Molecular Symmetry and Group Theory

Reprinted 1978, 1979, 1981, 1983, 1985, 1987, 1988, 1990, 1992, 1993, 1996 (twice), 1997, 1998.

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The first edition of this book was well received by both students and teachers. The second edition, therefore, has required only minor changes to the first seven chapters. In these I have put more emphasis on the idea of the basis of a reducible representation and have clarified a few small ambiguities which reviewers have pointed out. The diagrams have also been completely re-drawn. The major addition in this edition is a completely new chapter on linear combinations. This not only introduces the projection operator method as the rigorous approach to finding the form of vibrations, wave functions, etc., but goes on to develop a simplified approach to the subject making direct use of the character table. Again the emphasis is on the application of the techniques to real chemical problems rather than on the mathematics of the method. I hope that this will give readers an enthusiasm for symmetry methods and encourage them to learn more via the excellent advanced texts cited in the bibliography.

Finally I would like to thank the (often anonymous) reviewers whose comments have been helpful in the process of revision and all the staff at John Wiley & Sons for their patience as I failed to meet various deadlines.

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Programme 1

Symmetry Elements and Operations

Objectives

Assumed Knowledge

Symmetry Elements and Operations

1.1 The idea of symmetry is a familiar one, we speak of a shape as being “symmetrical”, “unsymmetrical” or even “more symmetrical than some other shape”. For scientific purposes, however, we need to specify ideas of symmetry in a more quantitative way.

1.2 If you said A, it shows that our minds are at least working along similar lines!

We can put the idea of symmetry on a more quantitative basis. If we rotate a piece of cardboard shaped like A by one third of a turn, the result looks the same as the starting point:

Since A and A′ are indistinguishable (not identical) we say that the rotation is a symmetry operation of the shape.

Can you think of another operation you could perform on a triangle of cardboard which is also a symmetry operation? (Not the anticlockwise rotation!)

1.3 Rotate by half a turn about an axis through a vertex i.e. turn it over

How many operations of this type are possible?

1.4 Three, one through each vertex.

We have now specified the first of our symmetry operations, called a PROPER ROTATION, and given the symbol C. The symbol is given a subscript to indicate the ORDER of the rotation. One third of a turn is called C₃, one half a turn C₂, etc.

What is the symbol for the operation:

1.5 C₄. It is rotation by of a turn.

A symmetry operation is the operation of actually doing something to a shape so that the result is indistinguishable from the initial state. Even if we do not do anything, however, the shape still possesses an abstract geometrical property which we term a symmetry element. The element is a geometrical property which is said to generate the operation. The element has the same symbol as the operation.

What obvious symmetry element is possessed by a regular six-sided shape:

1.6 C₆, a six-fold rotation axis, because we can rotate it by of a turn

One element of symmetry may generate more than one operation e.g. a C₃ axis generates two operations called C₃ and :

What operations are generated by a C₅ axis?

1.7

What happens if we go one stage further i.e. ?

1.8 We get back to where we started i.e.

The shape is now more than indistinguishable, it is IDENTICAL with the starting point. We say that , or indeed any , where E is the IDENTITY OPERATION, or the operation of doing nothing. Clearly this operation can be performed on anything because everything looks the same after doing nothing to it! If this sounds a bit trivial I apologise, but it is necessary to include the identity in the description of a molecule’s symmetry in order to be able to apply the theory of Groups.

We have now seen two symmetry elements, the identity, E, and a proper rotation axis C_n. Can you think of a symmetry element which is possessed by all planar shapes?

1.9 A plane of symmetry.

This is given the symbol σ (sigma). The element generates only one operation, that of reflection in the plane.

Why only one operation? Why can’t we do it twice – what is σ²?

1.10 σ² = E, the identity, because reflection in a plane, followed by reflection back again, returns all points to the position from which they started, i.e. to the identical position.

Many molecules have one or more planes of symmetry. A flat molecule will always have a plane in the molecular plane e.g. H₂O, but this molecule also has one other plane.

Can you see where it is?

AT THIS STAGE SOME READERS MAY NEED TO MAKE USE OF A KIT OF MOLECULAR MODELS OR SOME SORT OF 3-DIMENSIONAL AID. IN THE ABSENCE OF A PROPER KIT, MATCHSTICKS AND PLASTICINE ARE QUITE GOOD, AND A FEW LINES PENCILLED ON A BLOCK OF WOOD HAVE BEEN USED.

1.10a You were trying to find a second plane of symmetry in the water molecule:

1.11 σ is the plane of the molecule, σ′ is at right angles to it and reflects one H atom to the other.

The water molecule can also be brought to an indistinguishable configuration by a simple rotation. Can you see where the proper rotation axis is, and what its order is?

1.12 C₂, a twofold rotation axis, or rotation by half a turn.

A C₂ axis passing through space is the hardest of all symmetry elements to see. It will be much easier to visualise if you use a model of the molecule.

This completes the description of the symmetry of water. It actually has FOUR elements of symmetry – one of which is possessed by all molecules irrespective of shape. Can you list all four symmetry elements of the water molecule?

1.13 E C₂ σ σ′ Don’t forget E!

Each of these elements generates only one operation, so the four symbols also describe the four operations.

Pyridine is another flat molecule like water. List its symmetry elements.

1.14 E C₂ σ σ′ i.e. the same as water.

Many molecules have this set of symmetry elements, so it is convenient to classify them all under one name, the set of symmetry operations is called the C_2v point group, but more about this nomenclature later.

There is a simple restriction on planes of symmetry which is rather obvious but can sometimes be helpful in finding planes. A plane must either pass through an atom, or else that type of atom must occur in pairs, symmetrically either side of the plane. Take the molecule SOCl₂, which has a plane, and apply this consideration. Where must the plane be?

1.15 Through the atoms S and O because there is only one of each:

The molecule NH₃ possesses planes. Where must they lie?

1.16 Through the nitrogen (only one N), and through at least one hydrogen (because there is an odd number of hydrogens). Look at a model and convince yourself that this is the case.

A further element of symmetry is the INVERSION CENTRE, i. This generates the operation of inversion through the centre. Draw a line from any point to the centre of the molecule, and produce it an equal distance the other side. If it comes to an equivalent point, the operation of inversion is a symmetry operation, e.g. ethane in the staggered conformation:

N.B. The operation of inversion cannot be physically carried out on a model.

Which of the following have inversion centres

1.17 Only B and D e.g., for C, the operation i would take point x to point y which is certainly not equivalent:

An inversion centre may be:

a. In space in the centre of a molecule (ethane, benzene); or

b. At a single atom in the centre of the molecule (D above)

If it is in space, all atoms must be present in even numbers, spaced either side of the centre. If it is at an atom, then that type of atom only must be present in an odd number. Hence a molecule AB₃ cannot have an inversion centre but a molecule AB₄ might possibly have one.

Use this consideration to decide which of the following MIGHT POSSIBLY have a centre of inversion.

NH₃ CH₄ C₂H₂ C₂H₄ SOCl₂ SO₂C1₂

1.18 CH₄, C₂H₂, C₂H₄, SO₂C1₂ fulfil the rules, i.e. have no atoms present in odd numbers, or have only one such atom.

Which of these actually have inversion centres?

1.19 Only C₂H₂ and C₂H₄. Both have an inversion centre midway between the two carbon atoms.

What is the operation i²?

1.20 i² = E, for the same reason that σ² = E (Frame 1.10).

We now have the operations E, σ, C_n, i. Only one more is necessary in order to specify molecular symmetry completely. That is called an IMPROPER ROTATION and is given the symbol S, again with a subscript showing the order of the axis. The element is sometimes called a rotation-reflection axis, and this describes the operation very well.

The S_n operation is rotation by 1/n of a turn, followed by reflection in a plane perpendicular to the axis, e.g. ethane in the staggered conformation has an S₆ axis because it is brought to an indistinguishable arrangement by a rotation of 1/6 of a turn, followed by reflection:

N.B. Neither C₆ nor σ are present on their own.

₆