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

•determinism – Everything about a system’s future is known by solving for r(t) from Newton’s second law given initial conditions.

•continuous observables

•no restriction on the energy of systems

•wave nature of light

F=ma=md2

r

dt2

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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.

The Nature of Light

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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 nature of molecules and chemical bonds were among these “gaps”.

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 ‘small gaps’ came to be fundamental problems with Classical Physics and a radical new theory was needed to fill them.

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Blackbody Radiation - disobeyed classical physics

Observations that ‘Violated’ 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.

-As materials are heated to higher temperature, the radiation emitted tends toward higher frequencies.

•Indeed materials continually absorb and give off radiation. •In our everyday experience most radiation emitted is IR.

red hot < white hot < blue hot

There were several key experimental observations that could not be explained by classical physics that led to the development of quantum mechanics.

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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.

What is black body radiation?

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. 15

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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In 1900, Max Planck proposed a revolutionary idea. He proposed that the energy of the oscillators could only assume integer multiples of the frequency.

In classical physics, physical observables are allowed to take on a CONTINUUM of values! And the idea was not accepted.

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.

E = n h ν

Planck’s Interpretation and Idea of Quantization

n = 1, 2, 3…

h is a constant of proportionality

ν radiation’s frequency

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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 18

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.

Metal surface

e-

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.

kinetic energy of emitted electrons can be measured

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The Photoelectric Effect

Experimental Observations

• Kinetic energy of the electrons ejected are proportional to the frequency of light, not the amplitude/intensity.

Metal surface

e-

No matter how intense the light is, the kinetic energy of the ejected electrons remains the same!

• There was also experimentally observed that no electrons are ejected below a threshold frequency of the light.

• The intensity only increased the number of electrons ejected, not their kinetic energy.

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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.

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!

Metal surface

e- The same constant arose out of two completely different experiments!

He proposed that light itself exists in small packets of energy, or photons.

E=hυ=hc

λ

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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:

Electron Volt

When dealing with energies of individual photons units of electron volts are commonly encountered:

1 eV = 1.602 x 10-19 J

E = 3 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.

Balmer and Rydberg (and others) empirically derived a formula for the hydrogen line spectra:

n1, n2 = integers and n2>n1

Hydrogen spectra controlled by two integers! This further suggests some sort of quantization.

(656 nm) Series limit

(365 nm) (486 nm) (434 nm)

υ =1096801

n12

−1

n22

cm−1

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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.

Except for the assumption of angular momentum quantization, the above was derived with classical mechanics.

r = 0.529 Å Radius of ground state orbit

Bohr Model of the Hydrogen Atom

υ =me4

8εo2ch

3

1

n12

−1

n22

cm−1

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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:

Thus, any particle with a momentum p will travel with a wavelength λ

h is Planck’s constant again. λ=h

mv=

h

p

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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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Wave-particle duality of matter really only comes into play with very small masses.

Particle

Mass

(kg)

Speed

(m/s)

Wavelength

Electron accelerated

through 100 V

9.11x10-31 5.9x106 120 pm (atomic and

molecular distances)

Alpha particle ejected from

radium

6.68x10-27

1.5x107

6.6x10-3 pm (smaller than an atom)

Bullet 1.9x10-3

3.2x102

1.1x10-21 pm

(much smaller than a nucleus)

For macroscopic bodies, the wavelengths are completely undetectable and of no practical consequence.

λ=h

mv=

h

p

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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.

For us to ‘see’ the electron the photon must interact with the electron. But the photon has a momentum associated with it.

Thus, the very act of observing the electron leads to a change in its momentum.

One way to achieve this is to use light of wavelength l Δx.

p=h

λ

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Heisenberg’s Uncertainty Principle

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)

where

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.

“h-bar”

∆x∆p≈h

∆x∆p≥ℏ

2 ℏ=

h

2π

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e-

P+

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

Δx = 9.4 x10-28 m

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)

Δp = 1.3 x10-23 kg m/s

Δx = 50 x10-12 m

Δv = 1x107 m/s

(less than the radius of atomic nuclei)

p=mv

∆x∆p≥ℏ

2=5 x 10−35

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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.

• quantization of energy states with Planck’s constant popping up.

• Classical mechanics, combined ideas of quantization to reproduce experimental observations.

• wave - particle duality of both light and matter!!

• Uncertainty principle starts to hint at probabilities.

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Modern Physics: • Theory of Relativity

Theory of Relativity: Developed by Einstein in 1905, extended classical mechanics to high velocities and astronomical distances.

• Quantum Mechanics

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, Erwin Schrödinger discovered what is now called Schrödinger’s Wave Equation.

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

1929 - Dirac - Relativistic Quantum Mechanics

1941 - Feynman - Path Integral formulation of Quantum Mechanics

1926 - Schrödinger - Showed that Matrix and Wave Mechanics are equivalent

Introduced what we now call ‘spin’ as a degree of freedom, just like the position.

Matrix and wave mechanics did not take into account relativity.

Many tried to introduce relativity into QM, but had problems.

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.

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