206
8 Models of Chemical Bonding and “Empirical” Methods
8.2 Bond Lengths
It is often convenient to assume that an atom has a spherical form. Actually, an
atom does not have sharp boundaries, hence no clearly defined volume. However,
the atoms behave approximately as hard spheres. The distance between the nuclei
of two atoms cannot be much smaller than a given value characteristic of the nature
of the atoms. There is no universal definition of the radius of an atom. A covalent
radius is a measure of the size of an atom that forms part of a covalent bond. A
van der Waals radius is the radius of a sphere representing the distance of closest
approach for another non-bonded atom. The notion of ionic radius is used to discuss
the structure of ionic crystals. There are other definitions of the atomic radius, for
instance the ligand radius introduced in Sect. 8.4 of this chapter. The radius depends
on the electric charge. When an atom loses an electron, the remaining electrons are
more attracted to the nucleus, and the radius becomes smaller. Conversely, when an
electron is added, the size of the electronic cloud becomes larger.
8.2.1 Covalent Radii
The distance between two bonded atoms is largely determined by the size of the
atoms. As a consequence, the bond length r between two covalent atoms A and B
does not vary much. An acceptable approximation is to assume that r is the sum of
two covalent radii that are constant for a given atom
r (A − B) = r A + r B
(8.1)
For many atoms, the covalent radius is obtained by halving the homonuclear bond
distance r(E–E) (important exceptions are the electronegative atoms in F 2 , O 2 , and
N 2 )
r E = r (E − E)/2
( 8 . 2 )
For the other atoms, an atom of moderate electronegativity Y (as carbon) is chosen
as reference, and the radius of E is obtained by substracting r Y from r(Y–E). Pyykkö
et al. (2005) and Pyykkö and Atsumi (2009) determined consistent values by making
a least-squares fit of all available data. They obtained different values for single,
double, and triple bonds. Some values are reported in Table 8.1. Note that the atoms in
molecules (AIM)-bonding radius is identical to the covalent radius for homonuclear
diatomic molecules (Sect. 2.18).
Only homonuclear bonds are fully covalent. Therefore, differences from the values
calculated with (8.1) are expected because of the partial ionic character of the bond
and its fractional bond order (for instance for aromatic bonds).
8 Models of Chemical Bonding and “Empirical” Methods
8.2 Bond Lengths
It is often convenient to assume that an atom has a spherical form. Actually, an
atom does not have sharp boundaries, hence no clearly defined volume. However,
the atoms behave approximately as hard spheres. The distance between the nuclei
of two atoms cannot be much smaller than a given value characteristic of the nature
of the atoms. There is no universal definition of the radius of an atom. A covalent
radius is a measure of the size of an atom that forms part of a covalent bond. A
van der Waals radius is the radius of a sphere representing the distance of closest
approach for another non-bonded atom. The notion of ionic radius is used to discuss
the structure of ionic crystals. There are other definitions of the atomic radius, for
instance the ligand radius introduced in Sect. 8.4 of this chapter. The radius depends
on the electric charge. When an atom loses an electron, the remaining electrons are
more attracted to the nucleus, and the radius becomes smaller. Conversely, when an
electron is added, the size of the electronic cloud becomes larger.
8.2.1 Covalent Radii
The distance between two bonded atoms is largely determined by the size of the
atoms. As a consequence, the bond length r between two covalent atoms A and B
does not vary much. An acceptable approximation is to assume that r is the sum of
two covalent radii that are constant for a given atom
r (A − B) = r A + r B
(8.1)
For many atoms, the covalent radius is obtained by halving the homonuclear bond
distance r(E–E) (important exceptions are the electronegative atoms in F 2 , O 2 , and
N 2 )
r E = r (E − E)/2
( 8 . 2 )
For the other atoms, an atom of moderate electronegativity Y (as carbon) is chosen
as reference, and the radius of E is obtained by substracting r Y from r(Y–E). Pyykkö
et al. (2005) and Pyykkö and Atsumi (2009) determined consistent values by making
a least-squares fit of all available data. They obtained different values for single,
double, and triple bonds. Some values are reported in Table 8.1. Note that the atoms in
molecules (AIM)-bonding radius is identical to the covalent radius for homonuclear
diatomic molecules (Sect. 2.18).
Only homonuclear bonds are fully covalent. Therefore, differences from the values
calculated with (8.1) are expected because of the partial ionic character of the bond
and its fractional bond order (for instance for aromatic bonds).