For decades one has strived to synthesize a compound with the longest covalent C−C bond applying predominantly steric hindrance and/or strain to achieve this goal. On the other hand electronic effects have been added to the repertoire, such as realized in the electron deficient ethane radical cation in its D${}_{3d}$ form. Recently, negative hyperconjugation effects occurring in diamino-o-carborane analogs such as di-N,N-dimethylamino-o-carborane have been held responsible for their long C−C bonds. In this work we systematically analyzed CC bonding in a diverse set of 53 molecules including clamped bonds, highly sterically strained complexes such as diamondoid dimers, electron deficient species, and di-N,N-dimethylamino-o-carborane to cover the whole spectrum of possibilities for elongating a covalent C−C bond to the limit. As a quantitative intrinsic bond strength measure, we utilized local vibrational CC stretching force constants $k^a$(CC) and related bond strength orders BSO n(CC), computed at the $\omega$B97X-D/aug-cc-pVTZ level of theory. Our systematic study quantifies for the first time that whereas steric hindrance and/or strain definitely elongate a C−C bond, electronic effects can lead to even longer and weaker C−C bonds. Within our set of molecules the electron deficient ethane radical cation, in D${}_{3d}$ symmetry, acquires the longest C−C bond with a length of 1.935 Å followed by di-N,N-dimethylamino-o-carborane with a bond length of 1.930 Å. However, the C−C bond in di-N,N-dimethylamino-o-carborane is the weakest with a BSO n value of 0.209 compared to 0.286 for the ethane radical cation; another example that the longer bond is not always the weaker bond. Based on our findings we provide new guidelines for the general characterization of CC bonds based on local vibrational CC stretching force constants and for future design of compounds with long C−C bonds.

Hydrated calcium ion clusters have received considerable attention due to their essential role in biological processes such as bone development, hormone regulation, blood coagulation, and neuronal signaling. To better understand the biological role of the cation, the interactions between the Ca²⁺ ions and water molecules have been frequently investigated. However, a quantitative measure for the intrinsic Ca—O (ion–solvent) and intermolecular hydrogen bond (solvent–solvent) interactions has been missing so far. Here, we report a topological electron density analysis and a natural population analysis to analyze the nature of these interactions for a set of 14 hydrated calcium clusters via local mode stretching force constants obtained at the ωB97X-D/6-311++G(d,p) level of theory. The results revealed that the strength of inner Ca—O interactions for ${\left[\text{Ca}{\left({\text{H}}_2\text{O}\right)}_{\text{n}}\right]}^{2+}$ (n = 1–8) clusters correlates with the electron density. The application of a second hydration shell to ${\left[\text{Ca}{\left({\text{H}}_2\text{O}\right)}_{\text{n}}\right]}^{2+}$ (n = 6–8) clusters resulted in stronger Ca—O interactions where a larger electron charge transfer between lp(O) of the first hydration shell and the lower valence of Ca prevailed. The strength of the intermolecular hydrogen bonds, formed between the first and second hydration shells, became stronger when the charge transfers between hydrogen bond (HB) donors and HB acceptors were enhanced. From the local mode stretching force constants of implicitly and explicitly solvated Ca²⁺, we found the six-coordinated cluster to possess the strongest stabilizations, and these results prove that the intrinsic bond strength measures for Ca—O and hydrogen bond interactions form new effective tools to predict the coordination number for the hydrated calcium ion clusters.

Information resulting from a comprehensive investigation into the intrinsic strengths of hydrated divalent magnesium clusters is useful for elucidating the role of aqueous solvents on the Mg²⁺ ion, which can be related to those in bulk aqueous solution. However, the intrinsic Mg–O and intermolecular hydrogen bond interactions of hydrated magnesium ion clusters have yet to be quantitatively measured. In this work, we investigated a set of 17 hydrated divalent magnesium clusters by means of local vibrational mode force constants calculated at the $\omega$B97X-D/6-311++G(d,p) level of theory, where the nature of the ion–solvent and solvent–solvent interactions were interpreted from topological electron density analysis and natural population analysis. We found the intrinsic strength of inner shell Mg–O interactions for [Mg(H₂O)_n]²⁺ (n = 1–6) clusters to relate to the electron density at the bond critical point in Mg–O bonds. From the application of a secondary hydration shell to [Mg(H₂O)_n]²⁺ (n = 5–6) clusters, stronger Mg–O interactions were observed to correspond to larger instances of charge transfer between the lp(O) orbitals of the inner hydration shell and the unfilled valence shell of Mg. As the charge transfer between water molecules of the first and second solvent shell increased, so did the strength of their intermolecular hydrogen bonds (HBs). Cumulative local vibrational mode force constants of explicitly solvated Mg²⁺, having an outer hydration shell, reveal a CN of 5, rather than a CN of 6, to yield slightly more stable configurations in some instances. However, the cumulative local mode stretching force constants of implicitly solvated Mg²⁺ show the six-coordinated cluster to be the most stable. These results show that such intrinsic bond strength measures for Mg–O and HBs offer an effective way for determining the coordination number of hydrated magnesium ion clusters.

Substituent effects on the carbon–carbon bonds of hydrocarbons have been a topic of interest within the past seven decades as resultant information would enable one to tune the activity of CC bonds. However, current assessments of the C≡C, C=C, and C−C bond strength of acetylene, ethylene, and ethane as well as their derivatives rely on indirect measures such as bond length and bond dissociation enthalpy. In this work, we introduce a quantitative measure of the intrinsic strength of C≡C, C=C, and C−C bonds for a set of 40 hydrocarbon systems consisting of 3 parent structures, 36 hydrocarbon derivatives involving CF₃, CH₃, CHO, F, NH₂, or OH groups, and a conjugated system, based on vibrational spectroscopy. Local mode force constants $k^a$(CC) were computed at the CCSD(T)/cc-pVTZ level of theory for systems 1–32 and 34–40 and CCSD(T)/cc-pVDZ for 33. From $k^a$(CC), we derived related bond strength orders BSO n(CC) in order to provide quantitative measures of intrinsic bond strength. Topological electron density and natural population analyses were carried out as to analyze the nature of these bonds and complement bond strength measures. For substituted hydrocarbon systems, we found the strengthening/weakening of the CC bonds occurs as the covalent nature of the bond increases/decreases by means of varying charge delocalizations. Our findings provide new guidelines for desirably modulating C≡C, C=C, and C−C bond strength and for the design of prospective pathways for bond cleavage reactions. GRAPHICAL ABSTRACT

Reliable computational methodologies and basis sets for modeling x-ray spectra are essential for extracting and interpreting electronic and structural information from experimental x-ray spectra. In particular, the trade-off between numerical accuracy and computational cost due to the size of the basis set is a major challenge, since molecular orbitals undergo extreme relaxation in the core-hole state. To gain clarity on the changes in electronic structure induced by the formation of a core-hole, the use of sufficiently flexible basis for expanding the orbitals, particularly for the core region, has been shown to be essential. This work focuses on the refinement of core-hole ionized state calculations using the equation-of-motion coupled cluster family of methods through an extensive analysis on the effectiveness of “hybrid” and mixed basis sets. In this investigation, we utilize the CVS-EOMIP-CCSD method in combination and construct hybrid basis sets piecewise from readily available Dunning’s correlation consistent basis sets in order to calculate x-ray ionization energies (IEs) for a set of small gas phase molecules. Our results provide insights into the impact of basis sets on the CVS-EOMIP-CCSD calculations of K-edge IEs of first-row p-block elements. These insights enable us to understand more about the basis set dependence of the core IEs computed and allow us to establish a protocol for deriving reliable and cost-effective theoretical estimates for computing IEs of small molecules containing such elements.