Bloch-Redfield-Wangsness theory engine implementation using symbolic processing software

Abstract

We describe a general method for the automated symbolic processing of Bloch-Redfield-Wangsness relaxation theory equations for liquid-phase spin dynamics in the algebraically challenging case of rotationally modulated interactions. The processing typically takes no more than a few seconds (on a contemporary single-processor workstation) and yields relaxation rate expressions that are completely general with respect to the spectral density functions, relative orientations, and magnitudes of the interaction tensors, with all cross-correlations accounted for. The algorithm easily deals with fully rhombic interaction tensors, and is able, with little if any modification, to treat a large variety of the relaxation mechanisms encountered in NMR, EPR, and spin dynamics in general.

Design Principles and Theory of Paramagnetic Fluorine‐Labelled Lanthanide Complexes as Probes for 19F Magnetic Resonance: A Proof‐of‐Concept Study

“…Although the processes listed above do, in most cases, suffice, a complete description can only be claimed if the full relaxation super-operator treatment is performed [39,40] and the system stays within the validity range of the BRW theory.…”

Polynomially scaling spin dynamics II: Further state-space compression using Krylov subspace techniques and zero track elimination

“…If a computationally efficient way can be found to map out those subspaces, the size of the simulation is likely to be reduced. While the spin system level pruning and the resulting polynomially scaling simulation algorithm are a recent development [10], the trajectory level pruning (which, on its own, scales exponentially 1 ), has a long history, starting with the Lanczos/Arnoldi procedure proposed by Freed et al [13,20] and the general Krylov subspace techniques used in molecular spectroscopy, electrical circuit analysis and control systems theory [18,19]. We will now outline the connection between the algebraic operations performed in a spin dynamics simulation and the general theory of Krylov subspaces.…”

“…The current state of magnetic resonance theory can be described as comfortable-most pulse sequences and spin dynamics experiments in both solid and liquid state can be simulated with high accuracy [1][2][3][4][5][6][7][8], at least numerically [1,3,[6][7][8] and in many cases analytically [9,10]. The only hard limit is the available computing power-while many approximations succeed in reducing computation time by large factors [8,[11][12][13], and many special cases can be dealt with efficiently [14][15][16], the asymptotic scaling is in most cases exponential [11] and accurate simulations of arbitrarily coupled systems with more than $10 spins are difficult to perform.…”

Polynomially scaling spin dynamics simulation algorithm based on adaptive state-space restriction

“…The most popular approaches include the diagonalization [5,4], direct propagation [6,2], many-body theory [7,8], perturbation theory [9,10] and Campbell-Baker-Hausdorff series techniques [11,12], frequently adapted to include spatial [13], temporal [14] and permutation [15][16][17] symmetry, as well as a host of analytical [18,19] and numerical [1,20,21] techniques for pulse sequence design and spin system analysis. Such simulations have been successfully used for decades and excellent reviews exist, outlining the theory, implementation and programming in fine detail [2][3][4].…”