Future Theoretical Approaches in Nuclear Magnetic Resonance
Author(s)
Eugene Stephane Mananga*
ABSTRACT
Solving a time-dependent linear differential equation towards obtaining evolution operators is a central problem in solid-state nuclear magnetic resonance. To this end, average Hamiltonian theory and Floquet theory have been the two commonly used theoretically methods in spin dynamics of NMR. We recently introduced the Floquet-Magnus expansion approach and here, we present the methodology of potentials future theoretical approaches such as the Fer expansion, Chebyshev expansion and Cayley transformation that could be useful tools for numerical integrators and simulations of spin dynamics in NMR.
KEYWORDS
Solid-State NMR; Average Hamiltonian Theory; Floquet Theory; Floquet Magnus Theory; Fer Expansion
Solid-State NMR; Average Hamiltonian Theory; Floquet Theory; Floquet Magnus Theory; Fer Expansion
Cite this paper
Mananga, E. (2014) Future Theoretical Approaches in Nuclear Magnetic Resonance. Journal of Modern Physics, 5, 145-148. doi: 10.4236/jmp.2014.54024.
Mananga, E. (2014) Future Theoretical Approaches in Nuclear Magnetic Resonance. Journal of Modern Physics, 5, 145-148. doi: 10.4236/jmp.2014.54024.
References
[1] E. Schrodinger, Physical Review, Vol. 28, 1926, pp. 1049-1070. http://dx.doi.org/10.1103/PhysRev.28.1049 [2] W. Magnus, Communications on Pure and Applied Mathematics, Vol. 7, 1954, p. 649. [3] U. Haeberlen and J. S. Waugh, Physical Review, Vol. 175, 1968, p. 453. http://dx.doi.org/10.1103/PhysRev.175.453 [4] M. G. Floquet, Ann. Econ. Norm, Suppl. 12, 1883, p. 47. [5] J. H. Shirley, Physical Review B, Vol. 138, 1965, p. 979.
http://dx.doi.org/10.1103/PhysRev.138.B979 [6] Y. Zur, M. H. Levitt and S. Vega, Journal of Chemical Physics, Vol. 78, 1983, p. 5293.
http://dx.doi.org/10.1063/1.445483 [7] M. M. Maricq, Physical Review B, Vol. 25, 1982, p. 6622.
http://dx.doi.org/10.1103/PhysRevB.25.6622 [8] E. S. Mananga and T. Charpentier, Journal of Chemical Physics, Vol. 135, 2011, Article ID: 044109.
http://dx.doi.org/10.1063/1.3610943 [9] S. Blanes, F. Casas, J. A. Oteo and J. Ros, Physics Reports, Vol. 470, 2009, p. 151.
http://dx.doi.org/10.1016/j.physrep.2008.11.001 [10] E. S. Mananga, A. E. Reid and T. Charpentier, Solid State Nuclear Magnetic Resonance, Vol. 41, 2012, p. 32.
http://dx.doi.org/10.1016/j.ssnmr.2011.11.004 [11] E. S. Mananga and A. E. Reid, Molecular Physics, Vol. 111, 2013, pp. 243-257.
http://dx.doi.org/10.1080/00268976.2012.718379 [12] F. Fer, Bulletin de la Classe des Sciences, Academie Royale de Belgique, Vol. 44, 1958, p. 818. [13] P. K. Madhu and N. D. Kurur, Chemical Physics Letters, Vol. 418, 2006, p. 235.
http://dx.doi.org/10.1016/j.cplett.2005.10.134 [14] H. Tal-Ezer and R. Kosloff, Journal of Chemical Physics, Vol. 81, 1984, p. 3967.
http://dx.doi.org/10.1063/1.448136 [15] T. J. Rivlin, “Chebychev Polynomials,” 2nd Edition, Wiley, New York, 1990, p. 188. [16] C. B. Moler and C. F. Van Loan, SIAM Review, Vol. 45, 2003, p. 49. http://dx.doi.org/10.1137/S00361445024180 [17] A. Iserles, Foundations of Computational Mathematics, Vol. 1, 2001, pp. 129-160.
http://dx.doi.org/10.1007/s102080010003 [18] E. Süli and D. Mayers, “An Introduction to Numerical Analysis,” Cambridge University Press, Cambridge, 2003.
http://dx.doi.org/10.1017/CBO9780511801181 [19] E. M. Purcell, H. C. Torrey and R. V. Pound, Physical Review, Vol. 69, 1946, p. 37.
http://dx.doi.org/10.1103/PhysRev.69.37 [20] F. Bloch, W. W. Hansen and M. Packard, Physical Review, Vol. 70, 1946, p. 474.
http://dx.doi.org/10.1103/PhysRev.70.474 [21] A. Brinkmann, M. Eden and M. H. Levitt, Journal of Chemical Physics, Vol. 112, 2000, p. 8539.
http://dx.doi.org/10.1063/1.481458 [22] M. Hohwy, H. J. Jakobsen, M. Eden, M. H. Levitt and N. C. Nielsen, Journal of Chemical Physics, Vol. 108, 1998, p. 2686. http://dx.doi.org/10.1063/1.475661 [23] A. Brinkmann and M. H. Levitt, Journal of Chemical Physics, Vol. 115, 2001, p. 357.
http://dx.doi.org/10.1063/1.1377031 [24] M. Carravetta, M. Eden, X. Zhao, A. Brinkmann and M. H. Levitt, Chemical Physics Letters, Vol. 321, 2000, p. 205. http://dx.doi.org/10.1016/S0009-2614(00)00340-7 [25] R. Tycko, Journal of Chemical Physics, Vol. 126, 2007. [26] R. Tycko, “Lecture Notes,” First Winter School on Biomolecular SSNMR, Vermont, 2008. [27] M. Eden and M. H. Levitt, Journal of Chemical Physics, Vol. 111, 1999, p. 1516.
http://dx.doi.org/10.1063/1.479410 [28] B. C. Sanctuary and H. B. R. Cole, Journal of Magnetic Resonance, Vol. 71, 1987, p. 106. [29] M. H. Levitt, Journal of Magnetic Resonance, Vol. 48, 1982, p. 234. [30] R. Tycko, H. M. Cho, E. Schneider and A. Pines, Journal of Magnetic Resonance, Vol. 61, 1985, p. 90. [31] M. Leskes, P. K. Madhu and S. Vega, Progress in Nuclear Magnetic Resonance Spectroscopy, Vol. 55, 2010, p. 345. http://dx.doi.org/10.1016/j.pnmrs.2010.06.002 [32] I. Scholz, J. D. Van Beek and M. Ernst, Solid State Nuclear Magnetic Resonance, Vol. 37, 2010, p. 39.
http://dx.doi.org/10.1016/j.ssnmr.2010.04.003
[1] E. Schrodinger, Physical Review, Vol. 28, 1926, pp. 1049-1070. http://dx.doi.org/10.1103/PhysRev.28.1049 [2] W. Magnus, Communications on Pure and Applied Mathematics, Vol. 7, 1954, p. 649. [3] U. Haeberlen and J. S. Waugh, Physical Review, Vol. 175, 1968, p. 453. http://dx.doi.org/10.1103/PhysRev.175.453 [4] M. G. Floquet, Ann. Econ. Norm, Suppl. 12, 1883, p. 47. [5] J. H. Shirley, Physical Review B, Vol. 138, 1965, p. 979.
http://dx.doi.org/10.1103/PhysRev.138.B979 [6] Y. Zur, M. H. Levitt and S. Vega, Journal of Chemical Physics, Vol. 78, 1983, p. 5293.
http://dx.doi.org/10.1063/1.445483 [7] M. M. Maricq, Physical Review B, Vol. 25, 1982, p. 6622.
http://dx.doi.org/10.1103/PhysRevB.25.6622 [8] E. S. Mananga and T. Charpentier, Journal of Chemical Physics, Vol. 135, 2011, Article ID: 044109.
http://dx.doi.org/10.1063/1.3610943 [9] S. Blanes, F. Casas, J. A. Oteo and J. Ros, Physics Reports, Vol. 470, 2009, p. 151.
http://dx.doi.org/10.1016/j.physrep.2008.11.001 [10] E. S. Mananga, A. E. Reid and T. Charpentier, Solid State Nuclear Magnetic Resonance, Vol. 41, 2012, p. 32.
http://dx.doi.org/10.1016/j.ssnmr.2011.11.004 [11] E. S. Mananga and A. E. Reid, Molecular Physics, Vol. 111, 2013, pp. 243-257.
http://dx.doi.org/10.1080/00268976.2012.718379 [12] F. Fer, Bulletin de la Classe des Sciences, Academie Royale de Belgique, Vol. 44, 1958, p. 818. [13] P. K. Madhu and N. D. Kurur, Chemical Physics Letters, Vol. 418, 2006, p. 235.
http://dx.doi.org/10.1016/j.cplett.2005.10.134 [14] H. Tal-Ezer and R. Kosloff, Journal of Chemical Physics, Vol. 81, 1984, p. 3967.
http://dx.doi.org/10.1063/1.448136 [15] T. J. Rivlin, “Chebychev Polynomials,” 2nd Edition, Wiley, New York, 1990, p. 188. [16] C. B. Moler and C. F. Van Loan, SIAM Review, Vol. 45, 2003, p. 49. http://dx.doi.org/10.1137/S00361445024180 [17] A. Iserles, Foundations of Computational Mathematics, Vol. 1, 2001, pp. 129-160.
http://dx.doi.org/10.1007/s102080010003 [18] E. Süli and D. Mayers, “An Introduction to Numerical Analysis,” Cambridge University Press, Cambridge, 2003.
http://dx.doi.org/10.1017/CBO9780511801181 [19] E. M. Purcell, H. C. Torrey and R. V. Pound, Physical Review, Vol. 69, 1946, p. 37.
http://dx.doi.org/10.1103/PhysRev.69.37 [20] F. Bloch, W. W. Hansen and M. Packard, Physical Review, Vol. 70, 1946, p. 474.
http://dx.doi.org/10.1103/PhysRev.70.474 [21] A. Brinkmann, M. Eden and M. H. Levitt, Journal of Chemical Physics, Vol. 112, 2000, p. 8539.
http://dx.doi.org/10.1063/1.481458 [22] M. Hohwy, H. J. Jakobsen, M. Eden, M. H. Levitt and N. C. Nielsen, Journal of Chemical Physics, Vol. 108, 1998, p. 2686. http://dx.doi.org/10.1063/1.475661 [23] A. Brinkmann and M. H. Levitt, Journal of Chemical Physics, Vol. 115, 2001, p. 357.
http://dx.doi.org/10.1063/1.1377031 [24] M. Carravetta, M. Eden, X. Zhao, A. Brinkmann and M. H. Levitt, Chemical Physics Letters, Vol. 321, 2000, p. 205. http://dx.doi.org/10.1016/S0009-2614(00)00340-7 [25] R. Tycko, Journal of Chemical Physics, Vol. 126, 2007. [26] R. Tycko, “Lecture Notes,” First Winter School on Biomolecular SSNMR, Vermont, 2008. [27] M. Eden and M. H. Levitt, Journal of Chemical Physics, Vol. 111, 1999, p. 1516.
http://dx.doi.org/10.1063/1.479410 [28] B. C. Sanctuary and H. B. R. Cole, Journal of Magnetic Resonance, Vol. 71, 1987, p. 106. [29] M. H. Levitt, Journal of Magnetic Resonance, Vol. 48, 1982, p. 234. [30] R. Tycko, H. M. Cho, E. Schneider and A. Pines, Journal of Magnetic Resonance, Vol. 61, 1985, p. 90. [31] M. Leskes, P. K. Madhu and S. Vega, Progress in Nuclear Magnetic Resonance Spectroscopy, Vol. 55, 2010, p. 345. http://dx.doi.org/10.1016/j.pnmrs.2010.06.002 [32] I. Scholz, J. D. Van Beek and M. Ernst, Solid State Nuclear Magnetic Resonance, Vol. 37, 2010, p. 39.
http://dx.doi.org/10.1016/j.ssnmr.2010.04.003