# Bond order

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In chemistry, bond order, as introduced by Linus Pauling, is defined as the difference between the number of bonds and anti-bonds.

## Contents

The bond order itself is the number of electron pairs (covalent bonds) between two atoms. [1] For example, in diatomic nitrogen N≡N, the bond order between the two nitrogen atoms is 3 (triple bond). In acetylene H–C≡C–H, the bond order between the two carbon atoms is also 3, and the C–H bond order is 1 (single bond). In carbon monoxide C≡O+, the bond order between carbon and oxygen is 3. In thiazyl trifluoride N≡SF3, the bond order between sulfur and nitrogen is 3, and between sulfur and fluorine is 1. In diatomic oxygen O=O the bond order is 2 (double bond). In ethylene H2C=CH2 the bond order between the two carbon atoms is also 2. The bond order between carbon and oxygen in carbon dioxide O=C=O is also 2. In phosgene O=CCl2, the bond order between carbon and oxygen is 2, and between carbon and chlorine is 1.

In some molecules, bond orders can be 4 (quadruple bond), 5 (quintuple bond) or even 6 (sextuple bond). For example, potassium octachlorodimolybdate salt (K4[Mo2Cl8]) contains the [Cl4Mo≣MoCl4]4− anion, in which the two Mo atoms are linked to each other by a bond with order of 4. Each Mo atom is linked to four Cl ligands by a bond with order of 1. The compound (terphenyl)–CrCr–(terphenyl) contains two chromium atoms linked to each other by a bond with order of 5, and each chromium atom is linked to one terphenyl ligand by a single bond. A bond of order 6 is detected in ditungsten molecules W2, which exist only in a gaseous phase.

Bond order gives an indication of the stability of a bond. Isoelectronic species have the same bond order. [2]

In molecules which have resonance or nonclassical bonding, bond order may not be an integer. In benzene, the delocalized molecular orbitals contain 6 pi electrons over six carbons, essentially yielding half a pi bond together with the sigma bond for each pair of carbon atoms, giving a calculated bond order of 1.5 (one and a half bond). Furthermore, bond orders of 1.1 (eleven tenths bond), 4/3 (or 1.333333..., four thirds bond) or 0.5 (half bond), for example, can occur in some molecules and essentially refer to bond strength relative to bonds with order 1. In the nitrate anion (NO3), the bond order for each bond between nitrogen and oxygen is 4/3 (or 1.333333...). Bonding in dihydrogen cation H+2 can be described as a covalent one-electron bond, thus the bonding between the two hydrogen atoms has bond order of 0.5. [3]

## Bond order in molecular orbital theory

In molecular orbital theory, bond order is defined as half the difference between the number of bonding electrons and the number of antibonding electrons as per the equation below. [4] [5] This often but not always yields similar results for bonds near their equilibrium lengths, but it does not work for stretched bonds. [6] Bond order is also an index of bond strength and is also used extensively in valence bond theory.

bond order = number of bonding electrons - number of antibonding electrons/2

Generally, the higher the bond order, the stronger the bond. Bond orders of one-half may be stable, as shown by the stability of H+2 (bond length 106 pm, bond energy 269 kJ/mol) and He+2 (bond length 108 pm, bond energy 251 kJ/mol). [7]

Hückel molecular orbital theory offers another approach for defining bond orders based on molecular orbital coefficients, for planar molecules with delocalized π bonding. The theory divides bonding into a sigma framework and a pi system. The π-bond order between atoms r and s derived from Hückel theory was defined by Charles Coulson by using the orbital coefficients of the Hückel MOs: [8] [9] [ clarification needed ]

${\displaystyle p_{rs}=\sum _{i}n_{i}c_{ri}c_{si}}$,

Here the sum extends over π molecular orbitals only, and ni is the number of electrons occupying orbital i with coefficients cri and csi on atoms r and s respectively. Assuming a bond order contribution of 1 from the sigma component this gives a total bond order (σ + π) of 5/3 = 1.67 for benzene, rather than the commonly cited bond order of 1.5, showing some degree of ambiguity in how the concept of bond order is defined.

For more elaborate forms of molecular orbital theory involving larger basis sets, still other definitions have been proposed. [10] A standard quantum mechanical definition for bond order has been debated for a long time. [11] A comprehensive method to compute bond orders from quantum chemistry calculations was published in 2017. [6]

## Other definitions

The bond order concept is used in molecular dynamics and bond order potentials. The magnitude of the bond order is associated with the bond length. According to Linus Pauling in 1947, the bond order between atoms i and j is experimentally described as

${\displaystyle s_{ij}=\exp {\left[{\frac {d_{1}-d_{ij}}{b}}\right]}}$

where d1 is the single bond length, dij is the bond length experimentally measured, and b is a constant, depending on the atoms. Pauling suggested a value of 0.353 Å for b, for carbon-carbon bonds in the original equation: [12]

${\displaystyle d_{1}-d_{ij}=0.353~{\text{ln}}(s_{ij})}$

The value of the constant b depends on the atoms. This definition of bond order is somewhat ad hoc and only easy to apply for diatomic molecules.

## Related Research Articles

A chemical bond is a lasting attraction between atoms or ions that enables the formation of molecules and crystals. The bond may result from the electrostatic force between oppositely charged ions as in ionic bonds, or through the sharing of electrons as in covalent bonds. The strength of chemical bonds varies considerably; there are "strong bonds" or "primary bonds" such as covalent, ionic and metallic bonds, and "weak bonds" or "secondary bonds" such as dipole–dipole interactions, the London dispersion force and hydrogen bonding. Strong chemical bonding arises from the sharing or transfer of electrons between the participating atoms.

A covalent bond is a chemical bond that involves the sharing of electrons to form electron pairs between atoms. These electron pairs are known as shared pairs or bonding pairs. The stable balance of attractive and repulsive forces between atoms, when they share electrons, is known as covalent bonding. For many molecules, the sharing of electrons allows each atom to attain the equivalent of a full valence shell, corresponding to a stable electronic configuration. In organic chemistry, covalent bonding is much more common than ionic bonding.

In theoretical chemistry, a conjugated system is a system of connected p-orbitals with delocalized electrons in a molecule, which in general lowers the overall energy of the molecule and increases stability. It is conventionally represented as having alternating single and multiple bonds. Lone pairs, radicals or carbenium ions may be part of the system, which may be cyclic, acyclic, linear or mixed. The term "conjugated" was coined in 1899 by the German chemist Johannes Thiele.

In chemistry, aromaticity is a chemical property of cyclic (ring-shaped), typically planar (flat) molecular structures with pi bonds in resonance that gives increased stability compared to saturated compounds having single bonds, and other geometric or connective non-cyclic arrangements with the same set of atoms. Aromatic rings are very stable and do not break apart easily. Organic compounds that are not aromatic are classified as aliphatic compounds—they might be cyclic, but only aromatic rings have enhanced stability. The term aromaticity with this meaning is historically related to the concept of having an aroma, but is a distinct property from that meaning.

The octet rule is a chemical rule of thumb that reflects the theory that main-group elements tend to bond in such a way that each atom has eight electrons in its valence shell, giving it the same electronic configuration as a noble gas. The rule is especially applicable to carbon, nitrogen, oxygen, and the halogens; although more generally the rule is applicable for the s-block and p-block of the periodic table. Other rules exist for other elements, such as the duplet rule for hydrogen and helium, or the 18-electron rule for transition metals.

In chemistry, resonance, also called mesomerism, is a way of describing bonding in certain molecules or polyatomic ions by the combination of several contributing structures into a resonance hybrid in valence bond theory. It has particular value for analyzing delocalized electrons where the bonding cannot be expressed by one single Lewis structure.

In chemistry, polarity is a separation of electric charge leading to a molecule or its chemical groups having an electric dipole moment, with a negatively charged end and a positively charged end.

In chemistry, a lone pair refers to a pair of valence electrons that are not shared with another atom in a covalent bond and is sometimes called an unshared pair or non-bonding pair. Lone pairs are found in the outermost electron shell of atoms. They can be identified by using a Lewis structure. Electron pairs are therefore considered lone pairs if two electrons are paired but are not used in chemical bonding. Thus, the number of electrons in lone pairs plus the number of electrons in bonds equals the number of valence electrons around an atom.

In chemistry, molecular orbital theory is a method for describing the electronic structure of molecules using quantum mechanics. It was proposed early in the 20th century.

In chemistry, valence bond (VB) theory is one of the two basic theories, along with molecular orbital (MO) theory, that were developed to use the methods of quantum mechanics to explain chemical bonding. It focuses on how the atomic orbitals of the dissociated atoms combine to give individual chemical bonds when a molecule is formed. In contrast, molecular orbital theory has orbitals that cover the whole molecule.

In organic chemistry, Hückel's rule predicts that a planar ring molecule will have aromatic properties if it has 4n + 2 π electrons, where n is a non-negative integer. The quantum mechanical basis for its formulation was first worked out by physical chemist Erich Hückel in 1931. The succinct expression as the 4n + 2 rule has been attributed to W. v. E. Doering (1951), although several authors were using this form at around the same time.

In chemistry, orbital hybridisation is the concept of mixing atomic orbitals to form new hybrid orbitals suitable for the pairing of electrons to form chemical bonds in valence bond theory. For example, in a carbon atom which forms four single bonds the valence-shell s orbital combines with three valence-shell p orbitals to form four equivalent sp3 mixtures in a tetrahedral arrangement around the carbon to bond to four different atoms. Hybrid orbitals are useful in the explanation of molecular geometry and atomic bonding properties and are symmetrically disposed in space. Usually hybrid orbitals are formed by mixing atomic orbitals of comparable energies.

In organic chemistry, a bent bond, also known as a banana bond, is a type of covalent chemical bond with a geometry somewhat reminiscent of a banana. The term itself is a general representation of electron density or configuration resembling a similar "bent" structure within small ring molecules, such as cyclopropane (C3H6) or as a representation of double or triple bonds within a compound that is an alternative to the sigma and pi bond model.

A quintuple bond in chemistry is an unusual type of chemical bond, first reported in 2005 for a dichromium compound. Single bonds, double bonds, and triple bonds are commonplace in chemistry. Quadruple bonds are rarer but are currently known only among the transition metals, especially for Cr, Mo, W, and Re, e.g. [Mo2Cl8]4− and [Re2Cl8]2−. In a quintuple bond, ten electrons participate in bonding between the two metal centers, allocated as σ2π4δ4.

The Hückel method or Hückel molecular orbital theory, proposed by Erich Hückel in 1930, is a simple method for calculating molecular orbitals as linear combinations of atomic orbitals. The theory predicts the molecular orbitals for π-electrons in π-delocalized molecules, such as ethylene, benzene, butadiene, and pyridine. It provides the theoretical basis for Hückel's rule that cyclic, planar molecules or ions with π-electrons are aromatic. It was later extended to conjugated molecules such as pyridine, pyrrole and furan that contain atoms other than carbon and hydrogen (heteroatoms). A more dramatic extension of the method to include σ-electrons, known as the extended Hückel method (EHM), was developed by Roald Hoffmann. The extended Hückel method gives some degree of quantitative accuracy for organic molecules in general and was used to provide computational justification for the Woodward–Hoffmann rules. To distinguish the original approach from Hoffmann's extension, the Hückel method is also known as the simple Hückel method (SHM).

In chemical bonding theory, an antibonding orbital is a type of molecular orbital that weakens the chemical bond between two atoms and helps to raise the energy of the molecule relative to the separated atoms. Such an orbital has one or more nodes in the bonding region between the nuclei. The density of the electrons in the orbital is concentrated outside the bonding region and acts to pull one nucleus away from the other and tends to cause mutual repulsion between the two atoms. This is in contrast to a bonding molecular orbital, which has a lower energy than that of the separate atoms, and is responsible for chemical bonds.

In chemistry, Bent's rule describes and explains the relationship between the orbital hybridization of central atoms in molecules and the electronegativities of substituents. The rule was stated by Henry A. Bent as follows:

Atomic s character concentrates in orbitals directed toward electropositive substituents.

A molecular orbital diagram, or MO diagram, is a qualitative descriptive tool explaining chemical bonding in molecules in terms of molecular orbital theory in general and the linear combination of atomic orbitals (LCAO) method in particular. A fundamental principle of these theories is that as atoms bond to form molecules, a certain number of atomic orbitals combine to form the same number of molecular orbitals, although the electrons involved may be redistributed among the orbitals. This tool is very well suited for simple diatomic molecules such as dihydrogen, dioxygen, and carbon monoxide but becomes more complex when discussing even comparatively simple polyatomic molecules, such as methane. MO diagrams can explain why some molecules exist and others do not. They can also predict bond strength, as well as the electronic transitions that can take place.

Water is a simple triatomic bent molecule with C2v molecular symmetry and bond angle of 104.5° between the central oxygen atom and the hydrogen atoms. Despite being one of the simplest triatomic molecules, its chemical bonding scheme is nonetheless complex as many of its bonding properties such as bond angle, ionization energy, and electronic state energy cannot be explained by one unified bonding model. Instead, several traditional and advanced bonding models such as simple Lewis and VSEPR structure, valence bond theory, molecular orbital theory, isovalent hybridization, and Bent's rule are discussed below to provide a comprehensive bonding model for H
2
O
, explaining and rationalizing the various electronic and physical properties and features manifested by its peculiar bonding arrangements.

In theoretical chemistry, the bonding orbital is used in molecular orbital (MO) theory to describe the attractive interactions between the atomic orbitals of two or more atoms in a molecule. In MO theory, electrons are portrayed to move in waves. When more than one of these waves come close together, the in-phase combination of these waves produces an interaction that leads to a species that is greatly stabilized. The result of the waves’ constructive interference causes the density of the electrons to be found within the binding region, creating a stable bond between the two species.

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