2.3 Ionic Bonds
27
Fig. 2.11 Schematic
energy terms of the
benzene molecule
p z
sp 2
6
18
}
}
HOMO
LUMO
2.3 Ionic Bonds
Ionic crystals are made up from positively and negatively charged ions. The heteropolar or ionic bond
is the consequence of the electrostatic attraction between the ions. However, the possibly repulsive
character of next neighbors has to be considered.
For I–VII compounds, e.g. LiF or NaCl, the shells of the singly charged ions are complete: Li: 1s
2 2s
1
→ Li
+ : 1s
2 , F: 1s
2 2s
2 2p
5
→ F
− : 1s
2 2s
2 2p
6 . Compared to ions in a gas, a Na–Cl pair in the crystal
has a binding energy of 7.9 eV that mostly stems from the electrostatic energy (Madelung energy).
Van-der-Waals forces (cf. Sect. 2.6) only contribute 1–2%. The ionization energy of Na is 5.14 eV, the
electron affinity of Cl is 3.61 eV. Thus the energy of the NaCl pair in the solid is 6.4 (=7.9−5.1+3.6) eV
smaller than in a gas of neutral atoms.
The interaction of two ions with distance vector r i j is due to the Coulomb interaction
U
C
i j =
q i q j
4ππ 0
1
r i j
= ±
e
2
4ππ 0
1
r i j
(2.3)
and a repulsive contribution due to the overlap of (complete) shells. This contribution is typically
approximated by a radially symmetric core potential
U
core
i j
= λ exp(−λ/ρ)
(2.4)
that only acts on next neighbors. λ describes the strength of this interaction and ρ parameterizes its
range.
The distance of ions is denoted as r i j = p i j R, where R denotes the distance of next neighbors and
the p i j are suitable coefficients. The electrostatic interaction of an ion with all its neighbors is then
written as
U
C
i j = −α
e
2
4ππ 0
1
R
,
(2.5)
where α is the Madelung constant. For an attractive interaction (as in a solid), α is positive. It is given
(calculated for the i-th ion) as
α =
i j
±1
p i j
.
(2.6)
For a one-dimensional chain α = 2 ln 2. For the rocksalt (NaCl) structure (cf. Sect. 3.4.1) it is α ≈
1.7476, for the CsCl structure (see Sect. 3.4.2) it is α ≈ 1.7627, and for the zincblende structure (see
Sect. 3.4.4) it is α ≈ 1.6381. This shows that ionic compounds prefer the NaCl or CsCl structure.
27
Fig. 2.11 Schematic
energy terms of the
benzene molecule
p z
sp 2
6
18
}
}
HOMO
LUMO
2.3 Ionic Bonds
Ionic crystals are made up from positively and negatively charged ions. The heteropolar or ionic bond
is the consequence of the electrostatic attraction between the ions. However, the possibly repulsive
character of next neighbors has to be considered.
For I–VII compounds, e.g. LiF or NaCl, the shells of the singly charged ions are complete: Li: 1s
2 2s
1
→ Li
+ : 1s
2 , F: 1s
2 2s
2 2p
5
→ F
− : 1s
2 2s
2 2p
6 . Compared to ions in a gas, a Na–Cl pair in the crystal
has a binding energy of 7.9 eV that mostly stems from the electrostatic energy (Madelung energy).
Van-der-Waals forces (cf. Sect. 2.6) only contribute 1–2%. The ionization energy of Na is 5.14 eV, the
electron affinity of Cl is 3.61 eV. Thus the energy of the NaCl pair in the solid is 6.4 (=7.9−5.1+3.6) eV
smaller than in a gas of neutral atoms.
The interaction of two ions with distance vector r i j is due to the Coulomb interaction
U
C
i j =
q i q j
4ππ 0
1
r i j
= ±
e
2
4ππ 0
1
r i j
(2.3)
and a repulsive contribution due to the overlap of (complete) shells. This contribution is typically
approximated by a radially symmetric core potential
U
core
i j
= λ exp(−λ/ρ)
(2.4)
that only acts on next neighbors. λ describes the strength of this interaction and ρ parameterizes its
range.
The distance of ions is denoted as r i j = p i j R, where R denotes the distance of next neighbors and
the p i j are suitable coefficients. The electrostatic interaction of an ion with all its neighbors is then
written as
U
C
i j = −α
e
2
4ππ 0
1
R
,
(2.5)
where α is the Madelung constant. For an attractive interaction (as in a solid), α is positive. It is given
(calculated for the i-th ion) as
α =
i j
±1
p i j
.
(2.6)
For a one-dimensional chain α = 2 ln 2. For the rocksalt (NaCl) structure (cf. Sect. 3.4.1) it is α ≈
1.7476, for the CsCl structure (see Sect. 3.4.2) it is α ≈ 1.7627, and for the zincblende structure (see
Sect. 3.4.4) it is α ≈ 1.6381. This shows that ionic compounds prefer the NaCl or CsCl structure.