Physical Chemistry - Spectroscopy - Probing Molecules with Light_lecture34
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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 ...
SPECTROSCOPY: PROBING MOLECULES WITH LIGHT
In practice, even for systems that are very complex and poorly
characterized, we would like to be able to probe molecules and find out as
much about the system as we can so that we can understand reactivity,
structure, bonding, etc. One of the most powerful tools for interrogating
molecules is spectroscopy. Here, we tickle the system with electromagnetic
radiation (i.e. light) and see how the molecules respond. The motivation for
this is that different molecules respond to light in different ways. Thus, if
we are creative in the ways that we probe the system with light, we can hope
to find a unique spectral fingerprint that will differentiate one molecule
from all other possibilities. Thus, in order to understand how spectroscopy
works, we need to answer the question: how do electromagnetic waves
interact with matter?
The Dipole Approximation
An electromagnetic wave of wavelength λ, produces an electric field, E(r,t),
and a magnetic field, B(r,t), of the form:
E(r,t)=E0 cos(k·r – ωt) B(r,t)=B0 cos(k·r – ωt)
Where ω=2πν is the angular frequency of the wave, the wavevector k has a
magnitude 2π/λ and k (the direction the wave propagates) is perpendicular to
E0 and B0. Further, the electric and magnetic fields are related:
E0· B0=0 |E0|=c|B0|
Thus, the electric and magnetic
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fields are orthogonal and the
magnetic field is a factor of c (the
speed of light, which is 1/137 in
atomic units) smaller than the
electric field. Thus we obtain a
picture like the one at right, where
the electric and magnetic fields
oscillate transverse to the
direction of propagation.
Now, in chemistry we typically deal with the part of the spectrum from
ultraviolet (λ≈100 nm) to radio waves (λ≈10 m)1. Meanwhile, a typical molecule
1
There are a few examples of spectroscopic measurements in the XRay region. In these
cases, the wavelength can be very small and the dipole approximation is not valid.
, 5.61 Physical Chemistry Lecture #34 2
is about 1 nm in size. Let us assume that the molecule is sitting at the origin.
Then, in the 1 nm3 volume occupied by the molecule we have:
k·r ≈ |k| |r| ≈ 2p/(100 nm) 1 nm = .06
Where we have assumed UV radiation (longer wavelengths would lead to even
smaller values for k·r). Thus, k·r is a small number and we can expand the
electric and magnetic fields in a power series in k·r:
E(r,t)≈E0 [cos(k·0-ωt)+O(k·r)]≈E0 cos(ωt)
B(r,t)≈B0 [cos(k·0-ωt)+O(k·r)]≈B0 cos(ωt)
Where we are neglecting terms of order at most a few percent. Thus, in
most chemical situations, we can think of light as applying two time
dependent fields: an oscillating, uniform electric field (top) and a
uniform, oscillating magnetic field (bottom). This approximation is called
the Dipole approximation – specifically when applied to the electric
(magnetic) field it is called the electric (magnetic) dipole approximation. If
we were to retain the next term in the expansion, we would have what is
called the quadrupole approximation. The only time it is advisable to go to
higher orders in the expansion is if the dipole contribution is exactly zero as
happens, for example, due to symmetry in some cases. In this situation, even
though the quadrupole contributions may be small, they are certainly large
compared to zero and would need to be computed.
The Interaction Hamiltonian
How do these oscillating electric and magnetic fields couple to the molecule?
Well, for a system interacting with a uniform electric field E(t) the
interaction energy is
Hˆ E ( t ) = −µˆ iE ( t ) = −e rˆ i E ( t )
where µ is the electric dipole moment of the system. Thus, uniform electric
fields interact with molecular dipole moments.
Similarly, the magnetic field couples to the magnetic dipole moment, m.
Magnetic moments arise from circulating currents and are therefore
proportional to angular momentum – larger angular momentum means higher
circulating currents and larger magnetic moments. If we assume that all the
angular momentum in our system comes from the intrinsic spin angular
momentum, I=(Ix , Iy ,Iz), then the magnetic moment is strictly proportional to
I. For example, for a particle with charge q and mass m then
q gˆ
Hˆ B ( t ) = −m
ˆ iB ( t ) = − Ii B ( t )
2m
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