Monte Carlo simulation of DNMR spectra of coupled spin systems Zso ´fia Szalay, Ja ´nos Rohonczy * Department of Inorganic Chemistry, Institute of Chemistry, Eo ¨ tvo ¨s Lora ´ nd University, 112. Pf: 32, H-1518 Budapest, Hungary Received 26 July 2007; revised 4 December 2007 Available online 8 December 2007 Abstract A new program MC-DNMR is presented for the simulation of dynamic nuclear magnetic resonance spectra. The algorithm is a Monte Carlo type method based on the extension of single spin vector model to coupled spin systems. This extension is explained in detail and the theory is justified by examples. The main advantage of this program is the significantly smaller sizes of matrices than that in programs based on density matrix theory. So spectra of systems can be simulated that was impossible previously. Ó 2007 Elsevier Inc. All rights reserved. Keywords: DNMR; Dynamic NMR; Spectrum simulation; Monte Carlo method; Lineshape analysis 1. Introduction Methods for direct simulation of dynamic NMR spectra are well-known for systems with chemical exchange [1–7]. The most widespread simulation programs (DNMR5 [8–11], MEXICO [12–14], WinDNMR [15,16], Bruker’s TOPSPIN DNMR module [17]) are based on the calcula- tion of transitions from the density matrix. The most important limitation of this method is the huge computer memory requirement even for simple spin systems. For example in a non-mutual exchange of three conformers, each containing three coupled spins (the number of spins is N = 9), the size of the ‘supermatrix’ to be diagonalized (without any simplification) would be 2 4N =2 36 . This matrix blocks according to coherence level and these blocks are treated separately. The smaller blocks, the neglection of combinational transitions and the use of sparse matrix diagonalization methods [18,19] reduce the computer memory requirement radically, but the reduced matrix still can be too big for more complicated spin systems. The huge memory requirement of DNMR spectra simu- lation (as compared to static spectra simulation) originates mainly from the fact that the dimension of the matrix to be diagonalized (with all possible simplifications mentioned above) is proportional to s Æ 2 2n where s is the number of sites of non-mutual exchanges and n is the number of spins in one conformer. In this paper the theory and application of a new calcu- lation method MC-DNMR is presented for 1/2 spin nuclei. The main advantage of it is that the required memory is much less in the case of ‘multi-conformational’ systems than the RAM requirement of the methods based on the calculation of density matrix. The reduction is achieved by separating kinetic and scalar coupling. This solves the problem mentioned above: the size of the calculated system is kept at the number of spins in one conformer (n) and the time dependency is handled statistically so the density matrix need not have to be calculated. 2. Theory 2.1. Monte Carlo simulation of dynamic behaviour The description of the dynamic behaviour of the spin system in MC-DNMR program is based on the well- known vector model, where the fid is the result of the pre- cession of net magnetisation vector. When an exchange occurs at time point t ex , the Larmor frequency (x) of the 1090-7807/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jmr.2007.12.002 * Corresponding author. Fax: +36 1 3722 909. E-mail address: rohonczy@chem.elte.hu (J. Rohonczy). www.elsevier.com/locate/jmr Available online at www.sciencedirect.com Journal of Magnetic Resonance 191 (2008) 56–65