Symmetry, Spectroscopy and Computational Chemistry

Computational Chemistry (Symmetry and Spectroscopy)

The recommended texts for this course:

• Group Theory for Chemists, Kieran C. Molloy, Harwood Publishing, Chichester, either edition 1 or 2 is fine, an online version is accessible through the library or here
• Molecular Quantum Mechanics PW Atkins and RS Friedman, Oxford University Press, Oxford, wither 4th or 5th edition are fine, both books are available in the Library

Preparation

1. lecture notes on symmetry elements, flow chart pdf
2. lecture notes on rotation and character tables pdf
3. flow chart flow-chart
4. character tables character tables
5. 1pg revision notes for matrices: pdf

Exam preparation

1. exam format:
   • 50 marks divided into one 20 mark and two 15 mark questions
   • the format of the questions will be very similar to those provided in the end of lecture problems and tutorial
2. I have provided many questions with detailed model answers as part of in-lecture activities and end of lecture self-study problems as well as tutorial practice questions with answers
3. here is a copy of the 2024 exam and model answer 2024
4. at the end of each set of lecture notes are "Key Points" these give a strong indication of what you need to know!
5. if you want to e-mail me your answers to an old exam question I can give you feedback

Lecture 1

1. notes for the lecture:pdf
2. slides for the lecture:pdf
3. model answers to activities and probems: pdf
4. additional notes: optional
   • symmetry labels: pdf
5. reading: optional
   • Chapter 2: Groups and Representations, in "Group Theory for Chemists"
   • multiplying and using matrices: Chapter 4: Matrices and Matrix Algebra of "Maths for Chemists"
   • Section 5.5: Matrix Representations in Chapter 5 of "Molecular Quantum Mechanics"
6. resources related to this lecture optional
   • Heisenberg developed matrix mechanics (Wiki on Heisenberg) and won the nobel prize in physics in 1932 for the creation of quantum mechanics
   • Dirac noticed the connection between matrix mechanics and the Schrodinger's wave equation and put the two on an equal footing using matrices. Wiki on Dirac, Wiki on Schrodinger. Dirac and Schrodinger shared the nobel prize in physics in 1932 for the creation of quantum mechanics
   • connection to rotation matrices: Wiki on flight dynamics
   • on the importance of symmetry:
     • Wiki on symmetry in general
     • Wiki on Noether's theorem
     • Wiki on symmetry in physics
7. Clusters can have very high symmetry
   • in 1996 the chemistry nobel prize was awarded for the discovery of a new form of carbon, the fullerenes. These are carbon clusters that can form very high symmetry structures, the prototypical fullerene is Buckminsterfullerene C₆₀ which has the shape of a truncated-icosahedron.
   • the icosahedron has iscosahedral symmetry, and 120 symmetry operations. It is an example of a platonic solid, or three dimensional polygon. Each polygon is associated with a symmetry group (a polyhedral group) that leaves the polyhedron invariant. The polyhedral groups are the tetrahedral, octahedral and icosahedral.

Lecture 2

1. notes for the lecture:pdf
2. slides for the lecture:pdf
3. model answers to activities and probems: pdf
4. additional notes: optional
   • infinity groups: pdf
   • crystallography: pdf
   • C3 rotation matrices: pdf
   • equivalence: pdf
5. reading: optional
   • Chapter 2: Groups and Representations, in "Group Theory for Chemists"
   • Section 5.5: Matrix Representations in Chapter 5 of "Molecular Quantum Mechanics"
   • multiplying and using matrices: Chapter 4: Matrices and Matrix Algebra of "Maths for Chemists"
6. resources related to this lecture optional
   • crystal symmetry

Lecture 3

1. notes for the lecture:pdf
2. slides for the lecture:pdf
3. model answers to activities and probems: pdf
4. panopto recording:link
5. additional notes: optional
   • the great orthogonality theorem: pdf
6. reading: optional
   • Chapter 3: Reducible Representations, in "Group Theory for Chemists"
   • Section 10.15: Group theory and molecular vibrations in Chapter 10 of "Molecular Quantum Mechanics"
   • Chapter 4: Techniques of Vibrational Spectroscopy, in "Group Theory for Chemists"
7. resources related to this lecture optional
   • IR spectroscopy is utilised by astronomers to learn more about the universe. Water vapour exists in space and on far planets, and IR light can travel from these distant places to be analysed near earth. Water in the atmosphere blocks key parts of the spectrum, so the IR spectrometers need to be very high in the atmosphere (airborne observatories) or orbiting the earth (space telescopes).
     • SOFIA (Stratospheric Observatory for Infrared Astronomy) is a Boeing 747SP airliner modified to carry a telescope for IR observations at ~12 km in the stratosphere
     • Kuiper is a highly modified C-141A jet transport aircraft operated by NASA to support research in infrared astronomy.
     • Most of the optical telescopes launched into space (such as the Hubble Space Telescope) can also perform infrared observations. NASA's Spitzer Space Telescope is dedicated entirely to IR observations
   • Why is Water Blue?
     • Water absorbs in the red part of the visible spectrum and thus light which pass through and which is reflected from several meters of water appears blue. The red absorptions are due to high overtone and combination states of the vibrational spectrum of water which is shifted by the presence of H-bonding and just penetrates the red end of the visible spectrum. Deuterated water is colourless because the isotope effect is sufficient to shift this vibrational band out of the visible spectrum.
     • sourced from here and published in: J. Chem. Edu., 1993, 70(8), 612

Lecture 4

1. notes for the lecture:pdf
2. slides for the lecture:pdf
3. panopto recording:link
4. model answers to activities and probems: pdf
5. additional notes: optional
6. reading: optional
   • Chapter 5: Group Theory in "Molecular Quantum Mechanics"
   • Chapter 4: Techniques of Vibrational Spectroscopy, in "Group Theory for Chemists"
   • Appendix 1: Projection Operators, in "Group Theory for Chemists"
   • From "Molecular Quantum Mechanics", by Peter Atkins and Ronald Friedman.
     • revision Appendix 8 and 9: The radial and angular wavefunctions
     • revision Appendix 7: The harmonic oscillator: the standard solution
     • Section 10.8: The Vibrations of diatomic molecules
     • Section 10.13: Normal modes
     • Appendix 19: Normal modes: an example in "Molecular Quantum Mechanics"
7. resources related to this lecture optional
   • link to the lab instructions for building and optimising a molecule of NH₃
   • below are animations of the A₁ out-of-plane bending (umbrella) mode and totally symmetric stretching mode of NH₃
     
   • here is an online video combining all the things we have learned and applying it to a molybdenum carbonyl complex Mo(CO)₄[P(OPh)₃]₂, which can exist in the cis- and trans-forms

Assignment

• Assignment
• Released Mon 13th May
• Due (handed in to Chemistry office) by 5pm Fri 30th May
• Model answers

Tutorial 1

1. Tutorial question:pdf
2. model answers for the tutorial: pdf
3. Text book -Chapter 5: The Vibrational Spectrum of Xe(O)F₄, in "Group Theory for Chemists"
4. Relevant paper: Raman, X-ray and computational study: doi:10.1016/j.jfluchem.2011.05.010 "A Raman spectroscopic study of the XeOF₄/XeF₂ system and the X-ray crystal structure of alpha-XeOF₄ · XeF₂", M.J. Hughes, D.S. Brock, H.P Mercier and G.J. Schrobilgen, J. Fluorine Chem., 2011, 132(10), p660
5. A much older paper: doi:10.1063/1.1696273 "Vibrational Spectra and Valence Force Constants of the Square Pyramidal Molecules XeOF₄, IF₅, BrF₅, and ClF₅" G.M. Begun, W.H. Fletcher and D.F. Smith, J. Chem. Phys., 1965, 42 (6), p2236
6. the vibrations!
   

Lecture 5

1. notes from the lecture:pdf
2. slides from the lecture for printing: pdf
3. reading optional
   • From "Molecular Quantum Mechanics", by Peter Atkins and Ronald Friedman.
     • Chapter 6: Sections 6.2, 6.3 and 6.4 for Perturbation theory
     • those who wish to look at degenerate states or time-dependent perturbation theory can complete Chapter 6 : Techniques of Approximation advanced!
     • the Oxford University Press website has student resources from the text, they plot out the solutions to some of the perturbation equations, I'll leave you to play around with the worksheets
     • Section 6.17: The Einstein transition probabilities
     • Appendix 16: Electric dipole transitions
   • Molecular Spectroscopy, John M. Brown, Oxford University Press, Oxford. This is one of the Oxford University primers, so it is a small compact and excellently writen booklet. highly recommended

Lecture 6

1. notes from the lecture:pdf
2. slides from the lecture for printing: pdf
3. please complete the questionnaire
4. model answers to activities and probems: pdf
5. reading optional
   • Chapter 11: Symmetry and Selection Rules, in "Group Theory for Chemists", by Kieran Molloy
   • From "Molecular Quantum Mechanics", by Peter Atkins and Ronald Friedman.
     • Section 5.16: Vanishing integrals
     • Section 5.15: Direct product groups
     • Section 6.17: The Einstein transition probabilities
     • Section 10.8: The vibrational energy levels of diatomic molecules
     • Section 10.10: Vibrational selection rules
     • Section 10.12: Vibrational Raman transitions of diatomic molecules
     • Section 10.13: Normal modes
     • Section 10.14: Vibrational selection rules for polyatomic molecules
     • Appendix 16: Electric dipole transitions
6. resources related to this lecture optional
   • extra notes: extending the description of determining non-zero integrals to degenerate functions pdf
   • extra notes: reminder of the mathematical form of the electronic wavefunction pdf
   • Wiki on Shperical harmonics
   • Wiki on Hermite polynomials
   • Wiki on Laguerre polynomials

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