Numerical simulations of spin dynamics

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Jessica Carpenter
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1 Numerical simulations of spin dynamics Charles University in Prague Faculty of Science Institute of Computer Science Spin dynamics behavior of spins nuclear spin magnetic moment in magnetic field quantum mechanical problem density matrix Hamiltonian 1

2 Nuclear magnetic resonance local microscopic method with atomic resolution structure, chemical analysis, dynamics in solution as well as solid materials noninvasive imaging of living objects and much more... Numerical simulations when analytical tools fail or become too complicated Extracting structural and dynamical properties from spectra fitting Insight into the underlying spin dynamics magnetization flow in TROSY-ST2-PT Zn[SPhMe] 4 (Me 4 N) 2 Performance tests of new pulse sequences DQ filters Design of new pulse sequences 2

3 Optimal Control Spin-state-selective coherence transfer driven by dipolar interactions R O R O N C C N C C H H H H ubiquitin J CαC 55 Hz different samples different probes Kehlet et al., J Phys Chem Lett, 2011, 2, 543 SIMPSON virtual NMR spectrometer original paper: close to 750 citations SIMPSON: Bak, Rasmussen, Nielsen, J. Magn. Reson. 147, 296 (2000). SIMPSON with Optimal Control: Tosner, Vosegaard, Kehlet, Khaneja, Glaser, Nielsen, J. Magn. Reson. 197, 120 (2009). 3

4 NMR interactions and Hamiltonian molecule in solution in solid phase chemical shift J-coupling anizotropic chemical shift J-coupling anizotropic dipole-dipole interactions Hamiltonian usualy very sparse simple NMR experiments long periods of constant H Hamiltonian still quite sparse complicated NMR experiments always time-dependent H Basic approaches Hilbert space Liouville space Memory: density matrix for 15 spins 1/ = 16 GB CPU time: matrix exponential matrix matrix or matrix vector operations algorithms: reuse of propagators γ - COMPUTE block-diagonalization parallelization: powder averages mathematical operations 4

5 Matrix exponential Propagators diagonalization by eigen-decomposition golden standard, works always... H is real symmetric Padé approximation requires scaling & squaring time consuming matrix division Taylor expansion requires scaling & squaring only matrix multiplications, can be efficient when discarding small elements Chebyshev expansion requires scaling and shifting (eigenvalues between -1 and +1) only matrix multiplications (possibly sparse dense)?... Lanczos? Constant Hamiltonian Liouville space Hilbert space (similarity transform) evaluate in propagator eigenbasis Krylov propagation calculate propagator action without evaluating matrix exponential 5

6 Constant Hamiltonian Liouville space more tricks decomposition into non-interacting subspaces identify relevant subspaces for simulation L 1 L 2 smaller matrices skip empty/non-observable subspaces Time dependent Hamiltonian Hilbert space piece-wise constant approximation Magic angle spinning & complicated rf irradiation 1 H CP R decoupling t 1 13 C CP T/2 T/2 6

7 Hamiltonian periodicity No periodicity no way around re-evaluation of propagators propagators are re-used propagator of period efficiently evaluated at eigenbasis of period propagator Hamiltonian and MAS single interaction total interaction Hamiltonian Wigner coefficients can this structure be used? eigenvalues estimation evolution of eigenvectors interpolation combination with rf irradiation 7

8 Summary Most time consuming operation is calculation of propagators how to do it best? Propagators are most efficiently re-used in their eigenbasis eigendecomposition is necessary (or?) Calculations in Liouville space can be faster due to matrix-vector operations, possible subspace decomposition and exploiting high sparsity not used due to necessity of eigendecomposition 8

Numerical Simulation of Spin Dynamics

Numerical Simulation of Spin Dynamics Marie Kubinova MATH 789R: Advanced Numerical Linear Algebra Methods with Applications November 18, 2014 Introduction Discretization in time Computing the subpropagators