Simulation of steady-state NMR of coupled systems using Liouville space and computer algebra methods

A series of repeated pulses and delays applied to a spin system generates a steady state. This is relatively easy to calculate for a single spin, but coupled systems present real challenges. We have used Maple, a computer algebra program to calculate one- and two-spin symbolically, and larger systems numerically. The one-spin calculations illustrate and validate the methods and show how the steady-state free precession method converges to continuous wave NMR. For two-spin systems, we have derived a general formula for the creation of double-quantum signals as a function of irradiation strength, coupling constant, and chemical shift difference. The calculations on three-spin and larger systems reproduce and extend previously published results. In this paper, we have shown that the approach works well for systems in literature. However, the formalism is general and can be extended to more complex spin systems and pulses sequences.