Page 187 - Academic Press Encyclopedia of Physical Science and Technology 3rd Analytical Chemistry

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Encyclopedia of Physical Science and Technology EN007C-340 July 10, 2001 14:45

798 Infrared Spectroscopy

Symmetry elements other than the center of symmetry able, but has discrete values given from quantum mechan-
include planes of symmetry and two-fold or higher axes ics as
of symmetry. When a plane of symmetry is present, the 1
E = v + hν ◦ , v = 0, 1, 2,... (8)
plane can be thought of as a mirror. When each atom in 2
the molecule is moved to the position of its mirror im- Here E is the vibrational energy, h Planck’s constant, ν ◦ the
age, the resulting configuration is indistinguishable from classical vibrational frequency of the oscillator, and v the
the original. When a twofold axis of symmetry is present, quantum number, which can have only integer values. In
the molecule can be rotated by half a full circle to give a the classical oscillator, the lowest possible energy is zero
configuration indistinguishable from the original. A full when there is no vibration. In the quantum mechanical
1
discussion of symmetry and group theory cannot be given oscillator, the lowest possible energy is hν o , which is not
2
here. However, molecules that do not have a center of zero, so the molecule can never stop vibrating entirely.
symmetry may have IR-inactive vibrations as a conse- This state where v = 0 is called the ground vibrational
quence of these other symmetry elements. For example, state.
the tetrahedral sulfate ion SO 2− does not have a center of If the vibrational energy is to be increased, the quantum
4
symmetry, but the in-phase stretch of the four SO bonds number v must be increased. When the quantum number
is IR-inactive. The four negative oxygens move radially at is increased by 1, the energy change
E from the previous
the same time, but the symmetry requires that the center equation is
of their excess negative charge does not move relative to
E = hν o . (9)
the more positive sulfur.
A photon has an energy E given by
H. Quantum Mechanical Harmonic Oscillator E = hν p , (10)
The simplest classical harmonic oscillator is a single mass where ν p is the frequency of the photon. When the photon
m suspended from the ceiling by a spring that obeys electric field frequency ν p is equal to the classical dipole
Hooke’s law. If the mass is pulled down a distance x from moment oscillation frequency ν o for this vibration, the
its equilibrium point, the spring length minus its length at photon will have exactly the right energy (
E) needed to
equilibrium is x. A restoring force on the mass is gener- increase the vibrational quantum number by 1.
ated that is proportional to the spring length change. The The transition when the quantum number changes by
magnitude of the restoring force equals kx, where k is the 1 is called an allowed transition in a harmonic oscilla-
force constant. If the mass is held stationary at this point, tor. The most important of these is the transition where
the potential energy PE is the oscillator goes from the v = 0 level to the v = 1 level.
This is called the fundamental transition and is responsi-
2
1
PE = kx . (6)
2 ble for most of the strong bands in the IR spectrum. The
This is also the total energy for this condition since the ki- (v = 0 → v = 1) transition is much more probable than
the (v = 1 → v = 2) transition because at room tempera-
neticenergyiszero.Ifthemassisreleased,itmovestoward
ture many more oscillators exist in the low-energy v = 0
the equilibrium point and the kinetic energy increases as
state than in the v = 1 state (or higher states).
the potential energy decreases. At equilibrium, the energy
In a polyatomic molecule with 3n − 6 different nor-
is entirely kinetic; the mass overshoots the equilibrium
mal modes of vibration, each normal mode of vibra-
point and continues on until the energy is again entirely
tion can be treated separately. In the harmonic oscillator
potential at maximum amplitude, and the cycle is repeated
approximation,
again. Throughout the vibration the total classical energy

is unchanged and is E = v 1 + 1 hν 1 + ν 2 + 1 hν 2 + ··· (11)
2 2
1
E = kx 2 , (7) where each mode has its own quantum number v and fre-
2 max
quency ν. In the harmonic oscillator only one vibration
where x max is the maximum amplitude. In a classical vi-
may be excited at one time and the quantum number may
bration the maximum amplitude is continuously variable
change only by 1.
since one is free to pull out the spring to any length before
it is released to vibrate. This means that the energy of the
I. Effect of Anharmonicity
classical harmonic oscillator is continuously variable and
can have any value. In the single-mass harmonic oscillator discussed, the
Oscillators the size of molecules obey the laws of quan- restoring force is a linear function of the mass displace-
tum mechanics. The vibrational energy of the quantum ment and the potential energy is a squared function of
1
2
mechanical harmonic oscillator is not continuously vari- the mass displacement kx . Mechanical anharmonicity
2