Equilibrium CO bond lengths☆
Introduction
The most unequivocal geometrical structure of a molecule is its equilibrium structure, re, which corresponds to a minimum of the Born–Oppenheimer potential energy surface (PES) [1]. Estimates of equilibrium structures can be obtained experimentally via high-resolution spectroscopy and a few well-defined assumptions. Purely experimental determination of re structures is generally straightforward for diatomic molecules [2], but the level of difficulty increases rapidly with the number of atoms because it requires the analysis of, at least, all the fundamental vibrational states of all the isotopologues required for the structural determination. Furthermore, this analysis is often complicated by Coriolis interactions or anharmonic resonances [3]. For this reason, the number of purely experimental equilibrium structures for polyatomic molecules is quite small and is limited to molecules of moderate size.
Ab initio methods have become successful in accurately estimating equilibrium structures of even complex molecules. However, these determinations involve substantial computer resources because to achieve an accuracy similar to experimental techniques requires high-level electron-correlation treatments, such as coupled-cluster (CC) theory including single and double excitations (CCSD) [4] augmented with a perturbational estimate of the effects of connected triple excitations, CCSD(T) [5]. Furthermore, large basis sets, at least of quadruple zeta quality, are also required. For these reasons high-level wavefunction-based ab initio methods are also restricted to relatively small molecules [6].
The semiexperimental (SE) method, resulting in structures, uses equilibrium rotational constants obtained from experimental ground-state rotational constants and computed rovibrational corrections [7]. It is faster, simpler and often more accurate than either the purely experimental method or the high-level ab initio techniques for the determination of equilibrium structures [8]. Thanks to this method employing both experimental and theoretical information, the number of equilibrium structures that are accurately known, with uncertainty estimates on the order of 0.001–0.002 Å for bond lengths and 0.1–0.2° for angles, has grown exponentially during the last twenty years. The SE technique has been successfully used for molecular systems containing several atoms, such as the amino acids glycine [9], alanine [10], and proline [11]. However, this technique is still computationally too expensive for molecules containing more than about 20 atoms.
For large systems, lower-level electronic structure methods have to be used as, for instance, second-order Møller–Plesset perturbation theory (MP2) [12] or Kohn–Sham density functional theory (DFT) [13]. One problem with using these methods is that the computed bond lengths are subject to errors which can be substantial. Nevertheless, if the basis sets employed are large enough (note the less stringent demand in case of DFT), the errors generally turn out to be mainly systematic and can thus be estimated, as demonstrated, for instance, for the CH [14], NH [15], OH [16] and CC [17] bond lengths.
Besides the CC and CH bonds, the CO bond is another important bond in (organic) chemistry. It is involved in alcohols, aldehydes, ketones, acids, esters, and ethers, as well as in many heterocyclic molecules. Over the last few years, accurate equilibrium structures of many molecules containing CO bonds have been determined. One of the goals of this paper is to collect and analyze the accurately known CO bond lengths. Furthermore, such a collection of validated data allows to investigate whether it is possible to estimate the CO bond length employing either the MP2 or the even more widely applicable DFT methods, for example via Becke’s three-parameter hybrid exchange functional [18] and the Lee–Yang–Parr correlation functional [19], together denoted as B3LYP. Third, it is also important to know whether the variation of the CO bond lengths can be explained in simple terms, for example via the atoms-in-molecules (AIM) method.
Section snippets
Ab initio computation of Born–Oppenheimer (BO) equilibrium CO bond lengths
The CCSD(T) method of electronic structure theory is known to give accurate structures provided the basis sets employed are large enough [6], [20], [21]. For relatively small molecules, it is convenient to use the correlation-consistent polarized core–valence quadruple zeta basis set, cc-pwCVQZ, in brief, wCVQZ, [22], all electrons being correlated (AE). For larger molecules, the optimization is usually performed with the smaller correlation-consistent polarized quadruple-zeta basis set cc-pVQZ
Semiexperimental structures of F2CO and Cl2CO
The semiexperimental equilibrium rotational constants are obtained by correcting the experimental ground-state rotational constants with lowest-order vibration–rotation interaction constants (α) [33] determined quantum chemically. The vibration–rotation interaction constants can be estimated if the cubic force field of the molecule, expanded about a reference (usually the equilibrium) structure, is known. We chose the MP2 level of electronic structure theory to determine the anharmonic force
CO bond lengths
The length of the CO bond ranges from 1.11 Å, in HCO+ (note that this is shorter than in CO), to 1.43 Å, in HCOOCH3. See Table 5 for the complete list of 38 molecules studied here.
Using the values of the covalent radii for the C and O atoms, it is possible to roughly estimate the CO bond length: it is 1.12 Å for the triple bond [44], 1.24 Å for the double bond [45] and 1.38–1.42 Å for the single bond [46], [47].
The shortest bond is found for HCO+, r(CO) = 1.1056 Å, which is shorter than in CO, r = 1.1287
Approximate MP2 and DFT bond lengths
It is useful to check whether CO bond lengths could be estimated with reasonable accuracy at lower levels of electronic structure theory. We computed the structure of the molecules of Table 5 at two modest levels of electronic structure theory, MP2(FC)/VQZ and B3LYP/6-311+G(3df,2pd). The MP2 values are almost systematically too large, with a median deviation of −0.0044 Å for multiple bonds and −0.0005 Å for single bonds, whereas the B3LYP values are generally too small for multiple bonds and too
The case of OCSe
Taking into account the large residual found for OCSe, one may wonder whether the experimental re structure is accurate [56]. Actually, the two bond lengths r(CO) and r(CSe) were determined using the equilibrium rotational constants of seven isotopologues which significantly strengthens the reliability of the results. Furthermore, the accuracy was checked using two different empirical mass-dependent methods which are known to be reliable for such a small molecule without hydrogen [57]. Finally,
Summary and conclusions
The range of the CO bond is large but its variation can be qualitatively explained using the AIM theory. In particular, an almost linear relation is found between the bond length and the bond critical point density. Although this result is interesting, it does not permit to easily predict the variation of the bond length. It would be useful to have a simple intuitive way to roughly predict the variation. However, as several factors are involved, it is safer to only compare closely related
Acknowledgements
The work performed in Hungary was supported by the Scientific Research Fund of Hungary (OTKA, K72885 and NK83583). This project was also supported by the European Union and co-financed by the European Social Fund (Grant Agreement No. TÁMOP 4.2.1/B-09/KMR-2010-0003).
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In honor of Professor Jaan Laane for his many contributions to Science.