che-20028: physical & inorganic chemistry quantum chemistry: lecture 3 dr rob jackson office: lj...
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CHE-20028: PHYSICAL & INORGANIC CHEMISTRY
QUANTUM CHEMISTRY: LECTURE 3
Dr Rob Jackson
Office: LJ 1.16
http://www.facebook.com/robjteaching
CHE-20028 QC lecture 3
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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CHE-20028 QC lecture 3
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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CHE-20028 QC lecture 3
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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CHE-20028 QC lecture 3
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.
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William Rowan Hamilton (1805–1865)
CHE-20028 QC lecture 3
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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CHE-20028 QC lecture 3
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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CHE-20028 QC lecture 3
The kinetic energy operator
• The operator looks like:
• Which means: differentiate the wave function twice and multiply by
• means ‘h divided by 2’ and m is, e.g., the mass of the electron
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2 xmT
m2
2
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CHE-20028 QC lecture 3
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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CHE-20028 QC lecture 3
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.
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http://en.wikipedia.org/wiki/Harmonic_oscillator
Example of a harmonic oscillator: a diatomic molecule
• If one of the atoms is displaced from its equilibrium position, it will experience a restoring force F, proportional to the displacement.
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
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H ------ H
Restoring force and potential energy
• 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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221
2
22
2kx
xm
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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221
2
22
2kx
xm
CHE-20028 QC lecture 3
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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CHE-20028 QC lecture 3
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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CHE-20028 QC lecture 3
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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CHE-20028 QC lecture 3
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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CHE-20028 QC lecture 3
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 = -e2/40r (Coulomb’s Law)• Note negative sign - attraction
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CHE-20028 QC lecture 3
The Hamiltonian operator for hydrogen - 2
• The kinetic energy operator will be as before but in 3 dimensions:
• A shorthand version of the term in brackets is 2.
• We can now re-write Te and the full expression for H.
2
2
2
2
2
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2 zyxmTe
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CHE-20028 QC lecture 3
The Hamiltonian operator for hydrogen – 3
H = Te + Vne
• So, in full:
H = (-ħ2/2m) 2 -e2/40r
• The Schrödinger equation for the H atom is therefore:
{(-ħ2/2m) 2 -e2/40r} = 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: +e2/40r
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Energies and orbitals
• Solve Schrödinger’s equation using the Hamiltonian, and an expression for the wavefunction, :En = -RH/n2 (n=1, 2, 3 …)
(RH: Rydberg’s constant)
• The expression for the wavefunction is:
(r,,) = R(r) Y(, )
• s-functions don’t depend on the angular part, Y(, ); only depend on R(r).
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CHE-20028 QC lecture 3
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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CHE-20028 QC lecture 3
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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