*3.2. The Relationship between De and k*<sup>σ</sup> *in the B*··· *CO2, B*··· *N2O, and B*··· *CS2 Series*

It was established [31] for a wide range of hydrogen-bonded complexes B··· HX (X = F, Cl, Br, or I) and halogen-bonded complexes B··· XY (X and Y are halogen atoms) that their dissociation energies *D*<sup>e</sup> (as calculated ab initio at the CCSD(T)(F12c)/cc-pvdz-F12 level of theory) are directly proportional to their intermolecular stretching force constants *k*σ (as determined experimentally from centrifugal distortion constants *DJ* or Δ*<sup>J</sup>* obtained by measuring rotational spectra). The constant of proportionality was found to be 1.5(1) × <sup>10</sup><sup>3</sup> m2·mol<sup>−</sup>1. Later, it was shown for the B··· HF, B··· HCl, B··· F2, B··· Cl2, and B··· ClF series, where B is a Lewis base, N2, CO, HCCH, C2H4, HCN, H2S, H2O, PH3, or NH3, that the same constant of proportionality applies [32] when *k*<sup>σ</sup> was calculated ab initio at the CCSD(T)/aug-cc-pVTZ level of theory and *D*<sup>e</sup> was obtained via a CCSD(T)/CBS calculation, where CBS indicates a complete basis-set extrapolation using the aug-cc-pV*n*Z (*n* = T and Q) basis sets. The opportunity is taken here to investigate the corresponding relationship for the tetrel-bonded B··· CO2 complexes, the pnictogen-bonded B··· N2O complexes, and the chalcogen-bonded B··· CS2 complexes for the series of Lewis bases, B = CO, HCCH, H2S, HCN, H2O, PH3, and NH3, when both *k*σ and *D*e are calculated in the same way as described in Reference [32].

Values of *D*<sup>e</sup> and *k*<sup>σ</sup> so determined are recorded in Table 4, while Figure 7 shows a plot of *D*<sup>e</sup> as the ordinate and *k*<sup>σ</sup> as the abscissa for the B··· CO2, B··· N2O, and B··· CS2 series investigated here, with color coding of the points as red, blue, and yellow, respectively. For consistency with HCN··· CO2, of the isomers of HCN··· N2O, only the data for the T-shaped form are included in Table 4 and Figure 7. The calculation of *k*<sup>σ</sup> for the parallel isomer of N2O··· HCCH was prevented by convergence problems, as well as for H2S··· N2O because, as the N··· S distance was varied, there was a switch to the hydrogen-bonded arrangement N2O··· HSH. H3P··· CS2 was excluded because it does not involve a chalcogen bond, unlike the remaining B··· CS2 complexes. The results of a linear regression fit of the points in Figure <sup>7</sup> are as follows: gradient = 1.44(20) × 103 m2·mol−<sup>1</sup> and intercept on the ordinate = −0.32(124) kJ·mol−1. Thus, within the errors of the fit, *<sup>D</sup>*<sup>e</sup> and *<sup>k</sup>*<sup>σ</sup> are directly proportional, and the slope of the regression line agrees with those found previously for the B··· HF and B··· HCl series, and for the halogen-bonded series B··· F2, B··· Cl2, and B··· ClF [32] when calculations were conducted at identical levels of theory, namely 1.38(7) × 103 m2·mol−<sup>1</sup> and 1.49(5) × 103 m2·mol−1, respectively. Plots of *<sup>D</sup>*<sup>e</sup> versus *<sup>k</sup>*<sup>σ</sup> using *<sup>D</sup>*<sup>e</sup> values calculated at the CCSD(T)(F12c)/cc-pVDZ-F12 level of theory and experimentally available *k*σ [31], but with many more complexes in each of these two classes, gave almost identical slopes of 1.52(3) × 103 <sup>m</sup>2·mol−<sup>1</sup> and 1.47(3) × 103 m2·mol−1, respectively. Evidently, the same relationship between *<sup>D</sup>*<sup>e</sup> and *<sup>k</sup>*<sup>σ</sup> holds for hydrogen-bonded complexes B··· HX, halogen-bonded complexes B··· XY, the tetrel-bonded complexes B··· CO2, the pnictogen-bonded complexes B··· N2O, and the chalcogen-bonded complexes B··· CS2. This fact is visually established by the plot of *D*<sup>e</sup> versus *k*<sup>σ</sup> shown in Figure 8. The figure includes all B··· HF, B··· HCl, B··· F2, B··· Cl2, and B··· ClF complexes reported in Reference [32] and all the B··· CO2, B··· N2O, and B··· CS2 complexes included in Figure 7. Both sets of series were calculated in the same way, i.e., CCSD(T)/aug-cc-pVTZ for *k*<sup>σ</sup> and CCSD(T)/CBS for *D*e. The linear regression fit for all these data leads to 1.40(4) × 103 m2·mol−<sup>1</sup> for the slope and −0.42(46) kJ·mol−<sup>1</sup> for the intercept.


**Table 4.** Intermolecular dissociation energies *D*<sup>e</sup> and quadratic force constants *k*<sup>σ</sup> for B··· CO2, B··· N2O, and B··· CS2 complexes.

*<sup>a</sup>* Convergence problems when attempting to calculate the *<sup>E</sup>* (*<sup>r</sup>* − *<sup>r</sup>*e) versus (*<sup>r</sup>* − *<sup>r</sup>*e) curve to obtain *<sup>k</sup>*σ. *<sup>b</sup>* When attempting to calculate *k*σ, the geometry of the complex changes to the hydrogen-bonded isomer N2O··· HSH as (*<sup>r</sup>* − *<sup>r</sup>*e) increases. *<sup>c</sup>* The main non-covalent interaction in this complex is between P of PH3 and S of CS2, and it is a pnictogen bond, not a chalcogen bond.

**Figure 7.** Plot of *D*e calculated at the CCSD(T)/CBS level of theory (CBS indicates a complete basis-set extrapolation using the aug-cc-pV*n*Z (*n* = T and Q) basis sets) versus *k*σ calculated at the CCSD(T)/aug-cc-pVTZ level for B··· CO2, B··· N2O, and B··· CS2 complexes. See text for discussion.

**Figure 8.** Plot of *D*e calculated at the CCSD(T)/CBS level versus *k*σ calculated at the CCSD(T)/aug-cc-pVTZ level for B··· CO2, B··· N2O, and B··· CS2 complexes (this work; see also Figure 7), and B··· HF, B··· HCl, B··· F2, B··· Cl2, and B··· ClF complexes (see Reference [32] for the Lewis bases B involved and the values of *D*<sup>e</sup> and *k*<sup>σ</sup> for the B··· HX and B···XY series).

#### **4. Conclusions**

The series of B··· CO2, B··· N2O, and B··· CS2 complexes was investigated through ab initio calculations at the CCSD(T)/aug-pVTZ level of theory for the Lewis bases, B = CO, HCCH, H2S, HCN, H2O, PH3, and NH3. The atoms, except for some H, lie in a plane for all complexes. The intermolecular bonds in the B··· CO2 complexes are formed by interaction of the electrophilic region around the C atom of CO2 (see Figure 1) with n- or π-electron pairs (nucleophilic regions) carried by B and are, therefore, tetrel bonds. The geometry of each B··· N2O complex investigated (except perhaps for B = PH3) is similar to that of the corresponding member of the B··· CO2 series. Thus, the primary non-covalent interaction involves the central N atom of N2O with an n- or π-electron pair carried by B, but moderated by distortions that appear to arise from the secondary interaction of the electrophilic region of B (e.g., H atoms) with the O atom of N2O. The B··· CS2 series is geometrically distinct from the other two in that (apart from B = PH3) the primary non-covalent interaction is between the electrophilic region centered on the C<sup>∞</sup> axis of CS2 near to an S atom (see Figure 1) and an nor π-electron pair of B, leading to a linear (or nearly linear in the case ofB=H2S) C=S··· B system, and is, therefore, a chalcogen bond. These interpretations are electrostatic in origin and were applied previously to hydrogen bonds in B··· HX complexes [9] and halogen bonds in B··· XY complexes [40]. Consistent with the foregoing observations is the fact that the geometries of members of each of the three series, B··· CO2, B··· N2O, and B··· CS2, can be predicted by rules put forward some years ago for the same purpose for hydrogen-bonded complexes B··· HX and halogen-bonded complexes B··· XY. Moreover, this close relationship between hydrogen, halogen, tetrel, pnictogen, and chalcogen bonds is reflected in the recent generalized definition [43] proposed for non-covalent (E) bonds based on electrostatics, provided below.

*An* E *bond occurs when there is evidence of a net attractive interaction between an electrophilic region associated with an* E *atom in a molecular entity and a nucleophilic region (e.g., an n-pair or π-pair of electrons) in another, or the same, molecular entity, where* E *is the general name for an element of Group 1, 11, 14, 15, 16, or 17 in the Periodic Table.*

We note that some complexes investigated here can be described as of the σ-hole type, while others belong to the π-hole type.

Finally, we showed that the similarity between all of these types of non-covalent interaction extends to the direct proportionality of the dissociation energy *D*<sup>e</sup> and the quadratic intermolecular stretching force constant *<sup>k</sup>*σ, with a constant of proportionality 1.45(7) × <sup>10</sup><sup>3</sup> <sup>m</sup>2·mol−<sup>1</sup> describing all the series, B··· HF, B··· HCl, B··· F2, B··· Cl2, B··· ClF, B··· CO2, B··· N2O, and B··· CS2, when the two measures of binding strength are calculated at the CCSD(T)/CBS and CCSD(T)/aug-cc-pVTZ levels of theory, respectively. As discussed in Reference [31], a Morse function is an example of a potential energy curve for which the dissociation energy and the force constant are directly proportional.

**Supplementary Materials:** The supplementary materials are available.

**Author Contributions:** I.A. and A.C.L. are both contributed to the design of experiments, formal analysis and the writing of draft.

**Funding:** This work was carried out with financial support from the Ministerio de Economía y Competitividad (Project No. CTQ2015-63997-C2-2-P) and the Comunidad Autónoma de Madrid (S2013/MIT2841, Fotocarbon).

**Acknowledgments:** A.C.L. thanks the School of Chemistry, University of Bristol for a Senior Research Fellowship.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Sample Availability:** No samples are available from the authors.

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