RelCalc: symbolic evaluation of BWR theory relaxation rates in python, applications to TROSY effects in AX$$_n$$ spin systems

Relaxation phenomena greatly influence magnetic resonance experiments, from setting repetition rates and measurement times through to measuring internal motion of molecules. RelCalc provides a convenient and efficient means for symbolic and numerical evaluation of relaxation rates using Bloch, Wagness and Redfield (BWR) theory. Users supply interactions within a spin system arranged appropriately in the molecular frame. The effects of optional fluctuations around a symmetry axis, and then under either isotropic or anisotropic global tumbling are then computed. Relaxation rates linking pairs of Liouvillian basis operators are then returned symbolically, formatted as a sum of weighted reduced spectra density functions. Parameters are then supplied to replace symbolic quantities for values for rapid computation of numerical rates. We demonstrate the utility of the software by exploring relaxation interference (TROSY) effects and 'long-lived states' in [Formula: see text] and [Formula: see text] systems, which, for n=1,2, and 3, describe NH, NH[Formula: see text] and CH[Formula: see text] groups in proteins, and small molecules with n=4,6,8,12 and 20 arranged as platonic solids. The implementation is highly efficient with 65,536 relaxation rates required to perform complete Liouvillian simulation of [Formula: see text] methyl groups rotating about a symmetric axis in the presence of adjacent static 'external' spins being symbolically calculated in a few seconds on a single processor on a 2021 laptop. From this result, a complete set of numerical rates are then computed in under one second. RelCalc is implemented in python and is freely available.