Physical Spectroscopy

1
Physical Spectroscopy

• Lecture Course
• given by
• Dr. J. H. Clint

2
Contents

• Electromagnetic radiation quantum theory and
  interaction with molecules
• Rotational spectroscopy
• Vibrational spectroscopy
• Raman spectroscopy

• Atomic spectroscopy
• UV / Visible spectroscopy of diatomic molecules
• Nuclear Magnetic Resonance

3
Electromagnetic Radiation
Simple harmonic waves of interconnected electric
and magnetic fields
As waves in space they can be depicted as
electric (E) and magnetic (M) fields at right
angles. If the drawing represents a snapshot
at a fixed time, then the third axis is distance
(x) and the equation of the wave would be
A is the amplitude of the wave and l is the
wavelength, i.e. the distance between successive
peaks.
4
Electromagnetic Radiation
If the drawing represents the variation with time
(t) at a fixed point then the equation of the
wave will be
v frequency, i.e. the number of peaks passing
per second
Since the two equations refer to the same wave,
they must be related. The relation is through
the speed (c) of the radiation The speed of
all electromagnetic radiation in a vacuum is 3.0
x 108 m s-1
5
Variables and Units
Spectroscopists like to use an additional
quantity to describe electromagnetic radiation
the wavenumber, equal to the inverse of the
wavelength. It is therefore proportional to the
frequency
invariably expressed in units of cm-1
6
Plancks Quantum Theory

• In order to explain the dependence of the
  intensity of black body radiation on wavelength,
  Planck, in 1900, introduced the notion that
  energy could only be emitted in discreet amounts,
  or quanta. Their energy is proportional to their
  frequency
• E hv
• where h is Plancks constant.
• (h 6.626 x 10-34 J s)

Max Planck 1858 - 1947
7
Einstein and the Quantization of Radiation

• In 1905 Einstein extended Plancks ideas to
  suggest that electromagnetic radiation itself is
  quantized. He suggested that light could be
  considered as a beam of particles, each with
  energy hv
• In 1926 the American chemist G. N. Lewis proposed
  the name photon for these particles of radiant
  energy.

Albert Einstein 1879 - 1955
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Interaction of Electromagnetic Radiation with
Atoms and Molecules

• When an atom or molecule changes from a state of
  higher energy to one of lower energy, the energy
  it loses is radiated in the form of a photon.
• The frequency of the photon is proportional to
  the change in energy
• DE hv

9
Quantized Energy Levels in Atoms and Molecules

• Atoms and molecules can possess electronic
  energy. Molecules can also have vibrational and
  rotational energy.
• These energy states are quantized, i.e. can take
  on only certain discreet values.
• This forms the basis of spectroscopy. Molecules
  can change between one energy level and another
  only by absorbing (as in absorption spectroscopy)
  or emitting (emission spectroscopy) radiation of
  frequency
• v DE/h (E2 - E1)/h

E2
E1
Absorption Emission
10
Molecular Energy Levels

• The total energy of a molecule can be written
• E Eelectronic Evibrational
  Erotational
• The classification of molecular transitions leads
  naturally to a division of spectroscopy
  experiments into three groups
• Electronic spectroscopy (visible, UV)
• Vibration-rotation spectroscopy (infrared)
• Pure rotational spectroscopy (microwave)

11
Regions of the EM Spectrum
12
Pure Rotational Spectroscopy(Microwave
Spectroscopy)

• Rotational transitions are induced by an
  interaction between the incident radiation and an
  oscillating component of a rotating electric
  dipole.
• Hence a molecule must have a permanent dipole
  moment for it to show a pure rotational spectrum.
• Rotation about the centre of mass then produces
  the required oscillating component of the dipole.
• Rotational transitions are detected at
  wavenumbers in the range 0.1 - 100 cm-1 (i.e.
  microwave and far infrared).

13
Rotational Motion of Molecules
Consider an atom of mass m rotating at a
distance r from the axis of rotation
Angular velocity dq/dt w rad s-1 Linear
velocity v r.dq/dt rw Kinetic energy
½mv 2 ½mr2w2 If a molecule has several
atoms at different distances from the axis (all
with the same w of course), then the total
rotational energy of the molecule is
This gives us a definition of the moment of
inertia of the molecule mi mass of atom i ri
perpendicular distance from axis of rotation
14
Moments of Inertia of Various Molecules
Molecular shapes are classified by reference to
relative values of moments of inertia about
three axes at right angles
Spherical Tops IA IB IC
e.g. CH4 - non-polar therefore no microwave
absorption Linear Molecules IA 0 IB
IC e.g. CO2 Symmetrical Tops IA ? IB
IC e.g. CH3Cl Asymmetric Tops IA ?
IB ? IC e.g. H2O - have
very complex spectra
15
Diatomic Molecules - The Rigid Rotor Model
Only heteronuclear diatomics give a pure
rotational spectrum - i.e. those with a permanent
dipole The rigid rotor model for a diatomic
molecule assumes point masses m1 and m2 separated
by a fixed distance r, i.e. we ignore vibration
and bond stretching due to centrifugal forces.
The moment of inertia is given by where m is
known as the reduced mass
16
Diatomic Molecules - Allowed Energies
Solution of the Schrödinger equation for a rigid
rotor gives the following allowed energies
where J is the rotational quantum number which
can have values of 0, 1, 2, 3, etc. If we express
energies in wavenumbers then the allowed
energies are
where is the rotational constant of the
molecule
Erwin Schrödinger 1887-1961
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Diatomic Molecules - Absorption Spectrum
The selection rule for changes in rotational
energy for a diatomic molecule is
Thus for the absorption spectrum we can calculate
the position of the rotational lines by
considering the energy change when J becomes J1
By putting all the values of J into this equation
in turn, we can predict that the absorption
spectrum is a sequence of lines at wavenumbers
In other words, the pure rotational spectrum of a
diatomic molecule is predicted to be a series of
equally spaced lines in which the spacing is
18
Rotational Spectrum of CO
19
Diatomics - Calculation of Internuclear Distance
The line spacing in the microwave spectrum for
12C16O is 3.84 cm-1
Thus, the line spacing from the microwave
rotational spectrum of a diatomic molecule
enables us to determine the bond length of the
molecule with good precision.
20
Diatomics - Effect of Isotopes
No special experiments are needed. Natural
abundance of isotope is sufficient to allow
measurement. Isotopic substitution leaves
electronic binding, and therefore inter-nuclear
distance, r unchanged. But, change in mass ?
change in I
Additional lines with these spacings can be seen
in the spectrum for CO.
The top axis shows the expected positions of the
peaks for 13C16O and these are clearly seen. 13C
occurs as 1.1 of natural carbon.
21
Linear Polyatomic Molecules
Energy levels and selection rules are the same as
for diatomics. A dipole moment is required, hence
HCN and OCS have a microwave rotational spectrum
but not CO2 which is symmetrical- OCO. From
the line spacing we can determine the moment of
inertia I. The difficulty is to use this to
calculate bond lengths since there are two. The
answer is to use more than one isotopic form
(present naturally). If assume rA and rB are
not altered by isotopic substitution, can solve
simultaneous equations for these bond lengths.
E.g. from data for16O -12C -34S and 16O -12C
-32S we get the following bond lengths
m1
m3
m2
rA
rB
(34S is present at 4 in natural sulfur)
22
Stark Effect - behaviour in electric field
Application of electric field causes splitting of
spectral lines
Johannes Stark 1874-1957
J 1?2 rotational line for OCS
Splitting is proportional to ?2m2 where ?
field, m dipole moment. Hence method for
accurate determination of dipole moments.
23
Molecules in Outer Space
The construction of radio-telescopes with more
accurately parabolic dishes has extended
interstellar observations into the microwave
region.
Some Interstellar Molecules
The studies, usually in emission, are possible
because of activation of molecules by the
background radiation in the universe, equivalent
to a black body at a temperature of 2.7
K. Rotational lines have allowed detection of
many molecules in the dark clouds of dust and gas
within nebulae.
Many of these molecules would be unstable at
atmospheric pressure. Symmetrical molecules such
as N2, O2, H2 and HCCH are probably present
but are not detected (no dipole moment)
24
Molecules in Outer Space
Rotational spectrum of the Orion Nebula showing
molecules in the interstellar cloud
25
Molecules in Earths Atmosphere
Emission spectrum of atmosphere
Most lines are due to rotational transitions of
O3 Transitions J 1?0 for HF and J 2?1 for
H35Cl are also visible.
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Destruction of the Ozone Layer
The radical ClO acts as a catalyst in the
decomposition of ozone Cl O3 ? ClO
O2 ClO O ? Cl O2 High
concentrations of ClO can therefore be expected
at altitudes of maximum ozone depletion.
Computer analysis splits into two peaks which are
interpreted in terms of pressure broadening
Rotational line of ClO observed at ground level.
Conclusion ClO is mainly located at the inner
and outer edges of the stratosphere.
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Rotational (Microwave) Spectroscopy - Summary

• Only polar molecules absorb or emit.
• Very precise measurements allow calculation of
  moment of inertia I.
• For diatomics can calculate precise bond length.
• For triatomics up to four unknown bond lengths or
  angles can be calculated.
• Dipole moments accurately measured (Stark
  effect).
• Interstellar gases detected.

28
Vibrational (Infrared) Spectroscopy
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Vibrational (Infrared) Spectroscopy

• A commonly used laboratory technique (unlike
  microwave spectroscopy).
• Absorption of infrared energy increases the
  vibrational energy of molecules.
• Standard IR spectrometers cover the range of
  wavenumbers 400 - 4000 cm-1.
• Hence a typical wavelength is 10 mm which
  corresponds to DE 12 kJ mol-1 (compare this
  with bond dissociation energies which are in the
  range 100 - 1000 kJ mol-1).

30
A Filter-Grating IR Spectrometer
Wavelength is swept using a combination of
diffraction gratings (G1 and G2) and
filters. Absorbance is measured by moving the
attenuator until a null AC signal is found at the
detector.
Glass absorbs IR strongly so mirrors must be
front-silvered and sample cells made from
NaCl (transparent above 650 cm-1) KBr
(transparent above 400 cm-1)
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Diatomic Molecules - The Harmonic Oscillator
Imagine two masses, m1 and m2 joined by an open
spring (of length r) that obeys Hookes law. If
it is stretched or compressed, the restoring
force ( f ) is proportional to the extension, i.e.
re is the equilibrium length, k is the force
constant. The instantaneous potential energy (E)
in such a model is found by integrating this
force with respect to distance
Classical mechanics predicts the system would
oscillate with frequency
32
The Harmonic Oscillator - Allowed Energies
The quantum theory does not allow continuous
variation of energy as depicted on the previous
slide. Solution of the Schrödinger equation with
the parabolic energy/distance dependence of the
harmonic oscillator gives the allowed energies
where v 0, 1, 2, is the vibrational quantum
number.
ve is the vibrational frequency about the
equilibrium bond length re
or in wavenumbers
If the allowed energies are expressed in
wavenumbers (e in units of cm-1) we get
zero-pointenergy
i.e. equally spaced energy levels are predicted.
33
The Harmonic Oscillator - Selection Rules

• Gross Selection Rules
• Electric dipole moment of molecule must change as
  atoms are displaced during vibration
• For diatomics this means the molecules must have
  a permanent dipole moment
• hence only heteronuclear diatomics can absorb IR
  radiation to give a vibrational spectrum
• Specific Selection Rule
• Hence a single line in the spectrum is predicted
  since
• The observed spectrum for e.g. CO is not a single
  line but has at least two lines and each one
  appears double in low resolution

The double peaks are the result of rotational
fine structure (see later). The additional peak
at higher wavenumber is due to the failure of the
simple harmonic model
Vibrational spectrum for CO
34
Diatomic Molecules - The Anharmonic Oscillator

• The potential energy curve for real molecules
  differs from the parabola for a harmonic
  oscillator in two ways
• It is asymmetrical, the compression being steeper
  than the extension.
• At sufficiently high r the curve becomes
  horizontal due to the dissociation of the
  molecule into separate atoms.

A more realistic expression for the variation of
PE with bond length is the Morse potential
De is the dissociation energy measured from the
minimum in the curve. a is a measure of the bond
stiffness and is related to the force constant k.
35
The Anharmonic Oscillator - Allowed Energies
Solution of the Schrödinger equation using of the
Morse potential gives the allowed energies for a
vibrating diatomic molecule
where x is the anharmonicity factor
(typically about 0.01) and is the
wavenumber of the hypothetical harmonic
oscillator having the same force constant as the
Morse oscillator. There are two important
consequences of such energy levels

• Vibrational levels are not equally spaced but
  close up with increasing v. The spacing becomes
  0 at the dissociation limit.
• The selection rule Dv 1 is not strictly
  obeyed and one or two overtones appear with Dv
  2 and Dv 3

36
Calculation of anharmonicity constant x and
equilibrium oscillation wavenumber
For the fundamental absorption band v
0 ? v 1
For the 1st overtone absorption band v
0 ? v 2
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Calculation of anharmonicity constant x and
equilibrium oscillation wavenumber
For CO, fundamental absorption is at 2143 cm-1
1st overtone is at 4260 cm-1 So we can write
two simultaneous equations with two unknowns
(1)
(2)
Multiplying equation (1) by 3 gives
(3)
Subtracting equation (2) from equation (3) gives
Substituting this in equation (1) then gives
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Calculation of the force constant k
Since refers to the hypothetical harmonic
oscillator with the same force constant, we can
write
Re-arranging gives
Using our example of CO for which 2169
cm-1, c 3.0 x 1010 cm s-1
k is a measure of bond stiffness
So for CO
39
Diatomic Molecules - Dissociation Energy
At the dissociation limit, the difference between
successive energy levels becomes zero.
However, the lowest energy is the vibrational
ground-state (v 0). Measured from this, the
dissociation energy is
Therefore
The above refers to a single molecule and is
expressed in wavenumbers. Converting to joules
per mole we have
Substituting this into the first equation gives
the dissociation energy
Hence we can estimate bond energies from IR
spectroscopic observations.
i.e.
40
Diatomic Molecules - Dissociation Energy
Calculated from x and
More accurate values from the UV spectrum(see
later)
41
Diatomic Molecules - Rotational Fine Structure
High resolution plot of the fundamental
vibrational line at 2143 cm-1 for CO. The closely
spaced fine lines are due to simultaneous changes
in the rotational energy levels.
Missing Q branch
P branch
R branch
Absorbance
42
Diatomic Molecules - Rotational Fine Structure
Simplest treatment ignores anharmonicity and
centrifugal distortion
P branch
R branch
m 1, 2, 3,
where is the rotational constant of the
molecule
There are two series of bands with peaks at
approximately 2 spacing. Can then calculate
moment of inertia ( I ) and hence bond length ( r
) - in the same way as for pure rotational
(microwave) spectra - see slide 19.
Selection rules Dv 1 DJ 1 or
1
P branch R branch
There is no Q branch since DJ 0 is not allowed
for diatomic molecules
43
Diatomic Molecules - Rotational Fine Structure
D
D
J -1
J 1
J 5
J 4
v 1
J 3
J 2
J 1
Pictorial representation as an energy diagram
J 0
J 5
J 4
v 0
J 3
J 2
J 1
J 0
Corresponding spectrum
-1
Spectrum
cm
P branch
R branch
Q branch
R branch
44
Vibrations of Polyatomic Molecules
For diatomic molecules, vibration can occur in
only one way extension and compression of the
bond. For polyatomics with several bonds,
distortion can take place in several ways -
changes in lengths of bonds - changes in angles
between bonds. Consider a model for SO2 analogous
to that for diatomics - i.e. point masses and
ideal springs
The overall motion is then complex but can be
resolved into a superposition of simpler types of
motion the normal modes of vibration. In each
normal mode
- all atoms move in phase with the same
frequency - the centre of mass is fixed -
the motion is approximately simple harmonic
motion - the vibration energy is quantized.
Unlike diatomics, polyatomics vibrate at several
frequencies, approximately those of the normal
modes.
To resist changes in bond angle we need a third
spring
45
Polyatomic Molecules - Number of Normal Modes
Consider a molecule containing N atoms. Each atom
may move along any of three perpendicular axes.
Therefore the total number of such displacements
is 3N. Three of these displacements correspond to
movement of the centre of mass of the molecule,
i.e. to translation as a whole. The remaining 3N
3 displacements are internal modes. In
addition, three angles are needed to specify the
orientation of a non-linear molecule in space.
Therefore three of the internal displacements
leave all bond angles and lengths unchanged but
change the orientation of the molecule as a
whole. That leaves 3N 6 displacements to
represent the vibrational modes. A similar
calculation for a linear molecule, which requires
only two angles to specify its orientation in
space, gives 3N 5 vibrational
modes. Examples H2O (non-linear) has 3x3 6
3 normal modes, NH3 (non-linear) has 3x4 6
6 normal modes, CO2 (linear) has 3x3
5 4 normal modes. Each normal mode can be
thought of as an independent harmonic oscillator
with its own fundamental frequency.
46
Polyatomic Molecules - I.R. Active Vibrations
If the dipole moment of a molecule changes due to
a particular mode of vibration, absorption of IR
radiation due to that vibration can occur. Such
vibrations are said to be infra-red
active. Fundamental bands occur at the
frequencies of the normal modes. However,
because real molecules are somewhat anharmonic,
other weaker bands occur Overtones - when one
vibrational quantum number changes by more than
1. e.g. for a molecule with three fundamental
modes with frequencies v1, v2 and v3 with
vibrational quantum numbers v1, v2 and
v3 Combination bands when more than one
quantum number changes together
47
Non-linear Triatomic - e.g. H2O
Number of normal modes 3 x 3 6 3
Symmetrical stretch Angle bend
Asymmetric stretch
48
Linear Triatomic - e.g. N2O
N-N-O 1285 cm1 N-N-O 589
cm1 N-N-O N-N-O 2224 cm1
(degenerate)
?
?
?
All the observed bands can be approximately
accounted for as multiples or combinations of
these three fundamental bands.
49
IR Spectra of Complex Molecules

• Large molecules
• Complex spectrum
• Many overtones and combination bands
• Complete assignment impossible
• Technique
• Liquid or solid phase
• Hence no rotational structure
• Two types of vibration
• Group vibrations
• Skeletal vibrations

• Group Vibrations
• Involve small number of atoms
• Rest of molecule considered undisturbed
• Frequencies relatively unaffected by structure of
  rest of molecule
• Exact frequency depends on environment, hence
  additional structural information

50
Infrared Spectrum of a Complex Molecule
51
Characteristic IR Absorption Frequencies
52
Skeletal Vibrations - The Fingerprint Region

• Involve concerted motion of large numbers of
  atoms
• Bending and stretching of bonds
• Complex patterns so cannot de-convolute
• But are characteristic of certain skeletal types
• Useful fingerprint region 700 - 1400 cm-1

53
IR Spectrum of Phenol
OH stretch
Fingerprint Region
54
Vibrational (IR) Spectroscopy - Summary

• Detects changes in the vibrational energy of
  molecules
• Diatomic molecules can be treated as anharmonic
  oscillators which allows calculation of
• bond force constant
• bond dissociation energy
• Line-spacing in rotational fine structure allows
  calculation of the bond length
• Triatomic molecules show overtone and combination
  bands
• Polyatomic molecules with N atoms vibrate in 3N-5
  (for linear) or 3N-6 (for non-linear) normal
  modes
• Complex molecules show characteristic group
  frequencies and a fingerprint region