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Article

Non-Covalent Interactions Involving Alkaline-Earth Atoms and Lewis Bases B: An ab Initio Investigation of Beryllium and Magnesium Bonds, B···MR2 (M = Be or Mg, and R = H, F or CH3)

1
Instituto de Química Médica (IQM-CSIC), Juan de la Cierva, 3, E-28006 Madrid, Spain
2
School of Chemistry, University of Bristol, Cantock’s Close, Bristol BS8 1TS, UK
*
Authors to whom correspondence should be addressed.
Inorganics 2019, 7(3), 35; https://doi.org/10.3390/inorganics7030035
Received: 30 January 2019 / Revised: 15 February 2019 / Accepted: 26 February 2019 / Published: 5 March 2019
(This article belongs to the Special Issue Halogen Bonding: Fundamentals and Applications)

Abstract

:
Geometries, equilibrium dissociation energies (De), intermolecular stretching, and quadratic force constants (kσ) determined by ab initio calculations conducted at the CCSD(T)/aug-cc-pVTZ level of theory, with De obtained by using the complete basis set (CBS) extrapolation [CCSD(T)/CBS energy], are presented for the B···BeR2 and B···MgR2 complexes, where B is one of the following Lewis bases: CO, H2S, PH3, HCN, H2O or NH3, and R is H, F or CH3. The BeR2 and MgR2 precursor molecules were shown to be linear and non-dipolar. The non-covalent intermolecular bond in the B···BeR2 complexes is shown to result from the interaction of the electrophilic band around the Be atom of BeR2 (as indicated by the molecular electrostatic potential surface) with non-bonding electron pairs of the base, B, and may be described as a beryllium bond by analogy with complexes such as B···CO2, which contain a tetrel bond. The conclusions for the B···MgR2 series are similar and a magnesium bond can be correspondingly invoked. The geometries established for B···BeR2 and B···MgR2 can be rationalized by a simple rule previously enunciated for tetrel-bonded complexes of the type B···CO2. It is also shown that the dissociation energy, De, is directly proportional to the force constant, kσ, in each B···MR2 series, but with a constant of proportionality different from that established for many hydrogen-bonded B···HX complexes and halogen-bonded B···XY complexes. The values of the electrophilicity, EA, determined from the De for B···BeR2 complexes for the individual Lewis acids, A, reveal the order A = BeF2 > BeH2 > Be(CH3)2—a result that is consistent with the −I and +I effects of F and CH3 relative to H. The conclusions for the MgR2 series are similar but, for a given R, they have smaller electrophilicities than those of the BeR2 series. A definition of alkaline-earth non-covalent bonds is presented.

Graphical Abstract

1. Introduction

The non-covalent interactions of closed-shell molecules represent an important subject in many areas of chemistry and biology. The central position of the hydrogen bond in these disciplines is well known. Since the 1950s, there has been a rapid growth of interest in other non-covalent interactions. The halogen bond was first named and identified experimentally in the solid state in the 1950s by Hassel [1], and then in the gas phase as a weak interaction involving simple Lewis bases with di-halogen molecules in the 1990s [2]. The halogen bond was shown [2,3] to have properties similar to those of the hydrogen bond. Interest in the halogen bond has grown rapidly within chemistry, biology and materials science in the last two decades [4,5]. The comprehensive definitions of the hydrogen bond and the halogen bond by working parties set up by IUPAC were published in 2011 [6] and 2013 [7], respectively. The definition of the halogen bond explicitly invokes the interaction of a halogen atom (acting as an electrophile) with a non-bonding or π-bonding electron pair (the nucleophilic region) of, for example, a Lewis base. Tetrel bonds, pnictogen bonds, and chalcogen bonds are non-covalent interactions that have been investigated extensively in the gas phase [8] and condensed phase [9] since the 1970s, but were only named according to the group in the periodic table from which the atom acting as the electrophile originates (Groups 14, 15 and 16, respectively) in 2013 [10], 2011 [11], and 2009 [12], respectively. The IUPAC definitions of these newer types of interactions, similar to that of the halogen bond, are imminent [13]. However, the general applicability of such definitions based on electrostatics alone has been questioned in the case of some of the more unusual types of non-covalent interactions [14,15,16,17]. Other non-covalent interactions involving atoms of other groups in the periodic table acting as the electrophilic region can be identified. A recent example is the so-called coinage-metal bond B···MX, where B is a Lewis base and M is a Group 11 metal atom [18].
In this article, we report an investigation, by means of high-level ab initio calculations, of B···BeR2 and B···MgR2 complexes in which B is one of the six simple Lewis bases CO, H2S, PH3, HCN, H2O or NH3 and R is H, F or CH3. We will show that various Lewis acid molecules, BeR2 and MgR2, are linear, non-dipolar, and of geometry R–Be–R and R–Mg–R. In each case, we also show, from the molecular electrostatic surface potentials, that there is a positive belt around the central Group 2 atom which can act as the electrophilic region when forming either a beryllium or a magnesium bond [19] to the most nucleophilic region (a non-bonding electron pair) of the Lewis base. As well as the geometry optimizations of the complexes, we also calculate two measures of the binding strength, namely, the equilibrium dissociation energy, De, and the intermolecular stretching force constant, traditionally referred to as kσ [2]. The first is the energy required to remove the component molecules from the hypothetical equilibrium separation to infinite distance, while the second is a measure of the work required for a unit infinitesimal displacement from the equilibrium. It has been shown [20,21,22] that for a wide range of hydrogen-, halogen-, tetrel-, pnictogen- and chalcogen-bonded complexes, De is directly proportional to kσ and, moreover, that it is possible to reproduce the De values (and, therefore, the kσ values also) by assigning a set of electrophilicities, EA, to the Lewis acids, A, and nucleophilicities, NB, to the Lewis bases, B. An important aim of the present article is to discover whether this partitioning also applies to beryllium- and magnesium-bonded complexes.
Another aim of this study is to examine the effects of replacing both H atoms in H–Be–H and H–Mg–H, firstly by F and secondly by CH3 groups. According to the electronic theory of organic chemistry developed by Ingold [23] and in particular the inductive effect I, F removes electronic charge from the central atom relative to the hydride (the −I effect), while the methyl group pushes electrons towards the central atom through the +I effect. If so, the central Group 2 atom should become more electrophilic (EA should increase relative to that of the dihydride) in F–Be–F and F–Mg–F, but less electrophilic (decease of EA) in CH3–Be–CH3 and CH3–Mg–CH3. This conclusion is confirmed by the molecular electrostatic potential surfaces (MEPS) of F–Be–F, H–Be–H and CH3–Be–CH3. These were calculated for the 0.001 e/bohr3 electron density isosurface at the CCSD/aug-cc-pVTZ//CCSD(T)/aug-cc-pVTZ level of theory with the Gaussian-16 Program [24] and are shown in Figure 1. In each case, there is a blue belt that surrounds the central Be atoms. The deepest blue color corresponds to the most positive MEPS in each case and has a maximum value of 337, 167 and 119 kJ·mol−1 for F–Be–F, H–Be–H and CH3–Be–CH3, respectively. Thus, the blue belt surrounding the Be atom is the most electrophilic region in each molecule and the electrophilicity is greatest when F is the ligand and smallest when CH3 is the ligand, in agreement with the −I and +I inductive effects of F and CH3, respectively. Similar patterns are observed from the MEPSs of the Mg analogues (see Supplementary Material, Figure S1), except that for a given ligand, R, the maximum positive potential is higher for Mg than for Be, with values of 753, 321 and 280 kJ·mol−1, for R = F, H and CH3, respectively.

2. Results

2.1. Molecular Geometries

The molecular diagrams (drawn to scale) of the geometries of the three Lewis acids BeF2, BeH2 and Be(CH3)2 optimized at the CCSD(T)/aug-cc-pVTZ level of theory are shown in Figure 2. The geometries belong to the point groups D∞h, D∞h and D3d, respectively, and are consistent with two singly occupied sp hybrid orbitals on the central Be atom forming bonds with F, H or C, respectively. The similarly determined geometries for the three Mg analogues are isostructural with their Be counterparts, but are not shown. They are available from the Supplementary Material, which includes the optimized cartesian coordinates of atoms for all molecules investigated here.
Figure 3 displays the molecular diagrams (drawn to scale) of the six B···BeF2 complexes in which B = CO, HCN, H2O, NH3, H2S or PH3. The molecular diagrams of the corresponding sets of six B···BeH2 and B···Be(CH3)2 complexes are shown in Figure 4 and Figure 5, respectively. In each case, the fragment R2Be···L, where L is the atom of B involved in the intermolecular bond, is Y-shaped (local symmetry C2v). Thus, the angle, θ (which is defined in Figure 3), is zero in the BeR2 monomer molecules, but increases significantly in all B···BeR2 complexes investigated, as indicated by the values included in Table 1. The Y shape can be explained if it is assumed that, when the Lewis base, B, approaches R–Be–R and forms the complex, the hybridization at the central Be atom starts to change to sp2 and the third (empty) sp2 orbital receives the non-bonding electron pair of B with the result that a partial dative bond Be–L is formed with the acceptor atom of B. It is clear from Table 1 that the angles R–Be–R are all less than 180° in the B···BeR2 complexes but are greater than the ideal sp2 angles of 120° that would occur for a fully dative bond (i.e., 0° < θ < 30°). The BeCl3 anion [25] has three equivalent Be–Cl bonds and D3h symmetry, with ideal 120° angles. There are also increases δr in the distances r(R–Be) on formation of all B···BeR2 complexes considered here, as expected for the partial change from sp to sp2 hybridization at Be. The values of δr for all B···BeR2 complexes investigated are included in Table 1.
The observations about the angle θ and the increase δr in the distances r(R–Mg) also apply to the formation of the B···MgR2 complexes from the various MgR2 molecules. Table 2 includes these quantities for the 18 complexes that result from the interaction of the three MgR2 molecules (R = F, H or CH3) with the set of six Lewis bases, B = CO, HCN, H2O, NH3, H2S or PH3. The full geometries of these complexes are available in the form of the cartesian coordinates in the Supplementary Material. We note from Table 1 and Table 2 that the distance r(Mg···L) is correlated with the strength of the interaction in the Mg series, in the sense that shorter distances are associated with larger De values; the correlation is less clear in the Be series.

2.2. Relationship between De and kσ

The two measures (De and kσ) of the binding strength obtained through ab initio calculations for the 18 B···BeR2 complexes discussed in Section 2.1 are given in Table 1. The corresponding quantities for the 18 B···MgR2 are in Table 2. It should be noted, from Table 1 and Table 2, that these complexes tend to be more strongly bound according to both criteria (De and kσ) than those of a wide range of hydrogen-, halogen-, tetrel-, pnictogen- and chalcogen-bonded complexes with a similar set of Lewis bases previously investigated [20,21,22]. Typically, for the hydrogen- and halogen-bonded complexes considered in [22], for example, De ≈ 20 kJ·mol−1 and kσ ≈ 10 N·m−1. This larger binding strength of the B···BeR2 and B···MgR2 complexes is reflected in the significant geometrical distortions in BeR2 and MgR2 on complex formation noted in Section 2.1. Given the direct proportionality of De and kσ established in refs. [20,21,22] for hydrogen- and halogen-bonded complexes, it is of interest to examine whether a similar relationship between the two quantities holds for the B···BeR2 and B···MgR2 complexes discussed here.
Figure 6 shows a plot of De versus kσ for the 18 B···BeR2 complexes (B = CO, HCN, H2O, NH3, H2S or PH3; R = F, H or CH3). The result of a linear regression fit to the points is also shown. The points lie on a reasonably good straight line, which passes through the origin. Two minima at the CCSD(T)/aug-cc-pVTZ level were found for OC···Be(CH3)2. The first minimum occurs at a Be···C distance of 2.19 Å with De = 3.66 kJ·mol−1, while the second (and global) minimum is at 2.92 Å with De of 5.28 kJ·mol−1. The barrier between the two minima is less than 0.01 kJ·mol−1. Figure 7 is the plot of De versus kσ for the 18 B···MgR2 complexes. Thus, as found for a wide range of hydrogen-bonded B···HX complexes, halogen-bonded B···XY complexes and tetrel-, pnictogen- and chalcogen-bonded complexes [20,21], De is, in good approximation, directly proportional to kσ for both B···BeR2 and B···MgR2 series; that is, De = c’·kσ, where c’ is the constant of proportionality.
Although a single value of c’ = 1.40(4) × 103 m2·mol−1 was obtained by fitting all five types of complexes (hydrogen-, halogen-, tetrel-, pnictogen- and chalcogen-bonded) discussed in [20], the values of c’ obtained from the linear regressions in Figure 6 and Figure 7 for B···BeR2 and B···MgR2 are significantly smaller at 0.79(5) × 103 m2·mol−1 and 1.07(6) × 103 m2·mol−1, respectively. It should be noted, however, that the beryllium and magnesium bonds considered here are much stronger for a given B and the molecular distortions on formation of these bonds are greater than those for the other five types of non-covalent interactions listed. Plots of De versus kσ for B···BeR2 and B···MgR2 complexes for a given Lewis base, B, with a variation of the six Lewis acids (R = H, F and CH3) show much weaker correlation and are less informative. Oliveira, Kraka and Cremer [14,26] have published plots which show the variation of relative bond strength order versus local stretching force constant as a gentle, smooth curve for many halogen- and chalcogen-bonded complexes.

2.3. Nucleophilicities of B and Electrophilicities of BeR2 and MgR2 (R = F, H or CH3)

It has been shown that for complexes involving hydrogen bonds, halogen bonds, tetrel bonds, pnictogen bonds and chalcogen bonds, De can be represented by an equation of the type
De = cNBEA + d
where NB is the nucleophilicity of the Lewis base, B, EA is the electrophilicity of the Lewis acid, A, and c and d are constants. It is convenient to define c = 1.00 kJ·mol−1 so that NB and EA are dimensionless. Given the direct proportionality of De and kσ, Equation (1) can be recast with kσ as the subject and indeed it was with that version of the expression that NB and EA were first proposed for hydrogen-bonded complexes [27]. Here, we will use the version defined as Equation (1). It has also been established that the constant term, d, is usually small and can be negligible. Whether or not that is the case, the plots of De versus NB are usually good straight lines and it follows then that the gradient is dDe/dNB = cEA. In the earlier determinations of NB and EA for the B···HX complexes (X = F, Cl, Br, etc.), the following procedure was used. The values of NB were assigned to the various Lewis bases so that the plot of De (or kσ) versus NB for the B···HF complexes is a straight line through the origin. The sets of De for the B···HCl, B···HBr, etc., complexes were then plotted against NB values so defined to give good straight lines, the gradients of which then defined the electrophilicities of the various HX molecules. An alternative procedure, used in [20], is to assign NB and EA values by a global fit of the De values of 250 complexes held together by a wide range of non-covalent bonds. The graphical approach, however, is useful for illustrating systematic relationships between different series of complexes and is employed here for the six BeR2 and MgR2 series (R = F, H or CH3).
Figure 8 shows the plots of De versus NB for the series of B···MgF2, B···MgH2 and B···Mg(CH3)2 complexes when B = CO, HCN, H2O, NH3, H2S or PH3. The values of NB are those appropriate to the B···HF series when NNH3 is set to 7.5 to be consistent with its value reported in [20]. The remainder of NB are those chosen so that the points in a plot of De versus NB for all the B···HF complexes (data from [20]) lie on a straight line through the origin and are given in Table 3. This line for the B···HF is included in Figure 8 together with plots of De versus NB for B···HCl and B···HBr (De values from [20]) against the set of NB defined by B···HF. The straight lines for the B···MgR2 complexes are from least squares fits of the points (but with the points for B = H2O excluded for reasons given below) for each series and the gradients of the fits dDe/dNB = cEA lead to the EA values for A = MgF2, MgH2, Mg(CH3)2, HF, HBr and HCl listed in Table 3. The corresponding diagram for the B···BeR2 series is in Figure 9, in which the plots for B···HX (X= F, Cl and Br) are included. The points for H2O···BeR2 were again excluded from the linear regression fits. The values of EA derived from the gradients are in Table 3. The NB and EA values determined from the global fit of the De values of 250 hydrogen-, halogen-, tetrel-, pnictogen- and chalcogen-bonded complexes [20] are included in Table 3 for comparison. It is clear that there is reasonably good agreement between the NB values obtained here and those in ref. [20]. The same good agreement holds for the EA values of HCl and HBr. The reason for excluding the De values of the H2O···MgR2 and H2O···BeR2 complexes from Figure 8 and Figure 9, respectively, is that they imply NH2O values which significantly exceed those obtained from the B···HF data here (5.24) or from the global fit (4.89) in ref. [20] for H2O. If the value of De for each H2O···MgR2 were forced to lie on its appropriate regression line in Figure 8, the value NH2O ≈ 6.4 would be necessary for each R. A similar conclusion applies for the B···BeR2 complexes, implying that NH2O ≈ 6.1. Thus, H2O has a higher electrophilicity for the MR2 molecules than it does for HF. This could be related to the efficacy of water as a solvent for ions.
It is possible to estimate a value of the electrophilicity index, ω, as defined by the conceptual DFT method [28]. This index is given in terms of the energies of the lowest energy-unoccupied and the highest energy-occupied molecular orbitals (ELUMO and EHOMO), respectively, by the expression
ω ( E HOMO + E LUMO ) 2 / 8 ( E LUMO E HOMO )
When ELUMO and EHOMO are calculated at the CCSD/aug-cc-pVTZ//CCSD(T)/aug-cc-pVTZ level of theory, the results for ω are 1.97, 1.31 and 1.20 eV for BeF2, BeH2 and Be(CH3)2, respectively, and 1.92, 1.11 and 1.03 eV for MgF2, MgH2 and Mg(CH3)2, respectively. Figure 10 shows a plot of the EA values from the present work against ω. There is a reasonable correlation between the two measures of the electrophilicity of the six MR2.

3. Theoretical Methods

The equilibrium geometries, dissociation energies, De, and force constants, kσ, were obtained at the CCSD(T) computational level [29] for each B···BeR2 and B···MgR2 complex investigated. In the first step of the calculations, the geometry of the monomers and complexes was optimized with the aug-cc-pVTZ basis set [30] at the CCSD(T) level. A geometry scan of the intermolecular distance of ±0.1 Å from the optimized value, re, was then determined in steps of   ( r r e ) = 0.025 Å at the same computational level to yield the variation of the energy E ( r r e ) with the displacement ( r r e ) from equilibrium. As an example, the resulting curve for the OC⋯BeF2 complex is given in Figure 11. Such curves were then fitted by a third-order polynomial in ( r r e ) , from which kσ is obtained as the numerical value of the second derivative of E with respect to ( r r e ) evaluated at re. In order to obtain more accurate De values, complete basis set (CBS) extrapolation [CCSD(T)/CBS energy] was executed by using the CCSD(T)/aug-cc-pVTZ//CCSD(T)/aug-cc-pVTZ and CCSD(T)/aug-cc-pVQZ//CCSD(T)/aug-cc-pVTZ energies for all the systems [31,32]. Thus, the De values have been obtained as the difference of the CCSD(T)/CBS energy of the monomers and the complex. All ab initio calculations were performed with the MOLPRO-2012 program [33]. The molecular electrostatic potential surfaces of the various BeR2 and MgR2 monomers were calculated on the 0.001 e/bohr3 electron density isosurface at the CCSD/aug-cc-pVTZ//CCSD(T)/aug-cc-pVTZ level of theory by using the Gaussian-16 Program [24]. Table 1 and Table 2 include the De and kσ values for all complexes investigated here.

4. Conclusions

Ab initio calculations at the CCSD(T)/aug-cc-pVTZ level have yielded the geometries, intermolecular stretching force constants, kσ, and dissociation energies, De, of the 18 B···BeR2 complexes (B = CO, HCN, H2O, NH3, H2S or PH3 and R = F, H or CH3) and of the corresponding set of complexes in which Be is replaced by Mg. In all cases, De was determined by using the complete basis set extrapolation. The dissociation energies, De, reveal that, for a given R, the complexes involving Mg are more strongly bound than those involving Be—a conclusion that is consistent with the greater maximum positive MEPS for the former (see Figure 1 and Figure S1 of Supplementary Material). It has been shown that all the complexes have a Y shape that can be understood as follows. The free MR2 molecules are linear (see Figure 2). The following process may then be envisaged. The Lewis base, B, is assumed to approach MR2 so that the non-bonding electron pair of B (the most nucleophilic region of B) interacts with the belt of high electrophilicity that lies around the M atom (see blue regions in Figure 1) to give an initially T-shaped complex. As the Lewis base becomes closer, the linear R-M-R subunit distorts, with the R atoms/groups moving away from B to give the Y shape. One might envisage the following electronic description of the process. The two valence-shell electrons of the metal atom, M, in MR2 are assumed to singly occupy spz hybrids, which then form single bonds with F or H or C to give the linear molecules F–M–F, H–M–H and H3C–M–CH3, respectively. The electrophilic (relatively positive) belt around the metal atom, M, and perpendicular to the F–M–F line, is presumably a consequence of the empty npx and npy orbitals (n =2 for M = Be and n = 3 for M = Mg). As the non-bonding pair of B approaches and interacts with an empty px or py orbital, the hybridization at M changes gradually to take on some spm character. As m increases from 1 to 2, the angular deviation, θ (see Figure 3 for the definition of θ) from linearity, should increase from 0° to 30°, the latter corresponding to an R–M–R angle of 120°. We note from Table 1 and Table 2 that for a given M and R, the angle, θ, tends to increase as the binding strength (De or kσ) increases and about 20° for the most strongly bound complexes, namely, those involving H2O and NH3 with BeR2. Moreover, the lengthening δr(M–R) of the M–R bond tends to increase with binding strength. Both observations are consistent with a change from sp towards sp2 hybridization. Thus, it appears that the interaction of B and MR2 can be described as partly electrostatic and partly dative in character. It is noted that the dative bond character appears greater when M = Be than when M = Mg, with the non-linearities θ closer to 30°, with larger values of δr(M–R) and presumably values of m closer to 2 in the spm hybridization scheme. It appears, therefore, that these are not purely σ-hole/n-pair interactions.
The shapes of the B···BeR2 and B···MgR2 complexes can be predicted by a simple modification to a rule recently enunciated [20] for tetrel-bonded complexes of the type B···CO2, that is:
The equilibrium geometry of alkaline-earth bonded B···MR2 complexes (M = Be, Mg…) can be predicted by assuming that a radius of the most electrophilic ring around the M atom that is perpendicular to the MR2 line coincides with the axis of a non-bonding electron pair carried by B. Some deviation of MR2 from collinearity could occur.
For both sets of B···BeR2 and B···MgR2 complexes, it has been established that De is directly proportional to kσ to a good degree of approximation, as seen from Figure 6 and Figure 7. Moreover, as with more weakly bound complexes such as B···HX (X = F, Cl, Br), it has been possible to partition De into contributions from the individual molecules B and MR2, called the nucleophilicity, NB, of the Lewis base, B, and the electrophilicity, EA, of the Lewis acid, A, respectively. As may be seen from Table 3, the order of the EA values for both BeR2 and MgR2 sets when acting as Lewis acids is R = F > H ≥ CH3, which is the order expected from the −I inductive effect of F relative to H and the +I effect of the CH3 group relative to H, and is the order indicated by the MEPS in Figure 1. The −I effect of F is evidently greater than the +I effect of CH3. It is also clear from Table 3 that for a given R, the electrophilicity of BeR2 is greater than that of MgR2. This appears to be at variance with the MEPS, because the electrophilic (blue) belt around M is more positive for M = Mg than Be, with, for example, the maximum positive potentials for MgF2 and BeF2 at 753 and 337 kJ mol−1, respectively (see Figure 1 and Introduction). It is of interest that the order of electrophilicities given in Table 3 is BeF2 > BeH2.> Be(CH3)2 ~ MgF2 > MgH2 > Mg(CH3)2 >> HF > HBr ~ HCl, which indicates just how effective BeR2 and MgR2 are as Lewis acids. Various other scales of nucleophilicity and electrophilicity have been proposed. Some are based on the rate constants for organic reactions in solution [34], while others have been based on conceptual density functional theory (CDFT) [28]. A comparison of our results for the EA of MR2 with those estimated by the CDFT approach has been presented.
We have shown that the BeR2 and MgR2 Lewis acids discussed here undergo non-covalent interactions with a series of Lewis bases, all of which can provide a non-bonding electron pair to interact with the electrophilic belt that encircles the central metal atom in MR2. Evidently, these interactions can be described as beryllium bonds and magnesium bonds, respectively, by analogy with the recent definitions [6,7,18] of other non-covalent interactions such as halogen-, tetrel-, pnictogen-, chalcogen- and coinage-metal bonds. Therefore, we propose the following definition:
A alkaline-earth non-covalent bond occurs when there is evidence of a net attractive interaction between an electrophilic region associated with an atom of an element, E{II}, in a molecular entity and a nucleophilic region (e.g., a n-pair or π-pair of electrons) in another, or the same, molecular entity, where E{II} is an element of Group II in the periodic table.
Note that this definition is coherent with the IUPAC definition of the halogen bond [7].

Supplementary Materials

The following are available online at https://www.mdpi.com/2304-6740/7/3/35/s1, Figure S1: Molecular electrostatic surface potentials of the linear, non-polar molecules, MgF2, MgH2 and Mg(CH3)2 calculated at the 0.001 e/bohr3 electron density isosurface at the CCSD/aug-cc-pVTZ//CCSD(T)/aug-cc-pVTZ level of theory, Table S1: Optimized geometry, electronic energy and Variation of the energy E(r−re) as a function of the displacement (rre) from the global minimum at re at the CCSD(T)/aug-cc-pVTZ computational level.

Author Contributions

The conceptualization of the project, the calculations, the writing of the manuscript, the drawing of figures, checking of proofs, etc. were shared between I.A. and A.C.L.

Funding

This research was funded by Consejería de Educación e Investigación de la Comunidad de Madrid (P2018/EMT-4329 AIRTEC-CM) and Ministerio de Ciencia, Innovación y Universidades (PGC2018-094644-B-C22).

Acknowledgments

I.A. thanks Consejería de Educación e Investigación de la Comunidad de Madrid (P2018/EMT-4329 AIRTEC-CM) and Ministerio de Ciencia, Innovación y Universidades (PGC2018-094644-B-C22). A.C.L. thanks the University of Bristol for a Senior Research Fellowship.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Molecular electrostatic potential surfaces of the linear non-polar molecules, BeF2, BeH2 and Be(CH3)2 calculated at the 0.001 e/bohr3 electron density isosurface at the CCSD/aug-cc-pVTZ//CCSD(T)/aug-cc-pVTZ level of theory. The surface has been is made transparent to reveal the molecular model within. The most intense blue (and, therefore, the most electrophilic) belts centered on Be correspond to positive electrostatic potential energies of 337, 167 and 119 kJ·mol−1 for BeF2, BeH2 and Be(CH3)2, respectively, and confirm expectations based on the inductive effects of CH3 and F relative to H.
Figure 1. Molecular electrostatic potential surfaces of the linear non-polar molecules, BeF2, BeH2 and Be(CH3)2 calculated at the 0.001 e/bohr3 electron density isosurface at the CCSD/aug-cc-pVTZ//CCSD(T)/aug-cc-pVTZ level of theory. The surface has been is made transparent to reveal the molecular model within. The most intense blue (and, therefore, the most electrophilic) belts centered on Be correspond to positive electrostatic potential energies of 337, 167 and 119 kJ·mol−1 for BeF2, BeH2 and Be(CH3)2, respectively, and confirm expectations based on the inductive effects of CH3 and F relative to H.
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Figure 2. Geometries of BeF2, BeH2 and Be(CH3)2 optimized at the CCSD(T)/aug-cc-pVTZ level of theory (to scale).
Figure 2. Geometries of BeF2, BeH2 and Be(CH3)2 optimized at the CCSD(T)/aug-cc-pVTZ level of theory (to scale).
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Figure 3. Geometries (drawn to scale) of six B···BeF2 complexes optimized at the CCSD(T)/aug-cc-pVTZ level of theory, where B = CO, HCN, H2O, NH3, H2S and PH3.
Figure 3. Geometries (drawn to scale) of six B···BeF2 complexes optimized at the CCSD(T)/aug-cc-pVTZ level of theory, where B = CO, HCN, H2O, NH3, H2S and PH3.
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Figure 4. Geometries (drawn to scale) of six B···BeH2 complexes optimized at the CCSD(T)/aug-cc-pVTZ level of theory, where B = CO, HCN, H2O, NH3, H2S and PH3.
Figure 4. Geometries (drawn to scale) of six B···BeH2 complexes optimized at the CCSD(T)/aug-cc-pVTZ level of theory, where B = CO, HCN, H2O, NH3, H2S and PH3.
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Figure 5. Geometries (drawn to scale) of six B···Be(CH3)2 complexes optimized at the CCSD(T)/aug-cc-pVTZ level of theory, where B = CO, HCN, H2O, NH3, H2S and PH3.
Figure 5. Geometries (drawn to scale) of six B···Be(CH3)2 complexes optimized at the CCSD(T)/aug-cc-pVTZ level of theory, where B = CO, HCN, H2O, NH3, H2S and PH3.
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Figure 6. Variation of ab initio-calculated values of De with kσ for 18 B···BeR2 complexes (R = F, H or CH3; B = CO, HCN, H2O, NH3, H2S or PH3). For the linear regression, R2 = 0.939.
Figure 6. Variation of ab initio-calculated values of De with kσ for 18 B···BeR2 complexes (R = F, H or CH3; B = CO, HCN, H2O, NH3, H2S or PH3). For the linear regression, R2 = 0.939.
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Figure 7. Variation of ab initio calculated values of De with kσ for 18 B···MgR2 complexes (R =F, H or CH3; B = CO, HCN, H2O, NH3, H2S or PH3). For the linear regression, R2 = 0.952.
Figure 7. Variation of ab initio calculated values of De with kσ for 18 B···MgR2 complexes (R =F, H or CH3; B = CO, HCN, H2O, NH3, H2S or PH3). For the linear regression, R2 = 0.952.
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Figure 8. De versus the nucleophilicity, NB, for the B···MgR2 series and B···HX complexes (B = CO, HCN, H2O, NH3, H2S and PH3; R = F, H or CH3; X = F, Cl or Br). The NB values are defined by the B···HF straight line through the origin (see text for details). The points for H2O···MgR2 were excluded from the regression fits for reasons discussed in the text. The lines and points for B···HCl and B···HBr are almost coincident. (R2 = 0.994, 0.994, 0.990, 1.000, 0.993 and 0.988 for the Mg(CH3)2, MgH2, MgF2, HF, HCl and HBr lines, respectively).
Figure 8. De versus the nucleophilicity, NB, for the B···MgR2 series and B···HX complexes (B = CO, HCN, H2O, NH3, H2S and PH3; R = F, H or CH3; X = F, Cl or Br). The NB values are defined by the B···HF straight line through the origin (see text for details). The points for H2O···MgR2 were excluded from the regression fits for reasons discussed in the text. The lines and points for B···HCl and B···HBr are almost coincident. (R2 = 0.994, 0.994, 0.990, 1.000, 0.993 and 0.988 for the Mg(CH3)2, MgH2, MgF2, HF, HCl and HBr lines, respectively).
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Figure 9. De versus the nucleophilicity, NB, for the series B···BeR2 and B···HX complexes (B = CO, HCN, H2O, NH3, H2S and PH3; R = F, H or CH3; X = F, Cl or Br). The NB values are defined by the B···HF straight line through the origin (see text for details). The points for H2O···BeR2 were excluded from the regression fits for reasons discussed in the text. The regression lines and points for B···HCl and B···HBr are almost coincident. To avoid congestion, the regression line for the B···Be(CH3)2 points has been omitted. (R2 = 0.994, 0.996, 0.998, 1.000.0.993 and 0.988 for Be(CH3)2, BeH2, BeF2, HF, HCl and HBr lines, respectively).
Figure 9. De versus the nucleophilicity, NB, for the series B···BeR2 and B···HX complexes (B = CO, HCN, H2O, NH3, H2S and PH3; R = F, H or CH3; X = F, Cl or Br). The NB values are defined by the B···HF straight line through the origin (see text for details). The points for H2O···BeR2 were excluded from the regression fits for reasons discussed in the text. The regression lines and points for B···HCl and B···HBr are almost coincident. To avoid congestion, the regression line for the B···Be(CH3)2 points has been omitted. (R2 = 0.994, 0.996, 0.998, 1.000.0.993 and 0.988 for Be(CH3)2, BeH2, BeF2, HF, HCl and HBr lines, respectively).
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Figure 10. The relationship between the conceptual DFT electrophilicity index, ω, calculated from Equation (2) at the CCSD/aug-cc-pVTZ//CCSD(T)/aug-cc-pVTZ level of theory, and the EA determined here for various MR2 molecules (M = Be or Mg, R = F, H, or CH3).
Figure 10. The relationship between the conceptual DFT electrophilicity index, ω, calculated from Equation (2) at the CCSD/aug-cc-pVTZ//CCSD(T)/aug-cc-pVTZ level of theory, and the EA determined here for various MR2 molecules (M = Be or Mg, R = F, H, or CH3).
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Figure 11. Variation of the energy E ( r r e ) of OC···BeF2 as a function of the displacement ( r r e ) from the global minimum at re along the C2 axis of this Y-shaped complex. (F–Be–F) forms the arms of the Y and CO forms the stem. See Figure 3 for a molecular diagram. The geometry was re-optimized at each of the indicated points and the line through the points is the third-order polynomial curve from the regression fit to the points. The second derivative evaluated at re gives the intermolecular stretching force constant, kσ. The corresponding curves and the fitted polynomials for all B···BeR2 and B···MgR2 complexes (B = CO, H2S, PH3, HCN, H2O or NH3; R = F, H or CH3) investigated here are available in the Supplementary Material.
Figure 11. Variation of the energy E ( r r e ) of OC···BeF2 as a function of the displacement ( r r e ) from the global minimum at re along the C2 axis of this Y-shaped complex. (F–Be–F) forms the arms of the Y and CO forms the stem. See Figure 3 for a molecular diagram. The geometry was re-optimized at each of the indicated points and the line through the points is the third-order polynomial curve from the regression fit to the points. The second derivative evaluated at re gives the intermolecular stretching force constant, kσ. The corresponding curves and the fitted polynomials for all B···BeR2 and B···MgR2 complexes (B = CO, H2S, PH3, HCN, H2O or NH3; R = F, H or CH3) investigated here are available in the Supplementary Material.
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Table 1. Some ab initio calculated properties of the B···BeR2 complexes (R = F, H orCH3) for six different Lewis bases B a.
Table 1. Some ab initio calculated properties of the B···BeR2 complexes (R = F, H orCH3) for six different Lewis bases B a.
ComplexLewis Base BDe/kJ·mol−1kσ/N·m−1r(Be···A)/Å bAngle θcδr(Be–R)/Å d
B⋯BeF2CO26.7236.332.04015.00.024
NCH66.9887.591.81819.20.035
H2O95.94121.891.69718.70.040
NH3121.73133.191.77721.10.045
H2S43.5744.592.28916.90.029
PH341.5945.872.33717.70.035
B⋯BeH2CO21.2944.611.94216.30.019
NCH53.6785.381.79019.10.026
H2O80.94110.931.68818.00.030
NH3102.10123.111.78320.50.035
H2S34.5837.912.27016.00.021
PH334.0842.862.30517.00.023
B⋯Be(CH3)2CO5.282.002.9223.20.004
NCH32.7557.731.84418.10.035
H2O57.8282.211.72018.70.040
NH377.89104.241.80920.00.046
H2S16.9714.072.42514.10.025
PH314.1915.022.45614.80.027
a Calculations were performed at the CCSD(T)/aug-ccpVTZ level. De was obtained from a complete basis set (CBS) extrapolation. See Section 3 for details. b r(Be···A) is the distance between the Be atom and the nearest atom, L, of the Lewis base B. c The angle, θ, is the angular displacement of each group, R, in the complex from the straight line, R–Be–R defined in the free molecule (see Figure 3). d δr(Be–R) is the increase in the Be–R bond length (R = F, H or CH3) when B···BeR2 is formed from B and BeR2.
Table 2. Some ab initio calculated properties of the B···MgR2 complexes (R = F, H or CH3) for six different Lewis bases B a.
Table 2. Some ab initio calculated properties of the B···MgR2 complexes (R = F, H or CH3) for six different Lewis bases B a.
ComplexLewis Base BDe/kJ·mol−1kσ/N·m−1r(Mg⋯A)/Å bAngle θcδr(Mg–R)/Åd
B⋯MgF2CO36.6739.702.3968.70.011
NCH76.8072.722.17814.10.019
H2O99.3697.672.04611.40.021
NH3114.6990.212.16314.10.024
H2S56.0344.022.63110.80.016
PH353.0141.962.70311.70.017
B⋯MgH2CO18.5716.812.5677.60.008
NCH49.6245.082.26913.00.019
H2O70.8168.882.11111.30.023
NH382.0564.972.23314.00.028
H2S33.5923.742.7779.70.015
PH330.3321.812.8549.90.015
B⋯Mg(CH3)2CO16.5213.762.6096.50.006
NCH45.3341.102.28512.20.015
H2O64.5064.032.12411.10.019
NH375.7861.132.24513.50.023
H2S30.7920.722.8088.50.011
PH327.1218.852.8928.90.012
a Calculations were performed at the CCSD(T)/aug-ccpVTZ level. De was obtained from a complete basis set (CBS) extrapolation. See Section 3 for details. b r(Mg···L) is the distance between the Mg atom and the nearest atom, L, of the Lewis base B. c The angle, θ, is the angular displacement of each group, R, in the complex from the straight line, R–Mg–R defined in the free molecule (see Figure 3). d δr(Mg–R) is the increase in the Mg–R bond length (R = F, H or CH3) when B···MgR2 is formed from B and MgR2.
Table 3. Nucleophilicities of six Lewis bases, B, and electrophilicities of nine Lewis acids, A.
Table 3. Nucleophilicities of six Lewis bases, B, and electrophilicities of nine Lewis acids, A.
NucleophilicitiesElectrophilicities
Lewis Base BNB (This Work) aNB (From [20]) bLewis Acid AEA (This Work) cEA (From [20]) b
CO2.142.12BeF217.5(4)-
PH32.863.12BeH214.9(6)-
H2S3.023.43Be(CH3)213.5(6)-
HCN4.544.27MgF214.0(8)-
H2O5.244.89MgH211.5(5)-
NH37.507.52Mg(CH3)210.8(6)-
HF7.06.75
HBr5.1(3)4.59
HCl4.7(2)4.36
a Calculated by assuming that De = cNBEA with c = 1.00 k·Jmol−1 and NNH3 = 7.50 and that all De for the B···HF complexes (from ref. [21]) lie on a straight line through the origin. b Values from ref. [20] when determined by a global fit to De values of 250 complexes held together by various types of non-covalent bonds. c Obtained from the gradient dDe/dNB = cEA of the linear regression fit of each set of points in Figure 9 and Figure 10.

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Alkorta, I.; Legon, A.C. Non-Covalent Interactions Involving Alkaline-Earth Atoms and Lewis Bases B: An ab Initio Investigation of Beryllium and Magnesium Bonds, B···MR2 (M = Be or Mg, and R = H, F or CH3). Inorganics 2019, 7, 35. https://doi.org/10.3390/inorganics7030035

AMA Style

Alkorta I, Legon AC. Non-Covalent Interactions Involving Alkaline-Earth Atoms and Lewis Bases B: An ab Initio Investigation of Beryllium and Magnesium Bonds, B···MR2 (M = Be or Mg, and R = H, F or CH3). Inorganics. 2019; 7(3):35. https://doi.org/10.3390/inorganics7030035

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Alkorta, Ibon, and Anthony C. Legon. 2019. "Non-Covalent Interactions Involving Alkaline-Earth Atoms and Lewis Bases B: An ab Initio Investigation of Beryllium and Magnesium Bonds, B···MR2 (M = Be or Mg, and R = H, F or CH3)" Inorganics 7, no. 3: 35. https://doi.org/10.3390/inorganics7030035

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