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A rigorous theory of valence

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Abstract

Many sciences are described by two conceptually different but complementary theories, one visual and insightful the other better suited to computation. Chemistry has a powerful computational theory in quantum mechanics, but its visual and insightful theory of valence has never been properly developed. The rigorous theory of valence described here takes the atom as its fundamental particle. As the theory has no knowledge of the atom’s internal structure, the valence is not associated with the electron, hence it is free to adopt fractional values. Each atom is characterized by two experimental properties: its valence determined from the compositions of observed compounds, and its size determined from its observed coordination numbers. These two properties are able to describe a surprising amount of chemistry. The amount of valence an atom typically uses to form a bond determines both the conditions under which bonds can be formed and their resulting properties. A valence definition of electronegativity removes the need to distinguish between ionic and covalent bonds by defining the more nuanced concept of bond polarization, clearing up much of the confusion found in popular bonding models. The theory is compatible with the physical picture provided by quantum mechanics but not with the physically unrealistic ionic model. It can be used for analyzing, modeling, and teaching of chemical structure and reactivity, but having no knowledge of the internal structure of the atom it cannot be used to calculate energies or any of the properties that depend on the energy; for that quantum mechanics must be used.

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Fig. 1
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Adapted from Fig. 2.2 of Brown [15]

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Adapted from Fig. 2.3 of Brown [15]

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Reproduced from Fig. 3.1 in Brown [15]. Copyright 2016 Oxford University Press Reproduced with the permission of the licensor through PLSclear

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Adapted from Fig. 6.5 in Brown [15]

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Copyright 2016 Oxford University Press Reproduced with the permission of the licensor through PLSclear

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Copyright 2016 Oxford University Press Reproduced with the permission of the licensor through PLSclear

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Acknowledgements

I wish to acknowledge the stimulating discussions I have had with Frank Hawthorne and Olivier Gagné. I wish also to acknowledge the many inspiring discussions I have had over the years with my McMaster University students and colleagues.

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  1. Brockhouse Institute for Materials Research, McMaster University, Hamilton, ON, Canada

    I. D. Brown

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Dedicated to the memory of Jack Dunitz, Ron Gillespie and Richard Bader whose inspiration contributed to the ideas presented here.

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Glossary

a priori bond valence (‘Simple bonds’ section)

Bond valences calculated from the bond network using the network Eqs. 4 and 5. Using Eq. 7 it can be converted into an a priori bond length.

acid

See Lewis acid.

atom (‘Atoms’ section)

The basic particle of valence theory. The atomic number of an atom indicates which element it is. In valence theory, atoms are characterized by their valence and their average coordination number. They also possess the properties of bonding strength and electronegativity. Atoms are always spherical and electroneutral. Their physical structure is is most conveniently described by the core and valence shell model.

atomic valence V (‘First things: valence, coordination number, and equilibrium’ section)

The same as valence. This name is used when it is necessary to avoid confusion with the bond valence.

average coordination number <  N  > (‘First things: valence, coordination number, and equilibrium’ section)

The average of the experimentally measured coordination numbers of an atom with a given valence. Unless otherwise stated the ligands are assumed to be oxygen.

base

See Lewis base.

bond (‘Atoms’ section)

The attachment that links two atoms when their valence shells overlap. Bonds possess the properties of bond valence and polarization as well as bond length.

bond flux φ (‘Simple bonds’ section)

A physical description of the bond valence in terms of the electrostatic flux linking the cores of two bonded atoms to their shared valence It can be calculated using the ionic model and is expected to be the same as the bond valence.

bond length R (‘Simple bonds’ section)

The distance between the centers of the two atoms.

bond network (‘Simple bonds’ section)

A network of bonds linking the atoms that form a compound. In valence theory the bond network is the primary description of the structure of the compound. The properties of the bonds, such as their bond valence. bond polarization and bond length are largely determined by the topology of the bond network and the identity of the constituent atoms.

bond polarization ψ (‘Simple bonds’ section)

The degree of polarization of a bond, equal to the difference in the electronegativities of the two atoms that form the bond.

bond polarity (‘Simple bonds’ section)

The bond polarization describes the bond as directed from the atom with the larger to the atom with the smaller electronegativity.

bond valence s (‘First things: valence, coordination number, and equilibrium’ section)

The amount of valence an atom uses to form a given bond, hence it is a measure of the strength of the bond. In most cases both atoms contribute equal amounts of valence to form a bond. Depending on how its value is determined the valence is known as the a priori bond valence. the bond flux or the experimental bond valence.

bond valence vector s (‘Simple bonds’ section)

A vector directed along the line joining two bonded atoms having the magnitude of the bond valence.

bonding strength S (‘Atoms’ section)

The bond valence of a typical bond formed by an atom. It is equal to the valence of the atom divided by its corresponding average coordination number (Eq. 1). The bonding strength of the atom with the smaller electronegativity is also known as its Lewis acid strength, that of the atom with the larger electronegativity is also known as its Lewis base strength.

complex (‘Interesting Bonds’ section)

A collection of strongly bonded atoms including those traditionally called molecules or complex ions. The outer surface of the complex is normally composed of Lewis bases. Hydrogen atoms, if attached, act as its Lewis acid function. The bonding strengths of complexes are the external bonding strengths of the atoms on the surface of the complex.

compound

An aggregation of atoms held together by bonds.

coordination number N

The number of bonds formed by an atom.

coordination sphere

The atoms that are bonded to the given atom.

core (‘Atoms’ section)

In the core and valence shell model of an atom the core is the portion of an atom that is never used for bonding. It includes the atomic nucleus and the portion of the negative charge that is not part of the valence shell.

core and valence shell model (‘Atoms’ section)

The core and valence shell model provides a notional picture of an atom that is useful in understanding the atom’s properties. It is not a true physical picture of an atom, nor is it an essential part of valence theory.

deformation density (‘Atoms’ section)

The difference between the promolecule charge density and the experimentally observed negative charge density in a given compound.

effective Lewis acid strength S ae(‘Interesting bonds’ section)

The Lewis acid strength of the solvent water at a given pH.

effective Lewis base strength S be (‘Interesting bonds’ section)

The Lewis base strength of the solvent water at a given pH.

electronegativity χ (‘Atoms’ section)

In valence theory this is defined for Main Group elements as the bonding strength of the atom in its highest valence state. For hydrogen see ‘Hydrogen bonds and aqueous chemistry’ section and for transition metals see ‘Ammonia and the ammonium complex’ section. It is an intrinsic property of an atom.

experimental atomic valence (‘Simple bonds’ section)

The sum of the experimental bond valences of the bonds formed by the given atom.

experimental bond valence p (‘Simple bonds’ section)

The bond valence determined from the observed bond lengths using Eq. 7.

external valence (‘Interesting Bonds’ section)

The valence available to a complex for forming bonds to other atoms in the compound that are not part of the complex. It is equal to the sum of the valences of the atoms forming a complex, having regard to their signs. It is also equal to the sum of the valences of the external bonds formed by the individual atoms in the complex.

flux, electrostatic flux (‘Simple bonds’ section)

See bond flux.

Lewis acid (‘Simple bonds’ section)

In a bond, the atom with the lower electronegativity.

Lewis acid strength S a (‘Simple bonds’ section)

The bonding strength of a Lewis acid.

Lewis base (‘Simple bonds’ section)

In a bond, the atom with the higher electronegativity.

Lewis base strength S b (‘Simple bonds’ section)

The bonding strength of a Lewis base.

lone pair (‘Interesting Bonds’ section)

Two electron units of charge in the valence shell of an atom that are not used for bonding. Lone pairs become stereoactive if the atom is bonded to an atom with a larger bonding strength (Rule 10).

primary bond (‘Interesting Bonds’ section)

A bond that is significantly stronger then the average in a coordination sphere, particularly in the presence of stereoactive lone pairs.

promolecule (‘Atoms’ section)

The charge density (electron density) created by placing the charge densities of the isolated neutral atoms at the positions of the atoms in the given compound.

secondary bond (‘Interesting Bonds’ section)

A bond that is significantly weaker than the average in a coordination sphere, particularly in the presence of stereoactive lone pairs.

valence V (‘First things: valence, coordination number, and equilibrium’ section)

Defined by IUPAC (Rule 1), valence is a number that indicates the number of oxygen or hydrogen atoms with which it can combine to form a binary compound, thus valence can only be determined when an atom is part of a compound. According to this definition valence is a positive number, but in valence theory valence is identified with that portion of the negative charge of an atom that is used to form bonds. It is measured in valence units, each of which is equal to the electron charge. Valences of atoms are usually integers, though there are exception. The valence adopted by an atom depends on its atomic number and the electronegativities of its bonded neighbors as described by Rule 10. In the core and valence shell model the valence is always located in the valence shell. It may be referred to as the atomic valence to differentiate it from the bond valence.

valence shell (‘Atoms’ and ‘Interesting Bonds’ section)

The notional outer shell of an atom in the core and valence shell model. It contains all the charge representing the valence of the atom but may also contain non-bonding negative charge called lone pairs.

valence unit (vu) (‘First things: valence, coordination number, and equilibrium’ section)

The unit of valence associated with one electron unit of charge, see valence.

valence vector

See bond valence vector.

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Brown, I.D. A rigorous theory of valence. Struct Chem 34, 361–389 (2023). https://doi.org/10.1007/s11224-023-02128-w

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