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X An easy way to revise, perfect and test (!) Lewis structures, VSEPR and Symmetry Point Groups X X X X X X X
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X Working with Point Groups D. Nasipun, Stereochemistry of Organic Compounds The Sn Point Groups D. Nasipun, Stereochemistry of Organic Compounds Representa0ons of Point Groups Having a copy of the character tables for the point groups will be very helpful in this sec0on. They can be found in Miessler and Tarr Appendix C, or at the following website (this is only one of many possible sources): [email protected]://www.webqc.org/symmetry.php I have also posted a copy of some common tables on Blackboard Learn. Point group representa.ons show us how the coordinates of an object are aﬀected by the symmetry opera0ons within the point group to which the object belongs. We will look at three kinds of representa0ons for a point group: 1) Matrix representa0on 2) Irreducible representa0on 3) Reducible representa0on Simpliﬁed Matrix Algebra A matrix consists of an ordered array of numbers: Ⱥ1 6Ⱥ
Ⱥ9 4Ⱥ or [4 6 3 8]
Ⱥ
Ⱥ
Mul.plica.on of matrices: A requirement for mul0plying matrices is that the number of columns of the ﬁrst matrix be equal to the number of rows of the second matrix. Ⱥ A11
Ⱥ A
Ⱥ 21 A12 Ⱥ Ⱥ B11 Ⱥ Ⱥ A11B11 + A12 B21 Ⱥ
Ⱥ × Ⱥ B Ⱥ = Ⱥ A B + A B Ⱥ
A22 Ⱥ Ⱥ 21 Ⱥ Ⱥ 21 11
22 21 Ⱥ Cij = ∑ Aik × Bkj Sum, term by term, the products of each row of the ﬁrst matrix by the column of the second matrix. Example: Ⱥ1 6Ⱥ Ⱥ2Ⱥ Ⱥ (1)( 2) + (6)(5) Ⱥ Ⱥ32Ⱥ
Ⱥ9 4Ⱥ × Ⱥ5Ⱥ = Ⱥ(9)( 2) + (4)(5)Ⱥ = Ⱥ38Ⱥ
Ⱥ
Ⱥ Ⱥ Ⱥ Ⱥ
Ⱥ Ⱥ Ⱥ Simpliﬁed Matrix Algebra Another example: Ⱥ1 0 0 Ⱥ Ⱥ x Ⱥ Ⱥ (1)( x) + (0)( y ) + (0)( z ) Ⱥ Ⱥ x Ⱥ
Ⱥ0 − 1 0 Ⱥ × Ⱥ y Ⱥ = Ⱥ(0)( x) + (−1)( y ) + (0)( z )Ⱥ = Ⱥ− y Ⱥ
Ⱥ
Ⱥ Ⱥ Ⱥ Ⱥ
Ⱥ Ⱥ Ⱥ
Ⱥ0 0 − 1Ⱥ Ⱥ z Ⱥ Ⱥ(0)( x) + (0)( y ) + (−1)( z )Ⱥ Ⱥ − z Ⱥ
Ⱥ
Ⱥ Ⱥ Ⱥ Ⱥ
Ⱥ Ⱥ Ⱥ
We will be working with matrices that have non
zero values only along the diagonal. These are easier to work with. Matrix Representa0ons of Point Groups Every symmetry opera.on can be expressed as a transforma.on matrix. The transforma.on matrix can then be used to show the eﬀects of applying the symmetry opera0on to the coordinates x, y, and z. Our goal is not to develop an exhaus0ve treatment of group theory and character tables but to get an understanding of how to apply these ideas to make problems easier to solve and interpret. In order to get a [email protected] idea of how this works we will now look at a couple of examples. Don’t worry, the next 15 slides are a bit mathemaLcal but we will only use these as examples!! Matrix Representa0ons of Symmetry Opera0ons The representa0on of the symmetry opera0ons can be thought of in terms of a matrix. As an example, consider the transforma0ons of a general point x, y, z by the symmetry opera0ons in C3v (e.g., for NH3). Look at the opera0on of σxz x→x’ = x z, C3 y→y’ =
y z→z’ = z Or mathema0cally: x’ = 1x + 0y+ 0z...
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This note was uploaded on 03/28/2014 for the course CHM 2311 taught by Professor Richardson during the Winter '09 term at University of Ottawa.
 Winter '09
 richardson
 Mole

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