Transcript che-20028 QC lecture 3 - Rob Jackson`s Website

CHE-20028: PHYSICAL & INORGANIC CHEMISTRY
QUANTUM CHEMISTRY: LECTURE 3
Dr Rob Jackson
Office: LJ 1.16
[email protected]
http://www.facebook.com/robjteaching
Use of the Schrödinger Equation
in Chemistry
• The Schrödinger equation introduced
• What it means and what it does
• Applications:
– The particle in a box
– The harmonic oscillator
– The hydrogen atom
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Learning objectives for lecture 3
• What the terms in the equation
represent and what they do.
• How the equation is applied to two
general examples (particle in a box,
harmonic oscillator) and one specific
example (the hydrogen atom).
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The Schrödinger Equation
introduced
• The equation relates the wave function
to the energy of any ‘system’ (general
system or specific atom or molecule).
• In the last lecture we introduced the
wave function, , and defined it as a
function which contains all the available
information about what it is describing,
e.g. a 1s electron in hydrogen.
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What does the equation do?
• It uses mathematical techniques to
‘operate’ on the wave function to give
the energy of the system being studied,
using mathematical functions called
‘operators’.
• The energy is divided into potential and
kinetic energy terms.
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The equation itself
• The simplest way to write the equation
is:
H = E
• This means ‘an operator, H, acts on the
wave function  to give the energy E’.
– Note – don’t read it like a normal algebraic
equation!
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More about the
operator H
Remember, energy is divided
into potential and kinetic
forms.
H is called the Hamiltonian
operator (after the Irish
mathematician Hamilton).
The Hamiltonian operator
contains 2 terms, which are
connected respectively with
the kinetic and potential
energies.
William Rowan Hamilton
(1805–1865)
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Obtaining the energy
• So when H operates on the wave
function we obtain the potential and
kinetic energies of whatever is being
described – e.g. a 1s electron in
hydrogen.
• The PE will be associated with the
attraction of the nucleus, and the KE
with ‘movement’ of the electron.
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What does H look like?
• We can write H as:
H = T + V, where ‘T’ is the kinetic
energy operator, and ‘V’ is the potential
energy operator.
• The potential energy operator will
depend on the system, but the kinetic
energy operator has a common form:
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The kinetic energy operator
• The operator looks like:
2 2
T 
2m x 2
• Which means: differentiate the wave


function twice and multiply by 2m
•  means ‘h divided by 2’ and m is,
e.g., the mass of the electron
2
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Examples
• Use of the Schrödinger equation is best
illustrated through examples.
• There are two types of example,
generalised ones and specific ones, and
we will consider three of these.
• In each case we will work out the form
of the Hamiltonian operator.
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Particle in a box
• The simplest example, a particle moving
between 2 fixed walls:
A particle in a box is free to move in a space
surrounded by impenetrable barriers (red). When
the barriers lie very close together, quantum effects
are observed.
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Particle in a box: relevance
• 2 examples from Physics & Chemistry:
• Semiconductor quantum wells, e.g.
GaAs between two layers of AlxGa1-xAs
•  electrons in conjugated molecules,
e.g. butadiene, CH2=CH-CH=CH2
• References for more information will be
given on the teaching pages.
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Particle in a box – (i)
• The derivation will be explained in the lecture,
but the key equations are:
(i) possible wavelengths are given by:
 = 2L/n (L is length of the box), n = 1,2,3 ...
See http://www.chem.uci.edu/undergrad/applets/dwell/dwell.htm
(ii) p = h/  = nh/2L (from de Broglie equation)
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Particle in a box – (ii)
• (iii) the kinetic energy is related to p
(momentum) by E = p2/2m
• Permitted energies are therefore:
En = n2h2/8mL2 (with n = 1,2,3 ...)
• So the particle is shown to only be able
to have certain energies – this is an
example of quantisation of energy.
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The harmonic oscillator
The harmonic oscillator is a general
example
of
solution
of
the
Schrödinger equation with relevance
in
chemistry,
especially
in
spectroscopy.
‘Classical’ examples include the
pendulum in a clock, and the
vibrating strings of a guitar or other
stringed instrument.
http://en.wikipedia.org/wiki/Harmonic_oscillator
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Example of a harmonic oscillator:
a diatomic molecule
H ------ H
F = - kx
• If one of the atoms
is displaced from its • where x is the
displacement, and k
equilibrium position,
is a force constant.
it will experience a
restoring force F, • Note negative sign:
force is in the
proportional to the
opposite direction to
displacement.
the displacement
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Restoring force and potential
energy
• And by integration, • So we can write the
we can get the
Hamiltonian for the
potential energy:
harmonic oscillator:
• V(x) = k  x dx
• = ½ kx2
2
2


2
1
• H= 

kx
2m x 2 2
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1-dimensional harmonic oscillator
summarised
•
F = - kx
•
where x is the displacement,
and k is a force constant.
•
Note negative sign: force is in
the opposite direction to the
displacement
•
And by integration, we can get
the potential energy:
•
V(x) = k  x dx
•
= ½ kx2
•
So we can write the Hamiltonian
for the harmonic oscillator:
H=
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2 2 1 2

 2 kx
2
2m x
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Allowed energies for the harmonic
oscillator - 1
• If we have an expression for the wave
function of a harmonic oscillator
(outside module scope!), we can use
Schrödinger’s equation to get the
energy.
• It can be shown that only certain energy
levels are allowed – this is a further
example of energy quantisation.
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Allowed energies for the harmonic
oscillator - 2
En = (n+½) 
•  is the circular frequency, and n= 0, 1,
2, 3, 4
• An important result is that when n=0, E0
is not zero, but ½  .
• This is the zero point energy, and this
occurs in quantum systems but not
classically – a pendulum can be at rest!
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Allowed energies for the harmonic
oscillator - 3
• The energy levels
are the
allowed
energies for the
system, and are
seen in vibrational
spectroscopy.
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Quantum and classical behaviour
• Quantum behaviour (atomic systems) characterised by zero point energy,
and quantisation of energy.
• Classical behaviour (pendulum, swings
etc) – systems can be at rest, and can
accept energy continuously.
• We now look at a specific chemical
system and apply the same principles.
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The hydrogen atom
• Contains 1 proton and 1 electron.
• So there will be:
– potential energy of attraction between the
electron and the proton
– kinetic energy of the electron
• (we ignore kinetic energy of the proton Born-Oppenheimer approximation).
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The Hamiltonian operator for
hydrogen - 1
• H will have 2 terms, for the electron
kinetic energy and the proton-electron
potential energy
H = Te + Vne
• Writing the terms in full, the most
straightforward is Vne :
Vne = -e2/40r
(Coulomb’s Law)
• Note negative sign - attraction
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The Hamiltonian operator for
hydrogen - 2
• The kinetic energy operator will be as
before but in 3 dimensions:
2  2
2
2 
 2  2  2 
Te  
2m  x y z 
• A shorthand version of the term in
brackets is 2.
• We can now re-write Te and the full
expression for H.
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The Hamiltonian operator for
hydrogen – 3
H = Te + Vne
• So, in full:
H = (-ħ2/2m) 2 -e2/40r
• The Schrödinger equation for the H
atom is therefore:
{(-ħ2/2m) 2 -e2/40r}  = E 
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Hamiltonians for molecules
• When there are more nuclei and
electrons the expressions for H get
longer.
• H2+ and H2 will be written as examples.
• Note that H2 has an electron repulsion
term: +e2/40r
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Energies and orbitals
• Solve Schrödinger’s • The expression for
equation using the
the wavefunction is:
Hamiltonian, and an
expression for the (r,,) = R(r) Y(, )
wavefunction, :
• s-functions
don’t
En = -RH/n2
depend
on
the
(n=1, 2, 3 …)
angular part, Y(, );
(RH: Rydberg’s constant)
only depend on R(r).
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Conclusions on lecture
• The Schrödinger equation has been
introduced
(and the Hamiltonian
operator defined), and applied to:
– The particle in a box
– The harmonic oscillator
– The hydrogen atom
• In all cases, the allowed energies are
found to be quantised.
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Final conclusions from the Quantum
Chemistry lectures
• Two important concepts
have been introduced:
wave-particle duality,
and quantisation of
energy.
• In
each
case,
experiments
and
examples have been
given to illustrate the
development of the
concepts.
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