1 Chapter 1: Introduction Practical information: Lecturer: Majed.Chergui@epfl.ch, Office CH H1 625, Tel. 30457. Assistant: Dr Ahmed Ajdarzadeh Oskouei (ahmad.ajdarzadeh@epfl.ch), Office CH G0 598, Tel. 37598. Scope of the course: introduce fundamental concepts about electronic transitions in atoms and molecules. Present the experimental spectroscopic methods for investigating the molecular electronic properties. The treatment cannot be exhaustive, given the vastness of the subject, but at the end of the course you should have developed an idea of how to interrogate specific properties of a molecular system, how to describe it, and in general, where to start if you have to tackle a specific problem involving molecules, later on in your career. A few practical details: • Do not rely on class notes alone • Exercises: it is strongly advised to try and solve them by yourselves, as a way to verify your understanding of the subject. Questions on the exercises are often asked at the exam. • Exam: oral of max. 30 min with 15 mn preparation. Questions are on the topics treated in the course. • The following concepts are considered familiar: Hamiltonian, wave function, operators, angular momentum, matrices and matrix operations, elementary calculus (derivatives, integrals,…), hydrogen atom, perturbation theory. We also assume that you have taken Dr Drabbels courses on quantum chemistry and spectroscopy, eventually, also Dr Moser’s course on Photochemistry. • You are strongly encouraged to ask questions and discuss with the lecturer, the assistant and among yourselves, when the material presented is not clear, when you have a doubt as well as when it is just out of curiosity. Course material: The course is mostly based on:
2 Modern Spectroscopy by J. Michael Hollas (Wiley, 2004) However, elements of the course are also taken from (and more): • Atkins and Friedman, Molecular quantum mechanics • Harris and Bertolucci, Symmetry and spectroscopy This is not a course on quantum mechanics, but it makes extensive use of it. Any good book on quantum mechanics will be a useful reference. I will also hand out lecture notes after the lecture. Lectures are given on the black board. Exercices for the tutorials will be handed out on the last lecture, to be solved for the following week. It is mandatory that you solve them before coming to the lecture. The assistant will ask you in turn to solve them on the blackboard.
3 CONTENTS OF THE COURSE 0. INTRODUCTION 0.1. Presentation, scope and structure of the course, the exercises and the exam. 0.2. Units, length, time and energy scales of molecular physics. Comparison with kT , and other relevant quantities. 0.3. Wavelength and energy ranges of electronic transitions 1. REVIEW OF QUANTUM MECHANICAL CONCEPTS 1.1. Spectroscopy and quantum mechanics. 1.2. The Schrödinger equation and some of its solutions. 1.2.1 The Schrödinger Equation 1.2.2 The hydrogen atom 1.2.3 Electron spin and nuclear spin angular momentum 1.2.4 The Born–Oppenheimer approximation 1.2.5 The harmonic oscillator 2. ATOMIC SPECTROSCOPY 2.1 The periodic table 2.2 Vector representation of momenta and vector coupling approximations 2.3 Russell–Saunders coupling approximation 2.4 jj coupling 3. ELECTRONIC SPECTROSCOPY OF DIATOMIC MOLECULE 3.1. Molecular orbitals 2.2. Classi fi cation of electronic states 3.3. Selection rules of electronic transitions 3.4. Vibrational structure 4. ELECTRONIC SPECTROSCOPY OF POLYATOMIC MOLECULES 4.1. Group Theory 4.2. Molecular orbitals 4.3. Electronic and vibronic selection rules 5. CORE-LEVEL SPECTROSCOPIES 5.1. Photoelectron spectroscopies 5.2. Auger spectroscopy
4 5.3. X-ray fluorescence spectroscopy 5.4. X-ray absorption spectroscopy 5.5. Electron energy loss spectroscopy (EELS) 6. ULTRAFAST LASER SPECTROSCOPY 6.1. Coherence and concept of wave packets (rotational, vibrational, electronic, phonons) 6.2. The pump-probe method 6.3 Other experimental schemes
5 Orders of magnitude Units and constants It is important, from the start, to get an idea of what we are going to talk about and to develop a feeling for the energy, length, time… scales that we are going to be dealing with. It is also important to learn to estimate what the order of magnitude of the answer to a problem might be, before actually solving the problem itself. For example, 1 litre of water is about 1 Kg. • Divide by the molar mass (18 g) and get about 55 moles/litre • Multiply by Avogadro’s number ( N A ≈ 6 . 022 × 10 23 mol -1 ) and get ≈ 3 . 3 × 10 25 molecules/litre. • Divide 1 Kg by this number and get 3.0 × 10 - 26 Kg as the mass of a water molecule. • Divide the volume of one litre of water (1 × 10 -3 m 3 ) by the number of molecules ( ≈ 3 . 3 × 10 25 ) and get the approximate volume of a molecule ≈ 3 . 0 × 10 - 29 m 3 . • Take the cubic root of this volume and get ≈ 3 × 10 -10 m as the size of the molecule. • Divide by the number of bonds (2 for 3 atoms) and you get the average distance between atoms (bond length) ≈ 1 × 10 - 10 m. (the agreement with the real value is quite good, maybe a bit fortuitous). • Take Δ p Δ x ≈ ħ ( ≈ 1 . 05 × 10 -34 J s). Say Δ x for the electrons is of the order of the bond length. Then Δ p ≈ 1 × 10 -24 Kg m s − 1. • Divide Δ p by the electron mass (9.11 × 10 -31 Kg) and get an “electron velocity” of the order of 1 × 10 6 m s -1 ≈ 3 . 8 × 10 -3 c . In most cases, non- relativistic quantum mechanics should be a reasonable choice. • Take Δ p 2 / 2 m and get a corresponding kinetic energy of 6 × 10 -19 J • Divide by h and get the frequency of a photon with the same energy: ν ≈ 9 × 10 14 Hz ~ ≈ 3 × 10 4 cm -1 . This is the • Divide further by c and get the wavenumber number of waves within a cm, i.e. the inverse of the wavelength of the photon. Spectroscopists use this unit a lot, to the point that they often use it interchangeably for frequency and energy as well, which is not strictly correct, and you better not take this habit, but you should be aware of it nevertheless.
6 • Take the inverse and get the wavelength of the photon in vacuum: λ ≈ 330 nm. • Now, let’s say vibrational motion is confined to 1/10 of the bond. But the mass of a proton is ≈ 2000 times bigger than that of the electron. This ~ ≈ 1500 gives typical energies frequencies and wavenumbers 20 smaller ( cm − 1 ) and wavelengths 20 times larger ( λ ≈ 6 μ m). • Likewise, rotational motion is confined to something of the order of the bond. We still take the mass of a proton. That makes rotational energies etc. about 100 times smaller than vibrational energies ( ≈ 15 cm -1 ) and proportionally smaller for larger molecules. • amuse yourselves with calculating other interesting quantities (for example, the energy required to bring 1 Kg of molecules to the excited electronic state) Molecular sizes and masses are comprised between those of H 2 , 2 atoms, 2 a.m.u, 0.074 nm bond length, and those of DNA, 10 5 − 10 6 atoms, 10 6 − 10 7 a.m.u, 20 nm length. Energy units:
7 Effects of the electromagnetic radiation on transitions between quantum levels:
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