How to read a character table

This is for those who wish to understand basics of character table in symmetry and group theory. (in chemistry)

1. 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

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

3. 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.

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

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

6. 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?@#*

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

8. Water molecule belongs to C 2v point group that indicates that this type of structure has a specific symmetry properties. Indeed E, C 2 ,  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.

9.     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

10. 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 The similarity transformation is such that C -1 AC = B where C -1 C=E Similarity transformation

11. 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.)

12. 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??

13. 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.  

14. 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

15. What we generally know, The point group of the molecule.(for example for water like symmetries, C 2v 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

16. 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.

17. 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 .

18. 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.e.   here l 1 l 2, l 3 .. Are the dimensions of Irreducible representations.  

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

20. TGOT-3 3. The vectors whose components are the characters of two different irreducible representations are orthogonal Where  

21. 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

22. 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 [  ( C n )=-1 ]   3. Symmetry with respect to a C 2 normal to C n or to a  v is indicated by the subscript 1 ( A1, B1 etc.); anti-symmetry is indicated by the subscript 2 (A 2 , B 2 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 (A u )   6. E and T require some more labels but they are not easy to assign  

23. 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.

24. Let us examine a few character tables 1. Trans dicholoroethylene molecule of point group C 2h Total sym. Elements and operations=1+1+1+1=4= order of group No of classes=4, one of E, each operation makes a class Total irreducible representations=4 , A g , B g , A u and B u All of them are one dimensional. Note that characters of E (operation) =dimension of irreducible rep. Verify orthogonality, A 1 *A 2 =1*1*1 +-1*1*1+-1*1*1+ -1*1*1=0 Question- why the notations A g and A u and why not A 1 and B 1

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

26. 3 . BF 3 Molecule of point group D 3h Total sym. Elements and operations=1+2+3+1+2+3=12= order of group No of classes= 6 Total irreducible representations=6 , A 1 ’, A 2 ’ , E’, A 1 ”, A 2 ” and E” A1 and A2 are one dimensional and E as 2 dimensional Note that characters of E =dimension of irreducible rep. Verify orthogonality. Question- Why primes in notations and not B 1 , B 2 etc

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

28. 5. Octahedral Point group Observe yourself

29. 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

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

31. Now let us understand, how can we arrive at the basis function. Start with p x orbital for C 2v character table, for example Symmetry operations are- 1. Identity E- symmetric for all, Character +1 2. Rotation by180 = C 2 (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 C 2  xz  yz B 1 1 -1 1 -1 x

32. Now consider p y orbital for C 2v character table . Symmetry operations are- 1. Identity E- symmetric for all, Character +1 2. Rotation by180 = C 2 (along Z axis), P y 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 p y transforms as B 2 E C 2  xz  yz B 2 1 -1 -1 1 p y

33. similarly consider p z orbital as a basis for C 2v character table. Symmetry operations are- 1. Identity E- symmetric for all, Character +1 2. Rotation by 180 = C 2 (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 A 1 Thus p z transforms as A 1

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

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

36. 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.

37. So now all we have are,  (E)=1,(C 2 )=1, ( yz )=-1,  ( xz )=-1 These are the characters of A 1 irreducible representation. xy In similar fashion xz goes with B 1 and yz goes with B 2

38. Talking about d z 2 and dx 2 -y 2 E C 2  xy E C 2  xy  xz  xy Character for all operations remains unaltered. Thus dz2 and dx2-y2 orients as A1 representation.

39. 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.

40. Summary- contd. 8. this makes 1 st and 2 nd 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.

41. 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.

42. 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