How to read a character table

How to read a character
table!...............Part 1
DR. SOURABH MUKTIBODH
PROFESSOR OF CHEMISTRY
MJB GOVT. GIRL`S POST GRADUATE COLLEGE
INDORE (M.P.)
INDIA

Numbers also have
characters . It is
time now to decode
them.

What is a character table?
Chemistry student learns to develop a character table with or without using complex
mathematical functions.
But
The question is
What is this Character table??
And how to read/ understand it.

IS it as simple (or complex) as to read an
horoscope?
Only astrologers have the answers!!!

And are we really familiar with our
character table? Looks like an horoscope.

And if we understand the components of
character table, how to draw the inferences
for chemistry applications?
I don’t know what does
it mean. Hmmm a
character table?@#*

Well we know, now, that the molecules can be
classified into their peculiar point groups.(as
we are males and females)-

Water molecule belongs to C2v point group that indicates that this type
of structure has a specific symmetry properties. Indeed E, C2,v,v’are
four symmetry elements and operations of this point group. It forms a
group of order four.
Note that-
1. this set of operations forms a “Group” of order 4, i.e. it satisfies the postulates of group theory.
2. Each symmetry operation can be represented in the form of a matrix (matrix representation),
corresponding to Cartesian coordinate system. Thus we have set of matrices now.
3. these matrices can be decomposed to smaller dimensions using a method called as similarity
transformation technique.
4. If it is possible to decompose the matrices into lower dimensions than such representation is
termed as Reducible representation, and if the matrix can not be further decomposed into
smaller dimension than it is termed as irreducible representation.

1 0 0
0 1 0
0 0 1
1 0
0 1 1
3 dimensional
representation
2 dimensional
representation
1 dimensional
representation
Similarity
transformation
technique
Similarity
transformation
technique
Reducible representation Reducible representation Irreducible representation
For example 3d matrix for identity can be decomposed into smaller
dimensions

Matrix A transforms to Matrix B by
similarity transformation method
If a matrix belongs to a reducible representation it can be transformed
so that zero elements are distributed about the diagonal










333231
232221
131211
aaa
aaa
aaa










33
2221
2111
00
0
0
b
bb
bb
The similarity transformation is such that
C-1 AC = B where C-1C=E
Similarity
transformation

Irreducible representations
It is the Irreducible representation, that is of fundamental importance.
Irreducible representation can be of 1D, 2D, 3D etc. as this actually determines the dimensions
of matrices, which cannot be zero or fractional order.
Irreducible representations can be obtained by matrix manipulations.(which we will not discuss
here.)

Now we have a basis for character table.
A Character table is a collection of characters of irreducible representations., of a particular
point group.
Now What is character??

Character is simply the diagonal sum of the elements of
matrix representation.(This matrix representation may be
reducible or irreducible.)
For example, for the following matrices(representations), the characters are-
three, zero and minus one respectively.
1 0 0
0 1 0
0 0 1
1 0
0 −1
−1 0 0
0 1 0
0 0 −1

Lets again have a look on the character
table,(a part is ignored for now)
Notations for
irreducible
representations
characters
Symmetry
elements and
operations
Point group C2v

What we generally know,
The point group of the molecule.(for example for water like symmetries, C2v point gg group
group, and hence symmetry properties.)
What do we not know,
Number of irreducible representations and their dimensions, and of course their characters.
And to know this, What is required?
Matrix representation for every symmetry element of that point group, its decomposition to
smaller dimensions, to obtain its corresponding irreducible representations

Is it going to be cumbersome?
Yes of-course, if you do not have adequate
knowledge of matrix algebra.
Than what?
Not to worry!!!
Fortunately we have “ The great orthogonality theorm”
It allows to know all these things without involving complex mathematics.

The great orthogonality theorem (TGOT)
This theorem gives relation between the entries of the matrices of the irreducible
representations of a group.
It is used for the construction of character tables, i.e., tables of traces of matrices of an
irreducible representation.

Five important points of the theorem
1. The sum of the squares of the dimensions of the irreducible representations of a group is
equal to h (h=order=total sum of symmetry elements and operations)
𝑖=0
𝑖
𝑙2 = ℎ
i.e. 𝑙1
2 + 𝑙2
2 + 𝑙3
2 + ⋯ … … = h
here l1 l2,l3.. Are the dimensions of Irreducible representations.

TGOT-2
2. The sum of the squares of the characters of the irreducible representations of a group is also
equal to h
𝑖=0
𝑖
2 = ℎ
i.e.  1
2 + 2
2 + 3
2 + ⋯ … … = h
here  1 2,  3.. ……… Are the characters of Irreducible representations.

TGOT-3
3. The vectors whose components are the characters of two different irreducible representations
are orthogonal
0
𝑖
𝑖(𝑅)𝑗(𝑅) = 0
Where
𝑖 𝑅 = 𝑐ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟 𝑜𝑓 𝑖𝑡ℎ 𝑖𝑟𝑟𝑒𝑑𝑢𝑐𝑖𝑏𝑙𝑒 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑛𝑔 𝑡𝑜 𝑡ℎ𝑒 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑅
𝑗 𝑅 = 𝑐ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟 𝑜𝑓 𝑗𝑡ℎ 𝑟𝑒𝑑𝑢𝑐𝑖𝑏𝑙𝑒 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑛𝑔 𝑡𝑜 𝑡ℎ𝑒 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑅

TGOT-4 &5
4. In a given representation (reducible or irreducible) the characters of all matrices belonging to
operations in the same class are identical
5. The number of irreducible representations in a group is equal to the number of classes in the
group

Nomenclature for irreducible
representations
The nomenclature for Irreducible representations was introduced by R. S. Mulliken and
is based on the following rules:
1. 1D irreducible representations are indicated by A or B, 2D by E, 3D by T .
2. 1D symmetric [(Cn)=1] with respect to the principal Cn are indicated by A while by
B if antisymmetric [(Cn)=-1]
3. Symmetry with respect to a C2 normal to Cn or to a v is indicated by the subscript 1
(A1, B1 etc.); anti-symmetry is indicated by the subscript 2 (A2, B2 etc)
4. Symmetry with respect to h is indicated by primes (A’) while anti-symmetry is
indicated by double primes (E’’)
5. Symmetry with respect to i is indicated by g (Eg) while anti-symmetry is indicated by
u (Au)
6. E and T require some more labels but they are not easy to assign

Also note that-
1. character of +1 indicate that the basis function remains un- changed as a consequence of
symmetry operation.
2. character of -1 indicate that the basis function has reversed as a consequence of symmetry
operation.(that is x goes to –x etc.)
3. character of 0 indicate that the basis function has undergone more complex changes.

Let us examine a few character tables
1. Trans dicholoroethylene molecule of point group C2h
1. Total sym. Elements and operations=1+1+1+1=4= order of group
2. No of classes=4, one of E, each operation makes a class
3. Total irreducible representations=4 , Ag, Bg, Au and Bu
4. All of them are one dimensional.
5. Note that characters of E (operation) =dimension of irreducible rep.
6. Verify orthogonality, A1*A2=1*1*1+-1*1*1+-1*1*1+ -1*1*1=0
7. Question- why the notations Ag and Au and why not A1 and B1

2. Ammonia molecule of point group C3v
1. Total sym. Elements and operations=1+2+3=6= order of group
2. No of classes=3, one of E, 2of C3 & 3 of v
3. Total irreducible representations=3 , A1, A2 and E
4. A1 and A2 are one dimensional and E as 2 dimensional
5. Note that characters of E =dimension of irreducible rep.=2
6. Verify orthogonality, A1*A2=1*1*1+1*1*2+-1*1*3=0
7. Question- why the notations A1 and A2 and why not A and B

3. BF3 Molecule of point group D3h
1. Total sym. Elements and operations=1+2+3+1+2+3=12= order of group
2. No of classes= 6
3. Total irreducible representations=6 , A1’, A2’ , E’, A1”, A2” and E”
4. A1 and A2 are one dimensional and E as 2 dimensional
5. Note that characters of E =dimension of irreducible rep.
6. Verify orthogonality.
Question- Why primes in notations and not B1, B2 etc

4. Tetrahedral point group (Td)
1. Total sym. Elements and
operations=1+8+3+6+6=24= order of group
2. No of classes= 5
3. Total irreducible representations=5 , A1, A2 , E, T1
and T2
4. A1 and A2 are one dimensional and E as 2
dimensional and T1,T2 as 3dimensional irreducible
rep..
5. Verify orthogonality.
Question- Why characters of
irreducible rep. T1,T2 for
identity are 3 each.

5. Octahedral Point group
Observe yourself

Now we understand that to construct a character table, we
need not be a master of matrix algebra. The great orthogonality
theorem does it quiet easily.
However ,now, still two sections have not been understood yet. Cartesian co-ordinates and
binary product.
The functions to the right are called basis functions.
They represent mathematical functions such as orbitals,
rotations, etc.
Basis function

Orientation of p orbitals
Now we will take “p”
orbitals as a basis to
understand origin of
third column in
character table.

Now let us understand, how can we arrive at the basis function. Start
with px orbital for C2v character table, for example
Symmetry operations are-
1. Identity E- symmetric for all, Character +1
2. Rotation by1800= C2 (along Z axis), Px transforms, character -1
3. Reflection xz = of course x does not change, character +1
4. Reflection yz = of course x does change, character -1
Now see where are such characters, find characters of B1
Thus px transforms as B1
E C2 xz  yz
B1 1 -1 1 -1 x

Now consider py orbital for C2v character table.
2. Rotation by1800= C2 (along Z axis), Py transforms, character -1
3. Reflection xz = of course y not change, character -1
4. Reflection yz = of course y does change, character +1
Now see where are such characters, find characters of B1
Thus py transforms as B2
E C2 xz yz
B2 1 -1 -1 1 py

similarly consider pz orbital as a basis
for C2v character table.
2. Rotation by 1800= C2 (along Z axis), Pz remains unaltered
character +1
3. reflection xz = of course z does not change, character +1
4. reflection yz = of course z does not change, character +1
Now see where are such characters, find characters of A1
Thus pz transforms as A1

In the same way, we consider rotation along
x,y and z axis. Let us consider rotation along z
axis,Rz.
1. For identity character=+1
2. For C2, along z C2v axis, no change, character =+1
3. For xz rotation does change, z goes opposite way, character=-1
4. For yz also, same holds true. Character=-1
Observe the character table-
E C2  ’
A2 1 1 -1 -1 Rz
Similarly we can work out for other rotations also

Similarly binary product comes from
“d” orbitals. Take dxy for an example, for C2v character
table.
1. For identity character all characters=+1
2. For C2, along z axis perpendicular to xy plane no
change therefore character =+1
E, C2 No
change in
strructure

For  yz plane, orbital lobes are reversed. Character=-1
Operation yz
x
y
Similarly for  xz plane, orbital lobes are reversed. Character
is again equal to -1.

So now all we have are,  (E)=1,(C2)=1, (yz)=-1,  (xz)=-1
These are the characters of A1 irreducible representation.
xy
In similar fashion xz goes with B1 and yz goes with B2

Talking about dz
2 and dx2-y2
E
C2
xy
E
C2
xy
xz
xy
Character for all operations remains unaltered. Thus dz2 and dx2-y2 orients as A1
representation.

summary
A character table has the following components-
1. point group of the molecule whose character table is to be constructed.
2. symmetry elements and operations concerning to that point group.
3. classification of symmetry elements and operations in their classes.
4. Number of classes gives no. of irreducible representation.
5. dimensions of irreducible representations are obtained from point 1 of TGOT.*
6. Once dimensions are decided, characters for identity comes immediately. It is 1 for
1D, 2 for 2D and 3 for 3D.
7. other characters are decided from point 2 and 3 of TGOT.
*- TGOT- The great orthogonality theorem.

Summary- contd.
8. this makes 1st and 2nd column of the character table
9. last two columns can be developed by considering orbital symmetry. For
singular dimensions, take s, px, py and pz as a basis for symmetry and
transformation. Identity to which irreducible representation does this belongs.
Similarly for the other columns we take binary orbitals ( for example d
orbitals) as the basis and ternary orbitals (such as f orbitals) and so on. One
can now understand the theoretical importance of the character table.

Summary-contd
This allows us to develop a major part of the character table
Fortunately all such character tables are available and we really do
not need to poke our nose for constructing it. What is required is
to understand logic behind all such terms, to make predictions
concerning various physico-chemical properties of molecules, and
support arguments to theoretical chemistry.

Thanks
More to come- Applications of group theory/ character tables in-
1. Vibrational spectroscopy
2. Crystal field splitting
3. Hybridization
4. Molecular orbital theory and Huckels theory
5. Quantum mechanics

How to read a character table

Recommandé

Recommandé

Contenu connexe

Tendances

Tendances (20)

Similaire à How to read a character table

Similaire à How to read a character table (20)

Dernier

Dernier (20)

How to read a character table