21.5 The Valence-Bond Description of Polyatomic Molecules 881
21.17Using LCAOMOs made with hybrid orbitals, describe
the bonding and molecular shape of each of the molecules
or ions:
a.HNO 2
b.NO− 3
c.SO^24 −21.18Using LCAOMOs made with hybrid orbitals, describe
the bonding and molecular shape of each of the molecules
or ions:
a.C 2 H 2
b.C 2 N 2c.S^22 −21.5 The Valence-Bond Description
of Polyatomic Molecules
If the molecular orbital wave function of a molecule has no unpaired electrons and
no occupied antibonding orbitals it can be replaced by a simple valence-bond wave
function that is obtained by replacing each pair of bonding molecular spin orbitals
with a bonding factor such as that of Eq. (20.3-7). The hybrid orbitals were originally
defined to be used in valence-bond wave functions and are used in much the same way
as we have used them in LCAOMOs. The criteria for forming a good valence-bond
bonding factor are the same as those for forming a good bonding molecular orbital:
The two atomic orbitals should have the same symmetry around the bond axis, they
should have roughly equal energies, and they should have considerable overlap. In
early applications of the valence-bond theory the strength of a bond was assumed to
be proportional to the value of its overlap integral.
To give a simple description of the bonding in the water molecule using the valence-
bond method it is necessary only to specify that two nonbonding electrons occupy the
oxygen 1sspace orbital, four nonbonding electrons occupy two oxygen 2sp^3 hybrid
space orbitals, and four electrons occupy two bonding factors, each constructed from
an oxygen 2sp^3 hybrid and a hydrogen 1sorbital. The corresponding unnormalized
valence-bond wave function isΨVBψ 1 sO(1)ψ 1 sO(2)ψ 2 sp (^3) (1)(3)ψ 2 sp (^3) (1)(4)ψ 2 sp (^3) (4)(5)ψ 2 sp (^3) (4)(6)
×[ψ 2 sp (^3) (2)(7)ψ 1 sA(8)+ψ 1 sA(7)ψ 2 sp (^3) (2)(8)]
×[ψ 2 sp (^3) (3)(9)ψ 1 sB(10)+ψ 1 sB(9)ψ 2 sp (^3) (3)(10)] (21.5-1)
where we have omitted the spin factors. The subscript 1sA stands for the 1sorbital on
one hydrogen atom and the subscript 1sB stands for the 1sorbital on the other hydrogen
atom.
This wave function corresponds to nonpolar covalent bonds. We can add an ionic
term with both electrons on the oxygen atom:
ΨMVBcVBΨVB+cIΨI (21.5-2)
whereΨIis the completely ionic wave function in which each bonding factor is replaced
by a factor with both bonding electrons occupying the hybrid orbital on the oxygen
atom. The values of the coefficientscVBandcIwould be determined by minimizing
the variational energy.