Schematic illustration of the split propagation method (Algorithm B1) for the evaluation of the final state. The density matrix is factorized according to Eqs. (5) and (9) or (12); the two factors are then sliced and propagated independently. The matrix is reassembled at the head node at the end of the calculation. The best performance is in practice achieved with Eq. (12), which avoids computationally expensive factorizations.
Schematic illustration of the double transpose method (Algorithm B2) for the evaluation of the final state. The density matrix is sliced and propagated under the left side propagator, then transposed, redistributed and propagated under the right side propagator. Compared to the split propagation method, this algorithm requires less processing the head node, but has greater communication requirements.
An illustration of the fact that Eqs. (12)–(14) are exact transformations of the standard density operator propagation procedure in Eq. (1). The figure shows a pulse-acquire 1H NMR spectrum of 3-phenylmethylene-1H, 3H-naphtho-[1, 8-c, d]-pyran-1-one20 at 14.1 T. The results of a parallel calculation using Algorithm A, a single-threaded calculation using the same algorithm and the reference calculation using conventional propagation techniques are the same to machine precision.
Scaling behaviour of the parallel propagation algorithms.
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