3.1 Structure and spectrum

Through the calculations performed at the B3LYP/6-311 + +G(d,p) level of theory, Fig. 1 illustrates the optimized geometries of the C₂H₅⁺···Ar (I), C₂H₅⁺···He (II), C₂H₅⁺···Kr (III), and C₂H₅⁺···Ne (IV) complexes. The distance values of 2.5822 (I), 2.4116 (II), 2.6126 (III), and 2.3956 Å (IV) for the H···RG interactions are shorter than the corresponding sum of the van der Waals radii [60–62], whose amounts are 2.97 (H = 1.09 + Ar = 1.88), 2.490 (H = 1.09 + He = 1.40), 3.11 (H = 1.09 + Kr = 2.02), and 2.63 Å (H = 1.09 + Ne = 1.54). In conformity with these assertions, but in view of the applicability in investigations of chemical bonds and interatomic distances [63,64], the interaction distance at the B3LYP/6-311 + +G(d,p) of I is shorter than the values reported by Goswani and Arunan [65] regarding the H₂O···2Ar and H₂S···2Ar complexes at the MP2(full)/aug-cc-pVTZ level of theory. Absolutely, this is not enough to discriminate the efficiency between these levels of theory, but otherwise it also serves to show that ternary structures have not necessarily shorter intermolecular distances and higher interaction energies compared with binary ones [21,27,66,67].

In addition to H···RG, the values of the bond lengths of both C···H and π are also depicted in Fig. 1. Unlike many other intermolecular systems [68–70], the shorter interactions of II and IV are not responsible for the most intense molecular changes, in particular on the ethyl cation. Note that IV is a much shorter bound than III, although the corresponding values of Δr_π are –0.0006 and –0.0037 Å. Because the hyperconjugation controls the stabilization of C₂H₅⁺, this effect shows itself sensitive to the changes in the π···H and C···H contacts [71,72]. These bond lengths vary between 0.0199 and 0.0371 Å as well as between 0.0011 and 0.0130 Å; these profiles are not in line with the interaction strengths, obviously at this stage of the study regarding only the interatomic distances. However, all these results show a good concordance with the phenomenology of the infrared stretch modes, whose values are displayed in Table 1.

Although supported by the van der Waals radii, the long interatomic distances present weak stretch frequency modes, which oscillate between 62.8 and 89 cm⁻¹; their absorption intensities are not so low (e.g., 30.88 and 42.70 km·mol⁻¹ for II and III, respectively). As aforementioned, the π bonds are reduced and strengthened after the formation of the I–IV complexes, but now, the vibration analysis point out slight blue shifts with increasing absorption intensity ratios [73,74]. Bearing in mind that the ethyl cation is a proton donor, the distance shortening reaffirms that π···H possesses the same profile observed in a large number of systems stabilized via H-bond [75,76]. Similar to a selected group of systems [77,78], however, red and blue shifts also coexist in the ethyl cation. Just like mentioned by Yen et al. [79] in studies of complexes, such as H–Ng–F···H–F and H–F···H–Ng–F, actually there is a synergism between Δr_π···H and Δυ_π···H, although all computed red shifts of –51.7 (I), –2.1 (II), –108.9 (III), and –4.4 cm⁻¹ (IV) are not concordant with the profile of the interatomic strength. In other words, the intensities of the shifted frequencies are not affected by the interatomic distances. Note that the experimental value of 2217 cm⁻¹ obtained by Duncan et al. [80] was satisfactorily reproduced by the value of 2144.8 cm⁻¹ determined via B3LYP/6-311 + +G(d,p) calculations.

3.2 Energy and charge transfer: attraction, repulsion and eventual retro-donation

Some time ago, Salazar et al. [81] have mentioned the difficulty in studying complexes formed by rare gases. Especially about interaction energy, it is widely known that its quantity is negative if the geometry of the intermolecular complex falls at the minimum of the potential energy surface [82]. According to the values listed in Table 2, it can be perceived how I–IV complexes are weakly bound. As can be seen, the ΔZPE corrections outweigh BSSE, although in some cases, the BSSE values are negative [83], even though the interaction energy of –7.69 kJ·mol⁻¹ reveals that III is the most stable complex. Contrary to what one might point out, intermolecular interactions with low energies are not unusual, as can confirmed by a work by Lu et al. [84]. Of course, small BSSE amounts were also obtained, and above all else, this was also verified in weakly bound complexes modeled through the B3LYP/6-311 + +G(d,p) theoretical level [85]. Likewise, it may be highlighted that H···RG is a non-covalent interaction with low results of BSSE and ΔE^C, although this is not relevant because other intermolecular interactions also present a similar aspect [19]. What is quite clear in other works [86,87] is not evidenced here regarding the direct relationship between distances and energies of H···RG interactions.

One of the most applied procedures in the investigations of intermolecular interactions is the analysis of the charge-transfer mechanism, which has proven extremely efficient in several applications [88,89], although it is also neglected because the atomic charge has no physical basis and thereby is not included into the Schrödinger equation. Despite this, many are the algorithms implemented to compute atomic charges, namely those inserted into the ChelpG, NBO, Mulliken, and QTAIM schemes. From these approaches, Table 3 lists the values of the charge transfer (ΔQ_RG^ChelpG, ΔQ_RG^NBO, ΔQ_RG^Mull, and ΔQ_RG^QTAIM), whose mechanism was admitted as being initialized by donations of electronic density from rare gases towards the ethyl cation. In this manner, the positive values of ΔQ_RG reaffirm the transference of charge previously quoted, although it must be also asserted that the negative and small results of –0.00032 and 0.00088 au of II and IV are somewhat realistic because undoubtedly these systems are not only non-covalent but almost non-bonded. This scenario suggests that an electronic retro-donation from ethyl cation towards to neon and helium may be reliable.

Throughout the chemical literature with mention to hydrogen bonding [90,91], the great applicability of the charge transfers is concentrated in the characterization and/or justification of the shifting frequencies of the proton donors when these species bind with high density centers [92]. By contrasting all amounts of ΔQ_RG^ChelpG, ΔQ_RG^NBO, ΔQ_RG^Mull, and ΔQ_RG^QTAIM versus the values of Δυ_π···H, four relationships were equated as follows:

The role of charge transfer is well understood on the basis of the frontier molecular orbitals, in which the electron-density flows between them from the HOMO towards the LUMO of the proton acceptor and donor [93,94], respectively. Here, charge transfer is not entirely absolute because in the weakest bound systems, for instance II and IV, an electronic density retro-donating phenomenon (LUMO to HOMO) determined via QTAIM calculations is evidenced. Even though the linear and direct relationship between ΔQ_RG^NBO and Δυ_π···H was established, Fig. 2 shows that NBO is an efficient approach in view of the linear coefficient of –0.999.

3.3 NBO and QTAIM analysis: Bent rule and interaction strength

In a forward analysis of the molecular orbital, in Table 4 are listed other NBO parameters, such as the percentages of the s-character and p-character of carbon (π) and hydrogen (H···RG) in the ethyl cation. Increases of 0.03 and 0.09% as well as decreases of –0.02 and –0.01% in the s-character can be noted; these are closely related with the interaction strength of the I and III complexes as well as of the II and IV ones, respectively. For the first two complexes, which are the longer bound ones, conclusively the Bent rule is satisfied, whereas it is not for the last two complexes, because they are weakly bound and even energetically non-bonded. This analysis is not applied to interaction energy, but surely is treated as one additional justification to the frequency shifts of the proton donors [43]. In this context, Fig. 3 illustrates the relationship between Δυ_π···H and Δ%_s(C), whose linear profile with a coefficient of –0.997 yielded by the equation (5) can be considered highly satisfactory.

Another information must be registered regarding the blue shifts: the increase of %_s(X) in X reinforces the characterization of the hybridization effect as dominant rather than the hyperconjugation upon the formation of the Y···H−X H-bond. So, hybridization would be present in stronger H-bond systems containing blue shifts on their proton donors [43]. Nevertheless, it should be pointed out that none of these events was identified here. On the contrary, this work presents a unanimous scenario with weakly bonded systems with interactions varying between 0.343 up to –7.69 kJ·mol⁻¹ followed by red shifts in the stretch frequencies of π···H proton donors oscillators.

In Tables 5–7 are organized several topological parameters (ρ, ▿²ρ, G, U, –G/U, Δr^o, and є) obtained from QTAIM calculations. In addition to the first four terms previously presented, other QTAIM parameters were introduced, such as Δr^o and є. About Δr^o, it represents a measurement of the distance between the nucleus and the BCP along a bond path as well as how much this distance varies by the formation of the complex (r^o_π−BCP,c) in comparison with the monomer (r^o_π−BCP,m), which is often a proton donor in which the following subtraction is valid: Δr^o_π−BCP = r^o_π−BCP,c–r^o_π−BCP,m [43]. In accordance with the hybridization theory, it was admitted that the manifestation of the blue shifts on proton donors would not be caused only by the increase in the s-character of the orbital of X (elements bound to H to form H − X) supported by the Bent rule, but the increase in the H radius compared with the value of X also reinforces this statement. Through the values of Δr^o_π−BCP and Δr^o_BCP−H (see Table 5), the inference of π···H can be discussed, in which H is the proton donor site. In all complexes, QTAIM computation revealed that Δr^o_π−BCP increases while Δr^o_BCP−H is reduced, meaning then Δr^o_π−BCP > Δr^o_BCP−H, which undoubtedly justifies the red-shift frequencies of –51.7 (I), –2.1 (II), –108.9 (III), and –4.4 cm⁻¹ (IV).

Since we had a peculiar interest in assessing the sub-part of the proton donor, blue shifts on the frequencies of the π bonds were identified in opposition to the red shifts of π···H discussed above. Due to the symmetry of the ethyl cation, the negative values of Δr°_C−C (see Table 6) indicate a mutual movement of the carbon atoms toward the BCP located between them. Independently, the reductions of Δr°_C−C are consistent with the blue shift results of +1.8 (I), +0.2 (II), +3.0 (III), and +0.3 cm⁻¹ (IV). In agreement with the Bent rule [95], particularly the stronger bound complexes I and III are those in which the s-character of the carbon is enhanced. For the remaining complexes whose intermolecular stabilization is ruled by the lowest interaction energies, in other words as what has been observed in II and IV, there was no variation of the s-character of C and thereby the lowermost blue shift values of 0.2–0.3 cm⁻¹ could not be explained.

Furthermore, it should be stressed that either the strengthening or the weakness of the π bond and π···H interaction can be interpreted in the light of the ellipticity (є) [96,97], whose values are also displayed in Tables 5 and 6. Briefly speaking, it is well known that є mimics the unsaturated character of the C − C bond in the 0.0–0.45 range (from ethane, benzene, up to ethylene) [46], indicating that the bond will be strengthened if the profile of є is enlarged. Here, by focusing justly on π and π···H, the є values for these centers are enhanced and reduced after I–IV complexes have been formed. Because these events attest to stronger and weaker contacts, of course they can be also used as argument to justify the frequency changes, namely those shifted either to red or blue regions. As such, we plotted in Fig. 4, the relationship between the frequency shifts and ellipticity variations for π and π···H, and an excellent linear correlation was yielded by equation (6) below:

One of the foundations of the QTAIM is devoted to the characterization (See Fig. 5) of BCPs (w, y, and z) and Bond Paths (BP) along the chemical bonds [98] (through the BCP pointed by w) and interatomic interactions (through the BCP pointed by z), in which it can be formed by low, median or high charge density concentrations [99–103]. Such designations can be predicted through the obtained values of ▿²ρ listed in Table 7 (the relief map of the C₂H₅⁺···Ar complex is illustrated in Fig. 6). It can be seen that the ▿²ρ values for H···RG are positive, whereas they are negative for π···H and π (see Tables 5 and 6). In other words, the interaction H···RG is recognized as a closed-shell one and assumed to be non-covalent, albeit the very low energetic values have already revealed this before. In spite of this, the ratio values of G and U conduct the debate of the interaction strength as follows:

• • –G/U < 0.5 indicates a total covalent character;
• • –G/U < 1.0 indicates a partial covalent character;
• • –G/U > 1 a non-covalent character [15,16,26].

Through this, it can be seen that all H···Rg interactions have no covalent character, whereas π···H and π are totally covalent. This scenario corroborates directly the slight frequency shifts and the low interaction energies.

Meanwhile, even though the interactive system is located in an energy minimum with low attraction strength, the prediction of its interaction strength is absolutely allowed, and it can be performed in conformity with the equation presented below [104,105]:

As a first result, the relationship between the interaction energies and the values of the Δmol parameter is graphically illustrated in Fig. 7, in which a linear coefficient of –0.994 was yielded by equation (8):

In this, the vibration and topological properties used to generate the Δmol parameter were collected from the π···H contact. On the other hand, if the π bond is used instead of π···H, the value of the linear coefficient of –0.988 obtained from the analysis of Fig. 8 is symbolized by equation (9):

Regarding the construction of these two models, vibration modes (red and blue shifts) were used instead of structural parameters [26], although it has been demonstrated that the vibration frequencies produce relationships well satisfactorily [104]. In any event, it should be emphasized that the ethyl cation disposes of two centers to predict the interaction strength with rare gases, namely the π bond and the π···H interaction.