1. CHE-20028: PHYSICAL & INORGANIC CHEMISTRY QUANTUM CHEMISTRY: LECTURE 3
Dr Rob Jackson
Office: LJ 1.16

2. 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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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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4. 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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5. 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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6. 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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7. 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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8. 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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9. 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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10. 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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11. 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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12. 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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13. 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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14. See http://www.chem.uci.edu/undergrad/applets/dwell/dwell.htm
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
(ii) p = h/  = nh/2L (from de Broglie equation)
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15. 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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16. 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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17. Example of a harmonic oscillator: a diatomic molecule
H H
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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18. 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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19. 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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20. 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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21. 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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22. 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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23. 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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24. 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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25. 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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26. 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.
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27. 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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28. 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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29. (RH: Rydberg’s constant)
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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30. 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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31. 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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