Heteronuclear Diatomic Molecules
The preceding discussion concerned homonuclear diatomic
molecules, with the consequence that all of the molecular orbitals were
constructed from degenerate components, e.g., 2pA and 2pB. When we
consider heterogeneous molecules, the degeneracy is not present, and we
mustt rethink
thi k the
th results.
lt Thus
Th the
th terms
t
HAA and
d HBB are no longer
l
equal.
l
The previous LCAO derivation must be modified.
The general secular determinant
can be expanded out to
which has rather complex solutions for E,
IF, however, we can neglect overlap (S=0), then the secular determinant
simplifies to
H AA
E
H AB
H AB
H BB
0
E
Again expanding to
H AA . H BB
H AA . E
E . H BB
E
2
H AB
which has the following less complex solutions for E
1.
2
1.
2
H AA
H AA
1.
2
1.
2
H BB
H BB
1.
2
1.
2
H BB
H AA
H BB
H AA
2
2
4. H AB
2
2
4. H AB
IF in addition,|H
IF,
addition |HAA-H
HBB|>>HAB, the two solutions simplify to
H BB
H AA
H AB
H BB
H AA
H AB
H BB
2
2
H AA
2
0
This result corresponds
p
essentially
y to the energies
g
of
the parent atomic states being very different, i.e. having
very different ionization energies. Namely, the energies
of the molecular orbitals are little changed from those
of the parent, and there is very little bonding
interaction. The two molecular orbitals are essentially
th atomic
the
t i orbitals,
bit l ΨA and
d ΨB.
This result shows why
y only
y those orbitals that lie
close to each other in energy make significant
contributions to the bonding in heteronuclear molecules.
To be specific, consider now the heteronuclear
diatomic molecule HF. We approximate the coulomb integrals
HAA = αH = 13.6 eV, and HBB = αF = 19.6 eV.
We will let the resonance integral HAB = β= 2 eV.
These values are approximately correct.
Th generall secular
The
l determinant
d t
i
t now becomes
b
for
f this
thi HF case
13.6
2
E
2
18.6
E
which expands to
248.96
32.2 . E
E
2
0
with two solutions for E,,
19.301562118716424343
12.898437881283575657
898 3 88 835 565
These results are given pictorially below
0
The highest energy occupied atomic orbitals of H and F atoms and the
molecular orbitals they form.
form
When each of these values for E are substituted into the secular
equations, we obtain two wave functions,
Ψlower = 0.33
0 33ψ H + 0
0.94
94ψ F
and
Ψupper = 0.94ψ H − 0.33ψ F
Ψlower = 0.33ψ H + 0.94ψ F
and
Ψupper = 0.94ψ H − 0.33ψ F
One orbital is bonding and the other is antibonding. The bonding
orbital has 94% F character and the antibonding one has 94% H
character. There are two electrons to place in these orbitals, and they
will go into the lower energy one to form the ground state. It is
immediately clear why we refer to the HF ground state as being ionic,
with
ith structure
t
t
H+ F−. We
W also
l see that
th t the
th excited
it d state
t t represented
t d
here is not ionic.
We have implicitly considered only those orbitals that are close
to each other energetically.
energetically To get a feeling for the reasonableness of
this approximation, let us add a 2s orbital to the HF molecular orbitals
above.
We have previously paid attention only to the two states closest
in energy, the H 1s and the F 2p states. Were we justified to neglect
the other states? We find out by adding in the next closest state, the F
gy of the F 2s orbital is approximately
pp
y 40.2
2S state. The ionization energy
eV. If we couple this orbital to F 2p with a resonance integral of 1 eV,
then we obtain the secular determinant
which expands to the cubic equation in E,
9994.592
1542.4 . E
2
.
72.4 E
E
3
0
with three roots
40.246519404241607709
12.894404784997342795
2 . 10
( 20 ) .
i
19 259075810761049496
19.259075810761049496
2 . 10
( 20 ) .
i
Note how the positions of the two original levels are essentially unchanged
by the addition of the 2s level,
level and how the position of the new level is
essentially that of the F 2s level. This result is really our justification for
considering only those levels that are close to each other in energy.
To give a final example, consider the molecular orbitals of CO, another
interesting case.
We can construct LCAO-MO’s that are very
y similar
m
to those of
f
isoelectronic N2. Since the oxygen nuclear charge is greater than that
of carbon, all of the bonding LCAO-MO’s will have somewhat larger
coefficients for the oxygen orbitals and the antibonding ones will have
somewhat larger coefficients for the carbon component than for the
oxygen component.
In analogy with N2, the electron configuration of CO is
(σ 1s)2 (σ *1s)2(σ 2s)2 (σ* 2s)2 (π 2px)2(π 2py)2 (σ 2pz)2
I ha
have
e dropped the u and g des
designations,
gnat ons, ssince
nce the or
orbitals
ta s no longer
onger ha
have
e
inversion symmetry. I retain the * to indicate antibonding orbitals. The
bond order is three, as for N2. The bonding and antibonding aspects of
the lowest four orbitals essentially
y cancel, and we can equally
y well
represent the CO molecule in the alternate basis
(1s C)2 (1s O)2 (2s C)2 (2s O)2 (π 2px)2(π 2py)2 (σ 2pz)2