Ch 9

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Lecture 17 Intramolecular interactions
Ch 9 pages 493-497 442-444
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Summary of lecture 16

• We have partitioned the overall interaction
  energy between molecules or within molecules into
  terms that describe individual interactions as
  follows

The first term reflects interactions between
atoms within a molecule, the second term is the
Coulomb interaction between charged particles
the third reflects the strong repulsion between
atoms and molecules at short distance and the
fourth weak attractive forces between molecules
(London dispersion forces)
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Summary of lecture 16

• A very important interaction between biological
  molecules is the electrostatic potential
  expressed by Coulombs law, for two charges
  separated by a distance r

The Lennard-Jones potential describes weak
attractive and strong repulsive forces between
molecules.
For a given atom or molecular pair, it can be
written as
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Intramolecular Interactions
Thus far we have considered intermolecular
interactions in systems of molecules that have no
internal motions. However, except for monatomic
systems, all molecules display internal
motions For example, diatomic molecules undergo
whole-molecular rotation (i.e. rigid body
rotation) and changes in the length of the bond
(i.e. bond vibration).

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Intramolecular Interactions
When biological molecules interact with each
other or as we change temperature or denature
them, their chemical structure (configuration)
does not change, however, their three-dimensional
structure can and often does change Their
structure is determined by the rotation about
chemical bonds whose length and angles are fixed
by covalent properties of the chemical bonds that
hold these molecules together The conformation
of a molecule is dictated by the value of each of
the angles that are more or less free to rotate
(C-C double bonds, for example, are not free to
rotate).

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Intramolecular Interactions
For a large molecules like protein or DNA, there
are very many possible conformations, but many
are not populated because energetically
unfavorable For example, for proteins, only
certain regions of the conformational space in
the peptidic backbone are populated due to steric
clashes between amino acid side chains
(Ramachandrans plot) and similar considerations
apply to nucleic acids as well

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Intramolecular Interactions
Bond lengths and bond angles are determined by
interactions of electron and nuclei that define
the chemical bonds, and are eminently of quantum
mechanical nature. However, the forces that
change them can be treated by simple classical
physical models The energy of moving atoms so as
to stretch or compress a bond, or to change a
bond angle, depends (close to the value which is
most favored, or equilibrium value) on the square
of the change in bond length or the square of the
change in bond angles

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Intramolecular Interactions
Here the differences represent deviations from
the equilibrium value that is the bond length and
angle that are quantum mechanically most
favored The energy increases when the bonds are
perturbed from their equilibrium positions and,
as it happens, it takes far more energy to change
a bond length than a bond angle.

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Intramolecular Interactions
Rotations about single bonds cause large changes
in molecular conformations and dictate the
three-dimensional structure of biological
molecules. These rotations do not require much
energy (unless there are double bonds involved,
in which case the energy are substantial) For
example, the phenyl ring in phenylalanine in
proteins rotate rapidly around the bond
connecting it to the polypeptide chain at room
temperature, the barrier is only about 2.5
kJ/mole.

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Intramolecular Interactions
Torsion energies will have various minima
corresponding to most favored states, and can be
represented by a potential that has the following
form

Where f is the torsion angle. In order to rotate
a double bond, it takes much more energy because
the p bond has to be broken, while for a bond
with partial double character, like the C-N bond
in formamide or in peptides, the energy is
intermediate.
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Rigid Rotors
Let us consider rotations and vibrations in a
more formal context. A homonuclear diatomic
molecule (e.g. H2, N2, etc.) may be classically
represented as a dumbbell. Consider the dumbbell
as composed of two spheres with masses m, which
are intended to represent atoms, connected by a
mass-less bar, which is intended to represent a
chemical bond. The bond length is r.

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Rigid Rotors
If the dumbbell does not interact with other
dumbbells, the total energy is purely kinetic, as
usual, i.e
where Etrans is the kinetic energy of
translational motions and Erotate is the kinetic
energy associate with rotational motions

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Rigid Rotors
The translational energy is, in Cartesian
coordinates
The rotational motions are best represented in a
spherical coordinate system, in which angular
motions are described in terms of the angles q
and f, see diagram below. In this coordinate
system the angular kinetic energy is

Where I is the moment of inertia
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Vibrations and harmonic oscillators
The vibration of the bond of a dumbbell can be
represented as the motion of a harmonic
oscillator. The energy of a diatomic vibrator,
modeled as a classical harmonic oscillator is

Where m is again the reduced mass
If m1m2m, and k is the spring constant of the
bond
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The Trouble with Oscillators Black Body
Radiators Ultraviolet Catastrophe
Fundamental properties of nature like the
structure of the atom, atomic spectra, and the
distribution of radiant energy emitted by heated
materials cannot be properly explained by
classical physics A famous example, which was
very important in the early days of questioning
classical mechanics as an appropriate description
of matter and directly led to quantum mechanics
is provided by black body radiation.
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The Trouble with Oscillators Black Body Radiators
When a material, like a metal, is heated, it
first glows dull red (around T1100K), then as it
gets hotter its color shifts to the blue end of
the visible spectrum. This is the effect
responsible for the luminosity of light bulbs.
The graph of the intensity of emitted radiation
as a function of wavelength for several
temperatures is shown here
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The Trouble with Oscillators Black Body Radiators
A blackbody is an ideal version of a heated
object. To assure that the emitted radiation is
not simply being reflected from the body, the
body is painted black (i.e. so it does not
reflect radiation) and light is emitted only
through a pinhole. The radiation profile from
such object (also called a spectral distribution)
is called black body radiation.
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The Trouble with Oscillators Black Body Radiators
As the metal is heated, atoms in the metals
vibrate (they behave as harmonic oscillators) and
emit radiation into the hollow cavity (because
they consist of charged particles, which emit
radiation at the frequency of their vibrational
motion) When the density of energy in the cavity
equals the density in the metal, the black body
is at thermal equilibrium and emission into the
cavity stops. Classical physics predicted that
the intensity of radiation in the cavity as a
function of temperature and wavelength l is
given by an equation called Rayleigh-Jean Law
(ltEgt is the average energy per oscillator)
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The Trouble with Oscillators Black Body Radiators
The average oscillator energy can be calculated
from the vibrational partition function
The partition function for a classical harmonic
oscillator can be easily evaluated
Where n is the frequency of the vibrational
motion and
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The Trouble with Oscillators Black Body Radiators
Thus
The evaluation of the partition function has just
yielded the result expected from the
equipartition principle. This result substituted
into the Rahleigh-Jean Law gives
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The Trouble with Oscillators UV catastrophy
This result means that a common blackened
material, when heated, should emit intense levels
of radiation at short wavelengths, i.e. at very
high frequencies. This is contrary to common
experience. One cannot expect a heated material
to emit large quantities of ultraviolet
radiation, which is what classical physics
claims. The prediction of the Raleigh-Jean theory
is in such serious disagreement with experiment,
that it was called the ultraviolet catastrophe.