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Dual Lanczos simulation of dynamic nuclear magnetic resonance spectra for systems with many spins or exchange sites
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2.The classic implementations are documented in B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Y. Ikebe, V. C. Klema, and C. B. Moler, Matrix Eigensystem Routines-EISPACK Guide, 2nd ed. (Springer, New York, 1976).
2.See also W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. (Cambridge University, Cambridge, U.K., 1992).
3.The required asymptotic expansion of the gamma function is given in M. A. Spiegel, Mathematical Handbook (McGraw-Hill, New York, 1968), p. 102.
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6.For example, in the 1989, 2nd ed., Matrix Computations (Johns Hopkins University, Baltimore, 1989), Golub and Van Loan were “unaware of any successful applications of the unsymmetric Lanczos algorithm” on account of breakdown—see p. 503.
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15.Two groups of spins are treated as different type if all couplings between the constituent spins are weak—i.e., small compared with the chemical shift difference.
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22.Rate broadening—as a fullwidth at half maximum—is simply given by k when k is small compared with Δω. Division by 2π is required when the spectrum is given as a function of ν in Hz.
23.The overhead associated with underestimating the required number of Lanczos iterations is only a problem if is larger than or comparable to For example, in the case of spectra computed at frequencies for mutual exchange systems with 1 Hz line broadening, the direct spectrum portion of the computation constitutes a significant fraction of the total CPU time only for systems with fewer than nine spins. Since the simulation of a smaller spin system spectrum is reasonably fast, the elimination of overhead in this part of the computation is not critical to the usefulness of the methodology.
24.Note that for very small exchange rate, the number of iterations correlates with block size per site, N. In this case, the Liouvillian is almost block diagonal, with different blocks corresponding to different sites. The number of iterations to converge the spectrum is essentially determined by the number needed to converge the spectrum of a single site. However, even for a rate as small as 1 s−1, the number of Lanczos iterations required correlates with the total block size,
25.The Liouvillian block size, with account taken of coherence level conservation, is Compare this with Eq. (1) which gives the size of the largest sub-block of this Liouvillian. The sub-block is thus a factor, smaller. This factor exactly affords an additional spin—with the same computation time—when Note that the asymptotic form given here for the number of coherence level 1 transitions differs from that given in Sec. II of Ref. 1. The formula given there [the unlabeled equation after Eq. (14)] is incorrect. It is based on an inappropriately simplified Sterling formula. The conclusions of Ref. 1, based on this formula, are nevertheless still valid.
26.Eight spins is beyond the scope of the Householder method, implemented on our computer, due to the requirement of about 6 days of CPU time—this is not viable in a shared departmental facility. The memory requirements of the Householder method are not out of scope until nine spins are considered—more than 4 gigabytes random access memory is needed in this case.
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