Matrix four-spin cases

Together, these 16 product operators describe the 16 matrix elements in the 4 x 4 density matrix representation of a two-spin system (Chapter 10). In the matrix, each element represents coherence between (or superposition of) two spin states. As there are four spin states for a two-spin system (aid s i/3s, P as, and PiPs), there are 16 possible pairs of states, which can be superimposed or share coherence. The product operators are closer to the visually and geometrically concrete vector model representations, so in most cases they are preferable to writing down the 16 elements of the density matrix, especially as only a few of the elements are nonzero in most of the examples we discuss. 

In order to calculate the spin-angular parts of matrix elements of the two-particle operator (1) with an arbitrary number of open shells, it is necessary to consider all possible distributions of shells upon which the second quantization operators are acting. In [2] they are found to be grouped into 42 different distributions, subdivided into 4 different classes. This also explains why operator (1) is written as the sum of four complex terms. The first term represents the case when all second-quantization operators act upon the same shell (distribution 1 in [2]), the second describes the situation when these operators act upon the two different shells (distributions 2-10), third and fourth are in charge of the interactions upon three and four shells respectively (distributions 11-18 and 19-42). Such expression is particularly convenient to take into account correlation effects, because it describes all possible superpositions of configurations for the case of two-electron operator. 

If we return to the case of the diagonal matrix set, we can consider the isotopic species 2FtA(2FIx)3 (spins 1) and add a set of four nuclear quadrupole parameter matrices (traceless, say with Paz = 1-0FIz). It is found that here too, the NMR spectrum is independent of the coupling constants between the equivalent nuclei. 

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... 

The tensors Us and q are mutually orthogonal but do not form a complete set of basis tensors. Thus, only such tensors that are diagonal in the basis of the principal curvatures can be expressed as a linear combination of Us and q. If we take the analogous case of the Pauli spin matrices, we know that four 2x2 basis matrices are required to express an arbitrary 2x2 matrix. They are the 2x2 unit matrix e and the Pauli spin matrices Oy, and of which e and are diagonal, which means that only diagonal matrices can be expressed as linear combinations of e and alone. Thus, it was explicitly admitted in the... 

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