Design and Implementation of Runge–Kutta Methods for MAS NMR Lineshape Calculations

Abstract

The magic angle spinning (MAS) nuclear magnetic resonance (NMR) technique has proven successful for obtaining information about molecular structure and motion in solids. The most advanced description of the MAS NMR experiment involves the density operator and the stochastic Liouville–von Neumann equation. In this equation, the effects of the anisotropic nuclear spin interactions are represented by the Hamiltonian, while the motion involves a stochastic operator. For many systems, molecular motion may be described by a discrete Markov process. In order to obtain a solution for a discrete Markov process we have used a Lie algebra formalism to rewrite the stochastic Liouville–von Neumann equation in the form of a linear homogeneous system of coupled first-order differential equations. This system has periodically time-dependent coefficients and can only be solved by numerical methods. In this paper, we discuss the use of different Runge–Kutta methods to approximate the solution. These methods have the advantages of simplicity and relatively high efficiency and may be implemented with automatic stepsize control. The study involves the most important classes of Runge–Kutta methods including explicit, semi-implicit, and implicit schemes. The results have shown that the choice of the Runge–Kutta method depends on the motion. It is found that explicit Runge–Kutta methods are the most efficient schemes in the slow and intermediate motion regimes. However, in the fast motion regime, the stochastic Liouville–von Neumann equation becomes stiff. In this regime, we have used semi-implicit and implicit Runge–Kutta methods with better stability characteristics. For many semi-implicit or implicit Runge–Kutta methods, the stiff components are represented inaccurately, and the methods are shown to be relatively inefficient. In order to improve the efficiency we have designed and implemented stiffly A-stable Runge–Kutta methods. These have better stability and accuracy characteristics and are useful in the fast motion regime. In another approach, we have rewritten the stochastic Liouville–von Neumann equation in a form with a vanishing stiffness ratio. Based on this form we have designed modified explicit Runge–Kutta methods, which are stable and accurate in the fast motion regime. The results have shown that the Runge–Kutta methods improve the computational efficiency for small systems by a factor between two and five compared with the eigenvalue method. The efficiency increases with the dimension of the system, and for large systems the improvement approaches an order of magnitude. This is an important result because of the long computation times involved in MAS NMR lineshape calculations.