There are a lot of questions here on RG how to calculate bond orders but i want to know how to interpret them.
So what does it mean when for example:
- an sp3 carbon atom has a bond order significantly smaller than 4? say 3.8.
- a sulfur atom in a thio-ether Et-S-Et for example. has a bond order significantly larger than 2. say 2.3
- is an intramolecular hydrogen bond strong or soft when the bond order is around 0.05?
etc.
Lendvay G: Characterization of the progress of chemical reactions by ab initio bond orders, JOURNAL OF PHYSICAL CHEMISTRY US 98: pp. 6098-6104.
Lendvay G: BOND ORDERS FROM ABINITIO CALCULATIONS AND A TEST OF THE PRINCIPLE OF BOND ORDER CONSERVATION, JOURNAL OF PHYSICAL CHEMISTRY US 93: pp. 4422-4429.
Lendvay G: Bond order rearrangements during chemical reactions , Journal of Molecular Structure: THEOCHEM Volume 167, Issues 3–4, June 1988, Pages 331-338
Adam J. Bridgeman, Germán Cavigliasso,Luke R. Irelanda and Joanne Rothery, The Mayer bond order as a tool in inorganic chemistry, J. Chem. Soc., Dalton Trans., 2001, 2095-2108
I,. Mayer, Bond order and valence indices: A personal account, J. Comp. Chem, Volume 28, Issue 1,15 January 2007, Pages 204–221
Dear Jos,
The number of bonds between a pair of atoms is called the bond order. Bond orders can be calculated from Lewis structures, which are the heart of the valence-bond model. Oxygen, for example, has a bond order of two.
When there is more than one Lewis structure for a molecule, the bond order is an average of these structures. The bond order in sulfur dioxide, for example, is 1.5 the average of an S-O single bond in one Lewis structure and an S=O double bond in the other.
In molecular orbital theory, we calculate bond orders by assuming that two electrons in a bonding molecular orbital contribute one net bond and that two electrons in an antibonding molecular orbital cancel the effect of one bond. We can calculate the bond order in the O2 molecule by noting that there are eight valence electrons in bonding molecular orbitals and four valence electrons in antibonding molecular orbitals in the electron configuration of this molecule. Thus, the bond order is two.
Bond Order = Bonding Electrons - Antibonding Electrons/ 2 = 8-4/2 = 2
Although the Lewis structure and molecular orbital models of oxygen yield the same bond order, there is an important difference between these models. The electrons in the Lewis structure are all paired, but there are two unpaired electrons in the molecular orbital description of the molecule. As a result, we can test the predictions of these theories by studying the effect of a magnetic field on oxygen.
Atoms or molecules in which the electrons are paired are diamagnetic repelled by both poles of a magnetic. Those that have one or more unpaired electrons are paramagnetic attracted to a magnetic field. Liquid oxygen is attracted to a magnetic field and can actually bridge the gap between the poles of a horseshoe magnet. The molecular orbital model of O2 is therefore superior to the valence-bond model, which cannot explain this property of oxygen.
For more information, please see the following links:
http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch8/mo.html
https://www.boundless.com/chemistry/textbooks/boundless-chemistry-textbook/advanced-concepts-of-chemical-bonding-10/molecular-orbital-theory-82/bond-order-366-591/
Bond order in molecular orbital theory
In molecular orbital theory, bond order is also defined as half the difference between the number of bonding electrons and the number of antibonding electrons as per the equation below. This often but not always yields the same result. Bond order is also an index of bond strength and is also used extensively in valence bond theory.
Bond Order = Bonding Electrons - Antibonding Electrons/ 2
Generally, the higher the bond order, the stronger the bond. Bond orders of one-half can 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)
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 is experimentally described by:
8ij = exp [d1 - dij/b]
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:
d1 -dij = 0.353 log 8ij
The value of the constant b depends in the atoms.
The above definition of bond order is somewhat ad hoc and only easy to apply for diatomic molecules. A standard quantum mechanical definition for bond order has been debated for a long time.
Bruce Averill and Patricia Eldredge, Chemistry: Principles, Patterns, and Applications (Pearson/Prentice Hall, 2007), 409.
Pauling, Linus (March 1, 1947). "Atomic Radii and Interatomic Distances in Metals". Journal of the American Chemical Society. 69 (3): 542–553.
https://en.wikipedia.org/wiki/Bond_order
Hoping this will be helpful,
Rafik
Dear Jos,
counter-question: How you interpret the statement that "The average Belgian woman has 1.7 children"?
In physical reality, you deal with average electron densities around a set of atomic nuclei. The picture of single, double ... bonds is really an oversimplification of reality where you can gradually shift electron density between nuclei, instead of just "switching a bond on or off". Of course you can't split electrons, just like women don't give birth to 0.7 children, but if you look at their average distribution, that number emerges. So, to interpret bond orders: Do not think of it as a particle count, but rather as a charge density between atoms, expressed in elementary charges.
Also, as you have already noticed yourself, there are different ways to calculate bond orders. That is because you need to partition the overall electron density in your molecule into atom-centered and bonding contributions; but since there is no such strict distinction in reality, this partitioning is somewhat artificial, and different types of population analyses give you different results.
Dear Jos
Bond order is the number of chemical bonds between a pair of atoms. For example, in diatomic nitrogen N≡N the bond order is 3, 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. Bond order gives an indication of the stability of a bond. The element with bond order value 0 cannot exist. Those with fractional values exist but are not stable.
In molecules that have resonance or nonclassical bonding, bond order does not need to be an integer. In benzene, where 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. Furthermore, bond orders of 1.1, for example, can arise under complex scenarios and essentially refer to bond strength relative to bonds with order 1.
In molecular orbital theory, bond order is also defined as half the difference between the number of bonding electrons and the number of antibonding electrons as per the equation below. This often but not always yields the same result. Bond order is also an index of bond strength and is also used extensively in valence bond theory.
Bond order in molecular orbital theory
B.O. = number of bonding electrons − number of antibonding electrons 2 {\displaystyle {\text{B.O.}}={\frac {{\text{number of bonding electrons}}-{\text{number of antibonding electrons}}}{2}}\ }
Generally, the higher the bond order, the stronger the bond. Bond orders of one-half can be stable, as shown by the stability of H2+ (bond length 106 pm, bond energy 269 kJ/mol) and He2+ (bond length 108 pm, bond energy 251 kJ/mol).
You can also check this reference : Bruce Averill and Patricia Eldredge, Chemistry: Principles, Patterns, and Applications (Pearson/Prentice Hall, 2007), 409.
Cordialy
Marius
Bond order is the number of electrons exchanged between a pair of atoms in a material. Here, the term "exchanged" refers to quantum mechanical exchange, because an electron is a fermion. In a heuristid, non-formal sense, it is the number of electron "pairs" "shared" between two atoms in a material. However, it is important to keep in mind that chemical bonding can occur even when there is no electron pairing (e.g., [H2]+ ion).
Please find below a few comments on your specific questions:
1) an sp3 carbon atom has a bond order significantly smaller than 4? say 3.8.
--> First, I think you are referring to the sum of bond orders (SBO) for the C atom, which is often close to four because C has four electrons to share in covalent bonding. This (3.8 vs 4) difference is small (~5%), so I would not read too much into it.
2) - a sulfur atom in a thio-ether Et-S-Et for example. has a bond order significantly larger than 2. say 2.3
--> It is not uncommon for a sulfur SBO to be somewhat greater than 2. It doesn't indicate anything unusual. There are even cases (see paper referenced below) where sulfur atoms can hypercoordinate and achieve SBO near 6.
3) - is an intramolecular hydrogen bond strong or soft when the bond order is around 0.05?
--> This probably depends on the precise method you are using to compute the bond order. Using DDEC6 bond orders (see paper referenced below), the hydrogen bond order is often between 0.05 and 0.10, so your result doesn't seem unusual.
For more detailed description of related issues, see this paper on bond orders: T. A. Manz, "Introducing DDEC6 atomic population analysis: part 3. Comprehensive method to compute bond orders," RSC Advances, 7 (2017) 45552-45581 (http://dx.doi.org/10.1039/C7RA07400J ).
Jos I think you will be interested in read (Benzene on the basis of the three-electron bond):
Review. Benzene on the Basis of the Three-Electron Bond. (The Pauli exclusion principle, Heisenberg's uncertainty principle and chemical bond). http://vixra.org/pdf/1710.0326v4.pdf
https://dx.doi.org/10.2139/ssrn.3065288
The present work shows: 1. The present work shows the inapplicability of the Pauli principle to the chemical bond (pp. 104-106).
2. In addition, a new theoretical model of the chemical bond is proposed on the Heisenberg uncertainty principle (pp. 92-103).
3. That is, in fact, in this paper it is shown that modern concepts of the chemical bond can not be strictly considered theoretically true, but rather qualitative with empirical quantitative calculations.
See all works:
1. Structure of the benzene molecule on the basis of the three-electron bond. http://vixra.org/pdf/1606.0152v1.pdf
2. Experimental confirmation of the existence of the three-electron bond and theoretical basis ot its existence. http://vixra.org/pdf/1606.0151v2.pdf
3. A short analysis of chemical bonds. http://vixra.org/pdf/1606.0149v2.pdf
4. Supplement to the theoretical justification of existence of the three-electron bond. http://vixra.org/pdf/1606.0150v2.pdf
5. Theory of three-electrone bond in the four works with brief comments. http://vixra.org/pdf/1607.0022v2.pdf
6. REVIEW. Benzene on the basis of the three-electron bond (93 p.). http://vixra.org/pdf/1612.0018v5.pdf
7. Quantum-mechanical aspects of the L. Pauling's resonance theory. http://vixra.org/pdf/1702.0333v2.pdf
8. Quantum-mechanical analysis of the MO method and VB method from the position of PQS. http://vixra.org/pdf/1704.0068v1.pdf
9. Review (138 pages, full version). Benzene on the Basis of the Three-Electron Bond. (The Pauli exclusion principle, Heisenberg's uncertainty principle and chemical bond). http://vixra.org/pdf/1710.0326v4.pdf
10. Principle of Constancy and Finiteness of the Speed of Gravitational Interaction and Dark Matter. http://vixra.org/pdf/1806.0136v1.pdf
http://vixra.org/author/bezverkhniy_volodymyr_dmytrovych
https://papers.ssrn.com/sol3/cf_dev/AbsByAuth.cfm?per_id=2828345
P.S. One-electron bond in He2+.
And He2 + and H2 + are equally stable, because they are connected by a one-electron bond: compare the bond energies and the bond length (see page 262 L. Pauling, Nature of the chemical bond/translation from English by M. E. Dyatkina, edited by Prof. Ya. K. Syrkin. M. L .: GNTI Chemical Literature, 1947):
E (He2 +) = 58 kcal/mole
E (H2 +) = 61 kcal/mole
L (He2 +) = 1.09 Å
L (H2 +) = 1.06 Å
The helium atom has 2 electrons and one energy level (1S). Therefore in He2 + there are 3 electrons. The assumption that the bond in He2 + is three-electron is incorrect. The multiplicity of the bond is still determined by the fundamental rule of the octet. Therefore, it is easy to show that the bond in He2 + is one-electron (hence its multiplicity is 0.5). For this it is sufficient to apply the octet rule to He2 +, look at the image.
From the picture it is obvious that in order for the octet rule to be fulfilled in He2 +, the bond must be one-electron (only one electron can belong to both atoms simultaneously). Moreover, it can be concluded that the octet rule in a "condensed" form reflects the balance of the Coulomb repulsive-attraction forces between the electrons and the nuclei. Since, with the example of He2 +, it is obvious that, for a three-electron bond, the nuclei would remain without electrons, which led to a strong increase in the repulsion between the helium nuclei.
The fact that using the MO method in He2 + yields a three-electron bond (with a multiplicity of 0.5) should not be misleading, since the MO method contradicts quantum mechanics. See: http://vixra.org/pdf/1704.0068v1.pdf
“Quantum-Mechanical Analysis of the MO Method and VB Method from the Position of PQS”.
The MO method are analyzed using the principle of quantum superposition (PQS). It is shown that the main assumption of the molecular orbitals method (namely, that the molecular orbital can be represented like a linear combination of overlapping atomic orbitals) enters into an insurmountable contradiction with the principle of quantum superposition.
See the answer to the quorum: https://www.quora.com/Which-is-more-stable-He2%2B-or-H2%2B-and-why/answer/Volodymyr-Bezverkhniy?share=d6838bd2
If we do a similar procedure with oxygen in which there are two three-electron bonds, then it becomes immediately clear why the oxygen multiplicity is 2 (in spite of two three-electron bonds).
https://www.quora.com/What-is-a-three-electron-bond/answer/Volodymyr-Bezverkhniy?share=ae159a80
In molecular orbital theory, bond order is defined as half the difference between the number of bonding electrons and the number of anti-bonding electrons as per the equation below. This often but not always yields similar results for bonds near their equilibrium lengths, but it does not work for stretched bonds. Bond order is also an index of bond strength and is also used extensively in valence bond theory.
For example for poly atomic molecules, If there are more than two atoms in the molecule, follow these steps to determine the bond order:
1. Draw the Lewis structure.
2. Count the total number of bonds.
3. Count the number of bond groups between individual atoms.
4. Divide the number of bonds between atoms by the total number of bond groups in the molecule
Example: nitrate , NO−3
SOLUTION
1) Draw the Lewis structure.
2) Count the total number of bonds 4
The total number of bonds is 4.
3) Count the number of bond groups between individual atoms 3
The number of bond groups between individual atoms is 3.
4) Divide the number of bonds between individual atoms by the total number of bonds.
43=1.33(1.1)(1.1)43=1.33
The bond order is 1.33
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