CHEM3117 Symmetry

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CHEM3117 (Symmetry Quantum)

• OR
• How do we get something for nothing
• Dr Meredith Jordan Rm 241
• m.jordan_at_chem.usyd.edu.au
• 9351 4420

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Overview

• Lectures and topics
• Background, Introductions and Revision
• Lect 1 TWS, MJTJ, SHK
• Spectroscopy in the Born-Oppenheimer Limit
• Lect 1-6 Scott Kable
• Tute 1 SHK tutorial
• Symmetry, Quantum and Bonding
• Lect 7-14 Meredith Jordan
• Tute 2 MJTJ tutorial
• Spectroscopy outside the B-O Limit
• Lect 15-21 Tim Schmidt
• Tute 3 TWS tutorial
• Wrap-up, Revision, and a Glimpse of the Future
• Lect 22-23 TWS, MJTJ, SHK

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Copies of these Slides
These lectures are all available on my Web
site www.chem.usyd.edu.au/mjtj/CHEM3117 Ill
put links in to WebCT
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Assumed knowledge

• CHEM2401
• Topics Harmonic Oscillator
• LCAO theory
• Hückel theory
• MOs of ?-systems
• Equations HO energy levels
• Secular Equations

These lectures are all available on the Web site
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Assumed knowledge

• Scott Kables Lectures (1-6)
• Topics Franck-Condon Principle
• Selection Rules
• Normal Modes
• Equations
• Transition Moment Integral

These lectures are all available on the Web site
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Introduction to Symmetry

• How do we get something for nothing?

Where do spectroscopic selection rules come from?
Dv 1 (harmonic oscillator) DJ 1 (from
ang. momentum of photon) Franck-Condon Factor for
Vibronic excitations
Lectures 2 - 5
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Franck-Condon Factors (Lectures 2 3)
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Franck-Condon Factors (Lectures 2 3)

• 2. The transition moment is derived only from
  the electronic term.
• Consequences
• a) Whether a transition is allowed or not
  depends only on whether the ELECTRONIC transition
  is allowed or not
• b) The vibrational quantum numbers, v, do not
  constrain the transition (no Dv selection rule),
  although clearly the vibrational integral can be
  small or zero, e.g. by symmetry (ME).

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Selection Rules for Electronic Transitions (Lect
3)
There are no vibrational selection rules, so any
Dv is possible.
2
1
Relative vibrational intensities come from the FC
factor
v0
µ21 const. FC factor
v 0
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Selection Rules for Electronic Transitions
2
1
v0
v 0
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Selection Rules for IR Transitions (Lecture 3
2401 on web page)
The simple potential Harm. Osc.
V(x)
n4

• H.O. selection rule
• Dv 1

n3
n2
Dv 1 absorption Dv -1 emission
n1
n0
x
This selection rule can also be understood
intuitively as application of the resonance
condition.
Selection rules limit the number of allowed
transitions
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Selection Rules for IR Transitions (implicit in
Lectures 4 5)
Dv 1 (harmonic oscillator)
in one dimension
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Revision Harmonic Oscillator Wavefunctions
In an Harmonic Oscillator V ½ k y2 Let w
(k/m)½ and x (mw/?)½ y
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Overlap Integral

• Consider overlap of v 0 with the other
  wavefunctions

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Overlap Integral

• When is the Overlap Integral equal to zero?

wavefunctions are ORTHOGONAL
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Key Points

• Symmetry can tell you when an integral is zero
• Wavefunctions that are solutions for a particular
  problem (eg Harmonic Oscillator, Morse
  Oscillator, H atom/Rotational Motion) are
  ORTHOGONAL

• They form a COMPLETE SET of functions
• Any analytic function can be written as an
  expansion of any complete set of functions
  (Fourier Series decomposition)
• This is an extremely useful property

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7.1 Selection Rules for IR Transitions

• Consider transitions from state i 0 to state j
  0

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Selection Rules for IR Transitions

• Consider transitions from state i 0 to state j
  0

ODD x EVEN ODD
x x y0 y1
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Selection Rules for IR Transitions

• Consider transitions from state i 0 to state j
  0

Integral of ODD x EVEN 0 Dv ? 0 ?
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Selection Rules for IR Transitions

• Consider transitions from state i 0 to state j
  1

Good Overlap Integral of ODD x ODD ? 0 Dv 1 ?
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Selection Rules for IR Transitions

• Consider transitions from state i 0 to state j
  2

Integral of ODD x EVEN 0 Dv ? 2 ?
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Selection Rules for IR Transitions

• Consider transitions from state i 0 to state j
  3

Integral of ODD x ODD 0
?1 and ?3 are ORTHOGONAL Dv ? 3 ?
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Key Points

• Symmetry can tell you when an integral is zero
• This can be an extremely useful property
• Vibrations occur in 3-dimensional space
• Generalizing vibrations to polyatomics is tricky
  so go back to the electronic wavefunctions
  first...
• Then come back to polyatomic vibrations

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7.2 Electronic Wavefunctions

• Revision
• Idea of LCAOs and Molecular Orbital Theory
• Labels for Molecular Orbitals
• Lecture 8
• The Variational Principle
• Using The Variational Principle
• Matrix representations
• Symmetry Orbitals

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Key Ideas

• We can describe molecular orbitals and molecular
  wavefunctions in terms of the atomic orbitals,
  fi, on the atoms in the molecule
• y Si cifi
• This works because the atomic orbitals form a
  COMPLETE set, the atomic orbitals, fi, are all
  mutually orthogonal
• The atomic orbitals, fi, can be thought of as
  basis functions for the molecular wavefunction
• The molecular wavefunctions we derive, y, are
  approximations to the exact wavefunction

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CHEM1XXX revision

• Molecular orbitals are constructed as linear
  combinations of atomic orbitals

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CHEM1XXX revision
denotes antibonding

• Simplest picture

Can predict bond strengths qualitatively H2 Bond
Order 1
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CHEM1XXX revision
denotes antibonding

• eg C2, N2, O2, NO, OF etc
• F2

Can predict bond strengths qualitatively F2 Bond
Order 1
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CHEM1XXX revision

• Molecular orbitals may be classified according to
  their symmetry
• Looking end-on at a diatomic molecule, a
  molecular orbital may resemble an s-orbital, or a
  p-orbital.
• Those without a node in the plane containing both
  nuclei resemble an s-orbital and are denoted
  s-orbitals.
• Those with a node in the plane containing both
  nuclei resemble a p-orbital and are denoted
  p-orbitals.

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CHEM1XXX revision

• Molecular orbitals may be classified according to
  their contribution to bonding
• Those without a node between the nuclei are
  bonding.
• Those with a node between the nuclei are
  anti-bonding, denoted with an asterisk, e.g. s.

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Labels for Molecular Orbitals
g (gerade) u (ungerade ) reflect symmetry wrt
inversion

• sg and su pu and pg

asymmetric wrt inversion u
symmetric wrt inversion g
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Key Idea

• The labels we use for the molecular orbitals in a
  molecule reflect the symmetry of the molecule
• Well come back to this

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Copies of these Slides
These lectures are all available on my Web
site www.chem.usyd.edu.au/mjtj/CHEM3117 Ill
put links in to WebCT