Title: Introductory concepts: Symmetry
1Introductory conceptsSymmetry
2Outline
- 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
3Introduction
- 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.
4Symmetry operations
- What are the possible symmetry operations of a
molecule?
5Symmetry 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)
6Symmetry operations
- What additional operations are possible in a
crystal?
7Symmetry operations Space groups
- Translation
- Screw (translation and rotation)
- Glide (translation and reflection)
- Were focusing on point groups in this lecture.
8From 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?
9From 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?
10From 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
11From 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
12From atom to molecule H2
Yikes!
13From atom to molecule H2
14From atom to molecule H2
- These are the symmetry operations
15From 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
16From 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
17From atom to molecule H2
- These are the characters
- The character under the identity operation tells
you about the degeneracy of the IRep
18From 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)
19From 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)
20From 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.
21From 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!
22From atom to molecule H2
- Well now look at another operation which one
might be most useful?
23From atom to molecule H2
24From 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
25From atom to molecule HF
- What symmetry operations are lost relative to H2?
26From 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
27From 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.
28From 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?
29Correlation
- 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.
30Correlation 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).
31Correlation 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
32Correlation 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!
33Correlation 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?
34Direct 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?)
35Direct 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?
36Direct 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?
37Direct 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
38Direct 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
39Direct 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
40Direct 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?
41Direct 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.
42Direct products Electronic transitions
43Direct 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.
44Direct products Electronic transitions
- Are dipole forbidden transitions ever seen in
reality?
45Infrared 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.
46Infrared 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
47Infrared 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.
48Infrared 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.
49Infrared 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?
50Infrared and Raman modes CH4
- What is the IRep of this mode?
51Infrared 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.
52Local 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.
53Direct 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 ?
54Final 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