Download che-20028 QC lecture 2 - Rob Jackson`s Website

CHE-20028: PHYSICAL & INORGANIC CHEMISTRY
QUANTUM CHEMISTRY: LECTURE 2
Dr Rob Jackson
Office: LJ 1.16
[email protected]
http://www.facebook.com/robjteaching
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-20028 QC lecture 2
2
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?
CHE-20028 QC lecture 2
3
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.
CHE-20028 QC lecture 2
4
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).
CHE-20028 QC lecture 2
5
Schematic of Electron Diffraction
Experiment
Experimental setup
CHE-20028 QC lecture 2
Pattern
6
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.
CHE-20028 QC lecture 2
7
Electron Diffraction Experimental
set-up
CHE-20028 QC lecture 2
8
A typical electron diffraction pattern
The distances between the
rings are used to determine
structural information.
CHE-20028 QC lecture 2
9
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).
CHE-20028 QC lecture 2
10
LEED Experimental setup
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
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).
CHE-20028 QC lecture 2
12
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.
CHE-20028 QC lecture 2
13
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.
CHE-20028 QC lecture 2
14
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
CHE-20028 QC lecture 2
15
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-20028 QC lecture 2
16
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?
CHE-20028 QC lecture 2
17
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.
CHE-20028 QC lecture 2
18
Wave-particle duality - conclusion
http://abyss.uoregon.edu/~js/glossary/wave_particle.html
CHE-20028 QC lecture 2
19
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-20028 QC lecture 2
20
(Extra slide 1): The uncertainty
principle and zero point energy
• The following expression also applies:
E t  h/4
• This means that at the atomic level
there is never a state with zero energy
because of the uncertainty in the energy
value as given above. This in turn arises
from uncertainty in the electron position.
• See: http://www.calphysics.org/zpe.html
CHE-20028 QC lecture 2
21
(Extra slide 2): Demonstration of
ZPE: helium at low temperatures
• Experimental measurement of zero point
energy is difficult(!) but its consequences
can be seen:
– As temperature is lowered to absolute zero,
helium remains a liquid, rather than freezing
to a solid, because of its zero-point energy.
– Very high pressures are needed to freeze it.
CHE-20028 QC lecture 2
22
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!
CHE-20028 QC lecture 2
23
Electrons in atoms; how we represent
them (i)
• Orbits:
electrons
move round the
atom
following
defined paths.
• Not allowed by
Heisenberg’s
Uncertainty Principle
CHE-20028 QC lecture 2
24
Electrons in atoms; how we represent
them (ii)
• Orbitals
– only the volume and range of possible
positions occupied by the electrons can be
known:
CHE-20028 QC lecture 2
25
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.
CHE-20028 QC lecture 2
26
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.
CHE-20028 QC lecture 2
27
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 .
CHE-20028 QC lecture 2
28
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
CHE-20028 QC lecture 2
29
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
CHE-20028 QC lecture 2
30
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 , 
CHE-20028 QC lecture 2
31
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-20028 QC lecture 2
32