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288 9MOLECULAR STRUCTURE
are paired
σ(2, 1) = (1/2)1/2 {α(1)β(2) − β(1)α(2)}
In summary, the form of the VB wavefunction which results in a the lower-
ing of the energy, and hence bond formation, is symmetric with respect to
interchange of the electron labels, and in order to satisfy the Pauli principle
the associated spin wavefunction must be anti-symmetric with respect to this
exchange.�is wavefunction corresponds to one in which the spins are paired.
Solutions to exercises
E9A.1(b) �e nitrogen atoms in N2 are sp hybridized.�e σ bond is formed by the over-
lap of two sp hybrid atomic orbitals oriented towards each other along the inter-
nuclear axis. Using [9A.2–344], the spatial part of the valence-bond wavefunc-
tion of the σ orbital is writtenΨ(1, 2) = ψNspA(1)ψNspB(2)+ψNspA(2)ψNspB(1).
�is wavefunction is symmetric, therefore it has to be combined with the anti-
symmetric two-electron spin wavefunction given by [8B.3–319], σ−(1, 2).�e
resulting complete, asymmetric two-electron wavefunction satis�es the Pauli
principle and has the form of
Ψσ(1, 2) = [ψNspA(1)ψNspB(2)+ψNspA(2)ψNspB(1)]× [α(1)β(2)− β(1)α(2)]
�e two π orbitals are formed by the side-by-side overlap of two px and two py
orbitals. Following the same logic as above, the valence-bond wavefunction for
the 2pyπ orbital is written as
Ψπ(1, 2) = [ψNpyA(1)ψNpyB(2)+ψNpyA(2)ψNpyB(1)]× [α(1)β(2)− β(1)α(2)]
and similarly for the 2pxπ orbital.
E9A.2(b) �e resonance hybrid wavefunction constructed from one two-electron wave-
function corresponding to the purely covalent form of the bond and one two-
electron wavefunction corresponding to the ionic form of the bond is given in
[9A.3–346] as Ψ = Ψcovalent + λΨionic.�erefore the (unnormalized) resonance
hybrid wavefunction of N2 with two ionic structures is written asΨN2 = ΨN–N+
λΨN+N− + κΨN2−N2+ .
�e valence bonddescription of the triple bond inN2 is give inExampleE9A.1(b).
Omitting the spin functions for simplicity, the spatial wavefunction is a product
of that for the σ bond, and that for each of the π bonds
[ψNspA(1)ψNspB(2) + ψNspA(2)ψNspB(1)]
× [ψNpyA(3)ψNpyB(4) + ψNpyA(4)ψNpyB(3)]
× [ψNpxA(5)ψNpx B(6) + ψNpxA(6)ψNpx B(5)]
Somewhat arbitrarily electrons 1 and 2 have been assigned to the �rst valence-
bond wavefunction, 3 and 4 to the next and so on.
�e ionic structure N+N– can be thought of as arising when electrons 5 and
6 are both located in NpxB . In this case, the part of the wavefunction in the