Molecular Symmetry and Group Theory

1. Molecular Symmetry and Group Theory • Assuming that you recognize things with ‘symmetry’ vs those that don’t—we’ll now try to express this concept mathematically (as well as graphically) • Find and USE some models as you work your way through these examples and to help SEE various elements of symmetry. • Did I mention you should use a model kit? USE ONE

2. Some simple definitions…you need to know these to move forward • Symmetry Operation—a movement of a body such that. After the movement has been carried out, every point of the body is coincident with an equivalent point. • Rotate a molecule of BF3 by 120 °. What happens? Does it look the same? • Rotate a water molecule by 180 °. What happens? What if your axis is ‘off’?

3. Elements of Symmetry—5 of Them • There are five elements of symmetry. Some are easier to see than others. • Identity—symbol “E”. Included for the sake of mathematical completeness. • Rotation Axis (Proper Rotation)—symbol Cn; one or more rotations about the axis. • Plane of symmetry—symbol  (greek sigma). Reflection in the plane, like what you saw on the previous panel. • Center of inversion—symbol “i“, exactly what it sounds like. All atoms change X, Y, Z positions. • Improper rotation (tough to see)—Symbol “S”. Rotation followed by an reflection  to the plane of the rotation.

4. Look at this ‘graphically’ • The red element shows that the molecule has been rotated. But it’s STILL THE SAME!! Think about the ‘axis’ of rotation. Where IS the axis? Why does the axis matter SO MUCH? The bottom one ISN’T the same!!

5. Identity—the easiest element to see • The “E” element of symmetry. Do ABSOLUTELY nothing. The molecule looks exactly the same because you’ve done nothing.

6. Proper Rotation Axis n-Fold Rotations: Cn, where n is an integer rotation by 360°/n about a particular axis defined as the n-fold rotation axis. C2 = 180° rotation, C3 = 120° rotation, C4 = 90° rotation, C5 = 72° rotation, C6 = 60° rotation, etc. Rotation of H2O about the axis shown by 180° (C2) gives the same molecule back. Therefore H2O possess the C2 symmetry element.

7. Proper Rotation Axes Cont. • Other molecules—BF3 like we’ve seen already • BF3 has a 3-fold rotation axis, coming out of the plane • Also has THREE C2 axes IN the plane, as shown

8. More examples of Proper Rotation • platinum tetrachloride; PtCl4 adopts a square planar geometry. Two different rotation axes • Principle Axis—largest value of ‘n’ is called the principle rotation axis (we normally denote this as the z-axis). So the principle axis of PtCl4 is C4

9. Symmetry Elements—Mirror Planes “line” and “plane”…same thing. These represent PLANES of symmetry You can also sort of view them as C2 axes

10. Going back to water • There are two different types of mirror plane. • Parallel with principle rotation axis • Perpendicular to PRA • First graphic establishes the principle rotation axis (C2) • Second shows a mirror plane between 3 atoms, no obvious movement • Third—another mirror plane, looks a lot like the C2 axis (and has the same effect). • Because these mirror planes are parallel to C2 (PRA), they are ‘vertical’ mirror planes—v

11. Platinum tetrachloride—v and h • In addition to having a C4 and 4 C2’s, PtCl4 also has several mirror planes—4 v and another mirror plane IN the plane of the board. • This one is a little harder to see • It’s also  to our PRA (in/out plane of the board), so we denote that as h.

12. Inversion, i • Inversion—i—the inversion of ALL atoms through the center of the molecule • Note—the center of a molecule isn’t necessarily an atom • Consider ethane (the staggered form). Pardon the color explosion…but they distinguish each hydrogen and where they go.

13. Improper Rotation axis Sn • Harder to see…but this symmetry element combines two other operations—rotation followed by reflection  to the axis. • Again…consider ethane—staggered.

14. Symmetry Flow Chart—Point Groups