Use of a novel relationship between bond length and bond order to calculate accurate bond orders for carbon–carbon bonds

An exact relationship between bond length and bond order has been derived for the first time based on the concept of electron density. This relationship allows the calculation of sufficiently accurate bond orders and also determines the number of bond-forming electrons. According to this novel relationship between bond order and bond length, the bond order of the carbon–carbon bond in ethylene is 1.75, whereas it is 2.50 in acetylene. These bond orders are readily interpreted by the fragmentation of π-bonds and a consequent decrease in bond order, which is further supported by the chemical properties of these molecules. Assuming structure-specific fragmentation of π-bonds (i.e. one structural motif always adheres to one or two types of bond fragmentation scheme), the bond orders can be predicted for molecules containing multiple carbon–carbon bonds in excellent agreement with the experimental findings.     

A computationally efficient and reliable bond order measure

Bond order indexes are useful measures that connect quantum mechanical results with chemical understanding. One of these measures, the natural bond order index, based on the natural resonance theory procedure and part of the natural bond orbital analysis tools, has been proved to yield reliable results for many systems. The procedure's computational requirements, nevertheless, scales so highly with the number of functions in the basis set and the delocalization of the system, that the calculation of this bond order is limited to small or medium size molecules. We present in this work a bond order index, the first order perturbation theory bond order (fopBO), which is based on and strongly connected to the natural bond orbital analysis tools. We present the methodology for the calculation of the fopBO index and a number of test calculations that shows that it is as reliable as the natural bond orbital index, with the same weak sensitivity to variations among commonly used basis sets and, as opposed to the natural bond order index, suitable for the study of large systems, such as most of those of biological interest.     

The bond order—bond length relationship

The difference in length between two bond orders was reported by Pauling to be essentially the same, regardless of the atoms that make up the bond. To a first approximation these differences hold not only for bond orders 1, 2 and 3 but also for six membered aromatic rings containing all carbon, carbon-nitrogen, nitrogen-nitrogen, carbon-phosphorous, carbon-arsenic, and carbon-antimony bonds. An equation was developed (based upon these differences) that relates bond order and bond length. The output of this equation was compared with those of Gordy and Pauling. Our equation as well as the Gordy equation (with revised constants) return a bond length of 1.4 Å for bond order 1.67 which is consistent with theory. (This bond order was not used in developing either the equation or the revised Gordy constants.)     

Comparative studies on a maximum bond order principle

A recently proposed maximum bond order principle is studied with respect to choice of basis orbitals, choice of wavefunction and compared with other methods. Results for bond orders support the choice of Schmidt orthogonalized AO's with subsequent Löwdin orthogonalization. Differences between semiempirical andab initio wavefunctions in minimal basis sets usually have only minor effects on bond order values. For hydrocarbons bond order values are quite similar for Cohen's and this method. Finally, the dependence of bond orders on internal rotation and vibration is investigated in a few simple cases.     

Impact of bond order loss on surface and nanosolid mechanics

An analytical solution shows that a competition between bond order loss and the associated bond strength gain of the lower coordinated atoms near the edge of a surface dictates the mechanics of the surface and, hence, a nanosolid. Bond order loss lowers the activation energy for atomic dislocation, whereas bond strength gain enhances the energy density or mechanical strength in the region near the surface. Therefore, the surface is harder than the bulk interior at temperatures far below the melting point (T(m)), and the surface becomes softer at temperatures close to the surface T(m) that drops because of bond order loss. Matching predictions to measurements reveals that a transition happens to the Hall-Petch relationship for a nanosolid when the effect of bond order loss becomes dominant, and the critical size of the Hall-Petch transition depends intrinsically on the bond nature of the specimen and the ratio of T/T(m), where T is the temperature of operation.     

Exchange Charges and Bond Orders for Homonuclear Diatomic Molecules

A new bond order definition B = βS² is defined which relates the bond order B to the non-classical exchange charges in the bonding region. This definition is successful in correlating the bond order to its bond energy, when the overlap is calculated from a generalized valence-bond pair function and the parameter β is chosen to be the inverse of bond distance.     

On bond orders and valences in the Ab initio quantum chemical theory

The concept of bond order and valence indices calculated at the ab initio level is discussed in connection with their close relationship to the nonclassical exchange effects in bonding. An improved definition of bond order and free valence indices is given for the open-shell SCF (UHF ) case, and the generalization of the bond order and valence indices to correlated wave functions is also introduced.     

Bond length features of linear carbon chains of finite to infinite size: Visual interpretation from Pauling bond orders

Schemes for Kekulé structure counting of linear carbon chains are suggested. Mathematical formulas, which calculate the Pauling bond order P(k, N) of a chemical bond numbered by k, are given for the carbon chain with N carbon atoms. By use of the least‐squares fitting of a linearity, relationships between Pauling bond orders and bond lengths are obtained, and such correlation of the Pauling bond order–bond length can be qualitatively extended to the excited states. The relative magnitudes of Pauling bond orders in unsaturated carbon chains dominate C–C bond lengths a well as the bond length feature with the chain size increasing. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 94: 144–149, 2003     

Studies on organotin compounds using the Del Re method III. Variations in tin-chlorine,tin-carbon and tin-hydrogen stretching frequencies and tin-chlorine bond distance in organotin compounds

The concept of bond order has been extended to Del Re calculations, and the bond orders of tin-chlorine and tin-carbon bonds in Me_4-nSnCl_n (n = 1 to 4) type compounds have been calculated. The tin-chlorine bond order increases progressively from 0.922 in Me₃SnCl to 0.977 in SnCl₄, and correlates satisfactorily with the experimental tin-chlorine bond distances. The tin-carbon bond order, on the other hand, remains almost constant, in agreement with the constancy of tin-carbon bond distance in the series. The average tin-chlorine, tin-carbon and tin-hydrogen stretching frequencies in similar compounds vary linearly with the calculated bond polarities indicating variation in bond polarity to be the dominating factor. The unusually low values of the tin-carbon stretching frequency for the tin-vinyl bond compared to the tin-methyl bond in Me₃ViSn and Et₂Vi₂Sn can also be explained in terms of larger polarity of the tin-vinyl bond in these compounds.     

A CNDO/2 investigation of the CH3/CF3 substitution effect

The variation of the A-C bond lengths with substitution of methyl by perfluoromethyl in molecules of the kind A(CH₃ )_n is investigated using the CNDO/2 method. Calculations were performed with A as fluorine, oxygen, nitrogen, sulphur and phosphorous and n = 1, 2 or 3. The variation of the A-C bond length can be explained qualitatively by combining two effects, (1) changes in the covalent bond order and (2) changes in the ionic bond strength. While the covalent bond order decreases in all cases, the extent of the decrease depending largely on the electronegativity of A, the ionic bond order increases for fluorine, oxygen, nitrogen and sulphur and decreases in the case of phosphorous. The variations in the ionic bond strength are found to depend on the electronegativity of A as well as on the number of substituted methyl groups.     

Pauling bond orders in coronoid hydrocarbons: A general solution

Benzenoid and coronoid systems are considered. A bond order is defined in terms of the elements of the inverse of a skew-symmetric adjacency matrix. It is conjectured that it is identical with the Pauling bond order. A computer program was designed for computing Pauling bond orders of Kekuléan coronoids in general. Numerical examples are given. The skew-symmetric adjacency matrix was exploited for recognition of essentially disconnected coronoid systems. The 29 smallest essentially disconnected coronoids with the phenalene hole are depicted.     

Inverted-sandwich-type and open-lantern-type dinuclear transition metal complexes: theoretical study of chemical bonds by electronic stress tensor

We study the electronic structure of two types of transition metal complexes, the inverted-sandwich-type and open-lantern-type, by the electronic stress tensor. In particular, the bond order $b_\varepsilon$ measured by the energy density which is defined from the electronic stress tensor is studied and compared with the conventional MO-based bond order. We also examine the patterns found in the largest eigenvalue of the stress tensor and corresponding eigenvector field, the ??spindle structure?? and ??pseudo-spindle structure??. As for the inverted-sandwich-type complex, our bond order $b_\varepsilon$ calculation shows that relative strength of the metal-benzene bond among V, Cr, and Mn complexes is V?>?Cr?>?Mn, which is consistent with the MO-based bond order. As for the open-lantern-type complex, we find that our energy density-based bond order can properly describe the relative strength of Cr?CCr and Mo?CMo bonds by the surface integration of the energy density over the ??Lagrange surface?? which can take into account the spatial extent of the orbitals.     

Use of ELF and AIM parameter ratios for bond order characterization of two‐, three‐, and four‐coordinate gallium hydrides

Calculations at the B3LYP/6‐311 + G(d,p)//B3LYP/6‐311 + G(d,p) level involving the electron localization function (ELF) and atoms‐in‐molecules (AIM) methods have been carried out for two‐, three‐, and four‐coordinate gallium hydrides present in Na and Li salts and as the isolated dianionic species, for some isoelectronic germanium compounds, and for several neutral gallium hydrides. Using the ratio of delocalization indices and bond basin populations referenced to reasonable standards, formal bond orders are derived. While chemically expected bond orders are found in most cases, the situation in the [HGaGaH]²⁻ species appears to lie intermediate between bonds of order 2 and 3, and that for neutral trans‐bent HGaGaH is found to be best described as a bond of order 1. In these cases the larger bond order predicted by ELF bond basin populations evidently results from overlap of the bond basin into the lone pair (nonbonding) region of the molecule. © 2003 Wiley Periodicals, Inc. Heteroatom Chem 14:175–185, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/hc.10120     

Is it True That the Normal Valence‐Length Correlation Is Irrelevant for Metal–Metal Bonds?

The most intriguing feature of metal–metal bonds in inorganic compounds is an apparent lack of correlation between the bond order and the bond length. In this study, we combine a variety of literature data obtained by quantum chemistry and our results based on the empirical bond valence model (BVM), to confirm for the first time the existence of a normal exponential correlation between the effective bond order (EBO) and the length of the metal–metal bonds. The difference between the EBO and the formal bond order is attributed to steric conflict between the (TM)_n cluster (TM=transition metal) and its environment. This conflict, affected mainly by structural type, should cause high lattice strains, but electron redistribution around TM atoms, evident from the BVM calculations, results in a full or partial strain relaxation.     

Bond indices and heats of formation of some nitriles

The definition of bond index introduced by Ehrenson, and written in terms of bond order and overlap integrals, has been applied to a series of nitriles. A good correlation between heats of formation and a linear equation in the indices for each bond present is obtained. This was extended to include the results for a set of hydrocarbons.     

The value of the π-bond order–bond length relationship in geometry prediction and chemical bonding interpretation

The π-bond order–bond length relationship is reintroduced to the literature and extended to heteronuclear bonds by presenting graphs derived solely by theoretical methods. π-bond order and overlap population results for carbon–carbon, carbon–nitrogen, and carbon–oxygen bonds obtained from ab initio STO -3G calculations using theoretically-optimized geometries are reported for a series of pteridines and for a wide range of small organic molecules. The order–length correlation graphs are used in predicting the “intrinsic” single bond lengths for sp² – sp² and sp – sp hybridized C? C, C? N, and C? O bonds, and in evaluating the relative importance of hybridization, π-electron delocalization and bond polarization effects in causing bond shortening in conjugated and hyperconjugated molecules. The calculated value of the π-bond order for a given bond in a molecule is shown to be relatively insensitive to moderate geometry changes: Hence, a use for the correlation graphs in geometry prediction is suggested. Some results for the extended 4-21G basis set are also presented.     

Relations between length,order, and force constant for C-C bonds

A relation is derived between length and force constant for single and multiple carbon-carbon bonds, and also a relation between bond order and shortening consequent on delocalization of the π-electrons. The effect of the latter on the bond order is without appreciable effect on the length for linear conjugated molecules; the shortening of single bonds in compounds such as butadiene and diacetylene is due mainly to the state of hybridization of the carbon atoms. Constant length in the single bonds of a given hybrid type can not serve as a criterion for absence of shifts in electron density. The bond lengths in aromatic compounds are the most sensitive to π-electron delocalization.     

The shortest metal‐metal bond

The synthesis and isolation of stable bimetallic complexes that contain formally quintuply bonded transition metals is a novel and emerging field of science. Efforts have been undertaken in designing and tuning the ligands to achieve a very short (actually the shortest) metal‐metal bond. The motivation for these efforts arose from the expectation that an increasing bond order may go along with a shortening of the bond length. In consequence, formally quintuply bonded bimetallics could have shorter metal‐metal distances than quadruply bonded ones. A chromium homo‐bimetallic complex with a Cr‐Cr bond length of 1.7293(12) Å has been synthesized, and a formal bond order of five was assigned. This compound holds the record for the shortest metal‐metal bond in a stable molecule to date. At this stage, there is no evidence that additional shortening is impossible. © 2010 The Japan Chemical Journal Forum and Wiley Periodicals, Inc. Published online in Wiley InterScience ( www.interscience.wiley.com ) DOI 10.1002/tcr.201000028     

Quantum Monte Carlo Calculation of Correlation Effects on Bond Orders

We present a computational approach, using quantum Monte Carlo, that provides some insight into the effect of electron correlation on chemical bonding between individual pairs of atoms. Our approach rests upon a recently suggested relation between the bond order and charge fluctuations with respect to atomic domains. Within the present implementation we have taken a compromise between conceptual rigour and computational simplicity. In a first step atomic domains were obtained from Hartree-Fock (HF) densities, using Bader’s definition of atoms in molecules. These domains were used in a second step in quantum Monte Carlo calculations to determine bond orders for pairs of atoms. Correlation effects have been studied by comparison of HF bond orders with those obtained from pure diffusion quantum Monte Carlo calculations. We illustrate this concept for C–O and C–S bonds in different molecular environments. Our results suggest an approximate linear relation between bond order and bond length for these kinds of bonds.     

On the bond order of C2 and N2

The commonly adopted bond order values of C₂ and N₂ are critically investigated with a new bond order concept. Ab initio calculations with extended basis sets suggest that C₂ can be described by a double to triple bond closer to acetylene than to ethylene and N₂ by a triple bond. The basis set dependence is discussed. Also a relation between the number of basis functions, MO's and non-vanishing eigenvalues of the bond order parts of the density matrix is presented.