Download CHEM 347 Quantum Chemistry

CHEM 347
Quantum Chemistry
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Do not keep saying to yourself if you can possibly
avoid it, ‘But how can it be like that?’ because you will
get ‘down the drain’ into a blind alley from which
nobody has yet escaped. Nobody knows how it can be
like that”
R. P. Feynman
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Historical Development of Quantum Mechanics
At the end of the 19th century many physicists believed that all
principles of physics had been discovered. There were some major
successes.
•Newton’s Laws and classical mechanics
•Classical Thermodynamics
•Optics, electricity, magnetism
Key ideas of Classical physics
•continuous observables
•no restriction on the energy of systems
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d r
F=ma=m
2
dt
•determinism – Everything about a system’s future is
known by solving for r(t) from Newton’s second law given
initial conditions.
•wave nature of light
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The Nature of Light
1801
Thomas Young performed convincing experimental
evidence for the wave nature of light. Diffraction and
interference were observed.
interference pattern observed with
Young’s double slit experiment
equilvalent interference
pattern from waves of water
1860s Maxwell developed four equations that unified the laws
of electricity and magnetism.
The speed of an
electromagnetic wave predicted was the same as the
speed of light (c) that was experimentally measured.
Light was concluded to be an electromagnetic wave.
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The Physics of Chemistry was not as Well Developed
• We knew of the existence of nuclei and electrons.
• The periodic table had been developed. (empirically)
• Catalogues of chemical reactions were available.
• The nature of molecules and chemical bonds were among these
“gaps”.
By the beginning of the 19th century there were a number of
other experimental observations that could not be explained by
classical physics such as radioactivity!
The ‘small gaps’ came to be fundamental problems with Classical
Physics and a radical new theory was needed to fill them.
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Observations that ‘Violated’ Classical Physics
There were several key experimental observations that could not
be explained by classical physics that led to the development of
quantum mechanics.
Blackbody Radiation - disobeyed classical physics
One of the most important observed phenomena (from a
historical point of view) that made scientists question classical
mechanics was blackbody radiation.
What is black body radiation?
-All materials give off radiation when heated.
•Indeed materials continually absorb and give off radiation.
•In our everyday experience most radiation emitted is IR.
-As materials are heated to higher temperature, the
radiation emitted tends toward higher frequencies.
red hot < white hot < blue hot
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What is black body radiation?
A black body is a body that absorbs and emits all frequencies.
A black body is an idealization for any radiating material where
electrons are made to oscillate at the frequencies of the light
and because they oscillate they radiate back at that frequency.
The radiation emitted by a black body depends on its
temperature.
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A plot of the Intensity of the blackbody radiation
as a function of the wavelength of the radiation.
Many attempts were made to derive expressions consistent
with the above experimentally determined plots.
Most resulted in expressions that grew without bound (as
shown as the black line in the above plot.
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The classical expressions assumed that the radiation emitted
by a blackbody was due to oscillations of electrons (like
electrons in an antenna that give off radiation).
In Classical physics, these oscillating systems are allowed to
posses any energy and could radiate any frequency of light.
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Planck’s Interpretation and Idea of Quantization
In 1900, Max Planck proposed a revolutionary idea. He
proposed that the energy of the oscillators could only assume
integer multiples of the frequency.
E=nhν
n = 1, 2, 3…
h is a constant of proportionality
ν radiation’s frequency
In other words, Planck proposed that the energy of the
oscillators (the material!) was quantized and that only certain
quantities of light energy could be emitted.
In classical physics, physical observables are allowed to take on
a CONTINUUM of values! And the idea was not accepted.
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However, using the idea of quantization, Planck derived
expressions that beautifully reproduced the experimental
blackbody radiation plots if:
h = 6.626 x 10-34 J•s
This is now known as Planck’s constant It is a fundamental
constant of physics.
Nobel Prize 1918
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The Photoelectric Effect
Another phenomena of historic interest that “went against”
classical physics was the photoelectric effect.
Photoelectric effect: The ejection of electrons from the surface
of a metal by radiation.
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kinetic energy of emitted
electrons can be measured
Metal surface
The classical picture of light is that it is a oscillating electromagnetic
wave. Electrons at the surface oscillate with the changing electric
field so violently that they get knocked out or emitted.
This classical picture predicts that the kinetic energy of the
electrons should increase as the amplitude of the radiation
(intensity) increases.
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The Photoelectric Effect
eMetal surface
Experimental Observations
• Kinetic energy of the electrons ejected are proportional to
the frequency of light, not the amplitude/intensity.
No matter how intense the light is, the kinetic energy
of the ejected electrons remains the same!
• The intensity only increased the number of electrons
ejected, not their kinetic energy.
• There was also experimentally observed that no electrons are
ejected below a threshold frequency of the light.
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Einstein’s Interpretation of the Photoelectric Effect (1905)
In 1905, Einstein made a major conceptual
extension to Planck’s concept of quantization.
He proposed that light itself exists in small
packets of energy, or photons.
hc
E=hυ=
λ
The energy of a photon is therefore proportional to its frequency!
One amazing result that arose from Einstein’s explanation of the
photoelectric effect was that the calculated constant of
proportionality, h was in good agreement with Planck’s value
obtained from studies of blackbody radiation!
e-
Metal surface
The same constant arose out of
two completely different
experiments!
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Some energy magnitudes and units
Using Einstein’s relationship that relates the energy of a
photon to its wavelength we see that one photon of yellow light
(600 nm) contains:
E = 3 x 10-19 J
Electron Volt
When dealing with energies of individual photons units of
electron volts are commonly encountered:
1 eV = 1.602 x 10-19 J
Therefore a photon of yellow light has an energy of about 2 eV.
The O2 bond energy is about 498 kJ/mol which is about 5.2 eV.
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Discrete Emission Spectra of the Hydrogen Atom
(and other atoms too)
Discrete line spectra of atoms suggests energy quantization.
Series limit
(656 nm) (486 nm) (434 nm) (365 nm)
Balmer and Rydberg (and others) empirically derived a formula
for the hydrogen line spectra:
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1
υ=109680
−
2
n2
n
2
1
cm−1
n1, n2 = integers and n2>n1
Hydrogen spectra controlled by two integers! This further
suggests some sort of quantization.
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Bohr Model of the Hydrogen Atom
The classical mechanical picture of an electron and proton cannot
explain the discrete emission spectra of the hydrogen atom.
Bohr (1913) comes close with the Bohr model of the hydrogen atom.
Bohr assumed that the angular momentum of the electron in
hydrogen is quantized. This goes against the classical picture of
continuous properties, particularly angular momentum.
Using Planck’s constant and classical physically measured
constants Bohr was able to derive Rydberg’s empirically derived
expression for the hydrogen atom emission spectra.
me4
1
1
υ=
−
3 n 2 n2
8ε2
ch
o
2
1
cm−1
r = 0.529 Å
Radius of ground state orbit
Except for the assumption of angular momentum quantization,
the above was derived with classical mechanics.
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De Broglie’s Wave Nature of Matter
A conceptual break through came from the work of de Broglie in
1923.
By 1923 the dual wave-particle nature of light was accepted by
most leading physicists.
de Broglie extended and quantified the wave particle duality of
light to all particles (electrons, protons, baseballs).
Using special relativity, Einstein derived an expression for the
momentum of a photon (even though it has no mass). Using a
similar line of reasoning, de Broglie argued that the wave-length
of a particle was related to its momentum:
h h
λ=
=
mv p
h is Planck’s constant again.
Thus, any particle with a momentum p will travel with a wavelength λ
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In 1925, experimental verification of de Broglie’s wave hypothesis
came with electron diffraction experiments.
X-ray
electron
X-ray and electron diffraction
patterns through Al foil
Today, the wave property of matter is used routinely in chemistry
and biology.
electron microscopes and neutron diffraction - like X-ray structures
(e- microscopes have a higher resolution than optical microscopes)
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h h
λ=
=
mv p
Wave-particle duality of matter really
only comes into play with very small
masses.
Mass
(kg)
Speed
(m/s)
Wavelength
Electron
accelerated
through 100 V
9.11x10-31
5.9x106
120 pm
Alpha particle
ejected from
radium
6.68x10-27
Bullet
1.9x10-3
Particle
(atomic and
molecular distances)
1.5x107
6.6x10-3 pm
(smaller than an atom)
3.2x102
1.1x10-21 pm
(much smaller than a
nucleus)
For macroscopic bodies, the wavelengths
undetectable and of no practical consequence.
are
completely
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Wave-Particle Duality
Wave-particle duality refers to the fact that both light and
matter can exhibit either particle-like behavior or wave-like
behavior depending on how we observe them.
i.e. behavior depends on the nature of the
experiment.
• photons can behave like particles in a photo-electric experiment
•electrons and other particles can exhibit a wave-like
diffraction pattern
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Heisenberg’s Uncertainty Principle
mid 1920s
The wave-particle duality of both light and matter leads to some
very awkward results.
Consider the measurement of the position of an electron.
If we want to measure the electron within a distance Δx we must
use something of spatial resolution less than Δx.
One way to achieve this is to use light of wavelength l  Δx.
For us to ‘see’ the electron the photon must interact with the
electron.
h
But the photon has a momentum associated with it. p=
λ
Thus, the very act of observing the electron leads to a change in its
momentum.
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Developing this idea fully, Werner Heisenberg showed that it is not
possible to simultaneously determine the EXACT position and
velocity of a particle at the same time.
The greater the certainty we measure the position of a particle Δx,
the less certain we can be of the particles momentum Δp (vice-versa)
∆x∆p≈h
ℏ
∆x∆p≥
2
Heisenberg’s Uncertainty Principle
where
h
ℏ=
2π
“h-bar”
The uncertainty principle is not compatible with the deterministic
classical picture, since we can no longer specify exactly a particle’s
position and momentum simultaneously. We really can only talk
about probabilities.
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AGAIN the uncertainty principle, really only applies at the
microscopic scale.
h = 6.626 x 10-34 J.s
e.g. The uncertainty in the position of a baseball (145 g) thrown at 90
mph (40 m/s) if we measure the momentum to a millionth of 1.0%
(9x10-8 mph).
Δp = 5.6 x10-8 kg m/s
ℏ
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∆x∆p≥ =5 x 10
2
Δx = 9.4 x10-28 m
(less than the radius
of atomic nuclei)
e.g. The uncertainty in the momentum if we locate an electron within an
atom so that the uncertainty in its position is 50 pm. (Bohr radius)
Δx = 50 x10-12 m
Δp = 1.3 x10-23 kg m/s
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Δv = 1x107 m/s
P+
p=mv
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Historical Development of Quantum Mechanics
The stage is now set for the development of a new theory to
describe the microscopic world of electrons and nuclei.
• Classical mechanics, combined ideas of quantization to
reproduce experimental observations.
• quantization of energy states with Planck’s constant
popping up.
• wave - particle duality of both light and matter!!
• Uncertainty principle starts to hint at probabilities.
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Modern Physics:
• Theory of Relativity
• Quantum Mechanics
Theory of Relativity: Developed by Einstein in 1905, extended
classical mechanics to high velocities and astronomical distances.
Quantum Mechanics: developed over decades by many scientists.
Deals with the microscopic at the level of atoms, electrons and smaller.
Quantum Mechanics has had a profound effect on our
understanding of chemistry. There is a sub-discipline of chemistry
and quantum mechanics called ‘quantum chemistry’.
e.g. The covalent bond is a result of quantum mechanics and
cannot be adequately explained by classical physics.
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The Discovery of Quantum Mechanics
Quantum mechanics was discovered in 1925, by Heisenberg and
independently by Schrödinger a few months later.
Each
formulated QM in a different manner.
In 1925, during Christmas holidays,
Schrödinger discovered what is now
Schrödinger’s Wave Equation.
Erwin
called
He started from the idea that particles behave as
waves as quantified by de Broglie.
The formulation of Quantum Mechanics is known as Wave Mechanics
In quantum chemistry we most commonly deal with Schrödinger’s
formulation of quantum mechanics. Thus, quantum mechanics
equals wave mechanics for us.
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There are other notable formulations of quantum mechanics.
1925 - Heisenberg - Matrix Mechanics
1926 - Schrödinger - Showed that Matrix and Wave Mechanics are
equivalent
Matrix and wave mechanics did not take into account relativity.
Many tried to introduce relativity into QM, but had problems.
1929 - Dirac - Relativistic Quantum Mechanics
Introduced what we now call ‘spin’ as a degree of freedom,
just like the position.
Spin comes naturally in Dirac’s formulation but needs to be
added in an ad hoc manner in wave mechanics.
Dirac’s formulation is very complicated.
1941 - Feynman - Path Integral formulation of Quantum Mechanics
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