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Physical chemistry 266

only with an electron of opposite spin. This ensures that a molecular bond is created in
which the two electrons do not occupy the same quantum state, and so comply with the
Pauli exclusion principle (see Topic G6).
Bonds are formed from combinations of the atomic orbitals from each of the bonding
atoms, and the mathematical description of the molecular bond is therefore a function of
these atomic orbitals. It is a fundamental requirement that the electrons in the molecular
orbital are indistinguishable, and the simplest orbital function compatible with this is the
Heitler-London wavefunction:
ψ=ψ A(1)ψ B(2)+ψ A(2)ψ B(1)

The first product describes the case of electron 1 in orbital A (ψ A(1)) and that of electron
2 in orbital B (ψ B(2)), with the second product describing the complementary situation.
The two terms are not identical, as the electrons possess opposite spins. The resulting
wavefunction describes the condition where either electron may be found on either of the
bonded atoms.
In the simplest example, that of a hydrogen molecule, the atomic 1s orbitals are the
sole contributors to the bond, and the wavefunction takes the form:

The physical results of this mathematical expression are illustrated in Fig. 1a and 1b. The
resulting bond has cylindrical symmetry about the bond axis, and is termed a σ (sigma)
bond.

In elements with accessible p orbitals, such as oxygen or nitrogen, more complex
bonding may be obtained. The two atomic p orbitals which are parallel to the bonding
axis (the p z orbitals, by convention) may be combined so as to form a a bond (Fig. 1c),
but it is also possible for p orbital pairs which are perpendicular to the bonding axis (p x
on A and B or p y on A and B) to combine to give π (pi) bonds (Fig. 1d). The strength of
the π-bond is significantly less than that of the σ-bond, as the ‘sidelong’ overlap of the p
orbitals is less than that of the ‘direct’ overlap (the products and
being correspondingly reduced). Each pair of atomic p orbitals forms one
molecular π-bond, giving a maximum of three molecular bonds from each set of p
orbitals—one σ-bond and two mutually orthogonal π-bonds. The π-bonds do not have
cylindrical symmetry, having instead a nodal plane parallel with the bonding axis.