Introduction

Interaction between metals and π-electron systems of neutral aryl rings is an interesting bonding motif for supramolecular self-assembly [1]. For instance, various transition metals interacting with benzene rings form both simple π-complexes in the gas phase [2–4] and larger supramolecular structures with this bonding motif [5–7]. However, the knowledge on the complexes of the heavier metals of groups 13–16 with neutral arenes is rather limited [8]. Tin is a good example in this context. The results of crystallographic studies [9] indicate that intermolecular Sn · · · π interactions often involve low-valent tin atoms, and therefore the complexation of stannylenes [10] by neutral aryl rings has attracted much interest in recent years [11–14].

In this work, a series of five systems composed of a singlet tin(II) dihydride or dihalide SnX2 (X = H, F, Cl, Br, I) molecule and a benzene molecule is considered. From a computational viewpoint, such systems may be regarded as model systems that approximate real systems in which the complexation of stannylenes by the π-electron cloud of neutral aryl rings occurs [9]. We focus mostly on the energetic and electron density topological description of the interaction between SnX2 and C6H6 in the resulting complexes: SnH2 · · · C6H6 (1), SnF2 · · · C6H6 (2), SnCl2 · · · C6H6 (3), SnBr2 · · · C6H6 (4), and SnI2 · · · C6H6 (5). Such quantum-chemical theories as the symmetry-adapted perturbation theory [15, 16] in both its traditional formulation (HF-SAPT) and its variant based on the density functional theory (DFT-SAPT), the quantum theory of atoms in molecules (QTAIM) [17], and the noncovalent interactions (NCI) visualization index [18] are used to provide an in-depth insight into this interaction. The present investigation is an extension of our previous work [19] in which only limited characteristics of 1–5 were obtained because some basic geometrical and energetic parameters were sufficient for our benchmark assessment of the accuracy of the Møller–Plesset second-order perturbation theory (MP2) and the density functional theory (DFT). There is also another theoretical study of the interaction of SnX2 with C6H6 [20]. In that study, the properties of various SnX2 molecules were characterized by conceptual DFT reactivity indices and the complexation of SnX2 with a series of potential aromatic π-donors was examined. In particular, it was deduced from the results of natural bond orbital (NBO) calculations that the most important orbital interaction for this complexation was the overlap of the formally empty p-orbital on the Sn atom and the π-orbitals of the C6H6 molecule. Here, we employ not only a wide variety of first-principles methods for inspecting the energetics of the interaction between SnX2 and C6H6 but additionally we explore the interaction in the complexes 1–5 from the perspective of their electron density topology.

Computational details

The geometries of 1–5 are taken from our previous study [19] in which they were optimized at the ωB97X/aug-cc-pVTZ(-PP) level of theory [21–24]. These geometries are characterized by the absence of imaginary vibrational frequencies and they correspond to global minima on the potential energy surfaces of 1–5 (see also section S1 in Electronic Supplementary Material). Throughout this entire work, the aug-cc-pVTZ basis set [22] is ascribed to the atoms of H, C, F, Cl, and Br, whereas the aug-cc-pVTZ-PP basis set [23] is used for Sn and I. Their 28 core electrons are described by the corresponding energy-consistent Stuttgart/Cologne MDF pseudopotentials [24]. These pseudopotentials allow us to indirectly account for relativistic effects in Sn and I.

The formation of each complex is characterized by its complexation energy E complex that is defined as

$$ {E}_{\mathrm{complex}}={E}_{def}+{E}_{int} + \varDelta ZPVE $$
(1)

where E def is the so-called deformation energy needed to change the geometries of SnX2 and C6H6 from those exhibited by isolated molecules to those observed in the complex, E int is the interaction energy between SnX2 and C6H6 in the complex and ΔZPVE denotes the difference in the unscaled zero-point vibrational energies of the isolated SnX2 and C6H6 molecules and the SnX2 · · · C6H6 complex. E int can be computed using either supermolecular or perturbative approach. According to the former, subtracting the total energies of SnX2 and C6H6 in their geometries observed in the complex from the total energy of the SnX2 · · · C6H6 complex constitutes E int. In this work, both DFT methods (such as SVWN [25, 26], BLYP [27, 28], B3LYP [28, 29], M06-2X [30] and ωB97X [21]), and wave function theory (WFT) methods (such as HF [31, 32], MP2 [33], SCS-MP2 [34], CCSD [35], and CCSD(T) [35]) are used to obtain E int within the framework of the supermolecular approach. The E int energies yielded by the HF and DFT methods are corrected for the basis set superposition error using the full counterpoise method of Boys and Bernardi [36]. The “half-half” counterpoise correction [37] is included in the E int energies calculated using MP2, SCS-MP2, CCSD, and CCSD(T). The E int energies at the CCSD(T) level of theory are additionally extrapolated to the complete basis set (CBS) limit via a composite scheme (for details, see section S2). The MP2, SCS-MP2, CCSD, and CCSD(T) methods make use of the frozen core approximation in the treatment of core electrons. All the calculations described in this paragraph have been done with Gaussian D.01 [38] and TURBOMOLE 6.6 [39] (for the DFT and WFT methods, respectively).

The SAPT method is used to determine the E int energy between SnX2 and C6H6 in a perturbative manner. Within the framework of this method, E int is expressed as a sum of several energy terms that can be grouped into four principal components with clear physical meanings. The components covering exchange (E exch), electrostatics (E elst), induction (E ind), and dispersion (E disp) are assumed to include the following energy terms occurring in the SAPT expansion of E int:

$$ {E}_{\mathrm{exch}} = {E}_{\mathrm{exch}}^{(1)} $$
(2)
$$ {E}_{\mathrm{elst}} = {E}_{\mathrm{elst}}^{(1)} $$
(3)
$$ {E}_{ind}={E}_{\mathrm{ind}}^{(2)}+{E}_{\mathrm{exch}-\mathrm{i}\mathrm{n}\mathrm{d}}^{(2)} + \delta {E}_{HF} $$
(4)
$$ {E}_{\mathrm{disp}}={E}_{\mathrm{disp}}^{(2)}+{E}_{\mathrm{exch}-\mathrm{disp}}^{(2)} $$
(5)

The HF-SAPT [15, 16] and DFT-SAPT [40, 41] variants are employed. The gradient-regulated asymptotic correction of Grüning et al. [42] with the PBE0 functional [43] is introduced into the DFT-SAPT calculations of E int in order to improve the asymptotic behavior of the DFT-SAPT variant. All the SAPT calculations have been performed using the MOLPRO 2012.1 program [44, 45].

The QTAIM analysis of the topology of the electron density in 1–5 and the calculations of QTAIM properties have been done with the AIMAll 14.11.23 program [46]. The Multiwfn 3.3 program [47] has been used to perform the NCI analysis and to obtain a signature of electron pair distribution in terms of the electron localization function (ELF) [48]. The results are visualized using Jmol 14.2 [49], AIMStudio 14.11.23 [46], and VMD 1.9 [50].

Results and discussion

Structure and stability

Let us start by presenting the optimized geometries of 1–5. All these complexes exhibit a marked structural similarity in which the SnX2 molecule is arranged above the C6H6 ring (see Fig. 1a). The molecular plane of SnX2 is approximately parallel to the plane of the C6H6 ring. All five complexes are of C s symmetry. The Sn atom sits nearly on one of the C atoms, and the X atoms are directed outward the C6H6 ring. The distance between Sn and the C atom beneath (d Sn···C) falls in a narrow range between 2.919 and 3.076 Å (see Table S6 in Electronic Supplementary Material). This is much smaller than the sum of the Sn- and C-atom van der Waals radii (2.42 Å for Sn and 1.77 Å for C [51]). The values of the a Sn···C–H angle do not differ significantly from 90°, and therefore, they indicate the η 1 type of complexation for 1–5.

Fig. 1
figure 1

Optimized geometry of 1 with two characteristic geometrical parameters marked (a) or ELF isosurfaces shown (b). These isosurfaces are plotted with a contour value of 0.85

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The aforementioned, almost parallel orientation of the SnX2 and C6H6 molecular planes means that the formally empty p-orbital of the Sn atom is nearly perpendicular to the plane of the C6H6 ring. In such an orientation, the π-cloud overlaps the empty p-orbital effectively [20]. The lone electron pair of the Sn atom is positioned opposite the X atoms. Its localization is confirmed by the corresponding ELF isosurface depicted in Fig. 1b.

The comparison of the isolated SnX2 and C6H6 molecules with the corresponding molecular fragments of 1–5 reveals that SnX2 and C6H6 do not suffer any significant structural variations upon complexation (see Table S6). A minor effect of complexation on the geometries of SnF2 and C6H6 molecular fragments constituting a 1:1 complex was confirmed experimentally, using matrix IR spectroscopy [52]. The vibrational frequencies measured for this complex were close to those of SnF2 and benzene substrates, and on this basis, only a slight distortion of SnF2 and C6H6 geometries was deduced. The complexation of Sn(II) by a six-membered aromatic ring was also detected in several crystal structures [11–14]. The structures often revealed the η 6 type of Sn(II) complexation, with the distances between Sn(II) and the centroid of the interacting aromatic ring within the range from 3.2 to 4 Å [11, 12]. The lower end of this range is slightly larger than the d Sn···C values found for 1–5. Such a shortening of the distance between Sn(II) and the molecular plane of aromatic ring is a consequence of the η 1 type of complexation in 1–5.

Energetic effects associated with the complexation of SnX2 with C6H6 are summarized in Table 1. As evidenced by the values of E complex, the formation of 1–5 from the isolated SnX2 and C6H6 molecules is energetically favorable (that is, E complex < 0), although the resulting stabilization of the complexes does not exceed several kcal/mol. The complexation of SnH2 turns out to be slightly less energetically favorable than the complexation of tin(II) dihalides. The values of E def are small, which is a consequence of the minor structural reorganization of SnX2 and C6H6 upon complexation. For all five complexes, the interaction energy E int contributes most to E complex. The values of E int fall in a narrow range between −9.1 and −9.7 kcal/mol, which obviously suggests that the interaction between SnX2 and C6H6 in 1–5 is in fact quite similar. The difference in the energetics of the complexes becomes more significant with the ΔZPVE component included.

Table 1 Complexation energies and their components calculated using the ωB97X method
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The E int values shown in Table 1 indicate that the interaction between SnX2 and C6H6 can be classified into the category of weak intermolecular interactions. The interaction investigated here turns out to be much weaker than those involving benzene and alkali or alkaline-earth metal cations [53, 54] but stronger than the interactions in SF2 · · · C6H6 [55] and SeY2 · · · C6H6 (Y = H, F, Cl) [56].

The interaction energies of 1–5 have also been computed at various DFT and WFT levels, all using the geometries optimized by ωB97X/aug-cc-pVTZ(-PP). The calculated values of E int are reported in Table 2 (the E int values that are uncorrected by the counterpoise method can be found in Table S7). We now test the reliability of the E int energies obtained from less computationally expensive levels of theory against the CCSD(T)/CBS results.

Table 2 Interaction energies calculated using various methods
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The HF method predicts that the interaction between SnX2 and C6H6 leads to the stabilization of 1–5 but the resulting E int values are very small. Thus, the omission of electron correlation energy results in a major underestimation of the strength of the interaction between SnX2 and C6H6.

The DFT methods present diverse performance, depending on the generation of a given DFT method. The SVWN functional leads to a typical overbinding [57]. In contrast to SVWN, the BLYP and B3LYP functionals predict that the complexes 1–5 are bound too weakly or even unbound (BLYP produces E int > 0 for 5). The E int values obtained from B3LYP differ slightly from the corresponding B3LYP values reported in Ref. [20]. These differences originate from a greater number of core electrons described by the pseudopotential of Sn atom during the calculations carried out in Ref. [20]. A common method that can improve the performance of BLYP and B3LYP in predicting the interaction energy of noncovalent complexes is to include an empirical dispersion correction. Here, Grimme’s D3 term with Becke-Johnson damping (D3(BJ)) [58] is employed for BLYP and B3LYP. The application of the D3(BJ) term allows these two functionals to overcome the underestimation of E int for 1–5. However, both BLYP-D3(BJ) and B3LYP-D3(BJ) yield a minor overbinding in 1–5, which suggests that the D3(BJ) term generally tends to overestimate slightly the role of long-range electron correlation in these complexes. This extends findings reported in a prior study [59]. It was shown therein that B3LYP combined with Grimme’s dispersion correction yielded overbinding for hydrogen-bonded and ionic complexes. A slight overestimation of hydrogen-bond energy in DNA base pairs was also detected for the combination of BLYP with Grimme’s dispersion correction [60, 61]. More recent DFT generations, represented in Table 2 by M06-2X and ωB97X, produce the E int energies that mirror the CCSD(T)/CBS results fairly closely. It is known that the ωB97X functional has proved to be highly successful at providing accurate bond energies for compounds containing transition metals [62], as well as interaction energies for complexes of Sn(II) [19] and Sn(IV) [63, 64]. The good performance of M06-2X in predicting the E int energies of 1–5 seems to confirm the important role of medium-range electron correlation for these complexes [30, 65].

As for the SAPT method, its HF-SAPT variant overestimates the strength of the interaction between SnX2 and C6H6, while the DFT-SAPT variant demonstrates the opposite tendency. However, the former produces the E int energies with worse accuracy relative to the CCSD(T)/CBS results than the latter does. It is so because the HF-SAPT variant applied here makes use of the simplest SAPT formulation, usually called HF-SAPT0 [66], that neglects the effects of intramonomer correlation. The intramonomer correlation effects are accounted for within the framework of the DFT-SAPT variant. It has been recently reported that DFT-SAPT generally underbinds H-bonded complexes [67] and it also proves to be the case with the interaction between SnX2 and C6H6. The maximum DFT-SAPT underestimation of the strength of the interaction in 1–5 amounts to 1.2 kcal/mol — this is observed for 1.

The performance of the advanced WFT methods also demonstrates some characteristic features. The MP2 method systematically overestimates the strength of the interaction between SnX2 and C6H6 due to the omission of the repulsive intramolecular correlation correction within this method [68, 69]. The inclusion of Grimme’s spin-component scaling (SCS) scheme [34] in the MP2 correlation energy reduces the overbinding of 1–5 but the E int value of 5 is still overestimated by 1.4 kcal/mol. The overestimation occurring in SCS-MP2 results from the application of the “half-half” correction of the basis-set superposition error. However, this correction leads to better agreement with the CCSD(T)/CBS interaction energies than the use of either uncorrected or full counterpoise-corrected values (see Table S8). Similarly to SCS-MP2, the CCSD(T) method also shows an overbinding tendency in E int. Again, it is due to the application of the “half-half” correction. The E int values yielded by CCSD are too small, which points at the importance of triple excitations in the correlation energy of 1–5.

The CCSD(T)/CBS method, which is deemed to give the most accurate estimates of E int in 1–5, confirms that the interaction between SnX2 and C6H6 should be classified as weak, with the E int values between −10.0 and −11.2 kcal/mol. Comparing the CCSD(T)/CBS results with the ωB97X ones, we see that the former indicate a more diversified strength of the interaction in the investigated series of complexes. The SnCl2 molecule is bound strongest to the C6H6 molecule, whereas the bonding of SnH2 to C6H6 turns out to be particularly weak. For the complexes containing the tin(II) dihalides, no monotonous regularity in their CCSD(T)/CBS E int values is found while X becomes heavier and heavier.

Interaction energy decomposition

The next step of this work is to characterize the physical nature of the interaction between SnX2 and C6H6 using the SAPT method. We first focus on the DFT-SAPT interaction energy terms calculated for 1–5 and listed in Table 3. For all five complexes their electrostatic first-order term E (1)elst is negative, which can be naively understood as a result of the attraction between the π-cloud and the positively-charged Sn atom (its QTAIM charge ranges from 0.86 to 1.49 au in 1–5). The absolute values of E (1)elst are always noticeably smaller than the corresponding values of the first-order exchange term E (1)exch . The magnitude of the second-order induction E (2)ind (SnX2 → C6H6) is several times larger than its E (2)ind (SnX2 ← C6H6) analog. It means that the C6H6 molecule is easily polarized by the SnX2 molecule, whereas the polarization in the opposite direction is significantly less pronounced. The E (2)ind (SnX2 ← C6H6) term of each complex is practically counterbalanced by the corresponding E (2)exch ‐ ind (SnX2 ← C6H6) term, while E (2)ind (SnX2 → C6H6) is quenched up to ca. 66 % by its exchange counterpart. An even smaller percentage quenching of E (2)disp is attributed to E (2)exch ‐ disp . The δE HF term destabilizes all five complexes. This term collectively gathers mostly third- and higher-order induction and exchange-induction contributions. The values of δE HF decrease gradually in the sequence from 1 to 5, that is, with growing d Sn···C distance. This indicates that the higher-order effects become increasingly important at shorter distances. Even though the inclusion of δE HF into the DFT-SAPT E int values leads to the underestimation of the strength of the interaction in 1–5, the presence of δE HF is more beneficial for the accuracy of E int than omitting this term. This is a well-documented feature of SAPT computations involving pseudopotentials [70].

Table 3 SAPT interaction energy terms calculated using the DFT-SAPT variant
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Next, let us examine the relative importance of four principal components of E int in order to establish the physical origin of the interaction between SnX2 and C6H6. Table 4 summarizes the results of grouping either HF-SAPT or DFT-SAPT interaction energy terms into four principal components. We now take a closer look at the DFT-SAPT components but the findings we make using these components are also valid for the HF-SAPT components.

Table 4 Four principal components of SAPT interaction energies calculated using the HF-SAPT and DFT-SAPT variants
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For all five complexes, the DFT-SAPT decomposition of their E int energies yields three attractive components, namely E DFT ‐ SAPTelst , E DFT ‐ SAPTind and E DFT ‐ SAPTdisp . These three components provide sufficient stabilization to overcome the repulsive exchange component, and therefore, the resultant E int values are negative (see Table 2). Among the attractive components, electrostatics represents the most energetically favorable contribution, followed by dispersion and then by induction. The E DFT ‐ SAPTelst component provides slightly less than half of the total attraction between SnX2 and C6H6. However, this component does not surpass the E DFT ‐ SAPTexch one, and in consequence, the total contribution from the first-order DFT-SAPT energy terms remains repulsive. Therefore, the stability of 1–5 is due to the second- and higher-order DFT-SAPT energy terms, that are included in the E DFT ‐ SAPTind and E DFT ‐ SAPTdisp components. Of the two components, the latter plays the more important role. In relative terms, dispersion accounts for roughly one third of the total attraction between SnX2 and C6H6. The relevant role of the dispersion energy often occurs for complexes containing large, diffuse electron clouds, such as the π-electron system of benzene in our case. The stabilization arising mainly from the combined effect of electrostatics and dispersion has recently been detected also for some other benzene complexes, e.g. with HCN [71].

The absolute values of E DFT ‐ SAPTind essentially increase with the growing atomic number of the X atoms. However, there is a noticeable decrease of the magnitude of E DFT ‐ SAPTind when one goes from 4 to 5. This results from the usage of a pseudopotential for the core electrons of the I atoms. For 2–5, the magnitude of E DFT ‐ SAPTdisp and simultaneously the relative importance of this component increase while the halogen atoms get heavier. This is consistent with the growing polarizability of tin(II) dihalides [72]. In contrast to E DFT ‐ SAPTdisp , the other two attractive components of E int show a diminishing relative importance while moving throughout the series from 2 to 5.

Topological analysis of electron density

Complementary information about the interaction between SnX2 and C6H6 can be obtained from the QTAIM topological analysis of the electron density ρ in 1–5. The molecular graph determined for 1 is shown in Fig. 2. It is evident from the graph that there is a single bond path (BP) linking the Sn atom and the adjacent C atom of the C6H6 molecule. Such a feature of the topology of ρ is common to all five complexes 1–5. This suggests that the formation of a Sn · · · C contact between the Sn atom and its nearest neighboring C atom of C6H6 is associated with the existence of a bonding interaction between these atoms [73]. The existence of the BP linking the Sn atom and its nearest neighboring C atom also indicates that, from the perspective of the QTAIM, the complexes 1–5 should be described as η 1 complexes.

Fig. 2
figure 2

Molecular graph for 1. BCPs are denoted by small red circles, while the only ring critical point is marked by a small yellow circle. All atoms are colored the same as in Fig. 1

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The topological properties of ρ at the bond critical point (BCP) on the BP of the Sn · · · C contact in 1–5 provide a more detailed QTAIM characteristics of the interaction between SnX2 and C6H6. The values of several such properties are collected in Table 5. The values of ρ decrease with growing d Sn···C distance. The relationship between ρ and d Sn···C exhibits a good linear reverse correlation, with the respective coefficient of determination R2 being equal to 0.96. It is due to the fact that the range of the d Sn···C distances found for 1–5 is very narrow. For a broader range of distances a non-linear relationship should be rather expected, as it was demonstrated for other noncovalent interactions [74, 75]. The values of ρ at the BCP of the Sn · · · C contact in 1–5 are much smaller than typical ρ values at the BCP of a covalent Sn-C bond (e.g. 0.099 and 0.106 au for the Sn-C bonds of Sn(CH3)2 and Sn(CH3)4, respectively). Furthermore, the ρ values in Table 5 are smaller than those found for Sn · · · C contacts in the solid-state structure of dicationic tin-toluene complex [Sn(C7H8)3]2+ [13]. It is accompanied by an elongation of the Sn · · · C contact in 1–5 compared to the contacts of [Sn(C7H8)3]2+. The values of the Laplacian of the electron density ∇2 ρ in 1–5 are positive, which means that the charge density is locally depleted at the BCP relative to the neighboring points in space and, in consequence, it is locally concentrated in the basins of the Sn and C atoms of the contact. The total energy density H, which is the sum of the electron kinetic energy density G and the electron potential energy density V, adopts the negative sign but the values of H are very close to zero. This may suggest that there is only a minor covalent factor contributing to the nature of the interaction between SnX2 and C6H6. Moreover, the low values of ρ and ∇2 ρ > 0, together with –V/G > 1 and –λ 1/λ 3 < < 1, provide evidence for the lack of any appreciable covalency [17, 74]. The aforementioned criteria indicate that this interaction should be classed as the closed-shell, noncovalent interaction [17, 74]. The –λ 1/λ 3 criterion of the nature of interaction is calculated using the lowest λ 1 and highest λ 3 eigenvalues of the Hessian matrix of ρ. The QTAIM characteristics of the Sn · · · C bonding interaction in 1–5 is essentially similar to that found previously for metal-ligand bonds in some complexes of Mn [76, 77] and Zn [78].

Table 5 QTAIM properties at the BCP on the BP linking the Sn atom and the adjacent C atom in each of the complexes 1–5
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The standard QTAIM characterization presented above can be supplemented by the analysis of the NCI visualization index. The plots showing the NCI isosurfaces detected for 1, 2 and 5 are presented in Fig. 3. As can be seen in these plots, the interaction between SnX2 and C6H6 in the complexes is characterized by a blue isosurface located between the Sn atom and its nearest neighboring C atom. The region delineated by this isosurface illustrates the occurrence of the attractive interaction between SnX2 and C6H6. For the complexes containing the tin(II) dihalides, additional isosurfaces representing the interaction between SnX2 and C6H6 appear. Such isosurfaces are located between the Sn atom and the more distant C atoms of C6H6. These isosurfaces are colored in green in Fig. 3 and they are associated with the occurrence of an extremely weak, repulsive interaction. Some other NCI isosurfaces denoting the existence of a secondary attractive interaction can in turn be found between the I atoms and their nearest neighboring H atoms in 5. One may speculate that these isosurfaces can be ascribed to the occurrence of a very weak H-bonding, although no BP has been detected between the I and H atoms, and the distance of 3.651 Å between these atoms exceeds the sum of their van der Waals radii.

Fig. 3
figure 3

NCI isosurfaces for 1, 2 and 5. The isosurfaces are plotted with a reduced density gradient value of 0.35 au and they are colored from blue to red according to sign(λ 2)ρ ranging from −0.02 to 0.02 au. The colors denoting the H, C and Sn atoms are the same as in Fig. 1. The atoms of F and I are drawn in green and violet, respectively

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Electron density deformations and charge transfer

In the subsection presented above, the complexes have been examined mainly from the perspective of their ρ itself. Now, the comparison of this ρ with the electron densities of non-interacting SnX2 and C6H6 fragments is made in order to determine how the electron density adjusts to the interaction between SnX2 and C6H6. Such a comparison can conveniently be presented in the form of an electron density difference plot, as it is shown in Fig. 4 for 1, 2 and 5. For each of these complexes, its electron density difference has been computed as the difference between the ρ of the whole complex and the sum of the densities of individual SnX2 and C6H6 fragments in their geometries taken from the complex. The regions delineated by blue isosurfaces in Fig. 4 illustrate an increase in ρ arising from the interaction, while the red regions determine where ρ is reduced. The most relevant changes in the ρ distribution are detected for the Sn · · · C contact and its spatially closest neighborhood. There is a region of ρ reduction immediately below the Sn atom of SnX2, whereas an increase in ρ is observed above this atom. In other words, the ρ distribution around the Sn atom becomes polarized toward the π-electron cloud of C6H6 upon complexation. The regions of growing ρ are found around the X atoms and the ρ deformation around these atoms is enhanced while moving from X = I to X = F, that is, with the increasing electronegativity of X. The distribution of ρ within the C6H6 molecule also undergoes some changes upon the complexation with SnX2. The most prominent change can be perceived above the C atom involved in the contact with the Sn atom. The vast blue region of increased ρ spreads over the part of the adjacent C-H bond. This is accompanied by several regions of ρ loss around the H atoms on the periphery of C6H6.

Fig. 4
figure 4

Plots of the electron density difference calculated for 1, 2 and 5. The blue and red isosurfaces are plotted with contour values of 0.001 and −0.001 au, respectively. The colors coding individual elements are the same as in Fig. 3

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An essential aspect of the interaction between SnX2 and C6H6 is the magnitude of the charge transfer that possibly appears as a result of complexation. The charge transfer between SnX2 and C6H6 is estimated here using the QTAIM atomic charges calculated for 1–5. The QTAIM charges of atoms constituting the SnX2 fragment of the complexes are summed up, yielding the overall magnitude of charge transferred between SnX2 and C6H6. The magnitude of the charge transfer estimated in that manner adopts very small values, from 0.0388 au for 5 to 0.0578 au for 2. The complexes containing the tin(II) dihalides show a gradual decrease in the magnitude of charge transfer while going throughout the series from 2 to 5. For these complexes there is a strong linear association between the decreasing magnitude of charge transfer and the increasing value of d Sn···C (the resulting inverse correlation between these two quantities is quantitatively characterized by R2 = 0.99). The charge transfer in 1 is estimated to be 0.0415 au. The formation of all five complexes leads to a small flow of electron charge from the C6H6 molecule to the SnX2 molecule. Thus, the SnX2 fragment of 1–5 bears a slight negative charge, from −0.0388 to −0.0578 au. The detected very small charge transfer and its direction may point at the existence of a very weak donor-acceptor π → Sn contribution to the interaction in 1–5. It would be in line with the results of a previous computational study based on the NBO approach [20]. In that study, an electron donation from the π-type NBO orbitals of C6H6 to the formally empty p-NBO orbital on the Sn atom of SnX2 was indeed detected but the calculated charge transfers from C6H6 to SnX2 were even smaller than those reported here.

Analysis of vibrational frequencies

The formation of 1–5 introduces some characteristic changes in the frequencies of vibrations occurring for the SnX2 and C6H6 fragments of the complexes. Table 6 presents shifts in the frequencies of three vibrations upon complex formation. The three vibrations include asymmetrical and symmetrical stretching frequencies of Sn-X bonds (υ as,Sn-X and υ s,Sn-X) and out-of-plane deformation vibrations of C-H bonds in the C6H6 ring (δ C-H). The frequencies of these vibrations have been computed within the quantum harmonic oscillator approximation at the ωB97X/aug-cc-pVTZ(-PP) level of theory. They have not been scaled.

Table 6 Shifts in vibrational frequencies of 1–5 as the result of complex formation
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The negative values of Δυ as,Sn-X and Δυ s,Sn-X shown in Table 6 indicate that small red shifts are observed for the frequencies of Sn-X stretching vibrations. The occurrence of these red shifts is associated with a minor elongation of Sn-X bonds upon complexation (see Table S6). The elongation of Sn-X bonds and the accompanying red shifts of Sn-X stretching frequencies can be roughly explained by the charge transfers between SnX2 and C6H6. As it was discussed in the previous subsection, the SnX2 fragment of the complexes bears a slight negative charge and the plots of the electron density difference show regions of electron accumulation around the Sn and X atoms upon complexation. Such a charge transfer contributes to the red shifts. The NBO results reported in Ref. [20] confirm the existence of the charge transfer from the π-type orbitals of C6H6 to the empty orbitals of SnX2. Additionally, the charge transfer involving the π-type orbitals is reflected in the structural properties of the C6H6 ring. Our calculations reveal that the C-C bonds are elongated by ca. 0.002 Å upon the formation of SnX2 · · · C6H6. A region of growing ρ around the C atom adjacent to Sn has been clearly seen in Fig. 4. This implies that at the same time there is a reverse transfer toward C6H6, which also facilitates the red shifts. The NBO analysis has however established that the back-donation toward the π*-orbitals of C6H6 is negligible in comparison to the transfer toward the orbitals of SnX2 [20].

The calculated Δυ as,Sn-X, Δυ s,Sn-X and Δδ C-H of 2 can be compared with the corresponding experimental data taken from Ref. [52]. The calculated values demonstrate good agreement with the experimental data. The reliability of the presented computational predictions of shifts in vibrational frequencies can be further proven through the inclusion of additional complexes in this study. Two additional complexes composed of SnF2 and chlorobenzene or toluene have been considered because their Δυ as,Sn-F, Δυ s,Sn-F and Δδ C-H shifts were previously determined experimentally [52]. The results obtained for these two additional complexes are presented in detail in section S3. Suffice it to say here that the calculated shifts in the Sn-F and C-H vibrational frequencies of 2 and two additional complexes are in good agreement with experiment. Moreover, the calculated shifts of υ as,Sn-F and υ s,Sn-F reproduce the experimentally established trend in the magnitude of these shifts [52].

Conclusions

In this work a variety of quantum-chemical methods have been used to provide an insight into the intermolecular interaction occurring in the complexes of SnX2 with C6H6. By analyzing the results of energy and electron density topology calculations, we conclude with the following remarks.

  1. 1.

    The complexes are rather weakly bound and the E int energy between SnX2 and C6H6 turns out to be similar for all five complexes. A very small destabilizing effect associated with changes in the geometries of SnX2 and C6H6 appears during the formation of each complex. This effect does not exceed 4 % of the absolute values of E int.

  2. 2.

    The effects of electron correlation play a vital role in the proper description of the interaction in the five complexes. Among the DFT methods, those belonging to older DFT generations fail badly: both severe under- and overestimations of E int are possible. The inclusion of the D3(BJ) correction improves their performance, but it tends to overestimate the role of long-range electron correlation. Newer density functionals without empirical dispersion correction (M06-2X and ωB97X) acquit themselves reasonably well.

  3. 3.

    The SAPT analysis reveals that the electrostatics is the dominant attractive component of E int for all five complexes. However, the E elst component is compensated by the E exch component, and therefore, the stabilization of the complexes is determined to a great extent by the second-order component that is accountable for dispersion.

  4. 4.

    Based on the QTAIM and NCI results, the interaction between SnX2 and C6H6 can be classified as a closed-shell, noncovalent and attractive interaction. The formation of the complexes polarizes the electron density around the Sn atom toward the π-cloud of C6H6.

  5. 5.

    By integrating the electron density of the complexes over their QTAIM atomic basins, we have deduced a very small charge transfer from C6H6 to SnX2 for all five complexes.

  6. 6.

    The calculated shifts in the frequencies of Sn-X and C-H vibrations agree well with the available experimental data. The relatively small shifts of these vibrational frequencies upon complexation confirm that the interaction between SnX2 and C6H6 is rather weak and there is no appreciable change in the inner geometries of interacting SnX2 and C6H6.