Page 124 - Bruno Linder Elementary Physical Chemistry

August 18, 2010 11:37 9in x 6in b985-ch10 Elementary Physical Chemistry

Quantum Theory. The Chemical Bond 109

To a good approximation, the wave-function of the hydrogen-molecule
ion is

Ψ= ψ 1sA + ψ 1sB (10.5)
where the two wave-functions on the right are respectively the orbitals
centeredon A andB and Ψ is the molecular orbital. This type of approxi-
mation is referred to as linear combination of atomic orbitals (LCAO)and
the molecular orbital is referred to as LCAO-MO.
The molecular orbital in the above example is a σ-orbital because of
its cylindrical symmetry. The probability density of the molecule can be
written as
2
2
Ψ = ψ 2 1sA + ψ 1sB +2ψ 1sA ψ 1sB (10.6)
Here,
2
• ψ 1sA is the probability density of finding the electron on A.
2
• ψ 1sB is the probability density of finding the electron on B.
• 2ψ 1sA ψ 1sB is the overlap density arising from the constructive interfer-
ence of the atomic wave-functions. An electron can interact with both
nuclei, binding them.

10.4. Bonding and Anti-bonding Orbitals
From the linear combination of atomic orbitals, one can construct two
molecular orbitals, a bonding and an anti-bonding one.

Ψ= ψ A + ψ B bonding orbital (10.7a)
∗
Ψ = ψ A − ψ B anti-bonding orbital (10.7b)
Bonding orbitals, if occupied, result in lowering the (electron) energy, as
compared with the energy of the atom (causing an increase in cohesion).
Anti-bonding orbitals have higher energy and cause a reduction in cohesion.
Examples of MO representation diagrams are shown in Fig. 10.4
and 10.5. Note that these diagrams show energy levels of both separated and
combined atoms, depicting atomic and molecular orbitals. The principles
on which these results are based are the same as for atomic orbitals, namely
the Aufbau Principle, Hund’s Rule,etc.
(Here, as customary, anti-bonding orbitals are denoted by *). Each MO
can hold a maximum of two electrons having opposite spins. The π orbitals