22
2 Bonds
A
S
(a)
(b)
Fig. 2.1 Binding of the hydrogen molecule. a Dashed line: classical calculation (electrostatics), ‘S’, ‘A’: quantummechanical calculation taking into account Pauli’s principle (S: symmetric orbital, antiparallel spins, A: antisymmetric
orbital, parallel spins). The distance of the nuclei (protons) is given in units of the Bohr radius a B = 0.053 nm, the energy
is given in Rydberg units (13.6 eV). b Schematic contour plots of the probability distribution ( ∗ ) for the S and A
states
of the molecule is nonseparable and has the form B) = r (r A , r B ) ) σ (σ A , σ B ). The binding
state has a wavefunction with a symmetric orbital and antiparallel spins, i.e. r (r A , r B ) = r (r B , r A )
and σ (σ A , σ B ) = − σ (σ B , σ A ). The antisymmetric orbital with parallel spins is antibinding for all
distances of the nuclei (protons).
2.2.2 sp 3 Bonds
Elements from group IV of the periodic system (C, Si, Ge, . . .) have 4 electrons on the outer shell.
Carbon has the electron configuration 1s
2 2s
2 2p
2 . For an octet configuration bonding to four other
electrons would be optimal (Fig. 2.2). This occurs through the mechanism of sp
3 hybridization.
1 First,
one electron of the ns
2 np
2 configuration is brought into a p orbital, such that the outermost shell
contains one s, p x , p y , and p z orbital each (Fig. 2.3a–e). The energy necessary for this step is less than
regained in the subsequent formation of the covalent bonds. The four orbitals can be reconfigured into
four other wavefunctions, the sp
3 hybrids (Figs. 2.3f–i), i.e.
1 = (s + p x + p y + p z )/2
(2.1a)
2 = (s + p x − p y − p z )/2
(2.1b)
3 = (s − p x + p y − p z )/2
(2.1c)
4 = (s − p x − p y + p z )/2 .
(2.1d)
These orbitals have a directed form along tetrahedral directions. The binding energy (per atom) of the
covalent bond is large, for H–H 4.5 eV, for C–C 3.6 eV, for Si–Si 1.8 eV, and for Ge–Ge 1.6 eV. Such
energy is, for neutral atoms, comparable to the ionic bond, discussed in the next section.
1 It is debated in femtosecond chemistry whether the bond really forms in this way. However, it is a picture of overwhelming
simplicity.
2 Bonds
A
S
(a)
(b)
Fig. 2.1 Binding of the hydrogen molecule. a Dashed line: classical calculation (electrostatics), ‘S’, ‘A’: quantummechanical calculation taking into account Pauli’s principle (S: symmetric orbital, antiparallel spins, A: antisymmetric
orbital, parallel spins). The distance of the nuclei (protons) is given in units of the Bohr radius a B = 0.053 nm, the energy
is given in Rydberg units (13.6 eV). b Schematic contour plots of the probability distribution ( ∗ ) for the S and A
states
of the molecule is nonseparable and has the form B) = r (r A , r B ) ) σ (σ A , σ B ). The binding
state has a wavefunction with a symmetric orbital and antiparallel spins, i.e. r (r A , r B ) = r (r B , r A )
and σ (σ A , σ B ) = − σ (σ B , σ A ). The antisymmetric orbital with parallel spins is antibinding for all
distances of the nuclei (protons).
2.2.2 sp 3 Bonds
Elements from group IV of the periodic system (C, Si, Ge, . . .) have 4 electrons on the outer shell.
Carbon has the electron configuration 1s
2 2s
2 2p
2 . For an octet configuration bonding to four other
electrons would be optimal (Fig. 2.2). This occurs through the mechanism of sp
3 hybridization.
1 First,
one electron of the ns
2 np
2 configuration is brought into a p orbital, such that the outermost shell
contains one s, p x , p y , and p z orbital each (Fig. 2.3a–e). The energy necessary for this step is less than
regained in the subsequent formation of the covalent bonds. The four orbitals can be reconfigured into
four other wavefunctions, the sp
3 hybrids (Figs. 2.3f–i), i.e.
1 = (s + p x + p y + p z )/2
(2.1a)
2 = (s + p x − p y − p z )/2
(2.1b)
3 = (s − p x + p y − p z )/2
(2.1c)
4 = (s − p x − p y + p z )/2 .
(2.1d)
These orbitals have a directed form along tetrahedral directions. The binding energy (per atom) of the
covalent bond is large, for H–H 4.5 eV, for C–C 3.6 eV, for Si–Si 1.8 eV, and for Ge–Ge 1.6 eV. Such
energy is, for neutral atoms, comparable to the ionic bond, discussed in the next section.
1 It is debated in femtosecond chemistry whether the bond really forms in this way. However, it is a picture of overwhelming
simplicity.