also generally high and range between 0.155 (Ng = He) and 0.552 hartree e −1 (Ng = Kr).
The interactions occurring in HeSiF 2 2+ and NeSiF 2 2+ are both designated pCov(B t /H −/+ ), with rather high values of ρ s (ave) (0.0383 and 0.0434 e a 0 −3 , respectively), and BD s (ave) (0.155 and 0.159 hartree e −1 , respectively). The Ar-Si bond of ArSiF 2 2+ is designated pCov(B t /H − ), but its ρ s (ave) of 0.0743 e a 0 −3 points to an incipient covalent bond. True covalent bonds are, indeed, predicted for KrSiF 2 2+ and XeSiF 2 2+ , the occurring interactions being designated Cov(A). The bond occurring in RnSiF 2 2+ is pCov(A/H − ), but its ρ s (ave) of 0.0786 e a 0 −3 suggests a situation at the border of covalency. A comparison of the data obtained for the NgSiF 2 2+ and the NgSiF 3 + (vide supra) clearly uncovers that the increased charge at the Si atom dramatically enhances the degree of the interaction with the noble gas, shifting the Ng-Si bonds towards the domain of covalency. In order to explore the character of these bonds when Ng is inserted into the Si-F bond of SiF 3 + , we performed a bonding analysis of the FNgSiF 2 + . The obtained results are discussed in the following paragraph.

The FNgSiF 2 + (Ng = Kr, Xe, Rn): The Ng-Si Bond in Inserted Cations
We searched for the six FNgSiF 2 + (Ng = He-Rn), but only the heaviest congeners FKrSiF 2 + , FXeSiF 2 + , and FRnSiF 2 + were located as stationary points on the corresponding potential energy surface and characterized as true minima. The Ar congener FArSiF 2 + , located so far by Hopkinson, Bohme, and their coworkers [25] as an energy minimum at the DFT level of theory (B3LYP/DZVP), was not confirmed here at the MP2/aVTZ. This tendency of DFT methods to overestimate the stability of only marginally stable "inserted" noble gas compounds, especially those containing He, Ne, and Ar, is not surprising and is already documented in the literature [48,49].
The bonding situation of FNgSiF 2 + (Ng = Kr, Xe, Rn), not explored in the previous study [25], clearly emerges by examining the plots shown in Figure 5d-f and the data quoted in Table 3. All the Ng-F and Ng-Si bonds are of type A, and, over Ω s , the H(r) is invariably negative. The ρ s (ave) of any Ng-F bond is also definitely higher than 0.08 e a 0 −3 , and these interactions are safely designated Cov(A). The ρ s (ave) of the Ng-Si bonds are also rather high at around 0.07 e a 0 −3 but are still below the threshold of covalency; these interactions are, therefore, designated pCov(A/H − ). In any case, the corresponding BD s (ave) of 0.487 hartree e −1 (Kr-Si), 0.464 hartree e −1 (Xe-Si), and 0.441 hartree e −1 (Rn-Si) are invariably higher than the values predicted for the Kr-Si, Xe-Si, and Rn-Si bonds of the monocoordinated KrSiF 3 + , XeSiF 3 + , and RnSiF 3 + (0.431, 0.470, and 0.471hartree e −1 , respectively). In essence, the insertion of Ng into the Si-F bond of SiF 3 + produces Ng-Si bonds tighter than those occurring in the mono-coordinated NgSiF 3 + . This reflects the further stabilizing role of the electron-withdrawing F atom bound to Si.

Concluding Remarks
The purpose of the present study was to compare, using a uniform and accurate level of theory, the bonding situation of some exemplary noble gas-silicon cations. We re-examined, in particular, the previously reported NgSiX 3 + and Ar 2 SiX 3 + (X = H, F, Cl) [27] and extended the study to the still unexplored NgSiF 2 2+ , and FNgSiF 2 + . It was thus possible to gather a comprehensive view of the nature of the bonds occurring in the various experimentally observed species [24][25][26] and in their still unreported congeners. Ng-Si bonds generally feature a contribution of covalency, arising from the strong polarization of Ng by the Si atom. The effect is quite extensive for the systems containing Ar, Kr, Xe, and Rn but is also appreciable for those containing He and Ne. The small size of He promotes, in particular, peculiar effects not observed for the Ne congeners. As for the singly charged NgSiX 3 + , in keeping with the results of a previous theoretical study [27], we found that, for any X, the degree of the interaction and the role of covalency generally increase when going from Ng = He to Ng = Rn. In addition, for any Ng, these two factors progressively decrease in the order NgSiF 3 + > NgSiH 3 + > NgSiCl 3 + , this trend strictly mirroring the Lewis acidity of the cation decreases in the same order. The comparison between the NgSiF 3 + and NgSiF 2 2+ also uncovered the dramatic effect of the increased charge in enhancing the stability of the complexes and the degree of the interaction. The Ng-SiF 2 2+ bonds are, in fact, truly covalent in nature for Ng = Kr, Xe, and Rn. We also ascertained that the insertion of Kr, Xe, and Rn into the Si-F bond of SiF 3 + , so as to form FNgSiF 2 + , produces Ng-Si bonds appreciably tighter than those occurring in the corresponding NgSiF 3 + , being of incipient covalent character. We refer to this as the cooperative effect exerted by the adjacent F atom. Finally, the bonds occurring in the mono-coordinated xenon complexes invariably feature an appreciable contribution of covalency. This supports the conclusion reached in a previous study [22] of an incipient chemical bond between Xe and the cationic silicon cluster Si 4 + .