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9 Molecular Structure
9A Valence-bond theory
Answers to discussion questions
D9A.2 Promotion and hybridization are two modi�cations to the simplest version of
valence-bond (VB) theory, adopted to overcome obvious mismatches between
predictions of that theory and observations. In its simplest form VB theory
assumes that the functionsψA andψB that appear in aVBwavefunction, [9A.2–
344], are orbitals in free atoms occupied by unpaired electrons. For exam-
ple, such a theory would predict that carbon, with the electronic con�guration
2s22p2, would form two bonds on account if it having two unpaired electrons.
�is prediction is at odds with the characteristic valency of four shown by
carbon.
To account for the tetravalence of carbon it is supposed that one of the 2s
electrons is excited (‘promoted’) to the empty 2p orbital, giving a con�guration
of 2s12p3. �ere are now four unpaired electrons (in the 2s and 2p orbitals)
available for forming four valence bonds.
Hybrid orbitals are invoked to account for the fact that valence bonds formed
from atomic orbitals would have di�erent orientations in space than are com-
monly observed. For instance, the four bonds in CH4 are observed to be equiv-
alent and directed toward the corners of a regular tetrahedron. By contrast,
bonds made from the three distinct 2p orbitals in carbon would be expected to
be oriented at 90○ angles from each other, and those three bonds would not be
equivalent to the bond made from a 2s orbital. Hybrid atomic orbitals, in this
case sp3 hybrids, are formed by combining the atomic orbitals in such a way
that the hybrid orbitals have the required directional properties.
D9A.4 �e part of the VB wavefunction that depends on spatial coordinates is given
in [9A.2–344], Ψ(1, 2)space = ψA(1)ψB(2)+ψA(2)ψB(1).�e complete wave-
function includes a spin part, σ(1, 2)
Ψ(1, 2) = {ψA(1)ψB(2) + ψA(2)ψB(1)} σ(1, 2)
�e Pauli principle requires that the wavefunctionmust be antisymmetric, that
is it must change sign, upon interchange of particle labels: Ψ(2, 1) = −Ψ(1, 2).
�e spatial part is symmetric under this interchange of labels Ψ(2, 1)space =
+Ψ(1, 2)space, therefore the spin partmust be antisymmetric σ(2, 1) = −σ(1, 2).
�e antisymmetric spinwavefunction for two spins is the one inwhich the spins