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

CHE-20028: PHYSICAL & INORGANIC CHEMISTRY

QUANTUM CHEMISTRY: LECTURE 3

Dr Rob Jackson

Office: LJ 1.16

[email protected]

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

2


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).

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.

4


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.

5


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!

6

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.

CHE-20028 QC lecture 3 7

William Rowan Hamilton (1805–1865)


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.

8


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:

9


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

2

22

2 xmT

m2

2

10


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.

11


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.

12

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.



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)

14


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.

15

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

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


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 =


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 =


221

2

22

2kx

xm


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.

20



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!

21



• The energy levels are the allowed energies for the system, and are seen in vibrational spectroscopy.

22


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.

23


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).

24


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

25


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

22

2 zyxmTe

26


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

27

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


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).



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.

30


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.

31

Date post:	04-Jan-2016
Category:	Documents
Upload:	ann-turner
View:	222 times
Download:	4 times

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

Documents