CHE-20004: PHYSICAL CHEMISTRY QUANTUM CHEMISTRY: LECTURE 2 Dr Rob Jackson Office: LJ 1.16...

CHE-20004: PHYSICAL CHEMISTRY

QUANTUM CHEMISTRY: LECTURE 2

Dr Rob Jackson

Office: LJ 1.16

[email protected]

http://www.facebook.com/robjteaching

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Learning objectives for lecture 2

• To understand the interpretation of the electron diffraction experiment.

• To further understand wave-particle duality as applied to electrons, and apply the de Broglie equation.

• To understand what wave functions are and what information they provide.

CHE-20004 QM lecture 2

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Behaviour of electrons

• Having shown that light behaves as a particle at an atomic level, we turn to looking at electrons.

• What are electrons?– Subatomic particles, mass 9.11 x 10 –31 kg!

• But do they always behave as particles?


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The electron diffraction experiment - 1

• What happens if you ‘fire’ a beam of electrons at a crystal surface?

• This experiment was first performed in 1925 by Davisson and Germer, who used a nickel metal surface, and observed that the electrons were diffracted by the surface like light is when it passes through a prism.


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The electron diffraction experiment - 2

• When light passes through a prism or a diffraction grating, it is separated into different frequencies (colours), and a spectrum (diffraction pattern) is produced.

• The same thing happens when electrons are either shone at a crystal surface, or pass through a crystal (if thin enough).


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Schematic of Electron Diffraction Experiment


Experimental setup Pattern

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Why does electron diffraction happen?

• Electrons behave as waves at an atomic level, and their wavelength is comparable to the distances between atoms in a crystal – (what are these?).

• The regular array of atoms in the crystal then acts like a diffraction grating, and produces a diffraction pattern.


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Electron Diffraction Experimental set-up


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A typical electron diffraction pattern


The distances between the rings are used to determine structural information.

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Low Energy Electron Diffraction (LEED)

• Electrons do not penetrate far into crystals (why?), so they can be used to study the surfaces of crystals.

• This effect is exploited in low energy electron diffraction, where, provided the energies are low enough, surface features like adsorbed molecules can be detected (important in catalysis).


An electron beam is aimed at a crystal surface.

Electron energies in range 20-200 eV The electron gun is shown in green

The diagram (right) shows a crystal surface and the diffraction pattern obtained.

http://www.chem.qmul.ac.uk/surfaces/scc/scat6_2.htm#leed

LEED Experimental setup

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Wave-particle duality

• Taking the photoelectric effect, Compton effect and electron diffraction experiments together, it would appear that, at the atomic level, waves behave as particles, and particles as waves.

• This is called ‘wave-particle duality’.• Momentum and wavelength can be

related (a particle and a wave property).


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The de Broglie equation

• In 1924 (before the electron diffraction experiment was performed), the French scientist Louis de Broglie proposed that: = h/p

• Where is wavelength, p is momentum (= mv) and h is Planck’s constant.

• So we can calculate the wavelength of any moving object.


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Using the de Broglie equation – (i)

• How do the wavelengths of an electron and a bus compare?

• Suppose the electron is travelling at 106 ms-1, and the bus at 30 mph, ~ 13 ms-1

• me = 9.11 x 10–31 kg, mbus ~ 15000 kg• Calculate in each case, using the de

Broglie equation.


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Using the de Broglie equation – (ii)

• For an electron:

= 6.626 x 10-34/(9.109 x 10-31 x 106)

= 7.289 x 10-10 m(compare with distances between atoms)

• For a bus:

= 6.626 x 10-34/(15000 x 13)

= 3.398 x 10-38 m


Diffraction of other ‘particles’

• If electrons can be diffracted, what about larger objects?

• The current record is a C60 molecule, and even (apparently), C60F48 (!), see

http://www.univie.ac.at/qfp/research/matterwave/c60/index.html

• Calculate the wavelength of each of these molecules (assume v = 210 ms-1)

CHE-20004 QM lecture 2 16

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Planck’s Constant

• If Planck’s constant was larger, say by a factor of 10, quantum effects would be more of an issue.

• But it would have to be quite a lot larger before it affected us directly.– How much larger would it have to be for

the bus to have a wavelength of 1 Å?


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The identity of electrons – a family affair?

• The Thomson family seemed to have had ‘electrons in the blood’.

• J J Thomson discovered the electron, and won the Nobel Prize for showing it to be a particle.

• His son, G P Thomson, then won the Nobel Prize for showing it to be a wave.


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Wave-particle duality - conclusion

http://abyss.uoregon.edu/~js/glossary/wave_particle.html


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Heisenberg’s Uncertainty Principle

• Heisenberg’s Uncertainty Principle states that for a quantum particle, it is impossible to specify its position and momentum simultaneously. This is stated as:

x p h/4• One consequence of the Uncertainty

Principle is zero point energy.CHE-20004 QM lecture 2

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Quantum mechanics and electrons

• The planetary orbit model of electrons depends on being able to specify the trajectory of an electron.

• This means knowing its position and momentum simultaneously. – impossible with the Uncertainty Principle.

• So the orbit model is incompatible with the ideas of Quantum Mechanics!


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Electrons in atoms; how we represent them (i)

• Orbits: electrons move round the atom following defined paths.

• Not allowed by Heisenberg’s Uncertainty Principle

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Electrons in atoms; how we represent them (ii)

• Orbitals– only the volume and range of possible

positions occupied by the electrons can be known:

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What are orbitals?

• Orbitals replaced orbits as a way of trying to describe the location of electrons.

• A consequence of the wave behaviour of electrons is that their location can not be specified precisely, but only the volume in which they are found. This volume is an orbital.


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What are wave functions?

• If we treat an electron as a particle, we can say what its position and momentum (trajectory) is at any time.

• For wave behaviour, the trajectory is replaced by the wave function.

• The wave function provides all the possible information about the electron.


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An example of a wave function – the 1s electron in hydrogen

• All we can say about the position of the 1s electron in hydrogen is that it is located somewhere within the 1s orbital.

• The wave function is the mathematical function which, when plotted out, gives the 1s orbital.

• The wave function is usually represented by the Greek letter .


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Some hydrogen-like wave functions

The wavefunctions are labelled by the quantum numbers n, l and ml

e.g. for 3dxy, n=3, l=2 and ml = 1


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What other information can be obtained from wave functions?

• The most important property is probably the energy, E of the electron, and this is obtained from the wavefunction by solving the Schrödinger equation:

H = E• The equation, to be discussed in the

next lecture, involves the operation of H on the wavefunction to give E


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Mathematical representation of orbitals

• Orbitals can be represented by mathematical functions, which is important for later calculations.

• A general expression takes the form: = exp (-r) Y (, )– Where r, , are coordinates

• s orbitals only depend on r, but all other orbitals also depend on ,

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Conclusions from the lecture

• The idea of wave-particle duality has been completed by looking at the wave behaviour of the electron.

• The electron diffraction experiment and the de Broglie equation have been introduced.

• The idea of wave functions has been introduced and discussed.


CHE-20004: PHYSICAL CHEMISTRY QUANTUM CHEMISTRY: LECTURE 2 Dr Rob Jackson Office: LJ 1.16...

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