Multispin Cross-Correlated Transverse Dipolar NMR Relaxation in Solution

In this paper, we want to consider what would be involved in calculating the R (2) relaxation of amide protons in a protein caused by dipolar interactions with nearby protons, for which there are many. NMR textbooks give analytical equations and sometimes derivations for solution NMR relaxation due to dipolar interactions between two spins. There are also closed equations for dipolar interactions between three spins, which include relaxation interference, also known as cross-correlated cross-relaxation. We here derive an expression for interference between four spins. For larger systems, such as amide protons in a protein, we develop a local-field methodology, from which solution relaxation interference can be computed for a basically limitless number of interacting spins.