Genuine quadruple bonds between two main-group atoms. Chemical bonding in AeF⁻ (Ae = Be–Ba) and isoelectronic EF (E = B–Tl) and the particular role of d orbitals in covalent interactions of heavier alkaline-earth atoms

Abstract

Quantum chemical calculations of anions AeF⁻ (Ae = Be–Ba) and isoelectronic group-13 molecules EF (E = B–Tl) have been carried out using ab initio methods at the CCSD(T)/def2-TZVPP level and density functional theory employing BP86 various basis sets. The equilibrium distances, bond dissociation energies and vibrational frequencies are reported. The alkali earth fluoride anions AeF⁻ exhibit strong bonds between the closed-shell species Ae and F⁻ with bond dissociation energies between 68.8 kcal mol⁻¹ for MgF⁻ and 87.5 kcal mol⁻¹ for BeF⁻ and they show an unusual increasing trend MgF⁻ < CaF⁻ < SrF⁻ < BaF⁻. This is in contrast to the isoelectronic group-13 fluorides EF where the BDE continuously decreases from BF to TlF. The calculated dipole moments of AeF⁻ are very large between 5.97 D for BeF⁻ and 1.78 D for BaF⁻ with the negative end always at the Ae atom (Ae→F⁻). This is explained by the location of the electronic charge of the lone pair at Ae, which is rather distant from the nucleus. The analysis of the electronic structure of AeF⁻ suggests significant charge donation Ae←F⁻ into the vacant valence orbitals of Ae. A bonding analysis with the EDA-NOCV method suggests that the molecules are mainly covalently bonded. The strongest orbital interaction in the anions comes from the inductive polarization of the 2p_σ electrons of F⁻, which leads to a hybridization of the (n)s and (n)p_σ AOs at Ae. There are two degenerate π donor interactions Ae←F⁻ in all anions AeF⁻, which provide 25–30% to the covalent bonding. There is another σ orbital interaction in the anions, which is very weak in BeF⁻ and MgF⁻. In contrast, the second stabilizing σ orbital interaction in CaF⁻, SrF⁻ and BaF⁻ yields a strongly stabilizing σ orbital, because the Ae atoms use their (n − 1)d_σ AOs for bonding. The energy lowering of the second σ interaction in the latter anions is even stronger than the π bonding. The EDA-NOCV results suggest that BeF⁻ and MgF⁻ have three strongly polarized bonds, whereas CaF⁻, SrF⁻ and BaF⁻ have four bonding orbitals. The quadruple bonds in the heavier alkaline earth species are made possible because they use s/d valence orbitals like transition metals for covalent bonding. The EDA-NOCV analysis of the group-13 fluorides EF gives a conventional picture with one very strong σ bond and two rather weak π interactions.

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1 Introduction

In recent years, great progress has been made in the field of multiple bonds of atoms of groups 2 and 13 of the periodic table in a variety of compounds.¹ Experimental and theoretical studies have shown that there are stable molecules that have homoatomic or heteroatomic double or even triple bonds between alkaline earth or group 13 atoms and other elements of the periodic table. Prominent examples are compounds with a boron–boron triple bond L→B≡B←L with various donor ligands L such as CO² or N-heterocyclic carbenes.³ A gallium–gallium triple bond was suggested for the compound Na₂{GaC₆H₃-2,6-Trip₂}₂ (Trip = C₆H₂-2,4,6-ⁱPr₃), which has a trans-bent geometry,⁴ but the bond order was disputed by other researchers.⁵ Numerous compounds with homo- and hetero-atomic double bonds of the heavier group-13 atoms Al–In have been prepared,⁶ and the first stable molecule with a classical Al=Al double bond is clearly a milestone in synthetic main group chemistry following an earlier synthesis of a dialuminium complex with a formal triple bond.^7a,b A compound with a triple bond between Al and N has lately been prepared and structurally characterized by X-ray crystallography.^7c

Very recently, the compound HNBeCO featuring a triple bond between beryllium and nitrogen has been reported, where one bonding component is a dative bond.⁸ This is related to the triple bond in carbon monoxide |CO|, which is also best described with two electron-sharing π bonds and one σ dative bond.⁹ The relevance of dative bonds in main-group compounds, which goes back to original work by Lewis¹⁰ and Sidgwick,¹¹ has been pointed out in several review articles.¹² It became prominent with the discovery of carbones¹³ L→C←L and related species L→E_n←L (E = B–Tl, Si–Pb, and N–Bi; n = 1–3), where L is a donor ligand.¹⁴ The first compound with a beryllium–nitrogen double bond (CAAC)NBeR (CAAC = cyclic alkylaminocarbene; R = imino group) that is stable at ambient temperatures was recently synthesized by Gilliard.¹⁵ This came shortly after the isolation of the first stable compound with a beryllium–carbon bond featuring a double donation L₂CBe, which takes advantage of the appearance of two lone-pair orbitals at the C(0) atom of a carbone L₂.¹⁶

The examples show that the main group atoms of groups 2 and 13 can form molecules with double and triple bonds just like carbon, where alkene and alkyne compounds are ubiquitous. Molecules with multiple bonds between the heavier homologues of carbon are more difficult to synthesize, but even the heavy-atom group-14 homologues of alkynes REER (E = Si–Pb) could be isolated and were found to have trans-bent geometries.^17–19 Triple bonds between two main group atoms appear to be the maximum bond order that can be achieved between two atoms, which have a valence shell with (n)s and (n)p orbitals. A few years ago, it was suggested that C₂ has a quadruple bond that is stronger than the triple bond of acetylene,²⁰ which was rejected by other authors.²¹ The controversy was most recently settled by an experimental study using a high-resolution photoelectron imaging spectrometer, which revealed that the dominant contribution to dicarbon bonding in C₂ is a double-bonded configuration with two π-bonds and no accompanying σ-bond.²²

Higher bond orders than 3 may be achieved by transition metals, which have a valence shell with (n)s, (n)p and (n − 1)d orbitals where the latter usually dominate in covalent bonds.²³ There are numerous transition metal compounds known with the formal bond order 4 or even higher, because the (n − 1)d-AOs form additional bonding MOs in covalent bonds.²⁴ Very recently it was shown that a quadruple bond between two atoms may even be achieved when only one of the atoms is a transition metal. The diatomic anion RhB⁻ was experimentally detected and the transition to neutral RhB was studied by photoelectron spectroscopy.²⁵ The analysis of the spectra and quantum chemical calculations suggest that the neutral molecule in its ¹Σ⁺ electronic ground state possesses a quadruple bond Rh[quadruple bond, length as m-dash]B, which consists of two σ and two π bonds. The conclusion is that a quadruple bond between two atoms is possible when only one of the atoms is a transition metal.

Very recently it was shown that the heavier alkaline earth atoms Ae = Ca, Sr and Ba, which are main-group metals belonging to the s-shell elements of group 2, use mainly their (n − 1)d AOs for covalent bonding and that they exhibit the full scenario of transition-metal compounds. The octa-coordinated carbonyls Ae(CO)₈ (ref. 26) and the isoelectronic dinitrogen adducts Ae(N₂)₈ (ref. 27) as well as the tris–benzene complexes Ae(Bz)₃,²⁸ which fulfill the 18-electron rule for transition metals, could be prepared in the gas phase and in low-temperature matrices. A systematic study of the valence orbitals of the alkaline earth atoms in a variety of compounds showed that the lighter atoms Be and Mg use their (n)s and (n)p orbitals in polar covalent bonds, whereas the valence orbitals of the heavier atoms Ca, Sr and Ba comprise (n)s and (n − 1)d orbitals.²⁹ It was concluded that the alkaline earth metals beryllium and magnesium form covalent bonds like typical main-group elements, whereas calcium, strontium and barium covalently bind like transition metals.³⁰ The finding lead to a theoretical study of the diatomic anions AeB⁻ and the isoelectronic neutral species AeC (Ae = Ca, Sr, Ba), which have an electron triplet (³Σ⁻) ground state.³¹ The analysis of the bonding situation has shown that the six valence electrons occupy two bonding σ-orbitals with two electrons each and a degenerate π-orbital, in which each component has one electron. Accordingly, there are four occupied bond orbitals, but since there are only six valence electrons, the formal bond order is only three.³¹

In order to find a main-group molecule with a genuine quadruple bond where four bonding orbitals are doubly occupied, we searched for analogous systems with a larger number of valence electrons. A previous study of AeO showed that heavier molecules with Ae = Ca, Sr, Ba have only three doubly occupied MOs, because the oxygen atom is too electronegative for Ae←O σ-backdonation into a vacant d_σ AO of the Ae atom. Here we report about the anions AeF⁻ (Ae = Be–Ba), where the alkaline earth atoms are bonded to the electropositive fluorine anion. Since F⁻ has a completely filled valence shell, it can only serve as a donor. In contrast to the isoelectronic Ne, the fluorine anion should be a somewhat stronger donor. Our work was stimulated by the recent joint experimental and theoretical study of BeF⁻ by Heaven and co-workers,³² which showed that there is a strong dative bond between the closed-shell atoms Be and F⁻. The bond energy of BeF⁻ has been estimated with a lower bound of 81.9 kcal mol⁻¹, which shows that a dative bond between main-group atoms can be very strong.³² The nature of the bond in BeF⁻ was analysed with a valence bond (VB) model by Kalemos, who reported a similar bond energy of ∼83 kcal mol⁻¹ using high-level ab initio methods.³³ Very recently, the results of laser cooling and quantum chemical calculation of the beryllium anions BeX⁻ (X = F–I) were reported by Madi et al.³⁴ There are no studies of the heavier fluorides AeF⁻ (Ae = Mg–Ba) known to us. Here we report about the equilibrium geometries and bond dissociation energies (BDEs) of the full set of alkaline earth anions AeF⁻ and we present a thorough analysis of the bonding situation. We also calculated the neutral compounds of the group-13 atoms EF (E = B–Tl) in order to compare the chemical bonds between the isoelectronic molecules.

2 Methods

The bond lengths, vibrational frequencies and BDEs of AeF⁻ (Ae = Be–Ba) and EF (E = B–Tl) in the electronic singlet ground state were calculated at the CCSD(T)³⁵ and BP86 (ref. 36) levels in conjunction with the basis sets def2-TZVPP and def2-QZVPP.³⁷ The calculations were carried out with the program Gaussian 16.³⁸ The NBO calculations were carried out using the program NBO 7.0.³⁹ The QTAIM (Quantum Theory of Atoms in Molecules)⁴⁰ calculations of the Laplacian distribution ∇²ρ(r) were performed with the Multiwfn program.⁴¹

The bonding situation in the diatomic molecules was further analyzed by means of an energy decomposition analysis (EDA), which was introduced by Morokuma⁴² and by Ziegler and Rauk⁴³ in conjunction with the natural orbitals for chemical valence (NOCV)⁴⁴ method. The EDA-NOCV⁴⁵ calculations were carried out with the ADF 2018.105 program package⁴⁶ at the BP86-D3(BJ)⁴⁷ level with the Slater-type basis function of DZP quality⁴⁸ using the BP86/def2-QZVPP optimized geometries. DZP is a double-ζ quality basis set augmented by two sets of polarization functions. In this analysis, the intrinsic interaction energy (ΔE_int) between two fragments can be divided into four energy components as follows:

ΔE_int = ΔE_elstat + ΔE_Pauli + ΔE_orb + ΔE_disp. (1)

The electrostatic ΔE_elstat term represents the quasiclassical electrostatic interaction between the unperturbed charge distributions of the prepared fragments, and the Pauli repulsion ΔE_Pauli corresponds to the energy change associated with the transformation from the superposition of the unperturbed electron densities of the isolated fragments to the wavefunction, which properly obeys the Pauli principle through explicit antisymmetrization and renormalization of the production wavefunction. Since we used D3(BJ), this method also gives dispersion contribution (ΔE_disp). The orbital term ΔE_orb comprises the mixing of orbitals, charge transfer and polarization between the isolated fragments. The energy change involved in the latter step, which is the main difference between the Morokuma⁴² and Ziegler/Rauk⁴³ approaches, is calculated with an extension of Slater's transition state method⁴⁹ for energy differences. It is often referred to as the ETS method. The orbital term ΔE_orb can be further decomposed into contributions from each irreducible representation of the point group of the interacting system as follows:

The combination of the EDA with NOCV enables the partition of the total orbital interactions into pairwise contributions of the orbital interactions, which is very vital to get a complete picture of the bonding. The charge deformation Δρ_k(r), resulting from the mixing of the orbital pairs ψ_k(r) and ψ_−k(r) of the interacting fragments, presents the amount and the shape of the charge flow due to the orbital interactions (eqn (3)), and the associated energy term ΔE_orb provides the size of stabilizing orbital energy originated from such interaction (eqn (4)).

More details about the EDA-NOCV method and its application are given in recent review articles.⁵⁰

3 Bond lengths, vibrational frequencies and bond dissociation energies

Table 1 shows the calculated bond lengths and vibrational frequencies along with previous theoretical and experimental data. The bond distances at CCSD(T)/def2-TZVPP of BeF⁻ and EF (E = B–Tl) agree quite well with experimental results and previous calculations.⁵¹ The BP86/def2-QZVPP values show slightly larger deviations but the agreement with the ab initio values and the experimental data is sufficient to use DFT calculations for the bonding analysis, which is discussed below. It is interesting that the BP86/def2-QZVPP bond lengths of AeF⁻ (Ae = Ca, Sr, Ba) are slightly too short, whereas the interatomic distances for the isoelectronic molecules EF (E = Ga, In, Tl) are a bit too long, which might be due to the fact that spin–orbit coupling is neglected.

Table 1 Calculated and experimental bond lengths (Å) and vibrational frequencies (cm⁻¹) of AeF⁻ (Ae = Be–Ba) and EF (E = B–Tl)

Atom Ae or E	Bond length				Frequencies
	BP86^a	CCSD(T)^b	Exper.	Previous	BP86^a	CCSD(T)^b	Exper.	Previous
a Using a def2-QZVPP basis set. b Using a def2-TZVPP basis set. c Ref. 51a. d Ref. 51b. e Ref. 51c. f Ref. 51d. g Ref. 32. h Ref. 33. i Ref. 51e.
AeF ^−
Be	1.428	1.429	—	1.414^g	1019.7	1028.0	1059 ± 6^g	1073^h
Mg	1.847	1.840	—	—	548.3	577.7	—	—
Ca	1.992	2.104	—	—	517.3	485.6	—	—
Sr	2.134	2.237	—	—	432.8	410.6	—	—
Ba	2.234	2.399	—	—	411.7	408.3	—	—
EF
B	1.274	1.273	1.267^d	1.264^e	1346.4	1396.5	1402^d	1394^e
Al	1.683	1.665	1.654^d	1.663^c	752.1	796.8	802^d	800^c
Ga	1.805	1.780	1.774^d	—	586.8	633.9	622^d	—
In	2.018	1.988	1.985^d	2.01^f	501.5	545.0	535^d	555^f
Tl	2.122	2.096	2.084^d	2.101^i	449.3	482.6	477^d	—

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A related conclusion concerns the calculated vibrational frequencies, which will be helpful to identify the yet unknown anions AeF⁻ (Ae = Mg–Ba). The agreement with experimental results and previous calculations is quite good and the BP86/def2-QZVPP values deviate only slightly from the CCSD(T)/def2-TZVPP data.

Table 2 shows the calculated BDEs of the molecules along with experimental results and previous calculations.^51,52 The data for the alkaline earth anions AeF⁻ suggest that the dative interactions between closed-shell atoms are very strong not only in BeF⁻ but also in the heavier homologues. The trend of the BDEs for AeF⁻ is very interesting, because it does not show the regular decrease in bond strength normally observed from the top to the bottom of the periodic table. The CCSD(T)/def2-TZVPP values show that the BDE of AeF⁻ first decreases from Be to Mg, but then increases again and rises continuously with the unusual trend Ca < Sr < Ba. This is in striking contrast to the trend of the isoelectronic neutral species EF, where the BDE strongly decreases in the expected order B > Al > Ga > In > Tl. The BDE of TlF is only slightly stronger than one half of the BDE of BF, but the BDE of BaF⁻ is not much less than the BDE of BeF⁻. The BP86/def2-QZVPP values for AeF⁻ are very similar to the CCSD(T)/def2-TZVPP data. The BDEs for EF at BP86/def2-QZVPP are slightly higher than the more reliable CCSD(T)/def2-TZVPP values but the deviation is not substantial.

Table 2 Calculated and experimental bond dissociation energies (kcal mol⁻¹) D_e and zero-point vibrational energy corrected values D_o of AeF⁻ (Ae = Be–Ba) and EF (E = B–Tl)

Atom Ae or E	D e(Do)
	BP86^a	CCSD(T)^b	Exper.	Previous calc.
a Using a def2-QZVPP basis set. b Using a def2-TZVPP basis set. c Ref. 51b. d Ref. 51d. e Ref. 51c. i Ref. 52. g Ref. 32. h Ref. 33.
AeF ^−
Be	88.7 (87.3)	87.5 (86.0)	(≥81.4^g)	82.9^h
Mg	61.9 (61.1)	68.8 (68.0)	—	—
Ca	82.1 (81.4)	73.7 (73.0)	—	—
Sr	76.8 (76.2)	77.5 (76.9)	—	—
Ba	83.0 (82.4)	80.5 (79.9)	—	—
EF
B	185.6 (183.7)	177.5 (175.5)	182.1^c (180.1^c)	182.7^e
Al	163.1 (162.0)	157.4 (156.3)	160.0^c (158.9^c)	162.1^i
Ga	144.8 (144.0)	137.8 (136.9)	138.8^c (137.9^c)	—
In	133.5 (132.8)	124.7 (123.9)	121.9^c (121.1^c)	122.0^d
Tl	125.8 (125.2)	115.4 (114.7)	106.2^c (105.4^c)	—

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The size and the trend of the BDEs of AeF⁻ are surprising and they contradict common wisdom that chemical bonding between closed-shell atoms is weak.⁵³ The analysis of the interatomic interactions is discussed in the next section.

4 Bonding analysis

A pivotal piece of information about the chemical bonding in the diatomic molecules AeF⁻ and EF is given by the atomic partial charges, which indicates the electronic charge distribution. Table 3 shows the calculated atomic charges of the four different methods used, namely the NBO, Hirshfeld, Voronoi and CM5 partitioning. The NBO results for the anions AeF⁻ suggest that there is very little charge donation from F⁻ to Ae. The magnesium species MgF⁻ is a particularly interesting system, because it exhibits according to the NBO results no charge migration Mg←F⁻ at all although it has a BDE of D_e = 68.8 kcal mol⁻¹. In contrast to NBO, the Hirshfeld, Voronoi and CM5 methods indicate that there is a substantial donation of Ae←F⁻ for all systems, with the CM5 method giving smaller partial charges.

Table 3 Atomic partial charges (q) and bond orders P(A–B) of AeF⁻ (Ae = Be–Ba) and EF (E = B–Tl) calculated at the BP86/def2-QZVPP level

q					P(A–B)
Atom	NBO	Hirshfeld	Voronoi	CM5	Bond	Wiberg	Mayer
Be	−0.02	−0.65	−0.59	−0.49	Be–F	0.25	1.13
Mg	0.00	−0.51	−0.43	−0.41	Mg–F	0.14	0.58
Ca	−0.08	−0.55	−0.51	−0.40	Ca–F	0.30	0.70
Sr	−0.06	−0.52	−0.49	−0.36	Sr–F	0.25	0.66
Ba	−0.07	−0.54	−0.54	−0.37	Ba–F	0.28	0.66
B	0.54	0.05	0.03	0.00	B–F	0.85	1.41
Al	0.83	0.30	0.26	0.29	Al–F	0.34	1.05
Ga	0.80	0.33	0.32	0.34	Ga–F	0.39	0.93
In	0.83	0.38	0.36	0.41	In–F	0.34	0.83
Tl	0.82	0.41	0.40	0.44	Tl–F	0.35	0.78

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The negligible charge donation by the NBO method can be explained by its dubious way of division between valence and Rydberg orbitals and the bias in favour of the valence AOs in the algorithm. The NBO procedure considers only those AOs as valence orbitals, which are occupied in the electronic ground state of the atom, while vacant AOs are considered as Rydberg orbitals. It means that only the (n)s AOs of Be–Ba are treated as valence orbitals by the NBO method but not the (n)p and (n − 1)d AOs. The contributions of the latter AOs are neglected in the NBO calculation of bonding MOs and thus, the occupation of the (n)p and (n − 1)d AOs of Be–Ba due to Ae←F⁻ is mistreated. A related problem was recently reported for the systems ENi(CO)₃⁻ (E = Li–Cs),⁵⁴ where the covalent bond between Ni and the alkaline earth atoms E–Ni is neglected by the NBO method, because the (n)p AOs of Ni and E atoms are not considered as valence orbitals.⁵⁵

The NBO method calculates, in contrast to the anions AeF⁻, a much larger charge migration in the neutral systems EF (E = B–Tl) E→F than the Hirshfeld, Voronoi and CM5 methods. The difference is not due to the division of the AOs in valence and Rydberg orbitals, because the (n)s and (n)p AOs of B–Tl are both considered as valence AOs in the NBO method since both are occupied in the atomic ground states. The very small partial charges calculated for BF by the Hirshfeld, Voronoi and CM5 methods appear to contradict chemical intuition, because fluorine has a much higher electronegativity than boron. But the migration of the σ charge B→F is partly compensated for by the backdonation of the π charge from the two degenerate π orbitals of fluorine BF. It has been shown before that π donation from fluorine is not negligible.⁵⁶

Table 3 gives also the calculated bond orders P_AB using the Wiberg⁵⁷ and Mayer⁵⁸ approaches. The Wiberg bond order (WBO) method is based on the semiempirical CNDO method and it neglects the overlap of the AOs. The Mayer bond order (MBO) considers the overlap and it is better suited for polar bonds.⁵⁹ The MBO values in Table 3 are significantly larger than the WBO data and they are always bigger for neutral EF than for the isoelectronic anion AeF⁻. The numerical values appear to contradict the existence of multiple bonds particularly for AeF⁻. However, the bonds are very polar, which gives low values for the bond order.⁶⁰

Further information about the charge distribution in the molecules is provided by the shape of relevant molecular orbitals and by the Laplacian ∇²ρ(r). The atomic charges are only scalar quantities, which do not give any insight into the local charge concentration, which is of central importance for the physical and chemical properties of a molecule. Fig. 1 shows the shape of the highest occupied MOs (HOMO) of the molecules, which provides a first hint about the interatomic interactions. The HOMO of AeF⁻ has σ symmetry, which is essentially a lone-pair orbital at the Ae atom that is composed of the (n)s and (n)p_σ AOs of Be and Mg and the (n)s and (n − 1)d_σ AOs of Ca, Sr, and Ba. There is a small bonding contribution from the 2s and 2p_σ AOs of fluorine. The HOMO of EF has a similar shape, but the contribution of the 2s and 2p_σ AOs of F is larger and therefore the polarity of the orbital is smaller. Moreover, the AO part of the E atom is always a hybrid of the (n)s and (n)p_σ AOs with negligible (n − 1)d contribution, which emphasizes the peculiar covalent bonding of the heavier alkaline earth atoms.

Fig. 1 Shape of the HOMO of AeF⁻ (Ae = Be–Ba) and EF (E = B–Tl). The Ae/E atoms are on the left side and the fluorine atom on the right.

Fig. 2 shows the shape of the Laplacian distribution ∇²ρ(r) of the molecules. There is a local area of charge concentration ∇²ρ(r) < 0 in AeF⁻ in the lone-pair region of the Ae atom, which becomes more diffuse for the heavier atoms. In contrast, the charge concentration in AeF⁻ at F exhibits a very spherical shape with only a very weak droplet-like appendix in BeF⁻. The Laplacian distribution ∇²ρ(r) of EF shows a similar anisotropic charge distribution to that in AeF⁻, but the area of charge concentration ∇²ρ(r) < 0 at B and Al is more condensed and closer to the nucleus than at Be and Mg in the isoelectronic anions BeF⁻ and MgF⁻. Visual inspection of the Laplacian distribution of GaF, InF and TlF suggests that the local area of charge concentration is obscured and it becomes indistinguishable with the in-plane π charge distribution.

Fig. 2 Laplacian distribution ∇²ρ(r) > of AeF⁻ (Ae = Be–Ba) and EF (E = B–Tl) at the CCSD(T)/def2-TZVPP level. Red lines indicate the areas of charge concentration (∇ρ(r) < 0), while blue lines show the areas of charge depletion (∇²ρ(r) > 0). The solid lines connecting the atomic nuclei are the bond paths. Green dots are bond critical points.

Table 4 shows the calculated electric dipole moments of the molecules in comparison with previous work.⁶¹ The agreement between the calculated and experimental values is quite good. The experimental value for BF has a rather large uncertainty, because the experimental setup is difficult and it has been suggested that the experimental value is underestimated.⁶² Our values agree reasonably well with the most recent theoretical data. It is noteworthy that the BP86/def2-QZVPP values are in good agreement with the CCSD(T)/def2-TZVPP data.

Table 4 Calculated and experimental dipole moments [Debye] of AeF⁻ (E = Be–Ba) and EF (E = B–Tl) at the CCSD(T)/def2-TZVPP and BP86/def2-QZVPP levels. The arrow (→) indicates the dipole moment direction from the negative to the positive pole

	CCSD(T)/def2-TZVPP	BP86/def2-QZVPP	Previous	Experim.
a Ref. 61c. b Ref. 61b. c Ref. 61a. d Ref. 61d. e Ref. 61e. f Ref. 61f. g Ref. 61g. h Ref. 61h.
BeF^− (Be→F)	−5.97	−5.78	—	—
MgF^− (Mg→F)	−2.80	−2.90	—	—
CaF^− (Ca→F)	−3.27	−3.32	—	—
SrF^− (Sr→F)	−1.90	−1.86	—	—
BaF^− (Ba→F)	−1.78	−2.04	—	—
BF (B→F)	−0.82	−1.03	−0.877^c	−0.5 ± 0.2^g
AlF (Al←F)	1.37	1.49	1.407^a; 1.54^f	1.515^b
GaF (Ga←F)	2.25	2.27	2.22^h	2.45^d
InF (In←F)	3.24	3.24	2.86^h	3.40^d
TlF (Tl←F)	4.39	4.04	3.21^h	4.23^e

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The electric dipole moment of charged molecules depends on the origin of the coordinate system.⁶³ The values of AeF⁻ have been calculated with respect to the mass center of the nuclear charge, which is the conventional choice. The theoretically predicted dipole moments of AeF⁻ at the CCSD(T)/def2-TZVPP level range from an amazing value of 5.97 Debye for BeF⁻ to 1.78 Debye for BaF⁻ with the negative end always at the Ae atom. The highly anisotropic charge distribution in AeF⁻ represented by the Laplacian distribution (Fig. 2) explains the surprising size and orientation of the dipole moments, which are vector quantities reflecting the spherical distribution of the electronic charge. There is a local region of charge concentration in the lone pair region of Ae that is quite distant from the nucleus. In contrast, the negative charge on the fluorine in AeF⁻ is spherical around the nucleus, resulting in a comparatively small atomic contribution to the dipole moment. The interatomic distance in AeF⁻ increases, which leads to smaller absolute values of the dipole moment, because the bonding component to the dipole moment, which is dominated by the bond polarity toward fluorine, cancels part of the very large atomic component of Ae due to its anisotropic charge distribution. The increase of the dipole moment from MgF⁻ to CaF⁻ is likely caused by the change from sp hybridization at Mg to sd hybridization at Ca, which is discussed below.

Table 4 shows that the dipole moment at BF also has the negative pole at the less electronegative atom B, but the absolute value of 0.82 Debye is much smaller than that for isoelectronic BeF⁻. Moreover, the polarity of the dipole moment reverses and the absolute values increase with atomic size for the heavier systems AlF–TlF. The dipole moment of BF has been calculated before and the inverse polarity B→F has been discussed using VB and MO methods.^61a,i,k The studies underline the particular role of the lone pair MO at beryllium, which yields a large atomic contribution to the dipole moment. This becomes obvious from the shape of the Laplacian distribution of BF in Fig. 2, which displays a clear region of charge accumulation ∇²ρ(r) < 0 at boron. A comparison with the Laplacian of BeF⁻ shows that the charge accumulation at B in BF is more concentrated and closer to the nucleus than the charge accumulation at Be in BeF⁻. This explains why BF has a much smaller dipole moment than BeF⁻. The inverse polarity in AlF with respect to BF has been noted before^61c,f and was explained by the longer bond in the former molecule. Klein and Rosmus⁶⁴ showed that the dipole moments of BF and AlF have a similar slope as a function of the interatomic distance and that the dipole moment function of AlF also has a region with a polarity Al→F, but due to the larger equilibrium bond length the polarity in the vibrational ground state is Al←F. The contribution of the electronic charge in the bonding region of AlF, which is polarized towards F, overcompensates the atomic component of the anisotropic charge distribution at Al and yields a dipole moment with the polarity Al←F. This effect increases further when the bonds become longer and when the atoms E in EF become more electropositive. The decrease of the dipole moments of AeF⁻ and the increase of the dipole moments of EF can thus straightforwardly be explained when the spatial charge distribution of the electronic charge is considered. The atomic partial charges are not sufficient to explain the dipole moments, because they are scalar values whereas dipole moments are vector quantities.

We investigated the nature of the bonds in the AeF⁻/EF systems using the EDA-NOCV method, which gives a detailed description of the interatomic interactions in terms of electrostatic attraction, Pauli repulsion and pairwise orbital interactions. The goal was to provide a physically meaningful explanation for the surprisingly strong BDEs and the unusual trend of the bond strength in terms of a bonding model using a method, which has been proven to be very useful in a large variety of molecules.^50,65Table 5 gives the numerical results for BeF⁻ and MgF⁻ using the neutral atoms Ae and F⁻ as interacting moieties. The results for the heavier homologues CaF⁻, SrF⁻ and BaF⁻ show significant differences and are discussed separately.

Table 5 Results of the EDA-NOCV calculations of AeF(−) (E = Be, Mg) using the atomic fragments in the ¹S ground state at the BP86/DZP-D3(BJ)//BP86/def2-QZVPP level of theory. Energy values are given in kcal mol⁻¹

Energy term	Orbital interaction	Be (^1S) + F^− (^1S)	Mg (^1S) + F^− (^1S)
a The values in parentheses give the percentage contribution to the total attractive interactions ΔEelstat + ΔEorb + ΔEdisp. b The values in parentheses give the percentage contribution to the total orbital interactions ΔEorb.
ΔEint		−93.45	−72.64
ΔEPauli		186.70	93.03
ΔEdisp^a		−0.4 (0.1%)	−0.5 (0.3%)
ΔEelstat^a		−165.8 (59.2%)	−96.3 (58.1%)
ΔEorb^a		−114.0 (40.7%)	−68.8 (41.6%)
ΔEorb1^b	Ae←F^− σ polarization/induction	−77.8 (68.3%)	−47.7 (69.2%)
ΔEorb2^b	Ae←F^− π donation	−14.6 (12.8%)	−8.9 (13.0%)
ΔEorb3^b	Ae←F^− π donation	−14.6 (12.8%)	−8.9 (13.0%)
ΔEorb4^b	Ae←F^− σ polarization/induction	−6.3 (5.5%)	−2.7 (4.0%)
ΔErest^b		−0.6 (0.6%)	−0.6 (0.8%)

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The calculations suggest that the attractive interactions between Ae and F⁻ have ∼60% electrostatic character, whereas ∼40% comes from orbital (covalent) interactions ΔE_orb. It is noteworthy that the trend of the orbital interaction closely follows the trend of the total interaction energy ΔE_int. The breakdown of the total orbital term into pairwise interactions gives four major contributions ΔE_orb1–ΔE_orb4, which can be identified by inspecting the associated deformation densities and the connected orbitals. They are shown for BeF⁻ in Fig. 3. The deformation densities and orbitals of MgF⁻ are very similar and they are shown in Fig. S1 of the ESI.‡

Fig. 3 Plot of the deformation densities Δρ of the four most important orbital interactions in BeF⁻, which indicate the direction of the charge flow red → blue. Shape of the most important AOs of Be and F⁻ and MOs of BeF⁻. The eigenvalues ν give the relative size of the charge transfer.

The strongest orbital contribution ΔE_orb1 in AeF⁻ (Ae = Be, Mg) comes from a polarization of the (n)s AO of Ae through the inductive charge interaction of the 2p_σ AO of F⁻, which leads to a (n)s/(n)p_σ hybridization at Ae. The deformation density in Fig. 3 shows that there is mainly charge migration to the terminal end of Be, but also a small charge accumulation in the bonding region leading to energy lowering due to the donation of the 2p_σ AO of F⁻ into the vacant (n)p_σ AO of Ae. The HOMO of AeF⁻ is mainly a lone-pair orbital at Ae but it has an energetically significant Ae–F⁻ bonding contribution. The weaker orbital terms ΔE_orb2 and ΔE_orb3 come from the degenerate π bonding orbitals HOMO−1 and HOMO−1′, which are due to the dative π interaction (n)p_π(Ae)←2p_π(F⁻). The nature of the fourth orbital interaction ΔE_orb4 is interesting. The orbital HOMO−2 in AeF⁻ (Ae = Be, Mg) suggests that it is also a σ bonding orbital. The nature of this orbital in isoelectronic CO has been discussed in great detail and it was shown that the bonding character depends on the different electronegativities of the atoms.⁶⁶ The shape of the deformation density of Δρ₄ suggests some charge accumulation in the (n)p_σ AOs of the atoms originating from the corresponding depletion from F⁻ 2s/2p_σ, which yields a small energy lowering. The mixing of the vacant 3s AO of Be comes from the diffuse negative charge of the anion. But the stabilization of ΔE_orb4 in BeF⁻ and particularly in MgF⁻ is very small and it is too small to declare the molecules as quadruply bonded species. The analysis of the orbital interaction by the EDA-NOCV method agrees with the suggestion of dative bonding as originally suggested by Heaven et al.³²

The EDA-NOCV results for the heavier homologues CaF⁻/SrF⁻/BaF⁻ exhibit significant differences to the lighter systems BeF⁻/MgF⁻ and they show some peculiar features. The numerical data are shown in Table 6. The deformation densities Δρ and the connected fragment orbitals of CaF⁻ are shown in Fig. 4. The plots of the heavier systems SrF⁻ and BaF⁻ look very similar and are shown in Fig. S2 and S3 of the ESI.‡ The percentage contributions of covalent (orbital) interactions and coulombic forces are very similar in all five anions AeF⁻, but the pairwise orbital interactions in CaF⁻/SrF⁻/BaF⁻ differ from the lighter systems BeF⁻/MgF⁻. The strongest orbital term ΔE_orb1 in the heavier systems has a similar feature to that in the lighter homologues coming from the induction/polarization interaction between the occupied 2p_σ AO of F⁻ and the occupied and vacant AOs of Ae. In contrast to the lighter systems BeF⁻/MgF⁻, the second σ interaction ΔE_orb2 in the heavier systems (which is ΔE_orb4 in BeF⁻/MgF⁻) is quite strong and even stronger than the degenerate π bonding terms ΔE_orb3/4 (Table 6). The inspection of the σ AOs of the atomic fragments in the ΔE_orb2 of the heavier systems shows that the stabilizing interaction is due to Ae←F⁻ donation from the occupied 2p_σ AO of F⁻ into the vacant (n)d_σ AOs of Ae, which leads to the bonding orbital HOMO-2 (Fig. 4 and S2, S3‡). This is a profound difference from the lighter systems BeF⁻/MgF⁻, where the second σ-interaction provides only minor energy stabilization. Another difference concerns the AOs of the Ae atoms in the π bonding interactions. Fig. 4 and S2, S3‡ show that the degenerate π bonding terms ΔE_orb3/4 of the heavier atoms Ca, Sr, and Ba involve the (n)d_π AOs of the metals, whereas the π bonding terms ΔE_orb2/3 of BeF⁻/MgF⁻ comprise the (n)p_π AOs. These are further examples for the finding that the valence orbitals of the heavier alkaline earth atoms Ca, Sr, and Ba are the (n)s and (n)d AOs.^26–30 However, the most important conclusion is that the diatomic anions AeF⁻ (Ae = Ca, Sr, Ba) have four polarized bonding orbitals due to the presence of d-valence AOs and therefore can be considered as quadruply bonded molecules.

Table 6 Results of the EDA-NOCV calculations of AeF(−) (E = Ca, Sr, Ba) using the atomic fragments in the ¹S ground state at the BP86/DZP-D3(BJ)//BP86/def2-QZVPP level of theory. Energy values are given in kcal mol⁻¹

Energy term	Orbital interaction	Ca (^1S) + F^− (^1S) (^1S)	Sr (^1S) + F^− (^1S)	Ba (^1S) + F^− (1S)
a The values in parentheses give the percentage contribution to the total attractive interactions ΔEelstat + ΔEorb + ΔEdisp. b The values in parentheses give the percentage contribution to the total orbital interactions ΔEorb.
ΔEint		−92.94	−90.66	−101.47
ΔEPauli		108.59	97.57	104.78
ΔEdisp^a		−0.7 (0.4%)	−0.8 (0.4%)	−0.9 (0.4%)
ΔEelstat^a		−113.8 (56.5%)	−104.0 (55.2%)	−110.5 (53.5%)
ΔEorb^a		−87.0 (43.2%)	−83.5 (44.3%)	−94.9 (46.0%)
ΔEorb1^b	Ae←F^− σ polarization/induction	−44.7 (51.4%)	−40.6 (48.7%)	−39.0 (41.1%)
ΔEorb2^b	Ae←F^− σ donation	−16.2 (18.6%)	−17.5 (21.0%)	−25.5 (26.9%)
ΔEorb3^b	Ae←F^− π donation	−12.4 (14.3%)	−11.9 (14.3%)	−14.1 (14.8%)
ΔEorb4^b	Ae←F^− π donation	−12.4 (14.3%)	−11.9 (14.3%)	−14.1 (14.8%)
ΔErest^b		−1.3 (1.5%)	−1.4 (1.7%)	−2.2 (2.3%)

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Fig. 4 Plot of the deformation densities Δρ of the four most important orbital interactions in CaF⁻, which indicate the direction of the charge flow red → blue. Shape of the most important AOs of Ca and F⁻ and MOs of CaF⁻. The eigenvalues ν give the relative size of the charge transfer.

We also analyzed the chemical bonds in the group-13 fluorides EF (E = B–Tl) with the EDA-NOCV method. Table 7 gives the numerical results of the calculations. The deformation densities Δρ and the connected fragment orbitals of BF and GaF⁻ are shown in Fig. 5. The plots of the other systems AlF, InF and TlF look very similar and are shown in Fig. S4 of the ESI.‡ The stabilizing contributions of the orbital (covalent) term ΔE_orb are for all molecules ∼70% of the total attraction, which is a little higher than in the anions AeF⁻. As for the anions AeF⁻, the trend of the orbital interaction ΔE_orb of EF is the same as that for the total interaction energy ΔE_int. This is remarkable particularly for the heavier species showing the order GaF > InF > Tl, whereas the values of ΔE_orb and ΔE_int for BaF⁻ are higher than those for CaF⁻ (Tables 6 and 7). The breakdown of the total orbital interactions ΔE_orb of EF into pairwise interactions agrees with chemical intuition. The covalent bonding comes mainly from the electron-sharing σ interaction ΔE_orb1 due to pairing of the unpaired electrons, which provides > 85% of ΔE_orb. The contribution of the degenerate E←F π donation ΔE_orb2/3 is 10% in BF and it continuously decreases up to TlF, where it provides only 3% of ΔE_orb. The shape of the deformation densities Δρ and the connected fragment orbitals of EF⁻ show a conventional picture (Fig. 5, and S1–S3‡). The group-13 fluorides EF have strong σ single bonds reinforced by weak π bonds. All five occupied MOs of AeF⁻ and EF are provided in Fig. S5–S12.‡

Table 7 Results of the EDA-NOCV calculations of EF (E = B–Tl) using the atomic fragments in the ²P ground state at the BP86/DZP-D3(BJ)//BP86/def2-QZVPP level of theory. Energy values are given in kcal mol⁻¹

Orbital interaction	B (^2P) + F (^2P)	Al (^2P) + F (^2P)	Ga (^2P) + F (^2P)	In (^2P) + F (^2P)	Tl (^2P) + F (^2P)
a The values in parentheses give the percentage contribution to the total attractive interactions ΔEelstat + ΔEorb + ΔEdisp. b The values in parentheses give the percentage contribution to the total orbital interactions ΔEorb.
ΔEint	−183.42	−160.45	−141.71	−130.85	−123.37
ΔEPauli	486.98	246.13	197.47	151.46	125.83
ΔEdisp^a	−0.3 (0.0%)	−0.6 (0.1%)	−0.5 (0.2%)	−0.6 (0.2%)	−0.7 (0.3%)
ΔEelstat^a	−210.8 (31.4%)	−120.9 (29.7%)	−103.8 (30.6%)	−79.9 (28.3%)	−67.4 (27.0%)
ΔEorb^a	−459.3 (68.5%)	−285.1 (70.1%)	−234.9 (69.2%)	−201.7 (71.5%)	−181.1 (72.7%)
ΔEorb1^b	−396.1 (86.2%)	−261.7 (91.8%)	−218.9 (93.2%)	−191.1 (94.7%)	−173.3 (95.6%)
ΔEorb2^b	−26.2 (5.7%)	−9.4 (3.3%)	−6.2 (2.7%)	−4.0 (2.0%)	−2.8 (1.5%)
ΔEorb3^b	−26.2 (5.7%)	−9.4 (3.3%)	−6.2 (2.7%)	−4.0 (2.0%)	−2.8 (1.5%)
ΔErest^b	−10.7 (2.3%)	−4.7 (1.6%)	−3.5 (1.5%)	−2.7 (1.3%)	−2.3 (1.3%)

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Fig. 5 Plot of the deformation densities Δρ of the three most important orbital interactions in BF and GaF, which indicate the direction of the charge flow red → blue. Shape of the most important AOs of the atoms and MOs of EF. The eigenvalues ν give the relative size of the charge transfer.

The peculiar orbital interaction in the anions AeF⁻ and the unusual trend of the surprisingly large BDEs suggest that the polarizabilities of the Ae atoms might play a role in the calculated data. Table 8 shows the static dipole polarizabilities a_D of Be–Ba, which have been recommended through evaluation of experimental and calculated data.⁶⁷ The polarizability increases sharply as the atom becomes heavier, with a particularly large increase from Mg to Ca. This contributes to the interatomic attraction, which explains the unusual increase of the BDE of AeF⁻ from Mg to Ba. One reviewer pointed out that the calculated dissociation energies correlate nicely with a charge induced dipole model (α/r⁴) except for Sr. This supports the conclusion that the chemical bonds and the peculiar bond strength of the latter systems are largely due to the charge induced interactions.

Table 8 Static dipole polarizabilities α_D [a.u.] of the atoms Be–Ba^a

Be	Mg	Ca	Sr	Ba
a Ref. 67.
37.7	71.2	160.8	197.2	272

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Summary

The results of this work can be summarized as follows:

• The alkali earth fluoride anions AeF⁻ (Ae = Be–Ba) have very strong bonds between the closed-shell species Ae and F⁻. The theoretically predicted bond dissociation energies D_e at CCSD(T)/def2-TZVPP are between 68.8 kcal mol⁻¹ for MgF⁻ and 87.5 kcal mol⁻¹ for BeF⁻ and they exhibit an unusual increasing trend MgF⁻ < CaF⁻ (73.7 kcal mol⁻¹) < SrF⁻ (77.5 kcal mol⁻¹) < BaF⁻ (80.5 kcal mol⁻¹). This is in contrast to the isoelectronic group-13 fluorides EF (E = B–Tl), where the BDE continuously decreases from BF (177.5 kcal mol⁻¹) to TlF (115.4 kcal mol⁻¹).

• The calculated dipole moments of AeF⁻ are very large between 5.97 D for BeF⁻ and 1.78 D for BaF⁻ with the negative end always located at the Ae atom (Ae→F⁻). This can be explained by the location of the electronic charge of a lone pair at Ae, which is rather distant from the nucleus.

• The analysis of the electronic structure of AeF⁻ with different methods suggests a significant charge donation Ae←F⁻ into the vacant valence orbitals of Ae. In contrast, calculations with the NBO method give negligible charge transfer, because the (n)p AOs of the alkaline earth atoms are not considered as valence orbitals.

• The EDA-NOCV calculations suggest that ∼60% of the attractive forces in AeF⁻ come from the orbital interactions ΔE_orb, which have the same trend as the total interactions ΔE_int. The largest contribution ΔE_orb1 is in all anions due to the inductive polarization of the 2p_σ electrons of F⁻, which leads to a hybridization of the (n)s and (n)p_σ AOs at Ae. There are two degenerate π donor interactions Ae←F⁻ in all anions AeF⁻, which provide 25–30% to the covalent bonding. There is another σ orbital interaction ΔE_orb4 in the anions, which is very weak in BeF⁻ and MgF⁻. In contrast, the second stabilizing σ orbital interaction ΔE_orb2 in CaF⁻, SrF⁻ and BaF⁻ yields a bonding orbital, because the Ae atoms use their (n − 1)d_σ AOs for bonding. The energy lowering ΔE_orb2 in the latter anions is even stronger than the π bonding of ΔE_orb3/4. The EDA-NOCV results suggest that BeF⁻ and MgF⁻ have three strongly polarized bonds, whereas CaF⁻, SrF⁻ and BaF⁻ have four bonding orbitals. The quadruple bonds in the heavier alkaline earth species are made possible, because they use s d valence orbitals like transition metals for covalent bonding.

• The EDA-NOCV analysis of the group-13 fluorides EF gives a conventional picture with one very strong σ bond and two rather weak π interactions.

Data availability

Additional information is found in the ESI.‡

Author contributions

R. Liu, L. Qin, Z. Zhang and F. Sagan carried out the calculations and analyzed the results. L. Zhao supervised the work at Nanjing and M. Mitoraj supervised the work at Cracow. G. Frenking was responsible for the conceptual framework of the project, analyzed the results and wrote the first draft of the manuscript. L. Zhao, M. Mitoraj and G. Frenking prepared the final version of the work.

Conflicts of interest

The authors declare no conflict of interest.

Acknowledgements

LZ and GF acknowledge the financial support from Nanjing Tech University (No. 39837123 and 39837132), the National Natural Science Foundation of China (No. 21973044), and the Natural Science Foundation of the Jiangsu province (BK20211587), and the International Cooperation Project at Nanjing Tech University. We appreciate the high performance center of Nanjing Tech University for supporting the computational resources. We thank the PL-Grid Infrastructure and the Academic Computational Centre Cyfronet of the University of Science and Technology in Krakow for providing additional computational resources.

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Footnotes

† This paper is dedicated to Prof. Bernd Engels on the occasion of his 65th birthday.

‡ Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d3sc00830d

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