Introduction

Intermolecular interactions are vital in many aspects of chemistry and biology [1–3]. Without intermolecular interactions, no ordered or functioning materials—such as DNA—could form. The thermodynamic properties and kinetic characteristics of gases, liquids, and solids depend on the intermolecular potentials in which the molecules within the bulk move according to the thermal conditions. Transition states of elementary chemical reactions are also formed due to the interactions between reactants that collide [3]. There are different types of intermolecular interaction. When categorized by the nature of the interaction, there are (1) electrostatic interaction dominated interactions, such as hydrogen and halogen bonds [4], and attractive interactions between ions, (2) induction-dominated interactions, such as charge-transfer complexes, and (3) dispersion-dominated interactions, such as the attractions between noble gas atoms and π-stacking interactions between aromatic rings. Intermolecular interactions among π-electron-rich molecules are particularly interesting, because such molecules usually have large polarizabilities as well as high electron densities, meaning that they readily form dispersion-dominated complexes and attract positively charged atoms or molecules electrostatically [5–11]. A few types of interaction are associated with neutral π-electron-rich molecules. Among them, CH···π and π···π interactions are the most studied. CH···π interactions can be categorized as soft acid/soft base (SA/SB) interactions [12–14]. Dispersion usually dominates this type of interaction. In the case of activated CH···π interactions [15, 16], both electrostatic attraction and dispersion are important stabilizing forces. The strength and directionality of the CH···π interaction and the role of the electrostatic interaction are related to the acidity of the hydrogen donor. π-stacked structures were initially thought to be dominated by electrostatic interactions [17], since the preference for displaced parallel stacking over direct stacking could be understood by invoking quadrupole–quadrupole interactions. However, Podeszwa et al. later showed that the quadrupole–quadrupole interaction is not necessarily attractive for this configuration [6]. Studies of substituted complexes [18] and of SAPT components [6, 19, 20] have shown that dispersion is usually the largest contributor to the stabilization of these stacked structures.

Benzene is an archetypal π-electron-rich molecule with six π electrons that follows Hückel’s 4n + 2 rule of aromaticity. Therefore, complexes of benzene are often used to study the interaction between the π-cloud and other molecules or ions. It is well established that the tilted T-shaped (perpendicular or edge-to-face) dimer (Scheme 1) is the most stable structure. However, the displaced parallel stacked structure (Scheme 1) has a similar stability to the CH···π structure [21]. In both forms of the dimer, the dominant attractive force is dispersion [5, 6, 22–24]. Studies of the fused rings have shown that the displaced parallel dimer becomes more stable than the T-shaped structure as the molecule increases in size [25, 26]. In addition, substitutions on the aryl rings stabilize the displaced parallel dimer more than the T-shaped dimer [27].

Scheme 1
scheme 1

Structures of benzene and methane dimers

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Surprisingly, ethylene—the simplest π-system, with only one pair of π electrons—is less well studied than benzene. A total of 14 possible structures of ethylene dimer have been examined via several molecular orbital calculations [28–31]. Other than the displaced parallel (Ci symmetry) and CH···π (C2v symmetry) structures, another important geometry is a structure with four pairs of close dihydrogen contacts (D2d symmetry) (Scheme 1). This is the most stable structure found for ethylene dimers. Several databases for weak interactions, such as S22 [32] and PPS5 [33], have included this structure as a prototypical system. There is evidence that the dihydrogen contacts in organic molecules contribute attractively to intermolecular interactions, due to the topological properties of the electron density [34] and the fact that the interaction energy increases with the number of dihydrogen contacts in dimers of n-alkanes [35, 36]. However, it is not clear whether these CH···CH contacts in the ethylene dimer are stabilizing or not. The nature of the intermolecular interaction was analyzed for various ethylene dimer configurations by different energy partition schemes [20, 29–31]. Dispersion and electrostatic interactions were identified as the major attractive forces that stabilize these dimers. The importance of the electrostatic interaction seems to have been overlooked. Although the magnitude of the electrostatic stabilization is similar to that of the SAPT interaction energy, it is still considered less important than in some other complexes [20].

When studying the nature of the binding of complexes with π-electron-rich molecules, it would be useful to separate the contribution of π orbitals from that of the σ backbone, but such studies are rare. Based on a series of computations of aromatic compounds and their saturated counterparts, Grimme studied the role of π electrons in dispersion, and concluded that π···π stacking is missing in small π systems [37]. In this study, the contributions from π···π, σ···π, and σ···σ interactions to the MP2 correlation energy, instead of the dispersion energy, were separated. The electrostatic contribution was briefly discussed. The author examined only cyclic structures, not simpler linear molecules such as ethylene or butadiene.

Ethylene is the simplest system that has CH σ and π bonds. Therefore, the D2d, Ci, and C2v ethylene dimers are prototypical representations of CH···CH, π···π, and CH···π interactions, respectively. Due to its simplicity and symmetry, ethylene is a perfect model for separating σ and π contributions from the total electrostatic interaction and dispersion. The study reported in the present paper attempted to achieve such a separation for the three ethylene dimer configurations. The roles of σ and π electrons in the electrostatic interaction were investigated using distributed multipole analysis, and their roles in dispersion were analyzed by freezing some of the valence orbitals in the energy decomposition calculations. This study also attempted to further characterize the nature of the CH···CH interaction and to understand the importance of the π···π interaction in this small π-system.

Computational methods

The structures of the three ethylene dimers (D2d, Ci, and C2v) and several related systems were optimized at the MP2/aug-cc-pVDZ level with and without counterpoise correction (CP and non-CP). Vibrational frequencies were calculated at the same level of theory to establish the nature of each stationary point. The interaction energies were evaluated using the CCSD(T) theory at the complete basis set (CBS) limit. The CCSD(T)/CBS energies were calculated as the sum of the MP2/CBS energies and a CCSD(T) correction term (CCSD(T)), evaluated with the aug-cc-pVDZ basis set. This additivity scheme takes advantage of the fact that the CCSD(T) and MP2 methods exhibit approximately the same basis set dependence [38]. The MP2/CBS energies were calculated using the extrapolation scheme of Halkier et al. [39], via directly computed aug-cc-pVDZ and aug-cc-pVTZ counterpoised energies. When extrapolation was used with the aug-cc-pVTZ and aug-cc-pVQZ basis sets, in combination with CCSD(T) single-point energy at the aug-cc-pVTZ level, the interaction energy improved by 0.01 kcal mol−1 for the three ethylene dimers. Thus, the smaller basis sets were used for the CBS extrapolation performed in this study. All geometry optimizations, frequency, and single-point energy calculations were performed using the Gaussian 03 suite of programs [40].

To analyze the nature of the stabilization of the ethylene dimers, we carried out symmetry-adapted perturbation theory (SAPT) [41] calculations up to the third order using the aug-cc-pVDZ basis set. This method allows for the separation of intermolecular interaction energies into well-defined components, namely the electrostatic interaction (ES), induction (IND), dispersion (DISP), induction–dispersion (IN-DI), and exchange repulsion (EX) terms. These terms are all calculated as the sum of selected energies in the SAPT analysis, and expressions for them are given in the “Electronic supplementary material” (ESM). Since the IN-DI term is very small in all systems (e.g., −0.05 kcal mol−1 for the ethylene D2d dimer), it is not included in the tabulation. The total SAPT interaction energy (TOT) was calculated as the sum of various SAPT interaction components. For the larger complexes, such as substituted ethylene dimers and complexes involving benzene or butadiene, density functional theory–symmetry-adapted intermolecular perturbation theory (DFT-SAPT) [42–44] calculations with an aug-cc-pVDZ basis were performed instead of SAPT ones. The functional used was the PW91 correlation functional with 75 % PBE exchange mixed with 25 % HF exact exchange. To obtain the correct asymptotic behavior at long range, a shift was applied to the DFT-SAPT procedure [45]. The basis sets used for density fitting are the defaults employed in the MOLPRO program [46, 47]. The SAPT and density-fitting DFT-SAPT calculations were performed using the SAPT 2008 [48] and MOLPRO 2008 [46] programs, respectively.

It is important to note that the SAPT interaction energies calculated with the aug-cc-pVDZ basis set are known to be slightly underestimated with respect to (mainly) the dispersion energy [44]. Our own basis set benchmark calculations (Table 1) readily confirm this finding. The dispersion energy computed using aug-cc-pVDZ is underestimated by ∼10 % compared to the triple-zeta estimate, while all other energy terms are relatively basis-set independent. Thus, a better estimate of the dispersion contribution can be obtained by scaling the aug-cc-pVDZ value. This scaling factor was chosen to be 15 % based on benchmark calculations for the methane dimer, ethyne dimer, and methane···N2 complex [49]. It is also worth noting that the results obtained when using aug-cc-pVDZ result were better than those yielded by cc-pVTZ due to the importance of diffuse functions. As with previous benchmark studies on the DFT-SAPT method [44, 49], the total interaction energy of DFT-SAPT is more accurate than that of regular SAPT, while the dispersion energy is less favorable (Table 1). DFT-SAPT/aug-cc-pVDZ analysis of the ethylene D2d dimer has been reported by Kim et al. [20]. Their calculated energy components are similar to our computed values, except for the induction term (Table 1). This difference is due to the fact that a different DFT functional was used and a different mathematical expression was employed for the induction term (see [20] and the ESM).

Table 1 Comparison of the SAPT interaction energy contributions of the ethylene D2d dimer (in kcal mol−1) using correlation-consistent basis setsa. Calculated CCSD(T)/CBS interaction energy is −1.50 kcal mol−1
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To gain insight into the individual (electron pair)–(electron pair) contributions to the dispersion, we decomposed the dispersion energy into contributions from orbital pairs. The contribution of the i th and j th orbitals in monomers A and B, respectively, is calculated in the following way:

$$ E\left(i\kern0.5em \leftrightarrow \kern0.5em j\right)={E_{i-}}_{1,j-1}\kern0.5em -\kern0.5em {E_{i,j-}}_1\kern0.5em -\kern0.5em {E_{i-}}_{1,j}\kern0.5em +\kern0.5em {E}_{i,j}, $$

where E i,j is the dispersion energy (calculated from Eq. 4 in the ESM) with orbitals 1–i on A, and 1– j on B frozen. This method, which we call frozen-orbital SAPT (fo-SAPT), ensures that the sum of contributions from all pairs equals the total dispersion energy, which is a desired property for a partition scheme. The SAPT program uses the “infinite excitation energy approximation” [50], so the separation of the exchange–dispersion term is a reliable approach. The third-order dispersion and exchange–dispersion terms are more complicated than the second-order terms because they include double sums and even triple sums of the contributions of the occupied orbitals of each monomer. A double sum implies that there is no unique way to separate the contribution of each orbital. Since the frozen-orbital approach decomposes the orbital contributions in only one of the many ways possible, it may not be considered rigorous. Nevertheless, this approach does at least provide energy components that sum to the total dispersion energy. We consider our proposed approach an acceptable method for obtaining good qualitative results. It is important to note that the third-order dispersion and dispersion–exchange terms constitute only a very small portion of the total dispersion energy. The effect of these terms on the orbital-pair analysis is shown in Table S2 of the ESM. Essentially, our proposed fo-SAPT approach provides an estimate of the contribution of a particular (electron pair)–(electron pair) interaction between two monomers (e.g., σ−σ, σ−π, or π−π), towards the total dispersion energy. We will employ the symbol “↔” to denote this type of (electron pair)–(electron pair) interaction, e.g., σ↔π, throughout the text.

Topological analysis was performed with the MORPHY98 program [51] based on MP2/aug-cc-pVDZ densities. Distributed multipole analysis was performed with the GDMA program [52], and electrostatic interaction energies between the multipoles were calculated using the ORIENT program [53].

Results and discussion

Structures and interaction energies of ethylene dimers

In the study reported here, we focused on just three ethylene dimeric forms, namely the D2d, Ci, and C2v dimers. The optimized geometries are given in Fig. 1. The D2d structure has four sets of intriguing close dihydrogen contacts (2.55 Å). The best calculations reported for this D2d dimer include the W1 method [33] and the CCSD(T) calculation with the quadruple zeta basis set [32]. The optimized geometry obtained at the MP2/aug-cc-pVDZ level (Fig. 1) is very close to the best geometries reported in the literature (see Table S1 in the ESM). The calculated CCSD(T)/CBS interaction energy is −1.50 kcal mol−1, (−1.51 kcal mol−1 if two-point extrapolation with the aug-cc-pVTZ or aug-cc-pVQZ basis is used), in excellent agreement with the database value of −1.51 kcal mol−1 [32].

Fig. 1
figure 1

Optimized (MP2/aug-cc-pVDZ, non-CP) geometries of the D2d, C2v, and Ci ethylene dimers. Bond distances are in Å

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The stacked form of ethylene dimer is the simplest π-stacking structure. The ethylene sandwich dimer, with D2h symmetry, has been studied. However, it is not a bound structure because of the repulsive quadrupole interactions present [54–56]. Similar to the benzene dimer, displaced parallel ethylene dimer configurations are structures of considerable interest. Two such ethylene dimeric structures have been examined by Suzuki et al. [28]. One has a similar nature to the sandwich dimer, while the other possesses attractive electrostatic interaction energy, and is the structure (Ci symmetry) investigated in this study. At the MP2/aug-cc-pVDZ level, with and without counterpoise correction, the optimized Ci structure has one imaginary frequency. A thorough scan of the potential energy surface (PES) of this ethylene dimer was conducted. In our PES scan, the molecular planes of both ethylene molecules were kept parallel and the C=C and C–H bonds were fixed. The center of one molecule was fixed at the origin and the C=C bond was considered the z-axis. Hence, the position of the other ethylene molecule can be described by the three parameters r, θ, and ϕ in a spherical coordinate system (Fig. 2).

Fig. 2
figure 2

Coordinate setup for the potential energy scan of the Ci ethylene dimer

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The best optimized Ci structure calculated at the CCSD(T)/CBS level has r = 4.0 Å, θ = 30°, and ϕ = 70°, and an interaction energy of −1.13 kcal mol−1. It is worth noting that the extrapolated interaction energy for this geometry when using the aug-cc-pVTZ and aug-cc-pVQZ basis sets is also −1.13 kcal mol−1. In this best π-stacking structure, the intermolecular C···H distance, i.e., r(C···H), is 3.2 Å, and the distance between the center of one molecule and the nearest H atom in the other is also 3.2 Å. This suggests the possibility of a CH···π interaction in the Ci dimer.

The second most stable form of ethylene dimer is the C2v structure, which is characterized by two CH···π interactions. This C2v dimer has a slightly smaller interaction energy than the D2d dimer, despite the similar intermolecular distances of the two dimers. The optimized geometry of the C2v dimer calculated at the MP2/aug-cc-pVDZ level is also shown in Fig. 1. The changes observed in the C=C and C–H bond lengths upon transitioning from the monomer to the dimer are very small, less than 0.0001 Å. The intermolecular C···H distance (2.83 Å) is characteristic of typical/nonactivated CH···π interactions [8, 13, 57–60], The calculated CCSD(T)/CBS interaction energy of this C2v dimer is −1.08 kcal mol−1 (−1.09 kcal mol−1 with the larger basis), which is very close to that of the displaced parallel dimer (−1.13 kcal mol−1).

Symmetry-adapted perturbation theory

The SAPT energy contributions of the three ethylene dimers are listed in Table 2, together with some related complexes. A plot of various SAPT components of the D2d complex against intermolecular distance is shown in Fig. 3. The induction and induction–dispersion terms are always small; this is similar to what is seen for the benzene dimer [5, 24]. At all distances, all of the SAPT components are attractive except for exchange repulsion. Dispersion is the main source of stabilization; its magnitude is comparable to that of exchange repulsion. Due to their different decay rates, exchange repulsion is greater than dispersion when the intermolecular distance is less than the equilibrium distance, while dispersion is greater than exchange repulsion when the intermolecular distance is more than the equilibrium distance.

Table 2 SAPT interaction energy contributions (in kcal mol−1) for various ethylene dimers and related dimers
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Fig. 3
figure 3

Distance (rCM) dependence of SAPT components of the D2d ethylene dimer. Dispersion energy is scaled up by 15 %

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To provide further insight into why the D2d dimer is the most stable ethylene dimer, it is essential to determine the kinds of interactions that are present in the dimer. To this end, we calculated the (electron pair)–(electron pair) contributions to dispersion using the fo-SAPT method. For the D2d dimer, the results (Table 3) indicate that the electron pairs that contribute the most to dispersion are the CC π-electron and two CH σ-electron pairs. Three types of (electron pair)–(electron pair) interactions are possible: π↔π, σ↔π, and σ↔ σ interactions. A π↔π interaction does occur in the D2d dimer, but it is not as strong as the σ↔π and σ↔σ interactions. Close contact between two positively charged hydrogen atoms during interactions between C–H bonding electrons, e.g., #3↔#5 (Table 3), contributes significantly to attractive dispersion. This contribution is almost as large as that of the σ↔π interaction.

Table 3 (Electron pair)–(electron pair) contributions to the dispersion (in kcal mol−1) for the D2d, Ci, and C2v ethylene dimersa
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The optimum structural parameters for the Ci dimer at CCSD(T)/CBS level are rCM = 4.0 Å and r(C···H) = 3.2 Å, which are longer than those in the D2d dimer (rCM = 3.7 Å and r(C···H) = 3.1 Å). Therefore, all ofthe SAPT components in the Ci dimer are smaller than their corresponding terms in the D2d dimer (Table 2). Comparing the (electron pair)–(electron pair) contributions to the dispersion for the Ci (Table 3) and D2d structures, we observe that the Ci dimer makes a smaller contribution to the dispersion because the σ↔σ interaction is missing. A scanned SAPT decomposition with rCM kept at 4.25 Å (Fig. 4) reveals how the attractive and repulsive terms compensate for each other, resulting in a displaced parallel structure.

Fig. 4
figure 4

Plots of SAPT components for the Ci ethylene dimer at rCM = 3.7 Å: a electrostatic interaction, b exchange repulsion, and c dispersion. The hollow arrow indicates the lowest energy minimum and the solid arrow shows the face-to-face stack structure

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As demonstrated by Fig. 4a, the electrostatic interaction is repulsive if the two ethylene molecules are directly on top of each other (i.e., θ = ϕ = 90°); it becomes attractive when one molecule approaches the edge of the other molecule, similar to what is seen for the benzene dimer [17]. The electrostatic effect will be discussed in the next section.

Interestingly, the dispersion follows the same trend as the electrostatic interaction (Fig. 4c). Minimal dispersion occurs at θ =  ϕ= 90°. Direct stacking should favor π↔π interactions. Indeed, the contribution of π↔π to the dispersion in the direct stacking geometry (θ =  ϕ = 90°) is −0.46 kcal mol−1. At the optimum π-stacking geometry, the contribution of π↔π to the dispersion is only −0.16 kcal mol−1, which is significantly smaller than that in the direct stacking structure. Despite the favorable π↔π interaction in the direct stacking geometry, the dispersion is minimal for this geometry and increases as θ and ϕ decrease. When θ and ϕ are small, the σ↔π contribution is greater than the π↔π contribution because there is a better chance that the H and π electrons will be in close contact. The (electron pair)–(electron pair) contribution to the dispersion in the CCSD(T)/CBS optimized geometry support this view. The interaction between the CH bonding σ-electron pair and the π-electron pair is much larger than that between the two π-electron pairs (Table 3).

Both the electrostatic interaction and the dispersion favor smaller θ and ϕ values. However, at small θ and ϕ, the electron density overlaps greatly, which leads to very large exchange repulsion. This is how the displaced parallel structure comes about. The attractive electrostatic and dispersion interactions share the same trend, and together they offset the repulsive exchange, achieving the optimum structure for quadrupole interactions and σ↔π interactions.

The C2v ethylene dimer (Fig. 1) is considered the best configuration for quadrupole interactions [56]. This is readily reflected in the relatively large attractive electrostatic interaction term among the ethylene dimers. The proportions of dispersion and electrostatic interactions in the total SAPT interaction energy of the C2v ethylene dimer is similar to that of the benzene–methane complex (Table 2), so it is best considered a nonactivated CH···π system [15, 16]. The C2v ethylene dimer has similar electrostatic and dispersion components to the D2d dimer, but it has a smaller interaction energy due to the larger exchange repulsion (Table 2).

The (electron pair)–(electron pair) contributions (Table 3) show more details in the intermolecular interaction for the C2v dimer. The major contribution to the dispersion comes from the interaction between the π-electron pair from the H acceptor molecule and four (out of five) C–H σ-electron pairs of the H donor. Although the total number of significant interactions between electron pairs is smaller than in the D2d dimer, each interaction term has a larger stabilizing energy. As a result, the total contribution to dispersion is comparable to that of the D2d dimer. A smaller contribution to the dispersion comes from the interaction between the π-electron pairs, although this contribution is significantly larger than those in the Ci and D2d dimers (Table 3). This means that π↔π interactions are not restricted to the stacked geometry. Indeed, a π↔π contribution to the dispersion is common to all three ethylene dimers.

The plot of SAPT components versus intermolecular distance (Fig. 5a) is very similar to that seen for the D2d dimer, except that the point at which the dispersion attraction and exchange repulsion cancel each other out occurs at a larger distance. The fact that σ↔σ interactions and some σ↔π interactions are missing from the C2v dimer may suggest that more (electron pair)–(electron pair) interactions in dispersion help to cancel out the exchange repulsion. Next, we compared the D2d and C2v structures by connecting them through a rotated CH···π structure. If we define α to be the H-donor rotation angle, a rotation of α = 90° does not change the total interaction energy significantly (Fig. 5b). This is achieved through the canceling effect of decreasing the electrostatic interaction, dispersion, and exchange repulsion. When this rotated CH···π dimer is compared to the D2d dimer, most structural parameters—such as the intermolecular distance between the centers of mass, the distance between the H atom and the center of the other ethylene molecule—stay almost the same. The most significant difference is in the H···H distance; the rotated CH···π structure has a distance of 3.06 Å while the D2d structure has a shorter distance of 2.55 Å. Three (electron pair)–(electron pair) interaction terms contribute significantly to the dispersion in the rotated CH···π dimer, compared to nine in the D2d dimer. The total contribution to the dispersion in the D2d dimer is 0.39 kcal mol−1 larger than that in the rotated CH···π dimer, while the exchange is only larger by 0.19 kcal mol−1. Both the number of significant (electron pair)–(electron pair) interactions and the changes in exchange repulsion and dispersion indicate that the σ↔σ interaction does aid dispersion in the dispersion–exchange balance, leading to more intermolecular binding.

Fig. 5
figure 5

Plots of SAPT components of the C2v ethylene dimer with respect to a intermolecular distance (rCM) and b H-donor rotation angle (α). All dispersion energies are scaled up by 15 %

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The number of significant (electron pair)–(electron pair) interactions in dispersion is determined by the number of electron pairs that are in “contact” between the two monomers, and definitely affects the balance between dispersion and exchange. Here, we introduce the term “contact” as the number of significant (electron pair)–(electron pair) interactions, whereas “significant” is defined to be larger than twice the standard deviation of all the electron-pair interactions in dispersion. In the case of ethylene dimer, the magnitude of the ratio of exchange compared to dispersion decreases as the “contact” increases.

Table 4 summarizes the number of contacts in each of the three ethylene dimer configurations. If the number of contacts is large, the increase in exchange repulsion due to the short intermolecular distance will be compensated for by the dispersion interactions between more electron pairs. As a result, the two molecules in the complex are allowed to get closer to each other. For the ethylene dimers, the electrostatic interaction becomes more negative when the intermolecular distance shortens, leading to an increase in the total interaction energy. This argument readily explains the role of σ↔σ interactions in the intermolecular binding. The existence of an σ↔σ interaction results in a wider interaction area, allowing the monomers to approach each other more closely, which enhances the electrostatic attraction in the C···H interaction. Similar analyses (Table 5) performed for other related dimers, namely ethane and butadiene dimers, confirm that increasing the contact reduces the exchange/dispersion ratio. This lends further support to our argument.

Table 4 Comparison of the balance between dispersion and exchange repulsion with the number of contacts in ethylene dimers, according to SAPT analysis performed with the aug-cc-pVDZ basis set
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Table 5 Numbers of contacts, SAPT components, and total interaction energies (in kcal mol−1) of ethylene dimers, ethane dimers, and butadiene dimer
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The concept of contact has been used for alkane dimers in order to understand their stability [36], but this concept is mainly related to structural parameters rather than electron-pair interactions. One disadvantage of such a structure-based definition is that it is difficult to count the number of contacts because there is no well-defined cutoff distance that indicates whether a pair of atoms/groups are in contact. Although it may require tedious calculations, our electron-pair-based definition will usually give a definite value for the number of contacts, which can then be used to understand the stabilities of complexes.

Charge density analysis

The presence of an σ↔σ interaction in the D2d ethylene dimer is also supported by electron density topology analysis, based on Bader’s theory of atoms in molecules (AIM) [61, 62]. For each pair of H atoms in close contact, there is a bond path linking the two hydrogen atoms (Fig. 6b). Hence, there are four bond critical points in total (Fig. 6a). The positive sign of ∇2 ρ indicates that the interaction is closed shell in nature, e.g., a hydrogen bond [61, 62]. The small ρ and positive ∇2 ρ values (Table 6) are similar to the characteristic topological properties of a weak hydrogen bond, such as CH···O [63, 64] and CH···π [58, 65] interactions. An alternate geometry of the D2d dimer in which the CH2 terminal ends of the ethylene molecules face each other does not show any significant σ↔σ interaction. Accordingly, no bond critical point is found between the hydrogen atoms.

Fig. 6
figure 6

Plots of a bond critical points, b electron density contours, and c contours of the Laplacian of electron density for the D2d, C2v, and Ci ethylene dimers

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Table 6 Calculated topological properties (in atomic units) at bond critical points (BCP), based on AIM analysis using the MP2/aug-cc-pVDZ wavefunction
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As expected, the C2v ethylene dimer is characterized by two bond critical points (Fig. 6a). Each bond path links one hydrogen atom with one carbon atom of another ethylene molecule (Fig. 6b). The small ρ and positive ∇2 ρ values (Table 6) are comparable to those found in typical CH···π interactions [58, 65]. For the Ci dimer, one bond critical point is found (Fig. 6a). The ρ and ∇2 ρ values are fairly similar to those calculated for the D2d and C2v dimers (Table 6). The rather large ellipticity (ϵ) value readily confirms the strong interaction of the π electrons.

Electrostatic interaction

From the SAPT analysis, all three ethylene dimer conformations show attractive electrostatic interactions—even the D2d dimer with close dihydrogen contacts. This indicates that close contact of the positively charged hydrogen atoms does not necessarily imply electrostatic destabilization overall. The largest stabilizing component is dispersion. However, in the equilibrium geometry, this attractive term has almost the same magnitude as the exchange repulsion, making the net interaction energy close to the electrostatic stabilization contribution. To confirm the roles of dispersion and the electrostatic interaction in the intermolecular binding, several substituted CH···CH ethylene dimers (RHC=CHR)2, where R = NH2, OH, Cl, F, and CN, have been examined. The structures were optimized with the D2d symmetry constraint (expect for R = NH2 and OH, which were optimized with the D2 and C2 constraints, respectively). The optimized geometries and results of the SAPT analysis of these substituted dimers are given in Fig. S3 of the ESM and Table 7, respectively. Electron-donating groups (NH2 and OH) increase the interaction energy, and vice versa for electron-withdrawing groups (Cl, F, and CN). Intriguingly, there is a very strong correlation (R 2 = 0.99, Fig. S4 in the ESM) between the calculated interaction energy (CCSD(T)/CBS or DFT-SAPT) and the DFT-SAPT electrostatic term. On the other hand, the correlation with the dispersion component is rather poor, R 2 = 0.20 (see Fig. S4 in the ESM). Hence, we may conclude that dispersion and electrostatic interactions play different roles in dimer binding. Dispersion determines the nature of binding and is a more important influence on the interaction distance, while the electrostatic interaction plays an important role in determining the magnitude of the interaction energy at the equilibrium distance. This finding is in agreement with the results from a study on the benzene dimer performed by Hobza and Kim et al. [27].

Table 7 DFT-SAPT components and CCSD(T)/CBS interaction energies (in kcal mol−1) of various substituted ethylene dimers [(RHC=CHR)2, R = NH2, OH, H, Cl, F, and CN]
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Interestingly, the D2d dimer shows an electrostatic attraction as large as that of the C2v dimer. What is the origin of this significant attraction? To answer this question, we calculated electrostatic interaction energies using an approach based on the interaction of distributed multipole moments [66, 67] on each monomer. The multipole moments on all atoms are divided into three groups: charges and dipoles (group A), the quadrupole that describes the π-cloud (group C), and the other quadrupoles that describe the charge distribution in the molecular plane (group B). To avoid convergence problems, the dimers were maintained in the same relative orientation, but the intermolecular distance was scaled up to 5 Å. The calculated electrostatic interaction energies among these three groups are listed in Table 8. For the C2v and Ci dimers, the electrostatic attraction is clearly dominated by the quadrupole interaction of the π-cloud, and the interaction is larger for the C2v dimer than for the Ci dimer, which validates the use of the π-atom model for these two structures [17]. For the D2d dimer, the quadrupole interaction between the π-cloud is repulsive. However, the cross-interactions between quadrupoles of different types (those representing the π-cloud and those the molecular plane) increase the attractiveness of the total electrostatic interaction. Since the H atoms are on the outer rim of the ethylene molecular plane, they are the most accessible to intermolecular interactions. The quadrupole moments representing the π-cloud are mostly on C atoms. Therefore, such a cross-quadrupole interaction could be interpreted as a special type of C···H interaction. It appears that the proposed Coulomb interaction model [30] based on bond dipoles is insufficient to describe the electrostatic interaction in the D2d ethylene dimer.

Table 8 Electrostatic interaction energies (in kcal mol−1) decomposed to multipole interactions for ethylene dimers
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Conclusions

This study focused on the nature of the binding of each of the three ethylene dimer configurations D2d, Ci, and C2v. The D2d dimer, which is characterized by four pairs of dihydrogen contacts and the largest dispersion term, is the most stable form. The other two configurations, the Ci and C2v dimers, are almost equal in net interaction energy. Unlike the benzene dimer, the displaced parallel (Ci) dimer is not a stable structure. The stacking effect of π electrons is not significant in this very small π-containing system.

All three ethylene dimers have negative electrostatic interaction energies. Similar to benzene dimers, the largest stabilizing force in ethylene dimers is dispersion. However, this contribution to the stabilization is of almost the same magnitude as the exchange repulsion at the equilibrium distance. As a consequence, the electrostatic interaction is a significant influence on the interaction energies at the equilibrium distances of these ethylene dimers. The electrostatic stabilizations of C2v and Ci dimers are readily described by π atoms. In the case of the D2d dimer, it is important to consider the special attraction between C and H atoms to explain the attractive electrostatic term. As also seen in other neutral π-complexes, induction and induction–dispersion are very small in all three ethylene dimers.

For all three ethylene dimers, dispersion can arise from a π↔π, σ↔π, or σ↔σ (electron pair)–(electron pair) interaction. The σ↔π interaction is favored over the π↔π interaction, even in the displaced parallel dimer, probably because the system is too small. The σ↔σ interaction, which is unique to the D2d dimer, does indeed exist. It helps to stabilize the D2d dimer by increasing the number of electron-pair interactions in dispersion to alter the exchange–dispersion balance. A concept of “contact“ was introduced to describe such a balance. A larger number of contacts will help to reduce intermolecular distance, and in complexes where the attractive electrostatic interaction determines the degree of intermolecular binding strength, this usually means greater stability.