CHE-20004: PHYSICAL CHEMISTRY QUANTUM CHEMISTRY: LECTURE 2 Dr Rob Jackson Office: LJ

2 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 QM lecture 2

3 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 QM lecture 2

4 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 QM lecture 2

5 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 QM lecture 2

6 Schematic of Electron Diffraction Experiment CHE QM lecture 2 Experimental setup Pattern

7 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 QM lecture 2

8 Electron Diffraction Experimental set-up CHE QM lecture 2

9 A typical electron diffraction pattern CHE QM lecture 2 The distances between the rings are used to determine structural information.

10 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 QM lecture 2

An electron beam is aimed at a crystal surface. Electron energies in range eV The electron gun is shown in green The diagram (right) shows a crystal surface and the diffraction pattern obtained. LEED Experimental setup

12 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 QM lecture 2

13 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 QM lecture 2

14 Using the de Broglie equation – (i) How do the wavelengths of an electron and a bus compare? Suppose the electron is travelling at 10 6 ms -1, and the bus at 30 mph, ~ 13 ms -1 m e = 9.11 x 10 –31 kg, m bus ~ kg Calculate in each case, using the de Broglie equation. CHE QM lecture 2

15 Using the de Broglie equation – (ii) For an electron: = x /(9.109 x x 10 6 ) = x m (compare with distances between atoms) For a bus: = x /(15000 x 13) = x m CHE QM lecture 2

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 Calculate the wavelength of each of these molecules (assume v = 210 ms -1 ) CHE QM lecture 216

17 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 QM lecture 2

18 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 QM lecture 2

19 Wave-particle duality - conclusion CHE QM lecture 2

20 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 QM lecture 2

21 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 QM lecture 2

22 Electrons in atoms; how we represent them (i) Orbits: electrons move round the atom following defined paths. Not allowed by Heisenberg’s Uncertainty Principle

23 Electrons in atoms; how we represent them (ii) Orbitals –only the volume and range of possible positions occupied by the electrons can be known:

24 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 QM lecture 2

25 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 QM lecture 2

26 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 QM lecture 2

27 Some hydrogen-like wave functions The wavefunctions are labelled by the quantum numbers n, l and m l e.g. for 3d xy, n=3, l=2 and m l = 1 CHE QM lecture 2

28 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 QM lecture 2

29 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 , 

30 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 QM lecture 2