Introductory concepts: Symmetry

1
Introductory conceptsSymmetry

• Jon Goss

2
Outline

• Atomic orbitals to molecular orbitals
• Point group vs space group
• Point groups and diatomic MOs
• Correlation tables
• Little groups and k-point sampling
• Direct products
• Dipole selection rules for optical transitions
• Vibrational mode selection rules

3
Introduction

• From concepts of atomic orbital theory, we
  already have some understanding of the s, p, d,
• These orbitals are all spherically symmetric
• E.g. s, or (px, py, pz)
• For collections of atoms, to a first order
  approximation we can construct molecular orbitals
  which are linear combinations of atomic orbitals
  (LCAO approximation).
• This approach can be very informative.
• It is useful to be able to determine and even
  predict the overall symmetry of the MOs.

4
Symmetry operations

• What are the possible symmetry operations of a
  molecule?

5
Symmetry operations Point groups

• Reflection (sh, sv, sd)
• Rotation (Cn)
• Inversion (i)
• Improper rotation, which are combinations of
  rotation and perpendicular reflection (Sn)
• Also there is an identity operation (E)

6
Symmetry operations

• What additional operations are possible in a
  crystal?

7
Symmetry operations Space groups

• Translation
• Screw (translation and rotation)
• Glide (translation and reflection)
• Were focusing on point groups in this lecture.

8
From atom to molecule H2

• As the most simple example, well look at H2
• When the two H atoms are separated sufficiently
  far that we can treat them as atoms, the
  electrons on each can be considered as a
  spherically symmetric 1s state.
• As they move toward each other to form a bond,
  the two electrons can be modelled as forming
  linear combinations
• ?g(1sa1sb) and ?u(1sa-1sb)
• Which is the lower in energy, and why?

9
From atom to molecule H2

• To move on, we need to have a nomenclature for
  the symmetry of the molecule and those of the
  wave functions.
• First, what are the symmetry operations of H2?

10
From atom to molecule H2

• Identity
• Inversion
• Rotation about the bond through any angle
• Rotation by p about any axis perpendicular to the
  bond, passing through the mid-point of the bond
• Reflection through any plane containing the bond
• Rotation about the bond through any angle,
  followed by reflection in the plane perpendicular
  to the bond-axis containing the mid-point

11
From atom to molecule H2

• It turns out (through consultation with a good
  symmetry book, or http//www.staff.ncl.ac.uk/j.p.g
  oss/symmetry/) that the point group is D8h
• All homo-diatomic molecules (e.g. O2 and N2) have
  this symmetry
• We gain more information about the wave-functions
  from the character table

12
From atom to molecule H2
Yikes!
13
From atom to molecule H2

• Its not so hard

14
From atom to molecule H2

• These are the symmetry operations

15
From atom to molecule H2

• These are the irreducible representations
  (IRep)
• All aspects of the physical object (wave
  functions, normal modes etc) must be
  characterised by one of these

16
From atom to molecule H2

• These are the characters
• These are the traces of the representative
  transformation matrices, but we often use the
  values without explicit use of their origin

17
From atom to molecule H2

• These are the characters
• The character under the identity operation tells
  you about the degeneracy of the IRep

18
From atom to molecule H2

• These are the linear generating functions
• E.g. anything which is linear in z corresponds to
  an A1u (IRep)
• This gives information for dipoles (e.g.
  infrared-activity)

19
From atom to molecule H2

• These are the quadratic generating functions
• As with the linear functions, but corresponding
  to quadratic functions, telling us about second
  order functions including polarisability (Raman)

20
From atom to molecule H2

• For a wave functions of H2, we can determine the
  IReps by applying the symmetry operations to the
  function and determining the parity
• Look at inversion first.

21
From atom to molecule H2

• Remember, ?g(1sa1sb) and ?u(1sa-1sb)
• i ?g?g
• i ?u-?u
• Since both functions are non-degenerate, the
  IReps of a and b must be Ag and Au, respectively.
• However, we are yet to be precise!

22
From atom to molecule H2

• Well now look at another operation which one
  might be most useful?

23
From atom to molecule H2

• Lets look at C2

24
From atom to molecule H2

• Remember, ?g(1sa1sb) and ?u(1sa-1sb)
• C2 ?g?g
• C2 ?u-?u
• Therefore
• ?g corresponds to A1g many-body IRep
• ?u corresponds to A1u many-body IRep
• Hurrah!
• A one-electron picture

25
From atom to molecule HF

• What symmetry operations are lost relative to H2?

26
From atom to molecule HF

• Like all hetero-diatomic molecules, HF has C8v
  symmetry
• The electronic structure of HF is more
  complicated than that of H2 as there are more
  electrons involved

?
H 1s
F 2p
F 2s
F 1s
27
From atom to molecule HF

• We now have a more complicated problem as the
  atomic orbitals we start with include
  degeneracies.
• How does the loss of spherical symmetry in HF
  affect the 2p orbitals? (Choose the HF axis along
  z and consider px, py and pz.)
• This is an elementary example of a crystal field
  splitting.

28
From atom to molecule HF

• Let us assume that the molecule is ionic, HF-.
• The wave functions in order of increasing energy
  are
• F(1s)
• F(2s) ( a little H(1s))
• F(2pz)H(1s)
• F(2px)
• F(2py)
• What are their IReps?

29
Correlation

• If you look carefully at the character tables of
  the H2 and HF molecule examples, youll see that
  the latter is a subset of the former.
• The C8v group is a sub-group of D8h.
• The IReps of the sub-group are all correlated
  with IReps in the main-group.
• For example, the A1g IRep in D8h is correlated
  with A1 in C8v.
• This is a very useful relationship to know about.

30
Correlation Jahn-Teller

• For systems with orbitally degenerate many-body
  states, there is the potential for a reduction in
  the total energy by distorting the structure that
  removes the degeneracy.
• This is the Jahn-Teller effect, and this occurs
  in molecules, solids and importantly for us, in
  point defects.
• The simplest model for the J-T effect can be
  understood from the diagram, representing a
  positively charged vacancy in Si.

t2
EJT
Td
C3v
The ideal MOs can be obtained in same way that
those of H2 and HF were (LCAO).
31
Correlation Jahn-Teller

• The correlation of IReps tells us exactly what
  the IReps in the distorted case will be, but not
  their order.
• There is no need to go through a derivation for
  the IReps, as they are completely specified!

e
a1
Td
C3v
32
Correlation Little groups

• Correlation also serves us in the splitting of
  bands in the Brillouin-zone for non-zero k.
• The wave-functions at the G-point reflect the
  symmetry of the atomic geometry
• At other points, the wave-vector of the electron
  in general acts as distortion
• The symmetry of the wave-functions for a general
  k-point must be a sub-group of that at the
  G-point.
• Therefore the splitting of degenerate band along
  a high-symmetry branch in reciprocal space (such
  as those at the valence band top of a cubic
  material such as diamond, silicon, GaAs,) can be
  qualitatively predicted purely on symmetry
  grounds.
• For example, along the lt111gt branch of a cubic
  material, the little groups are trigonal triply
  degenerate bands are split into e and a.
• Looking at such features may help you spot
  problems in calculations!

33
Correlation Little groups
In the diamond band-structure along G-X, what do
you expect to happen to the four valence bands
which are a and t at the zone centre? Hint what
is happening along y and z?
34
Direct products

• In the final part of the lecture, well look at
  another use of the IReps determining which
  electronic and vibrational transitions are
  optically active.
• To do this we need to know how to combine IReps
  together (i.e. what is the IRep of a two
  functions for which we know the IReps?)

35
Direct products Electronic transitions

• The probability for a transition between
  electronic states ?0 and ?1 coupled by an
  electric dipole (photon) with electric field
  pointing along a given direction v is related to
  ??0v?1dV
• We have already seen how to determine the IReps
  of the wave functions, and actually, weve also
  seen how to get the IRep for the electric dipole
  field (the linear generating function).
• It can be shown that the integral will be exactly
  zero if the IRep of the product is other than
  even parity for all symmetry operations
  generally A, A1, Ag or A.
• This can be qualitatively understood by an
  extension of the idea that the integral between
  symmetric limits of an odd function is always
  zero.
• So, how do we obtain the IRep of the product?

36
Direct products Electronic transitions

• Youll be happy to learn that there is a simple
  method to determine the products simply from the
  character tables.
• Lets take the example of C3v point group.
• What is the direct product of A1 and A2?

37
Direct products Electronic transitions

• You start by calculating the sum over all
  operations of the products of the characters with
  each line in turn
• A1 1x(1x1x1) 2x(1x1x1) 3x(1x-1x1) 0
• A2 1x(1x1x1) 2x(1x1x1) 3x(1x-1x-1) 6
• E 1x(1x1x2) 2x(1x1x-1) 3x(1x-1x0) 0

38
Direct products Electronic transitions

• You divide each sum by the order of the group
  (the number of symmetry operations)
• A1 0/60
• A2 6/61
• E 0/60

39
Direct products Electronic transitions

• The product A1 x A2 contains each IRep this many
  times!
• A1 x A2 0 x A1 1 x A2 0 x E
• Its that easy ?
• In fact is always true that A1 x GX GX.
• Now try E x E

40
Direct products Electronic transitions

• We now have to include all three terms, ?0, v,
  and ?1.
• There are more terms, but the method is the same.
• Is an electric dipole transition allowed between
  two states with A1 and A2 symmetry?

41
Direct products Electronic transitions

• We already know that A1xA2 is A2, and we can see
  that the electric dipole will transform (in
  general) as (A1E).
• We need to see if (A2 x(A1E)) contains A1.
• The normal distributive laws apply, and the
  products commute
• A2(A1E)A2xA1A2xEA2A2xE
• We only need to see if A2xE contains A1
• It is easily shown that A2xEE, so it doesnt.
• A1 to A2 dipolar transitions are completely
  forbidden.

42
Direct products Electronic transitions

• What about A1 to E?

43
Direct products Electronic transitions

• The product of interest is
• (A1 x E x (A1E) ) (E x A1 E x E) E
  (A1A2E)
• Dipole allowed!
• Note, that if we had polarized light along z so
  that the dipole only transforms as A1, the
  transition would not occur only light with
  electric field amplitude in the x-y polarisation
  couples to A1-E transitions.

44
Direct products Electronic transitions

• Are dipole forbidden transitions ever seen in
  reality?

45
Infrared and Raman modes

• The final section is on vibrational mode
  characterisation.
• Vibrational modes are IR-active or Raman active
  depending upon symmetry.
• Formally, the IR-active mode selection rule is
  the same as that of the dipole transitions, but
  now were talking about vibrational wave
  functions, not electronic ones.
• Just like electronic problems, the
  characterisation of which modes can be seen
  experimentally is dependent (at least in part)
  upon the assignment of IReps to the modes of
  vibration.

46
Infrared and Raman modes H2O

• Let us look at the example of water
• Each O-H bond can be viewed as an oscillaor.
• There are two possible combinations (as with the
  two 1s electrons in H2) in-phase and anti-phase.
• We assign the point group first in the interests
  of brevity, Ill tell you that its C2v
• We now apply the operations to the displacement
  vectors

47
Infrared and Raman modes H2O

• Apply the C2 operation
• Then apply sv(xz) (the plane of the molecule)
• Note the symmetry of the molecule is never
  lowered.

48
Infrared and Raman modes H2O

• Again, apply the C2 operation
• Then apply sv(xz) (the plane of the molecule)
• Note, in general the symmetry of the molecule is
  less than C2v during the anti-symmetric stretch.

49
Infrared and Raman modes CH4

• The breathing mode is very simple as the symmetry
  of the molecule is Td at all times.
• A1 symmetry
• Is this IR-active?
• Raman active?

50
Infrared and Raman modes CH4

• What is the IRep of this mode?

51
Infrared and Raman modes CH4

• It turns out that you need three varieties to
  form a degenerate group.
• The symmetry operations map them into on-another,
  or to linear combinations of them.
• These may be tricky to characterise.
• See if you can show that these form a t2
  manifold.
• In IR-spectroscopy, it is this triplet of modes
  that are the high-frequency modes actually
  detected.

52
Local mode replica.

• The final part of this final part is the idea
  that vibrational modes may couple to an
  electronic transition, or convert a
  dipole-forbidden electronic transition into an
  allowed transition.
• We adapt the previous selection rule by adding
  the local mode symmetry to the product
• ??0?0v?1?1dV
• We assume (without any loss of generality) that
  the vibrational ground state is totally
  symmetric.
• We need the IRep product of the two electronic
  states, the dipole operator and the vibrational
  mode.

53
Direct products Mode assisted electronic
transitions

• What about a A1 to A2 transition in C3v with
  coupled to a vibrational mode with A2 symmetry?
• We simply take the product
• A1 x A2 x (A1E) x A2 A2 x (A1E) x A2 (A2
  E)xA2 A1E
• Hurrah allowed ?

54
Final summary

• From this introduction, you have seen some
  important ideas
• Point group symmetry
• The assignment of irreducible representations to
  electronic and vibrational wave functions
• The correlation of IReps
• Jahn-Teller
• Little groups in k-space
• The application of selection rules for
  spectroscopy