1. Lecture 37: Symmetry Orbitals
The material in this lecture covers the following in Atkins.
15 Molecular Symmetry
Character Tables
15.4 Character tables and symmetry labels
(a) The structure of character tables
(b) Character tables and orbital degeneracy
(c) Characters and operators
(d) The classification of linear combinations of orbitals
15.5 Vanishing integrals and orbital overlaps
(a) The criteria for vanishing integrals
(b) Orbitals with nonzero overlaps
(c) Symmetry-adapted linear combinations
Lecture on-line
Symmetry Orbitals (PowerPoint)
Handouts for this lecture

2. One dimensional irreducible
representations have the
character 1 for E.
They are termed A or B.
A is used if the character of the
principle rotation is 1.
B is used if the character of the
principle rotation is -1
A1 has the character 1 for all
operations

3. Irreducible representations with
dimension 2 are denoted E
Irreducible representations with
dimension 3 are denoted T
Number of symmetry species
(irreducible representations) =
Number of classes

4. A px orbital on the central atom
of a C2v molecule and the
symmetry elements of the group.
Epx = 1px; C2px = -1 px
svpx = 1px; sv’ px = -1 px
The irrep. is B1 and
The symmetry b1

5. A py orbital on the central atom
of a C2v molecule and the
symmetry elements of the group.
Epy = 1py; C2py = -1 py
svpy = - 1py; sv’ py = 1 py
The irrep. is B2 and
The symmetry b2

6. A pz orbital on the central atom
of a C2v molecule and the
symmetry elements of the group.
Epz = 1pz; C2pz = 1 pz
svpz = 1pz; sv’ pz = 1 pz
The irrep. is A1 and
The symmetry a1

7. A dxy orbital on the central atom
of a C2v molecule and the
symmetry elements of the group.
Edxy = 1dxy ; C2dxy = 1 dxy
svdxy = -1dxy ; sv’ dxy = - 1 dxy
The irrep. is A2 and
The symmetry a2

8. A 1s+ orbital on the two
terminal atoms
of a C2v molecule and the
symmetry elements of the group.
E 1s+ = 1 1s+ ; C2 1s+ = 1 1s+
sv 1s+ = 1 1s+ ; sv’ 1s+ = 1 1s+
The irrep. is A1 and
The symmetry a1

9. A 1s- orbital on the two
terminal atoms
of a C2v molecule and the
symmetry elements of the group.
E 1s- = 1 1s- ; C2 1s- = -1 1s-
sv 1s- = -1 1s- ; sv’ 1s- = 1 1s-
The irrep. is B2 and
The symmetry b2

10. A 2p- orbital on the two
terminal atoms
of a C2v molecule and the
symmetry elements of the group.
E 2p- = 1 2p- ; C2 2p- = 1 2p-
sv 2p- = -1 2p- ; sv’ 2p- = -1 2p-
The irrep. is A2 and
The symmetry a2

11. The value of an integral I (for example, an area) is independent
of the coordinate system used to evaluate it.
That is, I is a basis of a representation
of symmetry species A1 (or its equivalent).

14. 1. Decide on the symetry species of the
Individual functions f1 and f2 by
reference to the character table, and
write their characters in two rows in
the same same order as in the table

15. 1. Decide on the symmetry species of the
individual functions f1 and f2 by
reference to the character table, and
write their characters in two rows in
the same same order as in the table
2. Multiply the numbewrs in each column,
Writing the results in the same order
2py 1s
3. The new character must be A1
For the integral to be non-zero
2py1s
The symmetry species is B2

16. 2py 1s
2py1s
The symmetry species is A1

18. The character table of a group
is the list of characters of all its
irreducible representations.
Names of irreducible
representations: A1,A2,B1,B2.
Characters of irreducible
representations

19. The integral of the function f = xy
over the tinted region is zero.
In this case, the result is obvious
by inspection, but group
theory can be used to establish
similar results in less obvious
cases.

20. The integration of a function
over a pentagonal region.

21. Two symmetry-adapted linear
combinations of the p-basis
orbitals.
The two combinations each span a
one-dimensional irreducible
representation, and their symmetry
species are different.
Typical symmetry-adapted
linear combinations
of orbitals in a
C 3v molecule.

22. Construct a table showing the effect
of each operation on each orbital
of the original basis
To generate the combination of a
Specific symmetry species, take
Each column in turn and
(I) Multiply each member of the
Column by the character of the
Corresponding operator

23. (I) Multiply each member of the
Column by the character of the
Corresponding operator
(2) Add and divide by group order

24. (I) Multiply each member of the
Column by the character of the
Corresponding operator
(2) Add and divide by group order

25. (I) Multiply each member of the
Column by the character of the
Corresponding operator
(2) Add and divide by group order

26. (I) Multiply each member of the
Column by the character of the
Corresponding operator
(2) Add and divide by group order

27. (I) Multiply each member of the
Column by the character of the
Corresponding operator
(2) Add and divide by group order

28. (I) Multiply each member of the
Column by the character of the
Corresponding operator
(2) Add and divide by group order

29. What you should learn from this course
1. Be able to assign symmetries to orbitals from character
tables.
2. Be able to use character tables to determine whether
the overlap between two functions might be different
from zero.
3. Be able to use character table to construct symmetry
orbitals as linear combination of symmetry equivalent
atomic orbitals