Physical Chemistry -Molecular Orbital Theory_lecture28-30

5.61 Physical Chemistry Lecture #28 1




MOLECULAR ORBITAL THEORY­ PART I
At this point, we have nearly completed our crash­course introduction to
quantum mechanics and we’re finally ready to deal with molecules. Hooray!
To begin with, we are going to treat what is absolutely the simplest
molecule we can imagine: H+2 . This simple molecule will allow us to work
out the basic ideas of what will become molecular orbital (MO) theory.

We set up our coordinate r
system as shown at right, e­
with the electron positioned rB rA
at r, and the two nuclei
positioned at points RA and
RB, at a distance R from one HA HB
another. The Hamiltonian is RA R RB
easy to write down: H2+ Coordinates
∇2 ∇2 1 1 1
Ĥ = − 1 ∇r2 − A − B − − +
2 2M A 2MB ˆ − r̂
R ˆ − r̂
R ˆ −R
R ˆ
A B A B

Electron HA HB e­­HA e­­HB HA­HB
Kinetic Kinetic Kinetic Attraction Attraction Repulsion
Energy Energy Energy
Now, just as was the case for atoms, we would like a picture where we can
separate the electronic motion from the nuclear motion. For helium, we
did this by noting that the nucleus was much heavier than the electrons
and so we could approximate the center of mass coordinates of the
system by placing the nucleus at the origin. For molecules, we will make a
similar approximation, called the Born­Oppenheimer approximation.
Here, we note again that the nuclei are much heavier than the electrons.
As a result, they will move much more slowly than the light electrons.
Thus, from the point of view of the electrons, the nuclei are almost
sitting still and so the moving electrons see a static field that arises
from fixed nuclei. A useful analogy here is that of gnats flying around on
the back of an elephant. The elephant may be moving, but from the gnats’
point of view, the elephant is always more or less sitting still. The
electrons are like the gnats and the nucleus is like the elephant.

,5.61 Physical Chemistry Lecture #28 2



The result is that, if we are interested in the electrons, we can to a good
approximation fix the nuclear positions – RA and RB – and just look at the
motion of the electrons in a molecule. This is the B­Oppenheimer
approximation, which is sometimes also called the clamped­nucleus
approximation, for obvious reasons. Once the nuclei are clamped, we can
make two simplifications of our Hamiltonian. First, we can neglect the
kinetic energies of the nuclei because they are not moving. Second, because
the nuclei are fixed, we can replace the operators R ˆ and R ˆ with the
A B
numbers RA and RB. Thus, our Hamiltonian reduces to
∇2 1 1 1
el ( )
Hˆ R A ,R B = − r −
2
− +
R A − r̂ R B − r̂ R A − R B
where the last term is now just a number – the electrostatic repulsion
between two protons at a fixed separation. The second and third terms
depends only on the position of the electron, r, and not its momentum, so
we immediately identify those as a potential and write:
∇2 1
( ) 2
RA , RB
Hˆ el R A ,R B = − r +Veff ( r̂ ) +
RA − RB
This Hamiltonian now only contains operators for the electrons (hence the
subscript “el”), but the eigenvalues of this Hamiltonian depend on the
distance, R, between the two nuclei. For example, the figure below shows
the difference between the effective potentials the electron “feels” when
the nuclei are close together versus far apart:
R Small R Large




Veff(r)



Likewise, because the electron feels a different potential at each bond
distance R, the wavefunction will also depend on R. In the same limits as
above, we will have:
ψel(r)


R Small R Large

, 5.61 Physical Chemistry Lecture #28 3



Finally, because the electron eigenfunction, ψel, depends on R then the
eigenenergy of the electron, Eel(R), will also depend on the bond length.
Mechanically, then, what we have to do is solve for the electronic
eigenstates, ψel, and their associated eigenvalues, Eel(R), at many different
fixed values of R. The way that these eigenvalues change with R will tell us
about how the energy of the molecule changes as we stretch or shrink the
bond. This is the central idea of the Born­Oppenheimer approximation, and
it is really very fundamental to how chemists think about molecules. We
think about classical point­like nuclei clamped at various different positions,
with the quantum mechanical electrons whizzing about and gluing the nuclei
together. When the nuclei move, the energy of the system changes because
the energies of the electronic orbitals change as well. There are certain
situations where this approximation breaks down, but for the most part the
Born­Oppenheimer picture provides an extremely useful and accurate way to
think about chemistry.

How are we going to solve for these eigenstates? It should be clear that
looking for exact solutions is going to lead to problems in general. Even for
H2+ the solutions are extremely complicated and for anything more complex
than H2+exact solutions are impossible. So we have to resort to
approximations again. The first thing we note is that if we look closely at
our intuitive picture of the H2+ eigenstates above, we recognize that these
molecular eigenstates look very much like the sum of the 1s atomic orbitals
for the two hydrogen atoms. That is, we note that to a good approximation
we should be able to write:
ψ el ( r ) ≈ c11s A ( r ) + c2 1sB ( r )
where c1 and c2 are constants. In the common jargon, the function on the
left is called a molecular orbital (MO), whereas the functions on the right
are called atomic orbitals (AOs). If we write our MOs as sums of AOs, we
are using what is called the linear combination of atomic orbitals (LCAO)
approach. The challenge, in general, is to determine the “best” choices for c1
and c2 – for H2+ it looks like the best choice for the ground state will be
c1=c2. But how can we be sure this is really the best we can do? And what
about if we want something other than the ground state? Or if we want to
describe a more complicated molecule like HeH+2?