The Two-Spin System without Coupling

We now apply the concepts developed in the preceding section to the system of just two nuclei, first considering the case in which there is no spin coupling between them. We digress from our usual notation to call the two spins A and B, rather than A and X, because we later wish to use some of the present results in treating the coupled AB system. The four product basis functions are given in Eq. 6.1. We now compute the matrix elements needed for the secular determinant. Because there are four basis functions, the determinant is 4 X 4 in size, with 16 matrix elements. Many of these will turn out to be zero. For 3CU we have, from 

By the same procedure, the other three diagonal matrix elements (those on the principal diagonal) may be evaluated as 

In the absence of spin coupling, all off-diagonal elements are zero by virtue of the orthogonality of a and /3. This is a general theorem, not restricted to the case of two nuclei. We shall illustrate the result for 3 f12  

With all off-diagonal elements equal to zero, the secular determinant becomes 

It is well known that a determinant in the form of blocks connected with each other only by zeros may be written as the product of factors. Equation 6.16 can thus be factored into four 1X1 blocks, solutions of which are 

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