1.Symmetry and group theory (Notes 1).pdf - Notes on Symmetry and Group Theory On Monday in class we talked about symmetry elements and operations →

1.Symmetry and group theory (Notes 1).pdf - Notes on...

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Notes on Symmetry and Group Theory * On Monday in class, we talked about symmetry elements and operations These are great for classification, but to really take advantage of symmetry for a variety of applications, we need math * It turns out that symmetry operations that apply to any molecule collectively possess the properties of a mathematical group We will call these groups of symmetry elements "Point Groups", since they involve symmetry operations that leave one point in space unchanged As a result, we need “Group Theory” to describe and predict properties of molecules based on symmetry Mathematical Group = a collection of elements that are related according to certain rules: 4 Properties of a Mathematical Group 1) One group member must commute with all the others and leave them unchanged This is the identity operation ( E ) 2) Every element must have a reciprocal that is also a group member: A · A –1 = A –1 · A = E If R is the reciprocal of S , then S is the reciprocal of R An element can be its own reciprocal (e.g., E is always its own reciprocal) Related theorem: ( ABC ) –1 = C –1 B –1 A –1 o Proof: Say we have the product ABC = D Right-multiply by C –1 B –1 A –1 : ABCC –1 B –1 A –1 = DC –1 B –1 A –1 Simplify: ABCC –1 B –1 A –1 = AB ( E ) B –1 A –1 = A ( E ) A –1 = E = DC –1 B –1 A –1 If E = DC –1 B –1 A –1 , then C –1 B –1 A –1 = D –1 = (ABC) –1 3) The product of any two elements of the group is another element of the group If AB = C , then C is in a group with A / B Order matters! Commutation does not generally apply AB does not necessarily equal BA , so we describe AB as “ B left-multiplied by A ”, meaning we apply operation B first, then A (we move from right to left) 4) Multiplication is associative: ABC = ( AB ) C = A ( BC )
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Notes on Symmetry and Group Theory If we have a complete set of elements, we can write a group multiplication table showing how the different elements combine: Let’s take a real example (ammonia [NH 3 ] molecule), which is in the C 3 v point group C 3 v contains 6 operations: E , C 3 , C 3 2 , σ v (1) , σ v (2) , σ v (3) (assume C 3 rotations are clockwise) For the table below, assume that the column operation is applied first, followed by the row operation (analogous to saying that the column operation is left-multiplied by the row operation): C 3 v E C 3 C 3 2 σ v (1) σ v (2) σ v (3) E E C 3 C 3 2 σ v (1) σ v (2) σ v (3) C 3 C 3 C 3 2 E σ v (3) σ v (1) σ v (2) C 3 2 C 3 2 E C 3 σ v (2) σ v (3) σ v (1) σ v (1) σ v (1) σ v (2) σ v (3) E C 3 C 3 2 σ v (2) σ v (2) σ v (3) σ v (1) C 3 2 E C 3 σ v (3) σ v (3) σ v (1) σ v (2) C 3 C 3 2 E = Kind of like Sudoku Now let’s see whether the C 3 v point group’s operations really do meet the criteria for a group: 1) There is an identity element that commutes with all operations and leaves them unchanged there is an operation E that does this 2) Every element has a reciprocal that is also a group member this is easy to see simply by noting that every row and column contains an E , meaning that every
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