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     

Generalized bond orders

We introduce generalized bond orders defined in terms of weighted Kekule valence structures. The weights were determined by the contributions of linearly independent and minimal conjugated circuits in individual Kekule valence structure. When special values for the contributions of conjugated circuits of different size are assumed, one obtains quantities that show considerable similarity to the Pauling and the Clar's bond orders. Pauling bond orders are obtained when one assumes that all conjugated circuits make equal contribution to bond orders. © 1994 John Wiley & Sons, Inc.     

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.     

Ground-state properties of benzenoid hydrocarbons by simple bond orbital resonance theory approach

π-electron energies and bond orders of benzenoid hydrocarbons with up to five fused hexagons have been considered by the simple Bond Orbital Resonance Theory (BORT) approach. The corresponding ground states were determined according to four BORT models. In the first three models a diagonalisation of the Hückel-type Hamiltonian was performed in the bases of Kekulé, of Kekulé and mono-Claus and of Kekulé and Claus resonance structures, respectively. In the fourth model a simple BORT ansatz was used. According to this ansatz, the ground state is a linear combination of the positive Kekulé structures, all with equal coefficients. It was shown that π-electron energies and bond orders obtained by these models correlate much better with the PPP energies and bond orders than with the Hückel energies and bond orders. This indicates that a simple BORT approach is quite reliable in predicting the more sophisticated PPP results. Concerning the relative performance of the four BORT models, the best results were obtained with the BORT ansatz. The performance deteriorates with the expansion of the basis set. This is attributed to the fact that in these models the improvement of the basis set is not accompanied with the corresponding improvement of the Hamiltonian. Comparing the BORT-ansatz bond orders with the Pauling bond orders, it was shown that BORT-ansatz bond orders correlate much better with the PPP bond orders. This revised version was published online in July 2006 with corrections to the Cover Date.     

Electronegativity based on the simple bond charge model

A single physical interpretation of the various electronegativity scales of Pauling, Mulliken and Gordy is suggested, based on the simple bond charge (SBC) model of Parr and Borkman for the covalent bond. With a charge partition determined from vibrational frequencies, the SBC model is shown to account for the covalent bond energy in single-bonded homonuclear diatomic molecules and diamond-type crystals. The binding energy to the atom of a bond-electron in the single-bonded homonuclear diatomic molecules agrees with Mulliken's electroaffinity, and provides a definition for electronegativity. Gordy's empirical relation between the bond-stretching force constant and electronegativity is explained. It is then suggested that the physical effect underlying Pauling's thermochemical formula for electronagativity is the location of the bond charge in the heteronuclear molecule. The deviation of Pauling's formula from experiment in the case of the alkali hydrides can then be explained.     

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.     

Bond dissociation energies and bond orders for diatomic alkali halides

The bond dissociation energies for Alkali halides have been estimated based on the derived relations: $$\begin{gathered} D_{AB} = \bar D_{AB} + 31.973{\text{ e}}^{0.363\Delta x} {\text{ and}} \hfill \\ D_{AB} = \bar D_{AB} (1 - 0.2075\Delta xr_e ) + 52.29\Delta x, \hfill \\ \end{gathered} $$ where \(\bar D_{AB} = (D_{AA} \cdot D_{BB} )^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}} \) , Δx represents Pauling electronegativity differences x(_A ?x_B) and r _e is the internuclear distance. A simplified formula relating bond orders, q, to spectroscopic constants is suggested. The formula has the form q = 1.5783 × 10^?3 (ω _e ² r_e/ B_e)^1/2. The ambiguity arising from the Parr and Borkman relation is discussed. The present study supports the view of Politzer that q/(0.5r _e)² is the correct definition of bond order. The estimated bond energies and bond orders are in reasonably good agreement with the literature values. The bond energies estimated with the relations we suggested, for alkali halides give an error of 4.5% and 5.3%, respectively. The corresponding error associated with Pauling's equation is 40.2%.     

On inferences of bond character from bond length; the important role of nonbonded interactions

Although experimental molecular structure is one of the oldest and most frequently applied diagnostic tools in studies of the nature of chemical bonds, it is concluded that its application to all but the grossest details of bonds such as carbon-carbon bonds remains entirely speculative. The concept of bond length is discussed to emphasize the need for careful consideration of ambiguities associated with the natural indeterminacy of atomic positions. A theoretical basis is given to the empirical Schomaker-Stevenson rule relating bond length to electronegativity. The divergent views of Pauling and Walsh on hybridization, ionic character, and bond strength are reconciled to some extent with the aid of a simple model. It is shown that structural effects commonly attributed to conjugation, hyperconjugation, hybridization, and partial ionic character can be rationalized to a remarkable extent in terms of nonbonded interactions. It is suggested that these factors are not fundamentally as distinct from each other as they are often assumed to be, and that much more serious attention should be given to the role of nonbonded interactions.     

Pauling's electronegativity equation and a new corollary accurately predict bond dissociation enthalpies and enhance current understanding of the nature of the chemical bond

Contrary to other recent reports, Pauling's original electronegativity equation, applied as Pauling specified, describes quite accurately homolytic bond dissociation enthalpies of common covalent bonds, including highly polar ones, with an average deviation of +/-1.5 kcal mol(-1) from literature values for 117 such bonds. Dissociation enthalpies are presented for more than 250 bonds, including 79 for which experimental values are not available. Some previous evaluations of accuracy gave misleadingly poor results by applying the equation to cases for which it was not derived and for which it should not reproduce experimental values. Properly interpreted, the results of the equation provide new and quantitative insights into many facets of chemistry such as radical stabilities, factors influencing reactivity in electrophilic aromatic substitutions, the magnitude of steric effects, conjugative stabilization in unsaturated systems, rotational barriers, molecular and electronic structure, and aspects of autoxidation. A new corollary of the original equation expands its applicability and provides a rationale for previously observed empirical correlations. The equation raises doubts about a new bonding theory. Hydrogen is unique in that its electronegativity is not constant.     

Polycyclic aromatic hydrocarbons with five-membered rings: Modeling of experimental X-ray and neutron-diffraction structures

Several of the readily available theoretical programs are evaluated as tools for modeling the structures of polycyclic aromatic hydrocarbons with five-membered rings (CPAHs). The experimentally determined bond lengths and angles are compared to calculated values. Experimental bond lengths are also compared to Pauling and Huckel molecular orbital (HMO) bond orders. Previously published experimental X-ray and neutron-diffraction structures of acenaphthene, acenaphthylene, fluoranthene, cyclopent[o,p,q,r]benz[c]phenanthrene, and corannulene are modeled by the programs MMX, AM1, MNDO, and PM3, and previously reported STO-3G and 6-31G * data are also evaluated. In general, the error differences between the experimental and calculated results for all of the semiempirical programs were small. However, PM3 performed slightly better than AM1 and MMX, while MNDO generated structures which exhibited the largest deviation from experiment. Although the standard deviations for all programs are shown to be of comparable magnitude, a particular bond length or bond angle in any given theoretical calculation can exhibit significant error from the experimental data. The scatter in the bond order data computed from Huckel molecular orbital theory and valence bond theory is contrary to results obtained with alternant systems. It appears that these approaches are less successful at modeling accurately the nonalternant hydrocarbon systems described in this paper.     

Bond orders and electron delocalization indices for S–N,S–C and S–S bonds in 1,2,3-dithiazole systems

A parametric QTAIM-based (topological) model of bond orders and a modification of the Pauling bond order model are proposed for N,S-containing heterocycles, in particular, for 1,2,3-dithiazoles and 1,2,3-dithiazolium systems, which are prone to the formation of stable radicals and therefore are promising compounds in photovoltaics. These models have been parameterized for covalent S–N, S–C and S–S bonds using the electron delocalization indices. A modified Pauling’s bond order model uses turning radii, that is, the distances within which the potential acting on an electron in a system still tends to return that electron to the atomic basin, and avoids the need to choose the hybridization state of bound atoms arbitrarily.     

Generating reaction coordinates by the Pauling relation

The Pauling relation of bond order and bond length together with the BEBO postulate are utilized to generation reaction coordinates on potential energy surfaces of simple exchange reactions. A generalization of the Pauling relation where the constant is dependent on the equilibrium separation is proposed.     

A new approach to the relationship between bond energy and electronegativity

The familiar equation whereby Pauling related heteronuclear bond energies D_A–B to the electronegativity difference Δχ (=∣χ_A − χ_B∣) and the homonuclear bond energies D_A–A and D_B–B has been the subject of critical scrutiny for at least half a century. A modification to this equation that combines the concepts of electronegativity and hardness/softness can be rewritten in terms of two quantities x and y, both having absolute significance. Both homo- and heteronuclear bond energies can be rationalised from these new equations. The quantities x are linearly related to Pauling electronegativities, while y appears to be a measure of an atom’s intrinsic bonding potential, related to size and availability of valence orbitals.     

Examination of P-OR bridging bond orders in phosphate monoesters using (18)O isotope shifts in 31P NMR

Evidence indicates that phosphate monoesters undergo hydrolysis by a loose transition state with extensive bond fission to the leaving group. It has been proposed that part of the high dependence of the rate on the leaving group pKa (betalg = -1.23) arises from weaker ester bonds in the reactants as the pKa of the leaving group decreases, on the basis of X-ray structures and calculations. In contrast, IR and Raman studies suggest that the leaving group has little effect on the length of the P-OR bridging bond in solution. To gather additional data on this issue, we have used (18)O isotopic shifts in 31P NMR to monitor the bond order of P-O bonds in a range of phosphate esters with different leaving groups. Using this technique, we have been able to evaluate whether significant changes are observed in the P-O bond orders for the bridging and nonbridging positions of methyl, ethyl, phenethyl, propargyl, phenyl, and p-nitrophenyl phosphate using [(16)O(18)O] labeled species in deuterium oxide. The results indicate that the bridging and nonbridging bond orders to phosphorus in phosphate monoesters are not significantly altered by differences in the pKa of the leaving group or by the counterion of the phosphate ester dianion.     

Energy estimate of bond polarity

Conclusions The bond polarity, determined as the ionic component fraction of the bond energy, is related in a linear manner to the Pauling polarity value.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No, 10, pp. 2401–2402, October, 1982.The author thanks I. V. Stankevich for discussing the work.     

Balancing the Pauling bond orders in Kekuléan benzenoid hydrocarbons

We consider a cutting of the molecular graph B of a Kekuléan benzenoid molecule into two disconnected subgraphs, S and the other, by deleting from B certain edges. It is required that both subgraphs remain Kekuléan. The edges involved in this cutting are classified as starred and unstarred. A starred edge is incident to a starred carbon site of the subgraph S, whereas an unstarred edge to an unstarred carbon site of S. The following regularity is established: for any above-described cutting of any Kekuléan benzenoid system, the sum of the Pauling bond orders of the starred edges is equal to that for the unstarred edges.     

On heteroaromaticity of nucleobases. Bond lengths as multidimensional phenomena

Three hundred and nine carbon-carbon, carbon-nitrogen, and carbon-oxygen pi-bond lengths in high precision crystal structures of 31 purine and pyrimidine nucleobases were related to the Pauling pi-bond order, its analogues corrected to crystal packing effects, the numbers of non-hydrogen atoms around the bond, and the sum of atomic numbers of the bond atoms. Principal Component Analysis (PCA) and Hierarchical Cluster Analysis (HCA) demonstrated that the bond lengths in the nucleobases are three-dimensional phenomenon, characterized by nine distinct classes of bonds. Bond lengths predicted by Linear Regression models, Pauling Harmonic Potential Curves, Multiple Linear Regression, Principal Component, and Partial Least Squares Regression were compared to those calculated by molecular mechanics, semiempirical, and ab initio methods using PCA-HCA procedure on the calculated bond lengths, statistical parameters, and structural aromaticity indices. Incorporation of crystal packing effects into bond orders makes multivariate models to be competitive to semiempirical results, while further improvement of quantum chemical calculations can be achieved by geometry optimization of molecular clusters.     

Correlation of bond metallicity measures to electronegativity for binary oxides

We have investigated alkali, alkaline‐earth, and rutile binary oxides within density functional theory (DFT) and Bader's atoms‐in‐molecules theory, focusing on properties of bond and ring critical points, and their relations to band gap and Pauling electronegativity. We find linear relations of kinetic energy density, electron density, and the gap divided by kinetic energy density at the bond critical points to the difference of Pauling electronegativities of the cation and oxygen anion. At the ring critical points of rutile compounds, we also find that some bond metallicity measures are linearly related to the difference of electronegativities. This study extends our knowledge about the relations between bond critical points, band gap, and electronegativity, but also shows for the first time a quantitative relation between quantities at the ring critical points and global properties of the compounds.