Lecture 37: Symmetry Orbitals The material in this lecture covers the following in Atkins.

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) Symmetry Orbitals (PowerPoint) Handouts for this lecture

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

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

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

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

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

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

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

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

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

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).

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

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 1 -1 -1 1 1s+ 1 1 1 1 3. The new character must be A1 For the integral to be non-zero 2py1s+ 1 -1 -1 1 The symmetry species is B2

2py 1 -1 -1 1 1s- 1 -1 -1 1 2py1s- 1 1 1 1 The symmetry species is A1

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

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.

The integration of a function over a pentagonal region.

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.

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

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

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

Lecture 37: Symmetry Orbitals The material in this lecture covers the following in Atkins.