Page 287 - Instant notes

Molecular orbital theory of diatomic molecules I 273

The wavefunctions for two electrons, ψ(A) and ψ(B), in the atomic orbitals on two atoms,
A and B respectively, are combined to form a molecular orbital. The molecular orbital
wavefunction is given by:

where c m is a mixing coefficient, which is calculated for each orbital so as to minimize
the molecular energy as calculated using the Schrödinger equation. It is a further
requirement that the mixing coefficients should be normalized, so that . When
the bond is formed between two identical nuclei there can be no distinction between the
nuclei, and so the mixing coefficients (and therefore the orbital contributions), are equal.
The resulting molecular orbital becomes more accurately described as more of the
available atomic orbitals are included in the calculation. The variation principle states
that the lower the energy of the calculated orbital, the more accurately it describes the
actual molecular wavefunction.

Bonding and antibonding orbitals

The probability, p, of an electron in an orbital, ψ, being in an infinitesimal volume dτ at a
point r is obtained through the Born interpretation:

+
For H 2 , the molecular orbital probability function is then given by:
2
2
2
P(r)=(ψ(A)+ψ(B)) dτ=ψ(A) dτ+ψ(B) dτ+2ψ(A)ψ(B)dτ
This differs from the atomic orbitals as it represents the probability of finding an electron
in the two constituent atomic orbitals plus an additional term. This term has the effect of
increasing electron density in the internuclear region, and decreasing it in other regions
(Fig. 1a). Since electrons are waves, the enhanced electron density can be rightly
compared to the constructive interference of two waves. The enhanced electron density
between the nuclei enables the electrons to associate strongly with both nuclei
simultaneously, thereby reducing the free energy of the electrons, and so the molecule as
a whole.
The structure of a molecule depends upon the formation of bonds holding the nuclei in
their relative positions, and these result from occupied bonding orbitals (Fig. 1b).
Molecular bonding orbitals are derived from linear combination of atomic orbitals by the
addition of the component atomic orbitals, and act to decrease the free energy of the
+
molecule. The explicitly calculated result for H 2 reveals that the next highest energy
solution to the Schrödinger equation is an antibonding orbital corresponding to the
subtraction of one wavefunction from the other:
ψ(MO)=ψ(A)− (B)