Physical Chemistry - Modern Electronic Structure Theory_lecture32to33
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This course presents an introduction to quantum mechanics. It begins with an examination of the historical development of quantum theory, properties of particles and waves, wave mechanics and applications to simple systems — the particle in a box, the harmonic oscillator, the rigid rotor and the ...
MODERN ELECTRONIC STRUCTURE THEORY
At this point, we have more or less exhausted the list of electronic
structure problems we can solve by hand. If we were limited to solving
problems manually, there would be a lot of chemistry we wouldn’t be able to
explain! Fortunately, the advent of fast personal computers allows chemists
to routinely use more accurate models of molecular electronic structure.
These types of calculations typically play a significant role in interpreting
experimental results: calculations can be used to assign spectra, evaluate
reaction mechanisms and predict structures of molecules. In this way
computation is complementary to experiment: when the two agree we have
confidence that our interpretation is correct.
The basic idea of electronic structure theory is that, within the Born
Oppenheimer approximation, we can fix the M nuclei in our molecule at some
positions RI. Then, we are left with the Hamiltonian for the electrons
moving in the effective field created by the nuclei:
N N M
ZI N
1
Ĥ ≡ − 1
2 ∑ ∇ -∑∑
i=1
2
i
i=1 I =1
+∑
r̂i − R I i< j r̂i − r̂j Eq. 1
Where the first term is the kinetic energy of all N electrons, the second
term is the attraction between the electrons and nuclei and the third is the
pairwise repulsion between all the electrons. The central aim of electronic
structure theory is to find all the eigenfunctions of this Hamiltonian. As
we have seen, the eigenvalues we
Reaction get will depend on our choice of
Unstable
Barrier the positions of the nuclei –
intermediate
Eel(R1,R2,R3,…RM). As was the
case with diatomics, these
Eel(R1,R2)
energies will tell us how stable
the molecule is with a given
R2 Equilibrium configuration of the nuclei {RI} –
Conformation if Eel is very low, the molecule will
R1 be very stable, while if Eel is high,
the molecule will be unstable in
that configuration. The energy Eel(R1,R2,R3,…RM) is called the potential
energy surface, and it contains a wealth of information, as illustrated in the
picture at above. We can determine the equilibrium configuration of the
, 5.61 Physical Chemistry Lecture #32 2
molecule by looking for the minimum energy point on the potential energy
surface. We can find metastable intermediate states by looking for local
minima – i.e. minima that are not the lowest possible energy states, but which
are separated from all other minima by energy barriers. In both of these
cases, we are interested in points where ∇Eel = 0 . Further, the potential
surface can tell us about the activation energies between different minima
and the pathways that are required to get from the “reactant” state to the
“product” state.
Solving the electronic Schrödinger also gives us the electronic
wavefunctions Ψel(r1,r2,r3,…rN), which allow us to compute all kinds of
electronic properties – average positions, momenta, uncertainties, etc – as
we have already seen for atoms.
We note that while the Hamiltonian above will have many, many eigenstates,
in most cases we will only be interested in the lowest eigenstate – the
electronic ground state. The basic reason for this is that in stable
molecules, the lowest excited states are usually several eV above the ground
state and therefore not very important in chemical reactions where the
available energy is usually only tenths of an eV. In cases where multiple
electronic states are important, the
Hamiltonian above will give us
separate potential surfaces E1el, E2el, σ* potential
E3el … and separate wavefunctions surface
1 2 3
Ψ el, Ψ el, Ψ el. The different
potential surfaces will tell us about
the favored conformations of the
molecules in the different electronic σ potential
states. We have already seen this Erxn surface
for H2+. When we solved for the
electronic states, we got two
eigenstates: σ and σ*. If we put the electron in the σ orbital, the molecule
was bound and had a potential surface like the lower surface at right.
However, if we put the electron in the σ∗ orbital the molecule was not bound
and we got the upper surface.
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