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)

Abstract

Geometries, equilibrium dissociation energies (D_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···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, D_e, is directly proportional to the force constant, k_σ, 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, E_A, determined from the D_e 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.

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, 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.

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 (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 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 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.

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 C_2v). 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. Some ab initio calculated properties of the B···BeR₂ complexes (R = F, H orCH₃) for six different Lewis bases B ^a.

Complex	Lewis Base B	De/kJ·mol−1	kσ/N·m−1	r(Be···A)/Å b	Angle θ/° c	δr(Be–R)/Å d
B⋯BeF2	CO	26.72	36.33	2.040	15.0	0.024
	NCH	66.98	87.59	1.818	19.2	0.035
	H2O	95.94	121.89	1.697	18.7	0.040
	NH3	121.73	133.19	1.777	21.1	0.045
	H2S	43.57	44.59	2.289	16.9	0.029
	PH3	41.59	45.87	2.337	17.7	0.035
B⋯BeH2	CO	21.29	44.61	1.942	16.3	0.019
	NCH	53.67	85.38	1.790	19.1	0.026
	H2O	80.94	110.93	1.688	18.0	0.030
	NH3	102.10	123.11	1.783	20.5	0.035
	H2S	34.58	37.91	2.270	16.0	0.021
	PH3	34.08	42.86	2.305	17.0	0.023
B⋯Be(CH3)2	CO	5.28	2.00	2.922	3.2	0.004
	NCH	32.75	57.73	1.844	18.1	0.035
	H2O	57.82	82.21	1.720	18.7	0.040
	NH3	77.89	104.24	1.809	20.0	0.046
	H2S	16.97	14.07	2.425	14.1	0.025
	PH3	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 CH₃) when B···BeR₂ is formed from B and BeR₂.

. 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 D_3h 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. Some ab initio calculated properties of the B···MgR₂ complexes (R = F, H or CH₃) for six different Lewis bases B ^a.

Complex	Lewis Base B	De/kJ·mol−1	kσ/N·m−1	r(Mg⋯A)/Å b	Angle θ/° c	δr(Mg–R)/Åd
B⋯MgF2	CO	36.67	39.70	2.396	8.7	0.011
	NCH	76.80	72.72	2.178	14.1	0.019
	H2O	99.36	97.67	2.046	11.4	0.021
	NH3	114.69	90.21	2.163	14.1	0.024
	H2S	56.03	44.02	2.631	10.8	0.016
	PH3	53.01	41.96	2.703	11.7	0.017
B⋯MgH2	CO	18.57	16.81	2.567	7.6	0.008
	NCH	49.62	45.08	2.269	13.0	0.019
	H2O	70.81	68.88	2.111	11.3	0.023
	NH3	82.05	64.97	2.233	14.0	0.028
	H2S	33.59	23.74	2.777	9.7	0.015
	PH3	30.33	21.81	2.854	9.9	0.015
B⋯Mg(CH3)2	CO	16.52	13.76	2.609	6.5	0.006
	NCH	45.33	41.10	2.285	12.2	0.015
	H2O	64.50	64.03	2.124	11.1	0.019
	NH3	75.78	61.13	2.245	13.5	0.023
	H2S	30.79	20.72	2.808	8.5	0.011
	PH3	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 CH₃) when B···MgR₂ is formed from B and MgR₂.

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 D_e values; the correlation is less clear in the Be series.

2.2. Relationship between D_e and k_σ

The two measures (D_e and k_σ) 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 (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···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 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···BeR₂ and B···MgR₂ complexes discussed here.

Figure 6 shows a plot of D_e versus k_σ 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 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···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], D_e is, in good approximation, directly proportional to k_σ for both B···BeR₂ and B···MgR₂ series; that is, D_e = c’·k_σ, 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 D_e versus k_σ 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, D_e can be represented by an equation of the type

D_e = cN_BE_A + d

(1)

where N_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 BeR₂ and MgR₂ series (R = F, H or CH₃).

Figure 8 shows the plots of D_e versus N_B 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 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. Nucleophilicities of six Lewis bases, B, and electrophilicities of nine Lewis acids, A.

Nucleophilicities			Electrophilicities
Lewis Base B	NB (This Work) a	NB (From [20]) b	Lewis Acid A	EA (This Work) c	EA (From [20]) b
CO	2.14	2.12	BeF2	17.5(4)	-
PH3	2.86	3.12	BeH2	14.9(6)	-
H2S	3.02	3.43	Be(CH3)2	13.5(6)	-
HCN	4.54	4.27	MgF2	14.0(8)	-
H2O	5.24	4.89	MgH2	11.5(5)	-
NH3	7.50	7.52	Mg(CH3)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_BE_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.

. 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···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 dD_e/dN_B = cE_A lead to the E_A 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 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 H₂O···MgR₂ and H₂O···BeR₂ 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 H₂O. If the value of D_e for each H₂O···MgR₂ 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···BeR₂ complexes, implying that N_H2O ≈ 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 (E_LUMO and E_HOMO), respectively, by the expression

$$\omega \approx {\left(E_{\mathrm{HOMO}}+E_{\mathrm{LUMO}}\right)}^2/8\left(E_{\mathrm{LUMO}}-E_{\mathrm{HOMO}}\right)$$

(2)

When E_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 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 E_A 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, D_e, and force constants, k_σ, 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, 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⋯BeF₂ 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 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 D_e and k_σ 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, k_σ, and dissociation energies, D_e, 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, 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 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 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 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 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 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 sp^m 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 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 MR₂, 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 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 E_A 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].