General Procedures for Simulating Spectra

We know from Section 6.7 that the true wave function i/q,. . . , t/r4 are linear combinations of the basis functions. If we begin with symmetrized functions, such as and then each of the ip s can be formed exclusively from symmetric functions or exclusively from antisymmetric functions. Stated another way, functions of different symmetry do not mix. The result is that, like the situation with Fz, many off-diagonal elements of the secular equation must be zero, and the equation factors into several equations of lower order. We shall study an example of this factoring in Section 6.13, when we consider the A2B system. 

Simplification of the spectrum itself also results from the presence of symmetry, because transitions are permitted only between two symmetric or two antisymmetric states. We see in Section 6.13 that there is often a considerable reduction in the number of NMR lines. 

For the two-spin system the only symmetry operation is the interchange of the two nuclei, and the correct linear combinations, and could be constructed by inspection. When three or more symmetrically equivalent nuclei are present, the symmetry operations consist of various permutations of the nuclei. The correct symmetrized functions can be determined systematically only by application of results from group theory. We shall not present the details of this procedure. 

Nevertheless, it is valuable to investigate briefly a few spin systems beyond the AB system to observe their general spectral patterns and to derive some algebraic (as contrasted with digital) expressions that provide insight into the overall behavior of coupled spin systems. 

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