Average Interactions

In quantum mechanics, we think of each atom or molecule as having its own wave functions that describe the distribution of its electrons. The expected basis of interaction is that two atoms or molecules react to each other as dipoles, each atom s or molecule s dipolar electric field shining out as 1/r3 with distance r from the center. This dipole interaction averages to zero when taken over the set of electron positions predicted for the isolated atoms. However, when the isolated-atom electron distributions are... 

The major interaction between die nuclear spin magnetic moments in H2 and D2 is the dipolar interaction, equation (8.10). We should at least mention the existence of an electron-coupled scalar interaction this is very small compared with the dipolar interaction, and plays a very minor role in the gas phase measurements. In liquids, however, the dipolar interaction averages to zero, and the scalar coupling becomes the important observable interaction between nuclear spins. The power and range of applications of high-resolution n.m.r. in liquids depends ultimately upon the scalar shielding and spin-spin interactions. 

Modular FORTRAN programs for performing general ab initio multireference single- and double-excitation configuration interaction, averaged coupled-pair functional and linearized couple-cluster method calculations. Cray and other versions. 

We shall examine magnetic dipolar coupling in more detail in Chapter 7, where we show that when the interacting nuclei are in molecules that are in rapid, random motion, as are most small molecules in solution, this interaction averages almost completely to zero. In this chapter and in Chapter 6, we treat only situations in which magnetic dipolar interactions can be ignored. 

The next step is to define the Hamiltonian for the system. For NMR it is easy to write down the Hamiltonian for a set of nuclei in a molecule that is tumbling rapidly. We know that each nucleus interacts with the magnetic field as we discussed in Chapter 2, that direct dipolar interactions average to zero and need not be considered, and that indirect spin coupling between any pair of nuclei can be handled as a scalar product. All of these factors are included in the Hamiltonian given. 

MAS is the most common technique employed in solid-state NMR to remove line broadening. It involves fast mechanical sample rotation about an axis inclined at 54°44 (the value where 3 cos G -1 = 0, Equation 5.4) to the direchon of the external magnehc field. It can remove line broadening from dipolar interactions (averaged... 

For simplicity, only short-ranged two-body Lennard-Jones (LJ) interactions are considered, truncated at a short distance to impose mainly repulsive interactions. Average LJ parameters are used where appropriate in accordance with the conventional averaging... 

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