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POMA: A Complete Mathematica Implementation of the NMR Product-Operator Formalism
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| Journal of Magnetic Resonance, Series A |
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For weakly coupled spin systems the time evolution of the density operator in a pulsed NMR experiment can be calculated analytically by expressing the Hamiltonian and the density operator in the basis of Cartesian product operators. This approach, which combines a rigorous quantum mechanical treatment with an intuitive classical description, was the basis for the development of a wide variety of intricate NMR experiments. All calculations within the product-operator formalism proceed by repeated application of a small number of simple rules, which describe the time evolution of individual product operators under chemical shifts, scalar couplings, and radiofrequency pulses. Although these calculations are straightforward, the large number of terms arising can make them tedious, and computer support is highly desirable. Recently, a partial implementation of the product-operator formalism in Mathematica, a programming language designed for symbolic computation, has been reported. The present Communication describes a complete, highly flexible Mathematica implementation of the product-operator formalism for spin-I nuclei, POMA ("product operator formalism in Mathematica") which provides analytical results for the time evolution of weakly coupled spin systems under the influence of free precession, selective and nonselective pulses, and phase cycling. The complexity of the problems that can be treated is in practice limited only by the available computer time and memory. The complete Mathematica code is given in Fig. 1. It is apparent that Mathematica offers a particularly concise and elegant way to express the transformations rules of the product-operator formalism. If desired, the results of the calculations can be further processed using the built-in capabilities of Mathematica to handle mathematical expressions.
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