#
Non-Covalent Interactions Involving Alkaline-Earth Atoms and Lewis Bases B: An ab Initio Investigation of Beryllium and Magnesium Bonds, B···MR_{2} (M = Be or Mg, and R = H, F or CH_{3})

^{1}

^{2}

^{*}

## Abstract

**:**

_{e}), intermolecular stretching, and quadratic force constants (k

_{σ}) determined by ab initio calculations conducted at the CCSD(T)/aug-cc-pVTZ level of theory, with D

_{e}obtained by using the complete basis set (CBS) extrapolation [CCSD(T)/CBS energy], are presented for the B···BeR

_{2}and B···MgR

_{2}complexes, where B is one of the following Lewis bases: CO, H

_{2}S, PH

_{3}, HCN, H

_{2}O or NH

_{3}, and R is H, F or CH

_{3}. The BeR

_{2}and MgR

_{2}precursor molecules were shown to be linear and non-dipolar. The non-covalent intermolecular bond in the B···BeR

_{2}complexes is shown to result from the interaction of the electrophilic band around the Be atom of BeR

_{2}(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···CO

_{2}, which contain a tetrel bond. The conclusions for the B···MgR

_{2}series are similar and a magnesium bond can be correspondingly invoked. The geometries established for B···BeR

_{2}and B···MgR

_{2}can be rationalized by a simple rule previously enunciated for tetrel-bonded complexes of the type B···CO

_{2}. It is also shown that the dissociation energy, D

_{e}, is directly proportional to the force constant, k

_{σ}, in each B···MR

_{2}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, E

_{A}, determined from the D

_{e}for B···BeR

_{2}complexes for the individual Lewis acids, A, reveal the order A = BeF

_{2}> BeH

_{2}> Be(CH

_{3})

_{2}—a result that is consistent with the −I and +I effects of F and CH

_{3}relative to H. The conclusions for the MgR

_{2}series are similar but, for a given R, they have smaller electrophilicities than those of the BeR

_{2}series. A definition of alkaline-earth non-covalent bonds is presented.

## 1. Introduction

_{2}and B···MgR

_{2}complexes in which B is one of the six simple Lewis bases CO, H

_{2}S, PH

_{3}, HCN, H

_{2}O or NH

_{3}and R is H, F or CH

_{3}. We will show that various Lewis acid molecules, BeR

_{2}and MgR

_{2}, 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, D

_{e}, and the intermolecular stretching force constant, traditionally referred to as k

_{σ}[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, D

_{e}is directly proportional to k

_{σ}and, moreover, that it is possible to reproduce the D

_{e}values (and, therefore, the k

_{σ}values also) by assigning a set of electrophilicities, E

_{A}, to the Lewis acids, A, and nucleophilicities, N

_{B}, 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.

_{3}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 (E

_{A}should increase relative to that of the dihydride) in F–Be–F and F–Mg–F, but less electrophilic (decease of E

_{A}) in CH

_{3}–Be–CH

_{3}and CH

_{3}–Mg–CH

_{3}. This conclusion is confirmed by the molecular electrostatic potential surfaces (MEPS) of F–Be–F, H–Be–H and CH

_{3}–Be–CH

_{3}. These were calculated for the 0.001 e/bohr

^{3}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 CH

_{3}–Be–CH

_{3}, 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 CH

_{3}is the ligand, in agreement with the −I and +I inductive effects of F and CH

_{3}, 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 CH

_{3}, respectively.

## 2. Results

#### 2.1. Molecular Geometries

_{2}, BeH

_{2}and Be(CH

_{3})

_{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 D

_{3d}, 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.

_{2}complexes in which B = CO, HCN, H

_{2}O, NH

_{3}, H

_{2}S or PH

_{3}. The molecular diagrams of the corresponding sets of six B···BeH

_{2}and B···Be(CH

_{3})

_{2}complexes are shown in Figure 4 and Figure 5, respectively. In each case, the fragment R

_{2}Be···L, where L is the atom of B involved in the intermolecular bond, is Y-shaped (local symmetry C

_{2v}). Thus, the angle, θ (which is defined in Figure 3), is zero in the BeR

_{2}monomer molecules, but increases significantly in all B···BeR

_{2}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 sp

^{2}and the third (empty) sp

^{2}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···BeR

_{2}complexes but are greater than the ideal sp

^{2}angles of 120° that would occur for a fully dative bond (i.e., 0° < θ < 30°). The BeCl

_{3}

^{−}anion [25] has three equivalent Be–Cl bonds and D

_{3h}symmetry, with ideal 120° angles. There are also increases δr in the distances r(R–Be) on formation of all B···BeR

_{2}complexes considered here, as expected for the partial change from sp to sp

^{2}hybridization at Be. The values of δr for all B···BeR

_{2}complexes investigated are included in Table 1.

_{2}complexes from the various MgR

_{2}molecules. Table 2 includes these quantities for the 18 complexes that result from the interaction of the three MgR

_{2}molecules (R = F, H or CH

_{3}) with the set of six Lewis bases, B = CO, HCN, H

_{2}O, NH

_{3}, H

_{2}S or PH

_{3}. 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 D

_{e}values; the correlation is less clear in the Be series.

#### 2.2. Relationship between D_{e} and k_{σ}

_{e}and k

_{σ}) of the binding strength obtained through ab initio calculations for the 18 B···BeR

_{2}complexes discussed in Section 2.1 are given in Table 1. The corresponding quantities for the 18 B···MgR

_{2}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 (D

_{e}and k

_{σ}) 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, D

_{e}≈ 20 kJ·mol

^{−1}and k

_{σ}≈ 10 N·m

^{−1}. This larger binding strength of the B···BeR

_{2}and B···MgR

_{2}complexes is reflected in the significant geometrical distortions in BeR

_{2}and MgR

_{2}on complex formation noted in Section 2.1. Given the direct proportionality of D

_{e}and k

_{σ}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···BeR

_{2}and B···MgR

_{2}complexes discussed here.

_{e}versus k

_{σ}for the 18 B···BeR

_{2}complexes (B = CO, HCN, H

_{2}O, NH

_{3}, H

_{2}S or PH

_{3}; R = F, H or CH

_{3}). 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(CH

_{3})

_{2}. The first minimum occurs at a Be···C distance of 2.19 Å with D

_{e}= 3.66 kJ·mol

^{−1}, while the second (and global) minimum is at 2.92 Å with D

_{e}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 D

_{e}versus k

_{σ}for the 18 B···MgR

_{2}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], D

_{e}is, in good approximation, directly proportional to k

_{σ}for both B···BeR

_{2}and B···MgR

_{2}series; that is, D

_{e}= c’·k

_{σ}, where c’ is the constant of proportionality.

^{3}m

^{2}·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···BeR

_{2}and B···MgR

_{2}are significantly smaller at 0.79(5) × 10

^{3}m

^{2}·mol

^{−1}and 1.07(6) × 10

^{3}m

^{2}·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 D

_{e}versus k

_{σ}for B···BeR

_{2}and B···MgR

_{2}complexes for a given Lewis base, B, with a variation of the six Lewis acids (R = H, F and CH

_{3}) 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 BeR_{2} and MgR_{2} (R = F, H or CH_{3})

_{e}can be represented by an equation of the type

_{e}= cN

_{B}E

_{A}+ d

_{B}is the nucleophilicity of the Lewis base, B, E

_{A}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 N

_{B}and E

_{A}are dimensionless. Given the direct proportionality of D

_{e}and k

_{σ}, Equation (1) can be recast with k

_{σ}as the subject and indeed it was with that version of the expression that N

_{B}and E

_{A}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 D

_{e}versus N

_{B}are usually good straight lines and it follows then that the gradient is dD

_{e}/dN

_{B}= cE

_{A}. In the earlier determinations of N

_{B}and E

_{A}for the B···HX complexes (X = F, Cl, Br, etc.), the following procedure was used. The values of N

_{B}were assigned to the various Lewis bases so that the plot of D

_{e}(or k

_{σ}) versus N

_{B}for the B···HF complexes is a straight line through the origin. The sets of D

_{e}for the B···HCl, B···HBr, etc., complexes were then plotted against N

_{B}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 N

_{B}and E

_{A}values by a global fit of the D

_{e}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 BeR

_{2}and MgR

_{2}series (R = F, H or CH

_{3}).

_{e}versus N

_{B}for the series of B···MgF

_{2}, B···MgH

_{2}and B···Mg(CH

_{3})

_{2}complexes when B = CO, HCN, H

_{2}O, NH

_{3}, H

_{2}S or PH

_{3}. The values of N

_{B}are those appropriate to the B···HF series when N

_{NH3}is set to 7.5 to be consistent with its value reported in [20]. The remainder of N

_{B}are those chosen so that the points in a plot of D

_{e}versus N

_{B}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 D

_{e}versus N

_{B}for B···HCl and B···HBr (D

_{e}values from [20]) against the set of N

_{B}defined by B···HF. The straight lines for the B···MgR

_{2}complexes are from least squares fits of the points (but with the points for B = H

_{2}O excluded for reasons given below) for each series and the gradients of the fits dD

_{e}/dN

_{B}= cE

_{A}lead to the E

_{A}values for A = MgF

_{2}, MgH

_{2}, Mg(CH

_{3})

_{2}, HF, HBr and HCl listed in Table 3. The corresponding diagram for the B···BeR

_{2}series is in Figure 9, in which the plots for B···HX (X= F, Cl and Br) are included. The points for H

_{2}O···BeR

_{2}were again excluded from the linear regression fits. The values of E

_{A}derived from the gradients are in Table 3. The N

_{B}and E

_{A}values determined from the global fit of the D

_{e}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 N

_{B}values obtained here and those in ref. [20]. The same good agreement holds for the E

_{A}values of HCl and HBr. The reason for excluding the D

_{e}values of the H

_{2}O···MgR

_{2}and H

_{2}O···BeR

_{2}complexes from Figure 8 and Figure 9, respectively, is that they imply N

_{H2O}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 H

_{2}O. If the value of D

_{e}for each H

_{2}O···MgR

_{2}were forced to lie on its appropriate regression line in Figure 8, the value N

_{H2O}≈ 6.4 would be necessary for each R. A similar conclusion applies for the B···BeR

_{2}complexes, implying that N

_{H2O}≈ 6.1. Thus, H

_{2}O has a higher electrophilicity for the MR

_{2}molecules than it does for HF. This could be related to the efficacy of water as a solvent for ions.

_{LUMO}and E

_{HOMO}), respectively, by the expression

_{LUMO}and E

_{HOMO}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 BeF

_{2}, BeH

_{2}and Be(CH

_{3})

_{2}, respectively, and 1.92, 1.11 and 1.03 eV for MgF

_{2}, MgH

_{2}and Mg(CH

_{3})

_{2}, respectively. Figure 10 shows a plot of the E

_{A}values from the present work against ω. There is a reasonable correlation between the two measures of the electrophilicity of the six MR

_{2}.

## 3. Theoretical Methods

_{e}, and force constants, k

_{σ}, were obtained at the CCSD(T) computational level [29] for each B···BeR

_{2}and B···MgR

_{2}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, r

_{e}, was then determined in steps of $\left(r-{r}_{\mathrm{e}}\right)=$ 0.025 Å at the same computational level to yield the variation of the energy $E\left(r-{r}_{\mathrm{e}}\right)$ with the displacement $\left(r-{r}_{\mathrm{e}}\right)$ from equilibrium. As an example, the resulting curve for the OC⋯BeF

_{2}complex is given in Figure 11. Such curves were then fitted by a third-order polynomial in $\left(r-{r}_{\mathrm{e}}\right)$, from which k

_{σ}is obtained as the numerical value of the second derivative of E with respect to $\left(r-{r}_{\mathrm{e}}\right)$ evaluated at r

_{e}. In order to obtain more accurate D

_{e}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 D

_{e}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 BeR

_{2}and MgR

_{2}monomers were calculated on the 0.001 e/bohr

^{3}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 D

_{e}and k

_{σ}values for all complexes investigated here.

## 4. Conclusions

_{σ}, and dissociation energies, D

_{e}, of the 18 B···BeR

_{2}complexes (B = CO, HCN, H

_{2}O, NH

_{3}, H

_{2}S or PH

_{3}and R = F, H or CH

_{3}) and of the corresponding set of complexes in which Be is replaced by Mg. In all cases, D

_{e}was determined by using the complete basis set extrapolation. The dissociation energies, D

_{e}, 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 MR

_{2}molecules are linear (see Figure 2). The following process may then be envisaged. The Lewis base, B, is assumed to approach MR

_{2}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 MR

_{2}are assumed to singly occupy sp

_{z}hybrids, which then form single bonds with F or H or C to give the linear molecules F–M–F, H–M–H and H

_{3}C–M–CH

_{3}, 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 np

_{x}and np

_{y}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 p

_{x}or p

_{y}orbital, the hybridization at M changes gradually to take on some sp

^{m}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 (D

_{e}or k

_{σ}) increases and about 20° for the most strongly bound complexes, namely, those involving H

_{2}O and NH

_{3}with BeR

_{2}. 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 sp

^{2}hybridization. Thus, it appears that the interaction of B and MR

_{2}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 sp

^{m}hybridization scheme. It appears, therefore, that these are not purely σ-hole/n-pair interactions.

_{2}and B···MgR

_{2}complexes can be predicted by a simple modification to a rule recently enunciated [20] for tetrel-bonded complexes of the type B···CO

_{2}, that is:

The equilibrium geometry of alkaline-earth bonded B···MR_{2}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 MR_{2}line coincides with the axis of a non-bonding electron pair carried by B. Some deviation of MR_{2}from collinearity could occur.

_{2}and B···MgR

_{2}complexes, it has been established that D

_{e}is directly proportional to k

_{σ}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 D

_{e}into contributions from the individual molecules B and MR

_{2}, called the nucleophilicity, N

_{B}, of the Lewis base, B, and the electrophilicity, E

_{A}, of the Lewis acid, A, respectively. As may be seen from Table 3, the order of the E

_{A}values for both BeR

_{2}and MgR

_{2}sets when acting as Lewis acids is R = F > H ≥ CH

_{3}, which is the order expected from the −I inductive effect of F relative to H and the +I effect of the CH

_{3}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 CH

_{3}. It is also clear from Table 3 that for a given R, the electrophilicity of BeR

_{2}is greater than that of MgR

_{2}. 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 MgF

_{2}and BeF

_{2}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 BeF

_{2}> BeH

_{2}.> Be(CH

_{3})

_{2}~ MgF

_{2}> MgH

_{2}> Mg(CH

_{3})

_{2}>> HF > HBr ~ HCl, which indicates just how effective BeR

_{2}and MgR

_{2}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 E

_{A}of MR

_{2}with those estimated by the CDFT approach has been presented.

_{2}and MgR

_{2}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 MR

_{2}. 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.

## Supplementary Materials

_{2,}MgH

_{2}and Mg(CH

_{3})

_{2}calculated at the 0.001 e/bohr

^{3}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−r

_{e}) as a function of the displacement (r−r

_{e}) from the global minimum at r

_{e}at the CCSD(T)/aug-cc-pVTZ computational level.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Molecular electrostatic potential surfaces of the linear non-polar molecules, BeF

_{2}, BeH

_{2}and Be(CH

_{3})

_{2}calculated at the 0.001 e/bohr

^{3}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 BeF

_{2}, BeH

_{2}and Be(CH

_{3})

_{2}, respectively, and confirm expectations based on the inductive effects of CH

_{3}and F relative to H.

**Figure 2.**Geometries of BeF

_{2}, BeH

_{2}and Be(CH

_{3})

_{2}optimized at the CCSD(T)/aug-cc-pVTZ level of theory (to scale).

**Figure 3.**Geometries (drawn to scale) of six B···BeF

_{2}complexes optimized at the CCSD(T)/aug-cc-pVTZ level of theory, where B = CO, HCN, H

_{2}O, NH

_{3}, H

_{2}S and PH

_{3}.

**Figure 4.**Geometries (drawn to scale) of six B···BeH

_{2}complexes optimized at the CCSD(T)/aug-cc-pVTZ level of theory, where B = CO, HCN, H

_{2}O, NH

_{3}, H

_{2}S and PH

_{3}.

**Figure 5.**Geometries (drawn to scale) of six B···Be(CH

_{3})

_{2}complexes optimized at the CCSD(T)/aug-cc-pVTZ level of theory, where B = CO, HCN, H

_{2}O, NH

_{3}, H

_{2}S and PH

_{3}.

**Figure 6.**Variation of ab initio-calculated values of D

_{e}with k

_{σ}for 18 B···BeR

_{2}complexes (R = F, H or CH

_{3}; B = CO, HCN, H

_{2}O, NH

_{3}, H

_{2}S or PH

_{3}). For the linear regression, R

^{2}= 0.939.

**Figure 7.**Variation of ab initio calculated values of D

_{e}with k

_{σ}for 18 B···MgR

_{2}complexes (R =F, H or CH

_{3}; B = CO, HCN, H

_{2}O, NH

_{3}, H

_{2}S or PH

_{3}). For the linear regression, R

^{2}= 0.952.

**Figure 8.**D

_{e}versus the nucleophilicity, N

_{B}, for the B···MgR

_{2}series and B···HX complexes (B = CO, HCN, H

_{2}O, NH

_{3}, H

_{2}S and PH

_{3}; R = F, H or CH

_{3}; X = F, Cl or Br). The N

_{B}values are defined by the B···HF straight line through the origin (see text for details). The points for H

_{2}O···MgR

_{2}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. (R

^{2}= 0.994, 0.994, 0.990, 1.000, 0.993 and 0.988 for the Mg(CH

_{3})

_{2}, MgH

_{2}, MgF

_{2}, HF, HCl and HBr lines, respectively).

**Figure 9.**D

_{e}versus the nucleophilicity, N

_{B}, for the series B···BeR

_{2}and B···HX complexes (B = CO, HCN, H

_{2}O, NH

_{3}, H

_{2}S and PH

_{3}; R = F, H or CH

_{3}; X = F, Cl or Br). The N

_{B}values are defined by the B···HF straight line through the origin (see text for details). The points for H

_{2}O···BeR

_{2}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(CH

_{3})

_{2}points has been omitted. (R

^{2}= 0.994, 0.996, 0.998, 1.000.0.993 and 0.988 for Be(CH

_{3})

_{2}, BeH

_{2}, BeF

_{2}, HF, HCl and HBr lines, respectively).

**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 E

_{A}determined here for various MR

_{2}molecules (M = Be or Mg, R = F, H, or CH

_{3}).

**Figure 11.**Variation of the energy $E\left(r-{r}_{\mathrm{e}}\right)$ of OC···BeF

_{2}as a function of the displacement $\left(r-{r}_{\mathrm{e}}\right)$ from the global minimum at r

_{e}along the C

_{2}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 r

_{e}gives the intermolecular stretching force constant, k

_{σ}. The corresponding curves and the fitted polynomials for all B···BeR

_{2}and B···MgR

_{2}complexes (B = CO, H

_{2}S, PH

_{3}, HCN, H

_{2}O or NH

_{3}; R = F, H or CH

_{3}) investigated here are available in the Supplementary Material.

**Table 1.**Some ab initio calculated properties of the B···BeR

_{2}complexes (R = F, H orCH

_{3}) for six different Lewis bases B

^{a}.

Complex | Lewis Base B | D_{e}/kJ·mol^{−1} | k_{σ}/N·m^{−1} | r(Be···A)/Å ^{b} | Angle θ/° ^{c} | δr(Be–R)/Å ^{d} |
---|---|---|---|---|---|---|

B⋯BeF_{2} | CO | 26.72 | 36.33 | 2.040 | 15.0 | 0.024 |

NCH | 66.98 | 87.59 | 1.818 | 19.2 | 0.035 | |

H_{2}O | 95.94 | 121.89 | 1.697 | 18.7 | 0.040 | |

NH_{3} | 121.73 | 133.19 | 1.777 | 21.1 | 0.045 | |

H_{2}S | 43.57 | 44.59 | 2.289 | 16.9 | 0.029 | |

PH_{3} | 41.59 | 45.87 | 2.337 | 17.7 | 0.035 | |

B⋯BeH_{2} | CO | 21.29 | 44.61 | 1.942 | 16.3 | 0.019 |

NCH | 53.67 | 85.38 | 1.790 | 19.1 | 0.026 | |

H_{2}O | 80.94 | 110.93 | 1.688 | 18.0 | 0.030 | |

NH_{3} | 102.10 | 123.11 | 1.783 | 20.5 | 0.035 | |

H_{2}S | 34.58 | 37.91 | 2.270 | 16.0 | 0.021 | |

PH_{3} | 34.08 | 42.86 | 2.305 | 17.0 | 0.023 | |

B⋯Be(CH_{3})_{2} | CO | 5.28 | 2.00 | 2.922 | 3.2 | 0.004 |

NCH | 32.75 | 57.73 | 1.844 | 18.1 | 0.035 | |

H_{2}O | 57.82 | 82.21 | 1.720 | 18.7 | 0.040 | |

NH_{3} | 77.89 | 104.24 | 1.809 | 20.0 | 0.046 | |

H_{2}S | 16.97 | 14.07 | 2.425 | 14.1 | 0.025 | |

PH_{3} | 14.19 | 15.02 | 2.456 | 14.8 | 0.027 |

^{a}Calculations were performed at the CCSD(T)/aug-ccpVTZ level. D

_{e}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 CH

_{3}) when B···BeR

_{2}is formed from B and BeR

_{2}.

**Table 2.**Some ab initio calculated properties of the B···MgR

_{2}complexes (R = F, H or CH

_{3}) for six different Lewis bases B

^{a}.

Complex | Lewis Base B | D_{e}/kJ·mol^{−1} | k_{σ}/N·m^{−1} | r(Mg⋯A)/Å ^{b} | Angle θ/° ^{c} | δr(Mg–R)/Å^{d} |
---|---|---|---|---|---|---|

B⋯MgF_{2} | CO | 36.67 | 39.70 | 2.396 | 8.7 | 0.011 |

NCH | 76.80 | 72.72 | 2.178 | 14.1 | 0.019 | |

H_{2}O | 99.36 | 97.67 | 2.046 | 11.4 | 0.021 | |

NH_{3} | 114.69 | 90.21 | 2.163 | 14.1 | 0.024 | |

H_{2}S | 56.03 | 44.02 | 2.631 | 10.8 | 0.016 | |

PH_{3} | 53.01 | 41.96 | 2.703 | 11.7 | 0.017 | |

B⋯MgH_{2} | CO | 18.57 | 16.81 | 2.567 | 7.6 | 0.008 |

NCH | 49.62 | 45.08 | 2.269 | 13.0 | 0.019 | |

H_{2}O | 70.81 | 68.88 | 2.111 | 11.3 | 0.023 | |

NH_{3} | 82.05 | 64.97 | 2.233 | 14.0 | 0.028 | |

H_{2}S | 33.59 | 23.74 | 2.777 | 9.7 | 0.015 | |

PH_{3} | 30.33 | 21.81 | 2.854 | 9.9 | 0.015 | |

B⋯Mg(CH_{3})_{2} | CO | 16.52 | 13.76 | 2.609 | 6.5 | 0.006 |

NCH | 45.33 | 41.10 | 2.285 | 12.2 | 0.015 | |

H_{2}O | 64.50 | 64.03 | 2.124 | 11.1 | 0.019 | |

NH_{3} | 75.78 | 61.13 | 2.245 | 13.5 | 0.023 | |

H_{2}S | 30.79 | 20.72 | 2.808 | 8.5 | 0.011 | |

PH_{3} | 27.12 | 18.85 | 2.892 | 8.9 | 0.012 |

^{a}Calculations were performed at the CCSD(T)/aug-ccpVTZ level. D

_{e}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 CH

_{3}) when B···MgR

_{2}is formed from B and MgR

_{2}.

Nucleophilicities | Electrophilicities | ||||
---|---|---|---|---|---|

Lewis Base B | N_{B} (This Work) ^{a} | N_{B} (From [20]) ^{b} | Lewis Acid A | E_{A} (This Work) ^{c} | E_{A} (From [20]) ^{b} |

CO | 2.14 | 2.12 | BeF_{2} | 17.5(4) | - |

PH_{3} | 2.86 | 3.12 | BeH_{2} | 14.9(6) | - |

H_{2}S | 3.02 | 3.43 | Be(CH_{3})_{2} | 13.5(6) | - |

HCN | 4.54 | 4.27 | MgF_{2} | 14.0(8) | - |

H_{2}O | 5.24 | 4.89 | MgH_{2} | 11.5(5) | - |

NH_{3} | 7.50 | 7.52 | Mg(CH_{3})_{2} | 10.8(6) | - |

HF | 7.0 | 6.75 | |||

HBr | 5.1(3) | 4.59 | |||

HCl | 4.7(2) | 4.36 |

^{a}Calculated by assuming that D

_{e}= cN

_{B}E

_{A}with c = 1.00 k·Jmol

^{−1}and N

_{NH3}= 7.50 and that all D

_{e}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 D

_{e}values of 250 complexes held together by various types of non-covalent bonds.

^{c}Obtained from the gradient dD

_{e}/dN

_{B}= cE

_{A}of the linear regression fit of each set of points in Figure 9 and Figure 10.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

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···MR_{2} (M = Be or Mg, and R = H, F or CH_{3}). *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···MR_{2} (M = Be or Mg, and R = H, F or CH_{3}). *Inorganics*. 2019; 7(3):35.
https://doi.org/10.3390/inorganics7030035

**Chicago/Turabian Style**

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···MR_{2} (M = Be or Mg, and R = H, F or CH_{3})" *Inorganics* 7, no. 3: 35.
https://doi.org/10.3390/inorganics7030035