Symmetry Operations and Distinguishability

Before we can begin to predict the appearance of NMR spectra, we must be able to recognize when the nuclei (i.e., atoms) in a given structure will be distinguishable and when they will not. The test of distinguishability is based on symmetry relations among the nuclei, and it is these we will now explore. 

Perhaps the best one-word synonym for symmetry is balance, but our use of the term requires a more detailed analysis. A symmetry operation is defined as some actual or imagined manipulation of an object that leaves it completely indistinguishable from the original object in all ways, including orientation. Doing nothing to an object leaves it indistinguishable, but we will not consider this trivial identity operation further. Symmetrical objects are said to possess symmetry elements, which are centers (points), axes, or planes that help describe symmetry operations. An object that possesses no symmetry elements is labeled asymmetric. 

Here is another manipulation we could perform. Suppose we were to pass an imaginary mirror directly through the center of the sphere. Certainly the mirror image of the left half would be indistinguishable from the actual right half, and vice versa. Thus, a sphere is said to have a plane of symmetry 

No other objects are as perfectly symmetrical as a sphere. Still, many objects, and most molecules, have at least some symmetry. Let s discuss a few examples. 

Chloromethane, structure 4-1, has a nearly tetrahedral structure, where the chlorine and hydrogen atoms form the comers of a pyramid (tetrahedron) with carbon at its center. 

Notice also that if we were to rotate 4-1 by any angle other than an integer multiple of 120°, the molecule would be distinguishable from the original. For example, rotation of the 

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It is important to remember to distinguish between a symmetry operation and its representation by a matrix. The latter depends on the coordinate system adopted. 

It is important to distinguish between mmetiy properties of wave functions on one hand and those of density matrices and densities on the other. The symmetry properties of wave functions are derived from those of the Hamiltonian. The "normal" situation is that the Hamiltonian commutes with a set of symmetry operations which form a group. The eigenfunctions of that Hamiltonian must then transform according to the irreducible representations of the group. Approximate wave functions with the same symmetry properties can be constructed, and they make it possible to simplify the calculations. 

For each symmetry element of the second kind (planes of reflection and improper axes of rotation) one counts according to Eq. (1) the pairs of distinguishable ligands at ligand sites which are superimposable by symmetry operations of the second kind. 

This set of transformation operators Om associated with the symmetry operations of a given point group will therefore have a group table which is structurally the same as the one for the point group. In the next section we show that, if we introduce a coordinate system into the function space chosen for the 0M, we can define explicitly a set of 0M satisfying eqns (5-6.1) to (5-6.6). The reader is warned that since the correspondence between R and 0M is so close, many books (incorrectly) do not distinguish between them. 

This is a most useful result since we often need to calculate the inverse of a 3 x3 MR of a symmetry operator R. Equation (10) shows that when T(R) is real, I R)-1 is just the transpose of T(R). A matrix with this property is an orthogonal matrix. In configuration space the basis and the components of vectors are real, so that proper and improper rotations which leave all lengths and angles invariant are therefore represented by 3x3 real orthogonal matrices. Proper and improper rotations in configuration space may be distinguished by det T(R),... 

Here is the most important thing to remember. Atoms (i.e., nuclei) that are related by virtue of a symmetry operation are said to be symmetry (or chemically) equivalent. As such, they are indistinguishable in all respects, and they are isochronous that is, they precess at exactly the same frequency. Thus, symmetry-equivalent nuclei cannot be distinguished by NMR. Returning to the case of chloromethane, the three hydrogens are symmetry equivalent by virtue of both the C3 axis and the symmetry planes and therefore give rise to only one signal in the H NMR spectrum of the compound. 

All molecules can be described in terms of their symmetry, even if it is only to say they have none. Molecules or any other objects may contain symmetry elements such as mirror planes, axes of rotation, and inversion centers. The actual reflection, rotation, or inversion is called the symmetry operation. To contain a given symmetry element, a molecule must have exactly the same appearance after the operation as before. In other words, photographs of the molecule (if such photographs were possible ) taken from the same location before and after the symmetry operation would be indistinguishable. If a symmetry operation yields a molecule that can be distinguished from the original in... 

So called to distinguish them from certain microscopic symmetry operations with which we are not concerned here. The macroscopic elements can be deduced from the angles between the faces of a well-developed crystal, without any knowledge of the atom arrangement inside the crystal. The microscopic symmetry elements, on the other hand, depend entirely on atom arrangement, and their presence cannot be inferred from the external development of the crystal. 

Due to the inclusion of the spin-orbit coupling there is no need to distinguish between rotations in real and spin space. As can be seen in Table 5.2, the operations c r can be accompanied by the time reversal T. Because the operator T reverses the direction of the magnetic moments, time reversal T itself is of course no symmetry operation. 

Apart from the symmetry elements described in Chapter 3 and above, an additional type of rotation axis occurs in a solid that is not found in planar shapes, the inversion axis, n, (pronounced n bar ). The operation of an inversion axis consists of a rotation combined with a centre of symmetry. These axes are also called improper rotation axes, to distinguish them from the ordinary proper rotation axes described above. The symmetry operation of an improper rotation axis is that of rotoinversion. Two solid objects... 

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