This Physical Chemistry lecture uses graphs from the
following textbooks:
P.W. Atkins, Physical Chemistry, 7. ed., Oxford University
Press, Oxford 2003
G. Wedler, Lehrbuch der Physikalischen Chemie, 4. ed.,
Wiley-VCH, Weinheim 1997
Some Spectroscopic Techniques in Chemical Industry
Infrared (IR) Spectroscopy:
molecular rotation and vibration
(chemical identity, structure, and concentration)
Near Infrared (NIR) Spectroscopy:
molecular vibration
(chemical identity and concentration)
Raman Spectroscopy (RS):
molecular rotation and vibration
(chemical identity and structure)
Ultraviolet and Visible (UV/VIS) Spectroscopy:
(Atomic Absorption Spectroscopy (AAS))
electronic transitions in atoms and molecules
(chemical identity and concentration)
Nuclear Magnetic Resonance (NMR) Spectroscopy: nuclear orientation
(chemical identity and structure)
X-Ray Photoelectron Spectroscopy (XPS):
determination of surface composition
(surface stoichiometry and concentration)
Mass Spectrometry (MS):
determination of atomic and molecular masses
(chemical identity and structure)
Spectroscopy:
The analysis of the electromagnetic radiation
emitted, absorbed, or scattered by atoms or
molecules
The electromagnetic spectrum and the classification of the spectral regions. The band at the bottom indicates the
types of transitions that absorb or emit in the various regions (‘nuclear magnetism’ refers to the types of transition
used in NMR spectroscopy, ‘nuclear’ to transitions within the nucleus).
The origin of quantum mechanics
Spectroscopy is not understandable on the basis of classical mechanics, which
(1) predicts a precise trajectory for particles, with precisely specified locations and
momenta at each instant, and
(2) allows the translational, rotational, and vibrational modes of motion to be excited to
any energy simply by controlling the forces that are applied.
These conclusions agree with everyday experience. Everyday experience, however,
does not extend to individual atoms. In particular, experiments have shown that
systems can take up energy only in discrete amounts.
This observation is known as the
failure of classical physics
which means that classical mechanics fails when applied to the transfers of very small
quantities of energy and to objects of very small mass.
Most prominent examples are e.g. the black-body radiation, atomic and molecular
spectra, the photoelectric effect, and the diffraction of particles.
Black-body radiation
Empirical laws:
(1) Position of the maximum:
Tmax = const. = 0.288 Kcm
(Wien displacement law)
e.g. the sun:
T6000 K  max480 nm
An experimental representation of a
black body is a pinhole in an otherwise
closed container. The radiation is
reflected many times within the container
and comes to thermal equilibrium with
the walls at a temperature T. Radiation
leaking out through the pinhole is
characteristic of the radiation within the
container, and only a fundtion of
temperature.
(2) Exitance, i.e. the power emitted
by a region of surface divided
by the area of the surface:
M = T4
 = 5,6710-8 Wm-2K-4
(Stefan-Boltzmann law),
„T4 law“
The energy distribution in a black-body cavity at several temperatures.
Note how the energy density increases in the visible region as the
temperature is raised, and how the peak shifts to shorter wavelength. The
total energy density (the area under the curve) increases as the
temperature is increased (as T4).
Problem: Energy density distribution
with:
 
according to Maxwell‘s electro-dynamic theory:
E  d 

2
c
2
U  d
and :
frequency
E(): energy density distribution
U:
oscillator strength of the excited
electrons
c:
speed of light
classical treatment :
this implies that all frequencies should be
uniformly excited!

E  d 
2
c
2
 k  T  d

,
d 
c

2
d
follows

E  d 
c

4
 k  T  d
Rayleigh-Jeans law

agrees with
experiment only
for large 

theory predicts
E() for
 0

„classical
ultraviolet
catastrophe“
U = kT
(reason: 2 quadratic degrees of freedom for each
vibration)
c
The Planck distribution
The problem was solved by Max Planck. He could account for the observed
distribution of energy if he supposed that the permitted energies of an electromagnetic
oscillator of frequency  are integer multiples of h:
n = 0, 1, 2, …
E = n·h·
where h is a fundamental constant known as Planck’s constant (h=6.62608·10-34 J·s).
After introduction of a „mean oscillator energy“
U 
h 
e h / kT 1
a

  
b

a:
energy of oscillation with frequency 
1/b: probability for excitation of this particular oscillation
the Planck distribution could be derived:
E  d 
E  d 
2

c e

h 3
h / kT
hc 2
d
a) fits experimental curve very well
d
b) for h  0 the Planck distribution would
approach the Rayleigh-Jeans law

1

5 e hc / kT 1
Atomic and molecular spectra
The most compelling evidence for the quantization of energy comes from the observation of
the frequencies of radiation absorbed and emitted by atoms and molecules.
A region of the spectrum of radiation
emitted by excited iron atoms consists
of radiation at a series of discrete
wavelength (or frequencies).
When a molecule changes its state, it
does so by absorbing or emitting
radiation at definite frequencies. This
spectrum is part of that due to the
electronic, vibrational and rotational
excitation of sulphur dioxide (SO2)
molecules. This observation suggests
that molecules can possess only
discrete energies, not arbitrary energy.
Spectral lines can be accounted for if
we assume that a molecule emits a
photon as it changes between
discrete energy levels. Note that highfrequency radiation is emitted when
the energy change is large.
Wave-particle duality
The particle character of electromagnetic radiation
The observation that electromagnetic radiation
of frequency  can possess only the energies
0, h, 2h, … suggests that it can be thought of
as consisting of 0, 1, 2, … particles, each
particle having the energy h. These particles
of electromagnetic radiation are now called
photons. The observation of discrete spectra
from atoms and molecules can be pictured as
the atom or molecule generating a photon of
energy h when it discards an energy of
magnitude E, with E = h.
Further evidence for the particle-like character
of radiation comes from the measurement of
the energies of electrons produced in the
photoelectric effect. This effect is the ejection
of electrons from metals when they are
exposed to ultraviolet radiation. The
experimental characteristics of the
In the photoelectric effect, it is found that no electrons are
ejected when the incident radiation has a frequency below a
photoelectric effect are summarized on the next
value characteristic of the metal, and above that value, the
transparency.
kinetic energy of the photoelectrons varies linearly with the
frequency of the incident radiation.
Wave-particle duality
The particle character of electromagnetic radiation (cont’d)
1. No electrons are ejected, regardless of
the intensity of the radiation, unless the
frequency exceeds a threshold value
characteristic of the metal.
2. The kinetic energy of the ejected
electrons increases linearly with the
frequency of the incident radiation but is
independent of the intensity of the
incident radiation.
3. Even at low light intensities, electrons are
ejected immediately if the frequency is
above threshold.
These observations suggest an ejection of
the electron after collision with a particle-like
projectile that carries enough energy to eject
the electron from the metal. If we suppose
that the projectile is a photon of energy h,
then the conservation of energy requires that
the kinetic energy of the ejected electron
should obey
½ mev2 = h - 
In this expression  is a characteristic of the
metal called its work function, the energy
required to remove the electron from the
metal to infinity (Einstein, 1905).
The photoelectric effect can be explained if it is supposed that the
incident radiation is composed of photons that have energy proportional
to the frequency of the radiation. (a) The energy of the photon is
insufficient to drive an electron out of the metal. (b) The energy of the
photon is more than enough to eject an electron, and the excess energy
is carried away as the kinetic energy of the photoelectron (the ejected
electron
Wave-particle duality
The wave character of particles
fringe field electrode
In 1927, Davisson and Germer observed
diffraction of electrons by a crystal of nickel,
which acted as a diffraction grating.
Diffraction is a characteristic property of
waves because it occurs when there is
interference between their peaks and troughs.
Depending on whether the interference is
constructive or destructive, the result is a
region of enhanced or diminished intensity.
3 grids
sreen
2 MCPs
electron gun
single crystal
x 100 ... 10000
310 eV
Top: Exploded viwe of a modern low-energy electron
diffraction (LEED) apparatus and diffraction pattern
from CaF2(111).
Left: The Davisson-Germer experiment. The scattering of an electron
beam from a nickel crystal shows a variation of intensity characteristic of
a diffraction experiment in which waves interfere constructively and
destructively in different directions.
Wave-particle duality
The wave character of particles (cont’d)
Already in 1924 the French physicist Louis de
Broglie had suggested that any particle, not only
photons, travelling with a linear momentum p
should have a wavelength given by the de Broglie
relation:
h
 
p
That is, a particle with a high linear momentum
has a short wavelength (see figure). Macroscopic
bodies have such high momenta (even if they are
moving slowly) that their wavelength are
undetectably small, and the wave-like properties
cannot be observed.
Examples:
Electron, kinetic energy 100 eV:  = 1.22·10-10 m
Neutron, kinetic energy 300 K:  = 1.78·10-10 m
Man, m=75 kg, v=1 m·s-1:
 = 8.83·10-36 m
An illustration of the de Broglie relation between
momentum and wavelength. A wave is
associated with a particle (later this will be seen
to be the wavefunction of the particle). A particle
with high momentum has a short wavelength,
and vice versa.
Atomic structure and atomic spectra: First attempts
The spectrum of atomic hydrogen
Already in 1855 the Swiss schoolteacher Johann
Balmer pointed out that (in modern terms) the
wavenumbers of the emission lines which were
observed in the visible region when an electric
discharge is passed through gaseous hydrogen fit the
expression
 
1
22

1
n2
The lines this formula describes are now called the
Balmer series. When further lines were discovered in
the ultraviolet (Lyman series) and in the infrared
(Paschen series), the Swedish spectroscopist
Johannes Rydberg noted (in 1890) that all of them
were described by the expression
 1
1
  RH  2  2 
n
n2 
 1
RH  109 677 cm1
with n1=1 (the Lyman series), 2 (the Balmer series),
and 3 (the Paschen series), and that in each case
n2=n1+1, n1+2, … . The constant RH is now called the
Rydberg constant for the hydrogen atom.
The spectrum of atomic hydrogen. Both the observed
spectrum and its resolution into overlapping series are
shown. Note that the Balmer series lies in the visible
region.
Atomic structure and atomic spectra: First attempts
The spectrum of atomic hydrogen
The Rydberg formula strongly suggests that the
wavenumber of each spectral line can be written as
the difference of two terms, each of the form
Tn 
RH
n2
The Ritz combination principle states that the
wavenumber of any spectral line is the difference
between two terms:
  T1  T2
It is readily explained in terms of photons and the
conservation of energy. Thus, a spectroscopic line
arises from the transition of an atom from one energy
level (a term) to another (another term) with the
emission of the difference in energy as a photon (see
figure). This interpretation leads to the Bohr
frequency condition, which states that, when an
atom changes its energy by E, the difference is
carried away as a photon of frequency , where
E  h
Energy is conserved when a photon is emitted, so the
difference in energy of the atom before and after the
emission must be equal to the energy of the photon
emitted.
Atomic structure and atomic spectra: First attempts
The Bohr model of atomic hydrogen
• One of the most famous of the obsolete theories of the hydrogen atom, proposed by the
Swedish scientist Niels Bohr in 1913.
• Idea: The electron surrounds the nucleus like planets the sun.
• The Coulombic force of attraction (centripetal force) is balanced by the centrifugal
effect of the orbital motion.
• In addition to the frequency condition, Bohr introduced two postulates:
a) The electron does not radiate.
b) The angular momentum is limited to integral values of ħ (h/2): L=nħ.
• Under these conditions the energy and the radius of the electron’s orbit around the
nucleus can be calculated, and are found to be a function of the quantum number n:
rn 
0  h2  n2
  me  e
2
En  
me  e4
8  0  h  n
2
2
n  1,2,3,...
Atomic structure and atomic spectra: First attempts
The Bohr model of atomic hydrogen (cont’d)
• Result: discrete orbits and energy levels!!!
• Radius for n=1: r1=5.292·10-11 m  atomic length unit, Bohr radius a0.
• Comparison to Rydberg’s formula:
En  
EA
2
n
EA 
me  e 4
8  0  h
2

EA
RH 
hc
 excellent agreement between model and experiment!
• Advantage of Bohr’s model:
- very simple and intuitive description
- quantitative agreement
• Disadvantages of Bohr’s model:
- only good for one-electron systems (H, He+, Li2+, …)
- wrong assumptions (no well-defined orbits, … )
- agreement to experiment due to coincidence!
 quantum mechanical description required for exact treatment!
Quantum Theory: An Introduction
The Schrödinger equation
• Quantum mechanics acknowledges the wave-particle duality of matter by supposing
that, rather than travelling a definite path, a particle is distributed through space like a
wave. The mathematical representation of the wave that in quantum mechanics
replaces the classical concept of trajectory is called a wavefunction,  (psi).
• In 1926, the Austrian physicist Erwin Schrödinger proposed an equation for finding the
wavefunction of any system. The time-independent Schrödinger equation for a
particle of mass m moving in one dimension with energy E is

2
d2
2m dx
2
 V(x)    E  
The factor V(x) is the potential energy of the particle at the point x, and ħ (which is read
h-cross or h-bar) a modification of Planck’s constant: ħ = h/2.
• The Schrödinger equation should be regarded as a postulate, like Newton’s
equations of motion. However, it is (at least partially) possible to justify it.
• It can be shown that the term with the second derivative of the wavefunction with
respect to the coordinate in space corresponds to the classical kinetic energy, the
product V(x)· to the potential energy, and E· therefore to the total energy.
Quantum Theory: An Introduction
The Schrödinger equation (cont’d)
• For a three-dimensional system

2
2m
2  V(x,y,z)    E  
where 2 (‘del squared’ or ‘nabla squared’) is
 
2
2
x
2

2
y
2

2
z2
• In systems with spherical symmetry (in polar coordinates)
 
2
2
r 2
2 
1

 2 2
r r
r
and

2

1
2
sin2  2

1 

sin 
sin  

• In the general case the Schrödinger equation is written
H  E  
with
H  
2
2  V
2m
where H is the hamiltonian operator for the system (a Hamiltonian is a function which
expresses the energy in terms of its momentum and positional coordinates).
Quantum Theory: An Introduction
Born’s interpretation of the wavefunction
• In principle, the wavefunction contains all the dynamical information about the
system it describes. We will concentrate on the location of the particle.
• The Born interpretation focuses on the square of the wavefunction (or the square
modulus, ||2=*, if  is complex). For a one-dimensional system:
If the wavefunction of a particle has the value  at some point x, then the
probability of finding the particle between x and x+dx is proportional to ||2dx.
Thus, ||2 is the probability density, and to obtain the probability it must be multiplied by
the length of the infinitesimal region dx. The wavefunction  itself is often called the
probability amplitude.
• For a particle free to move in three dimensions (for example, an electron near a
nucleus in an atom), the wavefunction depends on the point dr with coordinates x, y and
z, and the interpretation of (r) is as follows:
If the wavefunction of a particle has the value  at some point r, then the
probability of finding the particle in an infinitesimal volume d = dx dy dz at that
point is proportional to ||2d.
Quantum Theory: An Introduction
Born’s interpretation of the wavefunction (cont’d)
• Thus, there is no direct significance in the negative (or complex) value of a
wavefunction: only the square modulus, a positive quantity, is directly physically
significant, and both negative and positive regions of a wavefunction may correspond to
a high probability of finding a particle in a region *.
The wavefunction  is a probability
amplitude in the sense that its
square modulus (* or ||2) is a
probability density. The probability of
finding a particle in the region dx
located at x is proportional to ||2dx.
The Born interpretation of the
wavefunction in three-dimensional
space implies that the probability of
finding the particle in the volume
element d = dx dy dz at some
location r is proportional to the
product of d and the value of ||2 at
that location.
The sign of a wavefunction has no
direct physical significance: the
positive and negative regions of this
wavefunction both correspond to the
same probability distribution (as
given by the square modulus of 
and dpicted by the density of
shafing).
* Later we shall see that the presence of positive and negative regions of a wavefunction is of great indirect interest, because
it gives rise to the possibility of constructive and destructive interference between different wavefunctions.
Quantum Theory: An Introduction
Heisenberg’s uncertainty principle
• It is impossible to specify simultaneously, with arbitrary precision, both the
momentum and the position of a particles (Werner Heisenberg, 1927).
Left: The wavefunction for a particle at a welldefined location is a sharply spiked function that
has zero amplitude everywhere except at the
particles position.
Right: The wavefunction for a particle with an illdefined location can be regarded as the
superposition of several wavefunctions that
interfere constructively in one place but
destructively elsewhere. As more waves are used
in the superposition (as given by the numbers
attached to the curves), the location becomes
more precise at the expense of uncertainty in the
particles momentum. An infinite number of
wavefunctions is needed to construct the
wavefunction of a perfectly localized particle.
• A quantitative version of this result is
p  q 
1
2
In this expression p is the ‘uncertainty’ in the linear momentum parallel to the
axis q, and q is the uncertainty in position along that axis.
Quantum Theory: An Introduction
Heisenberg’s uncertainty principle (cont’d)
• The uncertainties are precisely defined, for they are the root mean square
deviations of the properties from their mean values:
p 
p
2
 p
2

1/ 2
q 
q
2
 q
2

1/ 2
If there is complete certainty about the position of the particle (q=0), the only
way to satisfy the above equation is for p=, which implies complete uncertainty
about the momentum.
• The Heisenberg uncertainty principle is more general than the above equation
suggests. It implies to any pair of observables called complementary observables,
which are defined in terms of the properties of their operators.
Using Quantum Theory: Techniques and Applications
One-Dimensional Translational Motion
A particle in a box
Assume a box in which a particle is confined
between two walls at x=0 and x=L: The potential
energy is zero inside the box but rises to infinity at
the walls.
This model is an idealization of the potential energy
of a gas-phase molecule that is free to move in a
one-dimensional container. However, it is also the
basis of the treatment of the electronic structure of
metals and a primitive treatment of conjugated
molecules (like e.g. CH2=CH-CH=CH2).
The Schrödinger equation between the walls is the
same as for a free particle:

2
d2
 E
2m dx 2
The solution of this 2nd order differential equation is
simply:
k (x)  A sin(kx)  Bcos(kx)
k2 2
Ek 
2m
A particle in a one-dimensional region with
impenetrable walls. Its potential energy is zero
between x=0 and x=L, and rises abruptly to
infinity as soon as it touches the walls.
Since eikx=cos(x)±i·sin(x) this is
equivalent to k(x) = C·eikx + D·e-ikx
Using Quantum Theory: Techniques and Applications
One-Dimensional Translational Motion (cont’d)
A particle in a box: Boundary conditions
 For a free particle, any value of Ek is acceptable.
 When the particle is confined within a region, the acceptable wavefunctions must
satisfy certain boundary conditions, or constraints on the function at certain locations:
 It is physically impossible to find the particle with an infinite potential energy, so the
wavefunction must be zero where V is infinite, at x<0 and x>L, and
 the continuity of the wavefunction requires it to vanish just inside the well at x=0 and x=L,
so k(0)=0 and k(L)=0.
 For the wavefunction k(x) = A·sin(k·x) + B·cos(k·x) this implies that B=0, and kL=n,
with n=1,2,3,…: k(x) = A·sin(n··x/L)
 The integral of 2 over all the space available to the particle, i.e. the probability to find it
somewhere, is equal to 1. Therefore normalization yields the complete solution to the
problem:
L
L
1/ 2
!
nx
2
 dx  A sin
dx  1  A   
L
L
0
0

2
2

2
Using Quantum Theory: Techniques and Applications
One-Dimensional Translational Motion (cont’d)
The first five normalized wavefunctions of a particle
in a box. Each wavefunction is a standing wave,
and succsessive functions possess one more half
wave and a correspondingly shorter wavelength.
1/ 2
The allowed energy levels for a
particle in a box. Note that the
energy levels increase as n2,
and that their spearation
increases as the quantum
number increases. Note that
E=0 is impossible.
2
n (x)   
L
 nx 
sin 

 L 
for 0  x  L
and
n2h2
En 
8mL
(a)The first two wavefunctions,
(b) the corresponding probability
distributions, and (c)
representation of the probability
distributions in terms of the
darkness of the shading.
Using Quantum Theory: Techniques and Applications
Two-Dimensional Translational Motion
• Consider a two-dimensional version of the
particle in a box. The particle is now confined
to a rectangular surface of length L1 in xdirection and length L2 in y-direction. The
energy is zero everywhere except at the walls
where it rises to infinity. The Schrödinger
equation becomes now a function of both x
and y:
 d2 d2 


 E

2 

2m  dx 2
dy 
2
• This problem can be solved by the
separation of variables technique, which
divides the equation into (in this case) two
ordinary differentials, one for each variable:
A two-dimensional square well. The particle is
confined to the plane bounded by impenetrable
walls. As soon as it touches the walls its potential
energy rises to infinity.
(x,y)  X(x)  Y(y)

2
d2 X
2m dx
2
 Ex  X

2
d2 Y
2m dy
E  Ex  Ey
2
 Ey  Y
Using Quantum Theory: Techniques and Applications
Two-Dimensional Translational Motion (cont’d)
Each of the two previous differential equations is the same as the on-dimensional square-well
Schrödinger equation. Thus, the result for the latter can be directly adapted:
n1,n2 (x,y) 
2
L1L2 
1/ 2
 n x 
 n x 
sin  1  sin  2 
 L1 
 L2 
0  x  L1
0  y  L2
 n2
n2  h2
En1,n2   12  22 
L
L2  8m
 1
The wavefunctions of a particle confined to a rectangular surface depicted as contours of equal amplitude.
(a) n1=1, n2=1, (b) n1=1, n2=2, (c) n1=2, n2=2, and (d) n1=2, n2=2.
Using Quantum Theory: Techniques and Applications
Degeneracy
 An interesting feature of the solutions for a particle in a two-dimensional box is obtained
when the plane wave is square, with L1=L2=L.
 In this case, e.g. the wavefunctions with n1=1, n2=2 and n1=2, n2=1 have the same energy:
1,2
2
 x 
 2x 

sin 
sin 


L
 L 
 L 
E1,2 
5h2
8mL2
2,1
2
 2x 
 x 

sin 
sin 


L
 L 
 L 
E2,1 
5h2
8mL2
 Apparently different wavefunctions are degenerate, which means they correspond to the
same energy although their quantum numbers are different.
The wavefunctions of a
particle confined to a
square surface. Note that
one wavefunction can be
converted into the other by
a rotation of the box by
90°. The two functions
correspond to the same
energy. Degeneracy and
symmetry are closely
related.
Using Quantum Theory: Techniques and Applications
Tunneling
 If the potential energy of a particle does not rise to infinity when it is in the walls of the
container, and E<V, the wavefunction does not decay abruptly to zero.
 If the walls are thin (so that V falls to zero after a finite distance), the wavefunction oscillates
inside the box, varies smoothly within the wall, and oscillates again outside the box.
 The conditions of continuity inside the box, within the wall, and outside of the box enable
us to obtain the solution of the Schrödinger equation.
A particle incident on a barrier from the left has an
oscillating wavefunction, but inside the barrier there are
no oscillations (for E<V). If the barrier is not too thick,
the wavefunction is nonzero at its opposite face, and so
oscillations begin there (only the real component of the
wavefunction is shown).
The wavefunctions of a particle confined to a square surface.
Note that one wavefunction can be converted into the other by
a rotation of the box by 90°. The two functions correspond to
the same energy. Degeneracy and symmetry are closely
related.
Using Quantum Theory: Techniques and Applications
Tunneling (cont’d)
 The transmission probability, T, of a particle to travel through the wall is given by:


2

L
L
e

e


T  1 

16

1








1
where ħ=(2m(V-E)1/2) and =E/V. For high, wide barriers, in the sense that L»1, this
simplifies to:
T  16 1    e2L
The transmission probabilities for passage
through a barrier. The horizontal axis is the
energy of the particle expressed as a multiple of
the barrier height. The curves are labelled with
the values of L(2mV)1/2/ħ. The graph on the left
is for E<V and that on the right for E>V. Note
that T>0 for E<V, whereas classically T would
be zero. However, T<1 for E>V whereas
classically T would be 1.
Using Quantum Theory: Techniques and Applications
Tunneling (cont’d) and Microscopy
The wavefunction of a heavy particle decays more
rapidly inside a barrier than that of a light particle.
Consequently, a light particle has a greater probability
of tunneling through the barrier.
The central component in a scanning tunneling microscope
(STM) is an atomically sharp needle (Pt or W) which is scanned
across the surface of a conducting solid. When the tip is brought
very close to the surface, electrons tunnel across the intervening
space. In the constant-current mode of operation, the stylus
moves up and down corresponding to the topography of the
surface, which, including and adsorbates, can be mapped on an
atomic scale.
Using Quantum Theory: Atomic Spectra and Structure
The Hydrogen Atom
 For the case of separation of internal from external motion and using the Coulomb
potential energy of the electron in a hydrogen atom it is now straightforward to write down
its Schrödinger equation, i.e. for an electron orbiting a nucleus (in this case a single proton
with Z=1):
2
2
Ze2
Ze2
2
2
H  
 
and 
 
  E
2
40r
2
40r
 Due to the huge difference in mass between the nucleus and the electron, the reduced
mass µ can - in excellent approximation be replaced by the mass of the electron: µme.
 Because the potential energy is centrosymmetric (independent of angle), one can suspect
that the equation is separable into radial and angular components.
(r, , )  R(r)  Y(, )
Spherical polar
coordinates. A particle on
the surface of a sphere of
radius r, can be described
by its colatitude, , and the
azimuth, .
Using Quantum Theory: Atomic Spectra and Structure
Atomic Orbitals, Energy and Quantum Numbers
 An atomic orbital is a one-electron wavefunction for an
electron in an atom.
 Acceptable solutions for the hydrogen atom’s wavefunction can
be found only for integral values of a quantum number n, and
the allowed energies are:
En 
e 4
32202 2n2
n is the principal quantum number which determines the
energy of the electron, and can take the values n = 1, 2, 3, … .
 The radial contribution R(r) depends on one more quantum
number, l. An electron in an orbital with quantum number l has
an angular momentum of {l(l+1)}1/2ħ, with l = 0, 1, 2, 3, … n-1.
 The angular contribution depends on l and ml. An electron in an
orbital with quantum number ml has a z-component of angular
momentum mlħ with ml = 0, ±1, ±2, …, ±l.
 Orbitals with quantum numbers l = 0, 1, 2, 3, … are usually
called s, p, d, f … .
The energy levels of a hydrogen
atom. The values are relative to an
infinitely separated, stationary
electron and a proton
Using Quantum Theory: Atomic Spectra and Structure
Radial Wavefunctions
The radial wavefunctions for
the first few states of
hydrogenic atoms (i.e.
atoms with one electron
only) of atomic number Z.
Note that the s orbitals have
a finite and nonzero value
at the nucleus. The
horizontal scales are
different in each case:
orbitals with high principal
quantum numbers are
relatively distant from the
nucleus. Remember that s 
l=0, p  l=1, d  l=2 …
Using Quantum Theory: Atomic Spectra and Structure
Angular Wavefunctions
A representation of
the angular
wavefunctions for
l = 0, 1, 2, and 3.
The distance of a
point on the
surface from the
origin is
proportional to the
square modulus of
the amplitude of
the wavefunction at
that point.
Top: The permitted
orientations of angular
momentum when l=2. This
representation is too specific
because the azimuthal
direction of the vector (its
angle around z) is
undeterminable.
Right: (a) A summary of
above figure. However,
because the azimuthal angle
around z is undeterminable, a
better representation is (b)
where each vector lies on its
cone.
Using Quantum Theory: Atomic Spectra and Structure
The Hydrogen Atom: Shells and Subshells
 All the orbitals of a given value of n are said to form a single shell.
In a hydrogenic atom all orbitals belonging to the same shell have
the same energy. It is common to revere to successive shells by
letters:
n = 1 2 3 4…
K L M N…
 The orbitals with the same value of n but different values of l are
said to form a subshell of a given shell. These subshells are
generally also referred to by letters:
l = 0 1 2 3 4 5 6…
s p d f
g h i…
Left: The energy levels of the hydrogen atom
showing the subshells and (in square
brackets) the numbers of orbitals in each
subshell. In hydrogenic atoms, all orbitals of
a given shell have the same energy (this is
not the case in systems with more than one
electron !).
Right: The organization of orbitals
(white squares) into subshells
(characterized by l) and shells
(characterized by n).
Using Quantum Theory: Atomic Spectra and Structure
The Hydrogen Atom’s Orbitals: s Orbitals
 The orbital occupied in the ground state is the one with n=1
(and therefore with l=0 and ml=0). Its wavefunction is:
 
1
 a 
3
0
1/ 2
e
 r / a0
(a0 
40
me e
2
2
 52.9177 pm; Bohr radius)
 This wavefunction is independent of angle and has the same
value at all points of constant radius; that is, the 1s orbital is
spherically symmetrical. It has the maximum value at r=0. It
follows that the most probable point where the electron will be
found is the nucleus itself!
 All s orbitals are spherically symmetrical, but differ in the
number of radial nodes (0 for 1s, 1 for 2s, 2 for 3s, …).
Left: Representation
of the (a) 1s and (b)
2s hydrogenic
orbitals in terms of
their electron
densities (as
represented by the
density of shading).
Top: The variation of the mean radius of
a hydrogenic atom with the principal and
orbital momentum quantum numbers.
Note that the mean radius lies in the order
d < p < s.
Using Quantum Theory: Atomic Spectra and Structure
The Hydrogen Atom’s Orbitals: Radial Distribution Functions
 The wavefunction tells us, through the
value of 2, the probability of
finding an electron in any region.
Imagine a probe with a volume d and
sensitive to electrons, which can be
moved around the hydrogen atom. The
reading of this detector is shown in the
figure to the right.
 Now consider the probability of finding
the electron anywhere on a spherical
shell of thickness dr at a radius r. The
sensitivity volume is now the volume of
the shell, which is 4r2dr. Thus the
probability to find the electron in a
distance r is P(r)dr= 4r22dr, the result
of which is shown in the lower figure.
 For orbitals that are not spherically
symmetrical, the more general
expression r2R(r)2dr has to be used,
where R(r) is the radial wavefunction of
the orbital in question.
A constant-volume
electron sensitive
detector (the small
cube) gives its greatest
reading at the nucleus,
and a smaller reading
elsewhere. The same
reading is obtained
anywhere on a circle of
given radius: the s
orbital is spherically
symmetrical.
The radial distribution function P
gives the probability that the
electron will be found anywhere
in a shell of radius r. For a 1s
electron in hydrogen, P is a
maximum when r is eaual to the
Bohr radius a0 (!). The value of P
is equal to the reading that a
detector shaped like a spherical
shell would give as its radius is
varied.
Using Quantum Theory: Atomic Spectra and Structure
The Hydrogen Atom’s Orbitals: p Orbitals
 A p electron has nonzero angular momentum, which has a profound effect on the shape of the
wavefunction close to the nucleus, for p orbitals have zero amplitude at r=0. This effect can be
classically understood in terms of the centrifugal effect of the orbital angular momentum, which
tends to fling the electrons away from the nucleus.
 Since the solutions of Schrödinger’s equation usually contain imaginary contributions for p, d, f…
orbitals, they are usually represented as purely real linear combinations of the latter, since each of
these is a solution of the Schrödinger equation, too.
 These linear combinations are standing waves with no net angular orbital momentum around the zaxis, as they are superpositions of states with equal and opposite values. The px orbital has the
same shape as a pz orbital, but is directed along the x-axis; the py-orbital is similarly directed
along the y-axis.
Top: The boundary surface of p orbitals. A nodal plane passes through the nucleus and
separates the two lobes of each orbital.
Left: Close to the nucleus, p orbitals are proportional to r, d orbitals are proportional to r2, and f
orbitals are proportional to r3. Electrons are progressively excluded from the neighbourhood as
l increases. An s orbital has a finite, nonzero value at the nucleus.
Using Quantum Theory: Atomic Spectra and Structure
The Hydrogen Atom’s Orbitals: d Orbitals
 When n=3, l can be 0, 1, or 2. As a result, this shell consists of one 3s orbital, three 3p orbitals, and
53d orbitals.
 The five d orbitals have ml= +2, +1, 0, -1, -2, and correspond to five different angular momenta
around the z-axis (but the same magnitude of angular momentum around the z-axis, because l=2
in each case).
 As for the case of p orbitals, d orbitals with opposite sign of ml (and hence opposite sign of motion
around the z-axis) may be combined in pairs to give standing waves, whose boundary surfaces are
shown below.
 These real combinations are shown here as an example:
dxy  xyf(r)
dx 2 y2 
dyz  yzf(r)
1 2
(x  y2 )f(r)
2
dzx  zxf(r)


1

dz2  
3  3z2  r 2 f(r)
2

The boundary surfaces of d orbitals. Two nodal planes in each
orbital at intersect at the nucleus and separate the lobes of each
orbital. The dark and light areas denote regions of opposite sign
of the wavefunction.
Spectroscopy
What’s behind it?
 All changes on atomic and molecular scale correspond to transitions between discrete energy
levels.
 These energy levels, and the transition between them, can be described via the SCHRÖDINGER
equation.
 SPECTROSCOPY: Determination of the transition frequencies with the goal to obtain information
with respect to chemical identity and/or atomic or molecular structure.
Some Spectroscopic Techniques in Chemical Industry
Infrared (IR) Spectroscopy:
molecular rotation and vibration
(chemical identity, structure, and concentration)
Near Infrared (NIR) Spectroscopy:
molecular vibration
(chemical identity and concentration)
Raman Spectroscopy (RS):
molecular rotation and vibration
(chemical identity and structure)
Ultraviolet and Visible (UV/VIS) Spectroscopy:
(Atomic Absorption Spectroscopy (AAS))
electronic transitions in atoms and molecules
(chemical identity and concentration)
Nuclear Magnetic Resonance (NMR) Spectroscopy: nuclear orientation
(chemical identity and structure)
X-Ray Photoelectron Spectroscopy (XPS):
determination of surface composition
(surface stoichiometry and concentration)
Mass Spectrometry (MS):
determination of atomic and molecular masses
(chemical identity and structure)
Photoelectron Spectroscopies: XPS and UPS
Ionisation of atoms and
molecules with monochromatic
X-ray or ultraviolett radiation:
 Photoelectrons with welldefined, substance-specific
energy
Ekin    Eb  eSp
Extreme surface sensitivity:
2-3 layers only
An example: XPS* spectrum of
stainless steel
*ESCA (Electron Spectroscopy for Chemical
Analysis) is another acronym for XPS
- Chemical Identity
- Quantitative Chemical Analysis
- Oxidation States / Chemical Surroundings
- Adsorbate Species
IR
(ambient pressure)
VT-AFM/STM
XPS/UPS etc.
Load Lock
Preparation
Some Spectroscopic Techniques in Chemical Industry
Infrared (IR) Spectroscopy:
molecular rotation and vibration
(chemical identity, structure, and concentration)
Near Infrared (NIR) Spectroscopy:
molecular vibration
(chemical identity and concentration)
Raman Spectroscopy (RS):
molecular rotation and vibration
(chemical identity and structure)
Ultraviolet and Visible (UV/VIS) Spectroscopy:
(Atomic Absorption Spectroscopy (AAS))
electronic transitions in atoms and molecules
(chemical identity and concentration)
Nuclear Magnetic Resonance (NMR) Spectroscopy: nuclear orientation
(chemical identity and structure)
X-Ray Photoelectron Spectroscopy (XPS):
determination of surface composition
(surface stoichiometry and concentration)
Mass Spectrometry (MS):
determination of atomic and molecular masses
(chemical identity and structure)
General Features of Spectroscopy
Relation between frequency, wavelength, and wavenumber:
c
 


 
c
:
wavelength
:
frequency
:
wavenumber (usually measured in [cm 1])
c:
speed of light
• In emission spectroscopy, a molecule undergoes a transition from a state of high
energy E1 to a state of lower energy E2 and emits the excess energy as a photon.
• In absorption spectroscopy, the net absorption of nearly monochromatic (singlefrequency) incident radiation is monitored as the radiation is swept over a range of
frequencies.
• The energy, h, of the photon emitted or absorbed, and therefore the frequency of the
radiation emitted or absorbed, is given by the Bohr condition
h = E1 – E2
• Emission and absorption spectroscopy give the same information about energy level
separation, but absorption spectroscopy is more common in the lab.
Spectrometers: The Sources
Symbolic representation of the principal components used in obtaining an absorption
spectrum:
Sources: either polychromatic, i.e. spanning a wide range of frequencies, or
monochromatic, i.e. spanning a very narrow range of frequencies around a central
value
Many commercial spectrometers take advantage of black-body radiation from hot
materials, which can provide radiation spanning a wide range of frequencies
Common lab sources:
 Mercury lamp for far-infrared radiation (FIR; 35 – 200 cm-1)
 Globar (SiC, heated to 1500 K) for mid-infrared radiation
(MIR; 200 – 4000 cm-1)
 Quartz-tungsten-halogen lamp for near-infrared, visible
and near-ultraviolet radiation (NIR – VIS – UV;
320 – 2500 nm, i.e. 4000 - 30000 cm-1; T3000 K)
 Xenon discharge lamp for ultraviolet radiation (T6000 K)
A Special Source: Synchrotron Storage Rings
For certain applications, synchrotron
radiation from a synchrotron storage ring
is appropriate.
A synchrotron storage ring consists of an
electron beam travelling in a circular path
of several meters in diameter. As electrons
travelling in a circle are constantly
accelerated by the forces that constrain
them to their path, they generate radiation.
Synchrotron radiation spans a wide
range of frequencies, including the infrared
and X-rays.
Synchrotron radiation is much more intense
than can be obtained by most conventional
sources. The disadvantage of the source is
that it is so large and costly that it is
essentially a national facility, not a
laboratory commonplace (e.g. BESSY II in
Berlin-Adlershof).
A synchrotron storage ring. The electrons
injected into the ring from the linear accelerator
and booster synchrotron are accelerated to high
speed (almost the speed of light) in the main ring.
An electron in a curved path is subject to
constant acceleration, and an accelerated charge
radiates electromagnetic energy.
Spectrometers: Dispersing Elements
The dispersing element in most absorption
spectrometers operating in the ultraviolet to nearinfrared region of the spectrum is a diffraction
grating, which consists of a glass or ceramic plate
into which fine grooves have been cut and covered
with a reflective aluminium coating. The grating
causes interference between waves reflected from
its surface, and constructive interference occurs
when
n = d(sin  - sin )
where n=1, 2, … is the diffraction order,  is the
wavelength of the diffracted radiation, d is the
distance between grooves,  is the angle of
incidence of the beam, and  is the angle of
emergence of the beam.
In a monochromator, a narrow exit slit allows
only a narrow range of wavelength to reach the
detector. Turning the grating around an axis
perpendicular to the incident and diffracted beams
allows different wavelength to be analysed.
In a polychromator, there is no slit and a broad
range of wavelengths can be analysed
simultaneously by array detectors.
A polychromatic beam is dispersed by a diffraction grating
into three component wavelength 1, 2 and 3. In the
configuration shown only radiation with 2 passes through a
narrow slit and reaches the detector. Rotating the diffraction
grating in the direction shown by the double arrows allows
1 and 3 to reach the detector.
Spectrometers: Fourier Transform Techniques
Modern spectrometers, particularly those operating in the infrared and near-infrared, now almost always use
Fourier transform (FT) techniques of spectral detection and analysis.
The heart of a FT spectrometer is a Michelson interferometer. A Michelson interferometer works by
splitting the beam from the sample into two and introducing a varying path difference, p, into one of them
(moveable mirror). When the two components recombine, there is a phase difference between them, and
they interfere either constructively or destructively depending on the difference in the path lengths. The
detected signal oscillates as the two components alternately come in and out of phase as the path
difference is changed.
Left: A Michelson interferometer. The beam-splitting element divides the
incident beam into two beams with a path difference p that depends on the
location of the mirror M1. The compensator ensures that both beams pass
through the same thickness of material
Top: Interferograms produced as the path length p is changed
in the interferometer. In the left example only a singlefrequency component is present in the radiation, while in the
other one several (in this case, three) frequencies are present.
Spectrometers: Fourier Transform Techniques (cont’d)
The problem is to find I( ), the variation of intensity with wavenumber,
which is the spectrum one requires, from the record of values of intensity
as a function of path difference, I(p). This step is a standard technique of
mathematics, and is the ‘Fourier transformation’ step from which this
form of spectroscopy takes its name. Specifically:

1


I( )  4 I(p)  I(0)  cos 2pdp
2


0

where I(0) is the intensity for p=0. This integration is carried out in a
computer connected to the spectrometer.
A major advantage of the Fourier transform procedure is that all the
radiation emitted by the source is monitored continuously. This is in
contrast to a spectrometer in which a monochromator discards most of
the generated radiation. As a result, Fourier transform spectrometers
have a higher sensitivity than conventional spectrometers.
Another advantage of Fourier transform spectrometers is their superior
resolution. It is determined by the maximum path length difference, pmax,
of the interferometer:
 
1
pmax
To achieve a resolution of 0.1 cm-1 requires a maximum path length
difference of about 10 cm, corresponding to only 5 cm travel of the
moving mirror. This resolution would be very difficult to achieve with a
monochromator.
The three frequency components
and their intensities that account
for the appearance of the
interferogram on the previous
transparency. This spectrum is
the Fourier transfrom of the
interferogram, and is a depiction
of the contributing frequencies.
The intensities of spectral lines: The Beer-Lambert Law
The ratio of the transmitted intensity, I, to the incident intensity, I0, at a given frequency is called the
transmittance, T, of the sample at that frequency:
T 
I
I0
It is found empirically that the transmitted intensity varies with the length, l, of the sample and the
molar concentration, [J], of the absorbing species J in accord with the Beer-Lambert law:
I  I0  10[J]l
The quantity  is called the molar absorption coefficient (formerly, and still widely in use, the
‘extinction coefficient’). The molar absorption coefficient depends on the frequency of the incident
radiation and is greatest where the absorption is most intense.
It is sensible to introduce the absorbance, A, of a sample at a given wavenumber as
I0
A  log
I
or
A   logT
Then the Beer-Lambert law becomes
A  [J]l
The product [J]l was formerly known as the optical density of the sample. For known  the
measurement of A allows determination of the concentration of species J.
Molecular Vibrations: IR and NIR Spectroscopy
The wavenumbers of molecular vibrations are typically in the range from 400 to 4000 cm-1, and can be
excited by (mid-) infrared light. Vibrational frequencies above ~1000 cm-1 can in general be attributed
to specific functionalities (e.g. C-H, C-O, C-C or other bonds), those below ~1000 cm-1 are usually due
to more complex vibrations of the whole molecule. From the infrared spectrum unknown compounds
can be identified. The vibrational spectrum is a ‘fingerprint’ of the molecule. From comparison to
libraries even complex mixtures can be analysed.
0
20
40
Transmittance [%]
60
80
100
e,g, hexane C6H14 and xylene C8H10
4000
3500
3000
2500
2000
Wavenumber cm-1
1500
1000
500
Additional information is available in the near-infrared (NIR) region (~4000 - ~10000 cm-1), where
overtones and combination bands are found.
Molecular Vibrations: The Background
Ideal:
Hook's law:
F  D  x
Potential V:
F  
dV
dx
V 
1
D  x2
2
Combination with Newton‘s law:
d2x
 D  x  m 
dt 2
0 
1 D
2 m
or
Quantum Mechanics:
Real:
0 
d2
dx 2

1 D
2 
2 
1 2
E 
Dx    0
2 
2


1

E  h  0   v  
2

The vibrational frequency is absolutely
sensitive on



the force constants
the masses
Identification of different chemical
species
Quantitation using Beer-Lambert’s law
Analysis of complex mixture by
automatic comparison to spectra
libraries
How to do it: Fourier Transform Infrared Spectroscopy (FTIRS)
- measure the light intensity modulation as
function of mirror movement in an
interferometer
 simultaneous measurement of all
frequencies!
- calculation of the frequency spectrum
from the interferogram by Fourier
transformation (after apodisation,
zerofilling, and phase correction).

B() 
 I(x)cos(2x)dx


I(x) 
 B()cos(2x)d

Raman Spectroscopy
In Raman spectroscopy, molecular energy levels are explored by examining the frequencies present in
the radiation scattered by molecules. In a typical experiment, a monochromatic incident laser beam is
passed through the sample and the radiation scattered from the front face of the sample is monitored.
About 1 in 107 of the incident photons collide with the molecules, give up some of their energy which serves
to excite vibrations or rotations of the molecules, and emerge with a lower energy. These scattered photons
constitute the lower-frequency Stokes radiation from the sample. Other incident photons may collect
energy from the molecules (if they are already excited) and emerge as higher-frequency anti-Stokes
radiation. The component of radiation scattered into the forward direction without change of frequency is
called Rayleigh radiation.
Raman spectroscopy and regular absorption spectroscopy often give complementary information. In case of
molecules with a centre of inversion a molecular vibration is either Raman or infrared active.
A common arrangement adopted in Raman spectroscopy.
A laser beam first passes through a lens and than through
a small hole in a mirror with a curved reflecting surface.
The focused beam strikes the sample and scattered light is
both deflected and focused by the mirror. The spectrum is
analysed by a monochromator or an interferometer.
In conventional Raman spectroscopy, the incident radiation
does not match an absorption frequency of the molecule´,
and there is only a ‘virtual’ transition to an excited state.
Electronic Transitions: UV-VIS Spectroscopy
The energies needed to change the electron distributions of molecules are of the order of several
electronvolts (1 eV is equivalent to about 8000 cm-1 or 100 kJ mol-1). Consequently, the photons
emitted or absorbed when such changes occur lie in the visible and ultraviolet regions of the
spectrum. In some cases the relocation of electrons may be so extensive that it results in ionization or
dissociation of the molecule.
The nuclei in a molecule are subjected to different forces after an electronic transisition has occurred,
and the molecule may respond by starting to vibrate. The resulting vibrational structure of electronic
transitions can be resolved for gaseous samples, but in a liquid or solid the lines usually merge
together and result in a broad, almost featureless band. Superimposed on the vibrational transition
of a molecule in the gas phase is an additional structure that arises from rotational transitions.
Due to the broad absorptions, UV-VIS spectroscopy is usually
not the method of choice for the analysis of complex mixtures.
However, in case of compounds with rather specific absorption
frequencies it can be used for e.g. determination of
concentrations, control of purity et cetera.
The absorption spectrum of chlorophyll in the visible region. Note that it
absorbs in the blue and red regions, and that green light is not absorbed
appreciably.
Some Spectroscopic Techniques in Chemical Industry
Infrared (IR) Spectroscopy:
molecular rotation and vibration
(chemical identity, structure, and concentration)
Near Infrared (NIR) Spectroscopy:
molecular vibration
(chemical identity and concentration)
Raman Spectroscopy (RS):
molecular rotation and vibration
(chemical identity and structure)
Ultraviolet and Visible (UV/VIS) Spectroscopy:
(Atomic Absorption Spectroscopy (AAS))
electronic transitions in atoms and molecules
(chemical identity and concentration)
Nuclear Magnetic Resonance (NMR) Spectroscopy: nuclear orientation
(chemical identity and structure)
X-Ray Photoelectron Spectroscopy (XPS):
determination of surface composition
(surface stoichiometry and concentration)
Mass Spectrometry (MS):
determination of atomic and molecular masses
(chemical identity and structure)
Nuclear Magnetic Resonance (NMR) Spectroscopy
The Stern-Gerlach experiment provided evidence for electron spin. It turned out that many
nuclei also possess spin angular momentum. Orbital and spin angular momenta give rise
to magnetic moments, and to say that electrons and nuclei have magnetic moments means,
to some extent, that they behave like little bar magnets. Therefore, one can expect that the
application of a magnetic field should affect atoms and molecules.
The spin quantum number, I, of a nucleus is a fixed characteristic property and may be an
integer or a half-integer but is never negative. A nucleus with spin quantum number I has the
following properties:
1. An angular momentum of magnitude {I(I+1)}1/2ħ.
2. A component of angular momentum mIħ on an arbitrary axis, where mI = I, I-1, …, -I.
3. If I>0, a magnetic moment with a constant magnitude and an orientation that is
determined by the value of mI.
Number of protons
Number of neutrons
I
even
even
0
odd
odd
Integer (1, 2, 3, …)
even
odd
half-integer (1/2, 3/2, 5/2, …)
odd
even
half-integer (1/2, 3/2, 5/2, …)
Nuclear constitution and the nuclear spin quantum number
Several important nuclei, like 1H. 13C, 19F, and 31P have I = ½.
Nuclear Magnetic Resonance (NMR) Spectroscopy (cont’d)
If a nucleus with a magnetic moment is placed in a magnetic field of strength B0 in the z-direction, the (2I+1)
orientations of the nucleus have different energies, which are given by
EmI   mh
I L
L 
B0
2
Lis the so-called Larmor frequency,  the magnetogyric ratio for the nucleus under consideration.
The energy separation between the mI=+½ ( or ) and the mI=-½ ( or ) states of spin-½ nuclei, which
are nuclei with I=½, is
E  E  E   B0  hL
and resonant absorption occurs when the resonance condition h=ħB0 is fulfilled. That is, an 
transition occurs at =L.
The interactions between the ms states of an electron (or
mI states of a I=½ nucleus) and an external magnetic field
may be visualized as the precession of the vectors
representing the angular momentum on the cones drawn
here.
The nuclear spin energy levels of a spin-½ nucleus with
positive magnetogyric ratio (e.g. 1H or 13C) in a magnetic
field. Resonance occurs when the energy separation of the
levels matches the energy of the photons in the
electromagnetic field.
NMR: The Spectrometer
In its simplest form, nuclear magnetic resonance (NMR) is the study of the properties of molecules
containing magnetic nuclei by applying a magnetic field and observing the frequency of the resonant
electromagnetic field absorption. Larmor frequencies of nuclei at the fields normally applied lie in the
radiofrequency region of the electromagnetic spectrum, so NMR is a radiofrequency technique. For
example, at 12 T, protons come into resonance at about 500 MHz (the Larmor frequency of the
electromagnetic field). For such high magnetic fields, usually superconducting magnets are used.
NMR spectra have proved invaluable in chemistry (and in particular organic chemistry), for they reveal so
much structural information. A magnetic nucleus is a very sensitive, noninvasive probe of the surrounding
electronic structure.
The layout of a typical NMR spectrometer. The link from
the transmitter to the detector indicates that the high
frequency of the transmitter is subtracted from the high
frequency received signal to give a low frequency signal
for processing.
Nowadays the NMR technique is not only important in chemistry, where most of the modern instruments are
based on pulse techniques. One of the most striking applications is in medicine, where magnetic
resonance imaging (MRI) is a portrayal of the concentrations of protons in a solid object. A great
advantage of MRI is that it can display soft tissue, such as a cross-section through a patient’s head.
NMR: The Chemical Shift
Nuclear magnetic moments interact with the local magnetic field, which is influenced by the electron density
(electrons also have a magnetic moment) as well as the chemical surroundings with respect to other
magnetic nuclei. These contributions are proportional to the applied field, and it is conventional to write
B   B0
where the dimensionless quantity  is called the shielding constant of the nucleus.
Because the total local field is Bloc = B0 + B = (1-)B0 the Larmor frequency is
L 
Bloc
B
 (1  ) 0
2
2
It is convenient to express the resonance frequencies in terms of an empirical quantity called the chemical
shift, which is related to the difference between the resonance frequency, , and that of a resonance
standard, 0:
 
  0

0
 106
The standard for protons is the proton resonance 0 in tetramethylsilane (Si(CH3)4), commonly referred to
as TMS, which bristles with protons and dissolves without reaction in many liquids.
NMR: The Chemical Shift (cont’d)
The range of typical chemical shifts for (a) 1H resonances
and (b) 13C resonances.
The 1H NMR spectrum of ethanol. The bold letters denote the
protons giving rise to the resonance peak, and the step-like
curve is the integrated signal. The CH3 protons form one
group of nuclei with 1. The two CH2 protons are in a
different part of the molecule, experience a different local
magnetic field, and resonate at 3. Finally, the OH proton is
in another environment, and has a chemical shift of 4. The
increasing value of  (that is the decrease in shielding) is
consistent with the electron-withdrawing power of the O
atom, which reduces the electron density at the O atom most.
The integration of peak intensities yields 1:2:3, in agreement
with the (in this case known) chemical composition.
NMR: The Fine Structure
The splitting of resonances into individual lines is called the fine structure. It arises because each
magnetic nucleus may contribute to the local field experienced by the other nuclei and so modify their
resonance frequencies, The strength of the interactions is expressed in terms of the scalar spin coupling
constant, J, and reported in Hertz (Hz).
For the example ethanol (CH3CH2OH) which is shown below again, this means:
-
The three protons of the CH3 group split the resonance of the CH2 protons into a 1:3:3:1 quartet.
Likewise, the two protons of the CH2 group split the resonance of the CH3 protons into a 1:2:1 triplet
with the same splitting J.
All the lines mentioned so far are split into doublets by the OH proton, but the splitting cannot be
detected because the OH protons migrate rapidly from molecule to molecule and their effect averages
to zero.