5.61 Physical Chemistry Lecture #31 1
HÜCKEL MOLECULAR ORBITAL THEORY
In general, the vast majority polyatomic molecules can be thought of as
consisting of a collection of twoelectron bonds between pairs of atoms. So
the qualitative picture of σ and πbonding and antibonding orbitals that we
developed for a diatomic like CO can be carried over give a qualitative
starting point for describing the C=O bond in acetone, for example. One
place where this qualitative picture is extremely useful is in dealing with
conjugated systems – that is, molecules that contain a series of alternating
double/single bonds in their Lewis structure like 1,3,5hexatriene:
Now, you may have been taught in previous courses that because there are
other resonance structures you can draw for this molecule, such as:
that it is better to think of the molecule as having a series of bonds of
order 1 ½ rather than 2/1/2/1/… MO theory actually predicts this
behavior, and this prediction is one of the great successes of MO
theory as a descriptor of chemistry. In this lecture, we show how even a
very simple MO approximation describes conjugated systems.
Conjugated molecules of tend to be planar, so that we can place all the atoms
in the xy plane. Thus, the molecule will have reflection symmetry about the
zaxis:
z
Now, for diatomics, we had reflection symmetry about x and y and this gave
rise to πx and πy orbitals that were odd with respect to reflection and σ
orbitals that were even. In the same way, for planar conjugated systems the
orbitals will separate into σ orbitals that are even with respect to reflection
, 5.61 Physical Chemistry Lecture #31 2
and πz orbitals that are odd with respect to reflection about z. These πz
orbitals will be linear combinations of the pz orbitals on each carbon atom:
z
In trying to understand the chemistry of these compounds, it makes sense
to focus our attention on these πz orbitals and ignore the σ orbitals. The πz
orbitals turn out to be the highest occupied orbitals, with the σ orbitals
being more strongly bound. Thus, the forming and breaking of bonds – as
implied by our resonance structures – will be easier if we talk about making
and breaking π bonds rather than σ. Thus, at a basic level, we can ignore the
existence of the σorbitals and deal only with the πorbitals in a qualitative
MO theory of conjugated systems. This is the basic approximation of
Hückel theory, which can be outlined in the standard 5 steps of MO theory:
1) Define a basis of atomic orbitals. Here, since we are only interested
in the πz orbitals, we will be able to write out MOs as linear
combinations of the pz orbitals. If we assume there are N carbon
atoms, each contributes a pz orbital and we can write the µth MOs as:
N
π µ = ∑ ciµ pzi
i=1
2) Compute the relevant matrix representations. Hückel makes some
radical approximations at this step that make the algebra much
simpler without changing the qualitative answer. We have to compute
two matrices, H and S which will involve integrals between pz orbitals
on different carbon atoms:
H ij = ∫ p zi Hˆ pzj dτ Sij = ∫ pzi p zj dτ
The first approximation we make is that the pz orbitals are
orthonormal. This means that:
⎧1 i = j
Sij = ⎨
⎩0 i ≠ j
HÜCKEL MOLECULAR ORBITAL THEORY
In general, the vast majority polyatomic molecules can be thought of as
consisting of a collection of twoelectron bonds between pairs of atoms. So
the qualitative picture of σ and πbonding and antibonding orbitals that we
developed for a diatomic like CO can be carried over give a qualitative
starting point for describing the C=O bond in acetone, for example. One
place where this qualitative picture is extremely useful is in dealing with
conjugated systems – that is, molecules that contain a series of alternating
double/single bonds in their Lewis structure like 1,3,5hexatriene:
Now, you may have been taught in previous courses that because there are
other resonance structures you can draw for this molecule, such as:
that it is better to think of the molecule as having a series of bonds of
order 1 ½ rather than 2/1/2/1/… MO theory actually predicts this
behavior, and this prediction is one of the great successes of MO
theory as a descriptor of chemistry. In this lecture, we show how even a
very simple MO approximation describes conjugated systems.
Conjugated molecules of tend to be planar, so that we can place all the atoms
in the xy plane. Thus, the molecule will have reflection symmetry about the
zaxis:
z
Now, for diatomics, we had reflection symmetry about x and y and this gave
rise to πx and πy orbitals that were odd with respect to reflection and σ
orbitals that were even. In the same way, for planar conjugated systems the
orbitals will separate into σ orbitals that are even with respect to reflection
, 5.61 Physical Chemistry Lecture #31 2
and πz orbitals that are odd with respect to reflection about z. These πz
orbitals will be linear combinations of the pz orbitals on each carbon atom:
z
In trying to understand the chemistry of these compounds, it makes sense
to focus our attention on these πz orbitals and ignore the σ orbitals. The πz
orbitals turn out to be the highest occupied orbitals, with the σ orbitals
being more strongly bound. Thus, the forming and breaking of bonds – as
implied by our resonance structures – will be easier if we talk about making
and breaking π bonds rather than σ. Thus, at a basic level, we can ignore the
existence of the σorbitals and deal only with the πorbitals in a qualitative
MO theory of conjugated systems. This is the basic approximation of
Hückel theory, which can be outlined in the standard 5 steps of MO theory:
1) Define a basis of atomic orbitals. Here, since we are only interested
in the πz orbitals, we will be able to write out MOs as linear
combinations of the pz orbitals. If we assume there are N carbon
atoms, each contributes a pz orbital and we can write the µth MOs as:
N
π µ = ∑ ciµ pzi
i=1
2) Compute the relevant matrix representations. Hückel makes some
radical approximations at this step that make the algebra much
simpler without changing the qualitative answer. We have to compute
two matrices, H and S which will involve integrals between pz orbitals
on different carbon atoms:
H ij = ∫ p zi Hˆ pzj dτ Sij = ∫ pzi p zj dτ
The first approximation we make is that the pz orbitals are
orthonormal. This means that:
⎧1 i = j
Sij = ⎨
⎩0 i ≠ j
