Hydrogen-bond strength and structure
The results of our ZORA-BP86/TZ2P calculations are shown in Table 1 for a representative selection of hydrogen-, fluorine- and iodine-bonded model complexes DX⋅⋅⋅A−, covering D=F and I, X=H, F and I, as well as A−=F−, Cl−, Br− and I− (full data on all model systems can be found in Tables S1–S10 in the Supporting Information). In the first place, we note that all model reactions are associated with single-well potential energy surfaces (PES), that is, there is no separate energy minima for DX⋅⋅⋅A− and D⋅⋅⋅XA−. In the case where D=A, D∞h symmetric complexes with equal bond distances rDX=r are formed (see Table 1). Furthermore, bond enthalpies at 298 K (ΔH298) show the same trends as the electronic bond energies ΔE with differences in the order of one kcal mol−1 or less (see Tables S1–S5 in the Supporting Information). Thus, for clarity and a straightforward connection with our activation strain analyses, our discussion is based on the trends in electronic bond energies ΔE.
For example, from FH⋅⋅⋅F− to FH⋅⋅⋅I−, the hydrogen-bond strength (ΔE) is weakened from −53 to −18 kcal mol−1, while the r value increases from 1.159 to 2.319 Å, and the value of the bond stretch ΔrDH is reduced from 0.226 to 0.049 Å (see Table 1). This trend correlates with a systematic weakening of the halide’s proton affinity (PA) value of 373 kcal mol−1 for F− to 316 kcal mol−1 for I− (PA values at ZORA-BP86/QZ4P taken from Ref. 49). The effect is even more pronounced in the series from IH⋅⋅⋅F− to IH⋅⋅⋅I− along which ΔE weakens from a value of −81 to −22 kcal mol−1, the r value increases from 0.982 to 1.941 Å, and the ΔrDH value is reduced from 0.694 to 0.316 Å. Note that, on the other hand, from FH⋅⋅⋅F− to IH⋅⋅⋅F−, ΔE is strengthened from a value of −53 to −81 kcal mol−1, while the value of r decreases from 1.159 to 0.982 Å, and the stretch value ΔrDH is increased from 0.226 to 0.694 Å. The higher extent of deformation in the more strongly hydrogen-bonded complexes is also reflected by a more destabilizing strain energy (ΔEstrain; see Table 1). This trend correlates with a systematic weakening of the halogen–hydrogen bond from a homolytic bond dissociation energy (BDE) value of 144 kcal mol−1 in FH to 82 kcal mol−1 in IH (see Table 2). Furthermore, note that the bond distance r in DH⋅⋅⋅A− is in all cases longer than that of rHA in the diatomic HA molecule, as revealed by the corresponding difference in bond distances Δr=r−rHA. This difference Δr increases in value from 0.226 Å in FH⋅⋅⋅F− to 0.694 Å in FH⋅⋅⋅I− and from 0.049 Å in IH⋅⋅⋅F− to 0.316 Å in IH⋅⋅⋅I−.
We conclude that the DH⋅⋅⋅A− hydrogen bond becomes stronger and relatively shorter, while the DH bond becomes more elongated in the complex, as the A− anion is a stronger base and/or the DH bond is weaker.
Halogen-bond strength and structure
In part, the halogen bonds display similar trends to the hydrogen bonds, but there are also striking differences. In general and in agreement with ab initio results, the fluorine bonds are the weakest and the iodine bonds the strongest halogen bonds.21, 23, 24 The heavier DX⋅⋅⋅A− halogen bonds (i.e., X=Cl, Br and I) become weaker and longer as the accepting halide (A−) varies from F− to I−, similar to the corresponding hydrogen bonds. In the case of the iodine-bonded complexes DI⋅⋅⋅A−, for example, ΔE weakens from a value of around −70 kcal mol−1 for A−=F− to around −40 kcal mol−1 for X−=I− (see Table 1). However, the fluorine bonds DF⋅⋅⋅A− display a more complex dependency of ΔE upon variation of the accepting halide A−. From A−=F− to Cl−, the fluorine bond strength still weakens, similar to the situation for the hydrogen bonds and the heavier halogen bonds. However, thereafter, along A−=Cl−, Br− and I−, the fluorine-bond strength no longer continues to weaken but instead becomes stronger. This is most clearly seen in the series constituted by the complex FF⋅⋅⋅A− between a fluorine molecule and a halide ion. Here, ΔE for the fluorine-bond strength varies along A−=F−, Cl−, Br− and I− with values of −52, −43, −44 and −48 kcal mol−1, respectively (see Table 1).
Interestingly, variation of the donating atom D has opposite effects on the halogen bonds DX⋅⋅⋅A− and hydrogen bonds DH⋅⋅⋅A−. All halogen bonds studied here become weaker and longer as D is varied from F to I (see Table 1), whereas the hydrogen bonds were found to become stronger and shorter along this series (see above). For example, along the series from FF⋅⋅⋅F− to IF⋅⋅⋅F−, the fluorine-bond strength weakens from a ΔE value of −52 to only −24 kcal mol−1, the fluorine-bond distance r increases in value from 1.755 to 1.993 Å, and the stretch ΔrDX decreases in value from 0.335 to 0.181 Å.
Bond analyses with variation of A−
Our analyses show that the weakening of hydrogen bonds DH⋅⋅⋅A− and of heavier halogen bonds DX⋅⋅⋅A− (X=Cl, Br, I), as the accepting group varies from A−=F− to I−, is directly related to the concomitant reduction in electron-donating capacity of the np-type HOMO of the A− halide. The hydrogen and halogen bonds appear to have an electrostatic component (ΔVelstat) and a covalent component (ΔEoi) stemming mainly from the HOMO–LUMO interaction between the occupied halide np atomic orbital (AO) and the DH or DX antibonding σ* acceptor orbital, shown schematically in Figure 1. Both bonding components, ΔVelstat and ΔEoi, are weakened as the halide HOMO becomes more diffuse and effectively lower in energy50 from A−=F− to I− (see Tables S6, S8–S10 in the Supporting Information). Consequently, the interaction energy (ΔEint) and, thus, the net hydrogen- or halogen-bond strength ΔE becomes less stabilizing along A−=F− to I− (see Table 1 and Tables S1 and S3–S5 in the Supporting Information).
Figure 1. Simplified orbital-interaction diagrams for a) hydrogen-bonded complexes DH⋅⋅⋅A−, b) halogen-bonded complexes DX⋅⋅⋅A−, c) hydrogen halides DH, and d) dihalogens DX, as they emerge from our quantitative Kohn–Sham MO analyses.
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The key to understanding why fluorine bonds DF⋅⋅⋅A− show a more complex, partially opposite trend (i.e., the expected weakening from A−=F− to Cl− but thereafter a strengthening along A−=Cl−, Br− and I−) is contained in the counteracting effects evolving from DF bond stretching induced in the diatomic DF molecule as it interacts with the halide A−. Interestingly, activation strain analyses reveal that from an early until a relatively advanced stage of the complexation reaction, for a given point along the reaction coordinate ζ, we indeed recover the original trend in interactions, namely, that ΔEint(ζ) weakens from A−=F− to I−. This can be nicely seen in Figure 2 which, for six representative series, shows the activation strain diagrams along the entire reaction coordinate ζ projected onto the stretch ΔrDX of the complexation reaction between a DX molecule approaching the halogen-bond accepting A− (see section, Theoretical Methods). Each of the six activation strain diagrams in Figure 2 refers to one particular DH or DX molecule forming a hydrogen or halogen bond with A−=F−, Cl−, Br− and I−. The strain-energy curves (ΔEstrain) within each of these subgraphs coincide because they refer to the same diatomic molecule being stretched as the complexation reaction progresses. Consequently, the trend A−=F− to I− in the total DH⋅⋅⋅A− and DX⋅⋅⋅A− energy profiles ΔE in each subgraph is directly determined by the trend in the corresponding interaction-energy curves (ΔEint). Also, as can be seen in Figure 2, the ΔEint curve appears to be most stabilizing for A−=F− and then weakens along Cl−, Br− and I−, for any given diatomic molecule DH or DX, including all fluorine-bonded molecules DF.
Figure 2. Activation strain analyses along the reaction coordinate for DX + A− complexation as a function of A−=F−, Cl−, Br− and I−, projected onto the DX stretch ΔrDX for a) hydrogen bonds, b) fluorine bonds and c) iodine bonds, with donating groups D=F (left) and D=I (right). Energy profiles ΔE (solid lines) are decomposed into strain energy ΔEstrain (dashed lines above ΔE=0) and interaction energy ΔEint (dashed lines below ΔE=0).
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In other words, fluorine bonds DF⋅⋅⋅A− would also show a weakening in interaction ΔEint from A−=F− to I−, as the hydrogen bonds and all other halogen bonds, if it were not for the increasingly stretched DF bond in the fluorine-bond-donating diatomic molecule (see Table 1 and Figure 2). This structural phenomenon is promoted by a combination of factors: 1) a weak DX bond that is easily stretched; 2) a strong interaction with an approaching halide A−; and importantly, 3) a DX σ* acceptor orbital that quickly drops in energy as the DX bond elongates (see Figure 1). The latter generates a driving force for DX stretching in DX⋅⋅⋅A− because it enhances the orbital interactions and thus ΔEint (see Figures 1 and 2). Indeed, DX stretching is most pronounced if this bond in the diatomic fragment is weaker, that is, for the weaker halogen–hydrogen bonds (DX=IH) and the weaker halogen–halogen bonds (DX=FF; see Table 1). In the latter, it is able to affect the trend in overall bond strength ΔE. The DF stretching in fluorine-bonded complexes is most pronounced in the FF⋅⋅⋅A− series, along which the FF stretch ΔrDX increases from a value of 0.3 via 0.4 and 0.5 to 0.6 Å. This further stretch is able to induce the reversal of the trend in bond strength ΔE along the equilibrium structures FF⋅⋅⋅Cl−, FF⋅⋅⋅Br− and FF⋅⋅⋅I− (see Table 1).
Thus, fluorine-bond analyses in the DF⋅⋅⋅A− equilibrium geometries show that in most cases the interaction energy (ΔEint) between the stretched DF molecule and the halide A−, as well as its components ΔVelstat and ΔEoi, become more stabilizing along the entire series A−=F− to I−, that is, already from F− to Cl− (see Table 1, S2 and S7 in the Supporting Information). This is indeed most pronouncedly so in the series FF⋅⋅⋅A−, due to the FF bond in the DX fragment being relatively weak. Along the series FF⋅⋅⋅F−, FF⋅⋅⋅Cl−, FF⋅⋅⋅Br− and FF⋅⋅⋅I−, ΔEint increases in strength from a value of −75 to −78, −82, and −94 kcal mol−1, respectively. For comparison, along the corresponding series with the much stronger FI bond in the DX fragment, that is, FI⋅⋅⋅F−, FI⋅⋅⋅Cl−, FI⋅⋅⋅Br− and FI⋅⋅⋅I−, the ΔEint weakens from −81 to −56, −51, and −48 kcal mol−1.
The overall bond strength ΔE along the fluorine-bonded series shows the aforementioned initial weakening followed by a strengthening, because the DF stretching and the concomitant strain energy (ΔEstrain) becomes more destabilizing along the series and, from A−=F− to Cl−, dominates the strengthening in ΔEint (see Table 1).
We conclude that, in general, hydrogen bonds DH⋅⋅⋅A− and halogen bonds DX⋅⋅⋅A− become weaker along A−=F− to I− because the larger radii and lower np AO energies of the halides lead to weaker electrostatic attraction and weaker orbital interactions. Interestingly, for the same reason, F− is the halide with the strongest gas-phase basicity, the strongest alkyl cation affinity and the lowest barrier for SN2 reactions with halomethanes.49–51 The trend in DF⋅⋅⋅A− fluorine-bond strength is partially inverted, that is, ΔE becomes more stabilizing along A−=Cl−, Br− and I− because of a more subtle interplay of factors. Notably, a significant stretching of the relatively weak DF bond in the DF⋅⋅⋅A− equilibrium structures lowers the DF σ* acceptor orbital and thus amplifies the donor–acceptor orbital interactions, for example, along FF⋅⋅⋅Cl−, FF⋅⋅⋅Br− and FF⋅⋅⋅I−.
Bond analyses with variation of D
We recall that for the hydrogen bonds DH⋅⋅⋅A−, a heavier donating halogen D results in a stronger bond, whereas the same variation in D weakens the halogen bonds DX⋅⋅⋅A− (see Table 1 and S1–S5 in the Supporting Information). In both cases, the trend in bond strength ΔE is determined by the interaction energy ΔEint. For example, from FH⋅⋅⋅F− to IH⋅⋅⋅F−, ΔEint is strengthened from a value of −71 to −121 kcal mol−1, whereas from FI⋅⋅⋅F− to II⋅⋅⋅F− it is weakened from a value of −75 to −29 kcal mol−1 (see Table 1). The strain energy (ΔEstrain) is not negligible, but it does not alter the trend set by ΔEint. Our activation strain analyses explain the above differences between hydrogen and halogen bonds, but they also confirm once more that both are very similar in nature (see Figure 3).
Figure 3. Activation strain analyses along the reaction coordinate for DX + A− complexation as function of D=F, Cl, Br and I, projected onto the DX stretch ΔrDX for a) hydrogen bonds, b) fluorine bonds and c) iodine bonds, with accepting groups A−=F− (left) and A−=I− (right). Energy profiles ΔE (solid lines) are decomposed into strain energy ΔEstrain (dashed lines above ΔE=0) and interaction energy ΔEint (dashed lines below ΔE=0).
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Starting with some general observations, we find that for hydrogen as well as halogen bonds, the strain-energy (ΔEstrain) curves are most unfavorable when D=F and gradually become less destabilizing as the donating atom is varied along D=F, Cl, Br and I (see Figure 3). Furthermore, we find that for all DH⋅⋅⋅A− and DX⋅⋅⋅A− complexes, the interaction-energy (ΔEint) curves become less stabilizing along D=F, Cl, Br and I. The resulting energy profiles and, therefore, the stability and geometric properties of the complexes DH⋅⋅⋅A− and DX⋅⋅⋅A− depend on the balance between the ΔEstrain and ΔEint curves, which we first discuss individually.
The slope and shape of the ΔEstrain curves is of course directly related to the DX bond strength of the diatomic fragment, which in general becomes stronger as the polarity across the DH or DX bond increases (see Table 2). This is a well-known and understood phenomenon.52 From FH to IH, the halogen–hydrogen bond strength decreases significantly from a value of 143 to 82 kcal mol−1 (Table 2). The corresponding halogen–halogen bonds are all much weaker, and variations in the homolytic BDE are also much smaller. From FF to IF, the bond strength increases from 50 kcal mol−1 to 75 kcal mol−1, while for the fragments DX, where X is Cl, Br or I, the bond strength generally decreases from a value of around 70 kcal mol−1 for FX to around 50 kcal mol−1 for IX. Thus, for the hydrogen-bonded complexes, the ΔEstrain curves show a pronounced reduction in slope from FH to IH, which, in the corresponding hydrogen-bonded complexes FH⋅⋅⋅A− to IH⋅⋅⋅A−, translates into an increasing stretch ΔrDH of the diatomic fragment. As the stretch ΔrDH becomes larger from equilibrium structures FH⋅⋅⋅A− to IH⋅⋅⋅A−, the ΔEint curves have been able to descend further, to lower, more stabilizing energies. The final result is, thus, an increasing stability of the DH⋅⋅⋅A− complexes when the donating atom D is varied from F to I.
For the halogen bonds, the ΔEstrain curves are very similar and not decisive. The reason for the decreased stability of the DX⋅⋅⋅A− complexes upon the same variation of D from F to I is, therefore, that the ΔEint curves descend more gradually to overall less stabilizing values. ΔEint becomes less stabilizing from FX⋅⋅⋅A− to IX⋅⋅⋅A− because of decreasing electrostatic attractions (ΔVelstat) and, in some cases, also because of greater Pauli repulsions (ΔEPauli; see Tables S6–S10 in the Supporting Information). Both of these effects are easily explained considering the electronegativities of the halogens. Along the series FX to IX, the central atom X becomes relatively more electronegative, which will lead to a greater negative charge on this central atom, thus, reducing the electrostatic attraction with the anionic A−, while concomitantly the occupied orbitals will have more X character, which in turn introduces stronger Pauli repulsion.
Bond analyses with variation of X
A more direct comparison of hydrogen and halogen bonds DX⋅⋅⋅A− can be obtained by varying X along H, F, Cl and I, while keeping the donating atom (D) and the accepting halide (A−) constant (see Figure 4). We do this for four combinations of D and A−, giving rise to four subgraphs in Figure 4. There appears to be a regular trend of increasing strength from the fluorine bonds to the iodine bonds. This trend derives again directly from the electronegativity difference across the DX bond of the diatomic fragment; from DF to DI, the charge distribution on the DX fragment is increasingly polarized towards D, away from A− (see VDD atomic charges in Table 2), whereas the σ* acceptor orbital achieves a higher amplitude on X (see Figure 1). This results in a strengthening of the halogen bond DX⋅⋅⋅A− because of greater electrostatic attraction, less Pauli repulsion and more stabilizing donor–acceptor orbital interactions (see Tables S7–S10 in the Supporting Information).
Figure 4. Activation strain analyses along the reaction coordinate for DX + A− complexation as function of X=H, F, Cl and I, projected onto the DX stretch ΔrDX for bond donating groups a) D=F and b) D=I, and accepting groups A−=F− (left) and A−=I− (right). Energy profiles ΔE (solid lines) are decomposed into strain energy ΔEstrain (dashed lines above ΔE=0) and interaction energy ΔEint (dashed lines below ΔE=0).
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In analogy to the situation described above, hydrogen bonds might be expected to be much stronger than the halogen bonds due to the large and favorable polarization across the DH bond leading to a partially positively charged hydrogen atom in DH. For example, the VDD atomic charge on X in FH, FF and FI amounts to +0.20, 0.00 and +0.13 au, respectively (see Table 2). The decomposition of the interaction energy into its components shows indeed a stronger electrostatic attraction (ΔVelstat) to the bonding energy in the case of the hydrogen bonds (compare Tables S6–S10 in the Supporting Information). Note, however, that this does not imply that hydrogen bonds are always stronger than the corresponding halogen bonds, since in our model systems, the bonding mechanism is never purely, or even predominantly, electrostatic. The covalent or orbital-interaction term (ΔEoi) is relatively large and crucial for understanding the bonding in our model systems. For the hydrogen-bonded complexes DH⋅⋅⋅A−, the orbital-interaction term accounts for 40 to 66 % of the total bonding interactions (ΔVelstat+ΔEoi). The stabilization due to this term results predominantly from a charge transfer from the np orbitals of the halide into the σ* LUMO of the hydrogen halide (see Figure 1). For the halogen-bonded complexes DX⋅⋅⋅A−, the contribution from the orbital-interaction term ranges from 43 % for FI⋅⋅⋅F− to as much as 97 % for IF⋅⋅⋅F− at the other end of the spectrum. The larger covalent contribution, in the case of the halogen bonds, is the result of the low orbital energy of the empty dihalogen σ* orbital (e.g., −0.7 eV for FH and −6.2 eV for FF; see Table 2), which directly translates into a stronger donor–acceptor orbital interaction with the halide np orbital (compare Tables S6–S10 in the Supporting Information). Note that, percentagewise, ΔEoi in the halogen bonds appears even larger because of the aforementioned, less favorable electrostatic attraction ΔVelstat.
The nature of the strong hydrogen and halogen bonds discussed in this work strongly resembles that of the weaker, neutral hydrogen and halogen bonds, although dispersion interactions become relatively more important in the latter.2, 19, 24, 31 Preliminary results of dispersion-corrected ZORA-BP86-D3/TZ2P calculations on FI⋅⋅⋅FI (ΔE=−4.3 kcal mol−1), ClCl⋅⋅⋅ClCl (ΔE=−1.3 kcal mol−1) and II⋅⋅⋅II (ΔE=−6.6 kcal mol−1) show that the covalent component ΔEoi amounts to 43–59 %, whereas dispersion contributes 2–17 % to the total of all bonding interactions (ΔEoi + ΔVelstat +dispersion interaction; see Theoretical Methods and ref. 53 for the D3-dispersion correction as proposed by Grimme). The covalent contribution in these neutral model complexes stems from a donor–acceptor orbital interaction from an occupied π* orbital on one dihalogen fragment into the σ* orbital of the other dihalogen fragment.
We conclude that halogen bonds DH⋅⋅⋅A− and hydrogen bonds DX⋅⋅⋅A− have a very similar bonding mechanism consisting of both electrostatic and covalent contributions. The electrostatic attraction is less favorable in the halogen bonds due to a smaller and in some cases less favorably oriented polarization across the dihalogen molecule DX. Nevertheless, halogen bonds can become stronger than hydrogen bonds because of a more stabilizing covalent component in the former. The reason is the lower orbital energy of the empty σ* orbitals in dihalogen molecules DX leading to a stronger, more favorable donor–acceptor orbital interaction with the halide A− np orbital (see Table 2).