Symmetry and Group Theory - Lecture Notes | CHM 448, Lab Reports of Inorganic Chemistry

Download Symmetry and Group Theory - Lecture Notes | CHM 448 and more Lab Reports Inorganic Chemistry in PDF only on Docsity! Chapter 3 – Symmetry and Group Theory Symmetry Elements and Symmetry Observations All of you have an idea of what is meant by the word symmetry. It conveys a sense of order. We will now discuss a way of quantifying symmetry. Consider the following two figures: and . Both are fairly symmetrical, but is either more so? How can you tell? Before we go too far, we need to briefly address why this topic is important. A very simple consideration is melting points. Consider benzene and cyclohexane. Benzene melts at 4 ºC vs. -95° for cyclohexane. Why? The latter is a larger, heavier molecule, so using standard methods of prediction it should be easier to solidify (even taking into account the π-cloud). To answer this question first imagine the molecules in your mind's eye. Remember a crystal is a solid with ordered packing. Which will pack together most efficiently? As you soon will be able to determine quantitatively, the benzene is significantly more symmetrical. This, along with differences in intermolecular forces, causes benzene to freeze at a much higher temperature. A second and far more important use of group theory is in spectroscopy. You might be familiar with one aspect of this already. More symmetrical isomers tend to have simpler spectra. Consider the 1H NMR spectra of ortho- vs. para-xylenes for example. Most importantly, group theory allows spectral prediction. We won't go into this in detail but we’ll go over an example later on. The symmetry of a molecule can be described in total through the use of 5 operations. Each operation changes the molecule and the question is asked as to whether the new molecule is indistinguishable from the old. I'll take the operations in a different order from the book. In the text that follows and in Experiment 2 I attempt to provide two dimensional images to help you visualize these operations. There is an excellent website linked on the course webpage to provide you with 3-D pictures that you may find helpful. Identity, E- This operation does not change anything. All atoms remain in place. All molecules possess this symmetry element. This has mathematical significance and, so, must be included. Rotational axes, Cn- an imaginary line is passed through the molecule and the molecule is rotated. If the molecule can be stopped in a position such that it is indistinguishable from the original molecule it has 2 a rotation axis. For example, consider water (the hydrogen atoms have been labeled with different colors, but in reality are indistinguishable: H O H H O H 180º C2 C2 n = 360º/rotn = 360º/180º = 2
C2 axis (i.e. 2 rotations regenerate the original molecule) A C2 is called a 2-fold rotational axis. Molecules may have more than 1 Cn axis. e.g. PtCl42- There are also coincident C4 and C2 axes perpendicular to the molecular plane, passing through the platinum atom. the axis with largest n is called the principal rotation axis. Mirror planes, σ - An imaginary plane is passed through a molecule and all atoms are shifted to a spot on the plane. If the result is indistinguishable from the original, the plane has a mirror plane. There are 5 mirror planes in PtCl42- (i) passing through Pt and all four chlorine atoms, (ii) through Pt, Cl1, and Cl3 (⊥ to the molecular plane) (also through Pt, Cl2, and Cl4) (iii) through Pt and bisecting the angles made by Cl1-Pt-Cl2 and Cl3-Pt-Cl4 and ⊥ to the molecular plane (also Cl1-Pt-Cl4 and Cl2-Pt-Cl3) A mirror plane including the principle axis is designated σv, those perpendicular are σh. Inversion Center, i - This is a point in the center of the molecule. If all points x, y, z are inverted (i.e. go to -x, -y, -z) and the molecule retains its structure the molecule has an inversion center PtCl42- has an i, H2O doesn't. Improper Rotation Axis, Sn - The first 4 symmetry elements are relatively easy to imagine, while the improper rotation axis is more difficult. Here the molecule is rotated by some angle then reflected through a σh plane. The book has a good illustration of a S4 axis on page 52. Note S1 and S2 axes don't exist (σ & i, respectively). If a molecule has a point that remains unchanged during all symmetry operations, the molecule is said Pt ClCl Cl Cl C2 Pt Cl4 Cl1 Cl2 Cl3 2