Chemistry 445.
Lecture 4.
Molecular Orbital Theory
of diatomic molecules
The non-existent He2 molecule
(bond order = 0)
BO = (2-2)/2 = 0
The MO diagram for the He2 molecule is similar to that for the H2 molecule,
but we see that the energy drop of the pair of electrons in the σ1s orbital is
negated because the other pair in the σ*1s rises in energy by an equal amount.
There is thus no net stabilization, and so the He2 molecule does not exist.
The He2+ molecule/ion exists, bond order = ½
unpaired
electron so
is paramagnetic
BO = (2-1)/2 = ½
The logic of the MO diagram suggests that if we remove an electron
from the He2 molecule, we would obtain a stable [He2]+ cation, which
is true in the gas-phase. This illustrates the power of the MO approach,
since the Lewis dot diagram does not predict this.
The Li2 molecule. Bond order = 1
BO = (2-0)/2 = 1
Bond energy for
Li2 = 110 kJ.mol-1,
Compared to 436
kJ.mol-1 for H2.
Note. In drawing up an MO diagram, only the valence shells are
considered, so for the diatomic molecules from Li2 to F2, the overlaps
of the pairs of 1s orbitals are ingnored. This is valid because these
are filled, and they make no net contribution to the bonding.
The non-existent Be2 molecule,
bond order = 0
BO = (2-2)/2 = 0
Here again, MO theory predicts that Be2 does not exist, which
Lewis dot diagrams do not predict.
σ and π bonding
In the molecules we have considered so far, only σ overlaps have
been of importance. In the formal definition, a σ–bond is one which
lies along a rotational symmetry axis, (the rest of the molecule is
ignored). A π–bond does not lie along a rotational axis (symmetry
will be discussed later). In practical terms, a σ–bond lies along the
bond connecting the two atoms, whereas a π–bond does not.
z
z
+
pz
orbitals
π*(pz)
z
z
anti-bonding π* MO
π(pz)
+
bonding π MO
O2 molecule, bond order = 2
molecules with
unpaired electrons
are paramagnetic
The ability
to predict
the number
of unpaired
electrons in
molecules is
where MO
excels, and
Lewis-dot
fails.
BO = (6-2)/2 = 2
(disregardiing
overlap of
2s orbitals)
O
O2
O
F2 molecule, bond order = 1
F2 has no unpaired
electrons, and so
is diamagnetic
BO = (6-4)/2 = 1
Variation of the energies of the 2s and 2p
orbitals in crossing the periodic table from Li
to F. (H&S Fig. 1.22)
Energy levels of first-row homonuclear diatomic
molecules (H&S Fig 1.23)
crossover point
Molecules Li2, Be2, B2,C2 and N2
have π(2p) lower in energy than σ(2p)
Molecules O2, and F2 have π(2p)
higher in energy than σ(2p)
Be2 molecule, bond order = 0
(BO = 0, means does not exist)
BO = (2-2)/2 = 0
B2 molecule, bond order = 1
BO = (2-0)/2 = 1
N2 molecule, bond order = 3
diamagnetic
BO = (6-0)/2 = 3
N
N2
N
C2 molecule, bond order = 2
diamagnetic
BO = (4-0)/2 = 2
Singlet oxygen (1O2)
BO = (6-2)/2 = 2
Singlet Oxygen
is an excited
state of the
ground state
triplet 3O2
molecule. It is
much more
reactive, and
will readily
attack organic
molecules.
O
O2
O
The O2 molecule in its excited singlet state which is 25 kcal/mol in energy
above the ground triplet state. Irradiation with IR light causes excitation to
the singlet state, which can persist for hours because the spin-selection rule
(see later) inhibits transitions that involve a change of spin state.
Orbital parity – gerade (g) and ungerade (u)
Symmetry of orbitals and molecules is of great importance, and we
should be able to determine whether orbitals are gerade (g) or ungerade
(u) (from German for even or odd). This is because in the spectra of
inorganic compounds whether absorption of a photon to produce an
electronic transition can occur is determined by whether the two orbitals
involved are g or u. According to the Laporte selection rules, transitions
from gu and ug are allowed, but gg and uu are forbidden. An
orbital is g if it has a center of inversion, and u if it does not. So looking
at atomic orbitals, we see that s and d are g, while p orbitals are u: (see
next page for definition of center of sym.)
s-orbital
gerade (g)
p-orbital
ungerade (u)
d-orbital
gerade (g)
Orbital parity – gerade (g) and ungerade (u)
Symmetry of orbitals and molecules is of great importance, and we
should be able to determine whether orbitals are gerade (g) or
ungerade (u) (from German for even or odd). This is because in the
spectra of inorganic compounds whether absorption of a photon to
produce an electronic transition is determined by whether the two
orbitals involved are g or u. According to the Laporte selection rules,
transitions from gu and ug are allowed, but gg and uu are
forbidden. An orbital is g if it has a center of inversion, and u if it
does not. So looking at atomic orbitals, we see that s and d are g,
while p orbitals are u: (see next page for definition of center of sym.)
center of
inversion
a=b
a
a
b
s-orbital
gerade (g)
not a
center of
inversion
b
a≠b
p-orbital
ungerade (u)
a
b
d-orbital
gerade (g)
Parity (g or u) of molecular orbitals:
not a center
of inversion
a≠b
(sign of wavefunction is
opposite)
center of
inversion
a=b
a
a
b
b
σ*(1s)u
π*(2p)g
a
a
b
σ(1s)g
b
center of
Inversion
a=b
not a center
of inversion
a≠b
π(2p)u
The test for whether an MO is g or u is to find the possible center of inversion
of the MO. If two lines drawn out at 180o to each other from the center, and of
equal distances, strike identical points (a and b), then the orbital is g.
Energy levels of the N2 molecule
see if
you can
decide
which are g
or are u, and
bonding or
anti-bonding
Energy levels of the N2 molecule
(calculated using semi-empirical MO theory)
σ*2pu
π*2pg
σ2pg
σ*2su
π*2pu
σ2sg
Labeling molecular orbitals
as g or u:
The following little table will help you to label molecular
orbitals as g or u. For σ-overlap, the bonding orbitals are
g, while the antibonding orbitals are u, while for π–
overlap the opposite is true:
bonding MO
anti-bonding MO
σ-bonding
g
u
π-bonding
u
g
To summarize:
• A bonding molecular orbital has overlap of the two
atomic orbitals, and has no nodal plane. An antibonding orbital has a nodal plane between the two
atoms forming the bond.
• g orbitals have even parity, and have a center of
inversion. u orbitals have odd parity and have no
center of inversion
• In drawing up an MO diagram, you should fully label
all atomic orbitals and MO’s (indicate atomic orbital
MO is derived from ( 1s, 2p, etc.), g or u, σ or π,
bonding or non-bonding (*) ), indicate number of
unpaired electrons, diamagnetic or paramagnetic,
and bond order.