MO theory and MO diagrams for linear molecules
1.
Concepts of MO theory – LCAO method, bonding and antibonding MOs
The nature of bonding in molecules can be described theoretically by molecular orbital theory (MO theory).
Molecular orbitals (MOs) are the result of an overlap of atomic orbitals (AOs) from the atoms that form the
bond. Therefore, MOs can be, in a good approximation, constructed as a superposition of the contributing
atomic orbitals:
where the
are weighting coefficients and the
are the respective AOs. This approximation is called
linear combination of atomic orbitals or LCAO method.
The first case,
, is called a bonding MO, which lies lower in energy than the
constructing AOs.
The second case,
, is called an antibonding MO. It lies higher in energy than the
constructing AOs.
Principle: An overlap between AOs is only possible if they have similar symmetry!
2.
Construction of MO energy level diagrams for diatomic molecules
For diatomic molecules (e.g. H2, O2, HF…), MO schemes can be constructed very easily with some rules:
a) Draw the levels of the valence shell AOs of the respective atoms on an energy scale. If the atoms are
the same, then their AOs have the same energy. If the atoms are different, those of the more
electronegative one lie lower in energy. Denote the AOs by their names (1s, 2s, 2p etc.).
b) N AOs will yield N MOs in total. Draw the bonding MOs lower in energy relative to the AOs and the
antibonding MOs higher in energy. Non-bonding MOs remain at the position of the AOs which do
not overlap with other ones due to symmetry lack or too different energies.
c) In homonuclear diatomic molecules (A2), the MOs have the same distance to both AOs forming the
corresponding MO. In heteronuclear diatomic molecules (AB), the bonding MOs lie closer to the
AO of the more electronegative atom and the antibonding MOs lie closer to the AO of the more
electropositive atom, respectively.
d) Denote the MOs by symbols. In molecules of the type A2, one uses
,
(due to the center of
are used. Antibonding MOs are
inversion), whereas in molecules of the type AB, the symbols ,
often also denoted by an asterisk (e.g.
). -orbitals are one-fold degenerate,
-orbitals are
two-fold degenerate. -orbitals generally lie lower in energy than -orbitals.
e) Fill the MOs with electrons. Take care of Pauli’s principle and Hund’s rule.
f) The bond order can be determined with
). n is the number of electrons in
bonding, n* the number of electrons in antibonding MOs. The bond order is an effective number of
bonds and thus a measure for the strength of a bond.
3.
Literature
[1]
P. Atkins, T. Overton, J. Rourke, M. Weller, F. Armstrong, Shriver & Atkins’ Inorganic Chemistry,
5th edition, 2010, Oxford University Press, Oxford (pp. 42-58).
[2]
P. Atkins, R. Friedman, Molecular Quantum Mechanics, 4th edition, 2005, Oxford University Press,
Oxford (pp. 249-286).
4.
Further tasks
1)
Draw a schematic MO energy level diagram for N2 in its ground state using the rules shown above
(see Shriver/Atkins for the sequence of MOs). Determine the bond order of N2. Would you expect N2
to be highly reactive?
2)
Molecular oxygen, O2, is a paramagnetic gas. This cannot be explained by its usual Lewis structure.
Try to account for that property by constructing its MO scheme. What spin multiplicity (2S+1) does
O2 have in its ground state?