Average Hamiltonian theory

A discussion about the effects of the higher-order van Vleck correction terms will be given at a later stage, where we deal with the BMFT approach. However, a comment about Average Hamiltonian Theory (AHT) must be made at this point. This powerful theoretical approach is valid for single frequency periodic Hamiltonians. 

The Average Hamiltonian Theory (AHT) [23,26] is applied to express the propagator of a periodic Hamiltonian T-Lmtit + krt) = Hintit) in terms of a time independent effective Hamiltonian %). When the Hamiltonian is also cyclic and Uint kTt) = Uint(0) = 1, the propagator gets the generalised form 

This effective Hamiltonian is again not unique but can be chosen such that its eigenvalue differences are smaller than l/2o t. Maricq [100-102] and others [14, 103] have demonstrated that the Magnus expansion of the effective Hamiltonian in AHT and the van Vleck transformation approach of the Floquet Hamiltonian are equivalent. At the time points krt the Floquet solution for the propagator in Eq. 24 has the form 

and considering only the first two terms of the Taylor expansion 

This expression is identical to the zero and first AHT terms obtained when the first-order Magnus expansion terms are calculated using the integral expressions and the Fourier expansion of Hint t)- Thus the effective Hamiltonian to first-order in AHT differs from the first-order van Vleck expansion, Eq. 61b. This difference has been discussed by Goldman [98], Mehring [14] and others [103, 104] and it was shown that the additional term in H should be discarded 

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The inclusion of average Hamiltonian theory derived pulse sequence building blocks specifically designed for the measurement of RDCs is highly desirable. The homonuclear dipolar decoupling sequences used in ref. 190 lead to the less complex multiplet structures observed in isotropic samples with the chemical shift resolution reduced by a factor of 2-3, depending on the multiple pulse... 

Important guidelines for the construction of a multiple-pulse sequence with desired properties are provided by average Hamiltonian theory (see Section IV). The effective Hamiltonian created by the sequence must meet a number of criteria (see Section IX). Most importantly, spins with different resonance frequencies, that is, with different offsets and Vj from a given carrier frequency, must effectively be energy matched in order to allow Hartmann-Hahn transfer. This can be achieved if the derivative of the effective field with respect to offset vanishes, which is identical to the Waugh criterion for efficient heteronuclear decoupling... 

Multiple-pulse sequences can be conveniently analyzed and classified with the help of average Hamiltonian theory, which also provides valuable... 

The efficient combination of RF pulses with MAS requires theoretical tools. While the initial multiple-pulse sequences were based on the average Hamiltonian theory (AHT) [14,25,26,32,33], some of the later schemes were more effectively formulated on the basis of Floquet theory [52], combinations of both AHT and Floquet theory [51], synchronisation arguments [42,43,46,55], and numerical methods [50]. A combination of RF pulses with MAS was also visualised in terms of certain symmetry conditions based on AHT [77]. An attempt to present all these in a unified picture is perhaps not out of place for a better understanding of the underlying phenomena. 

We now calculate the perturbation to the Zeeman field due to the quadrupolar interaction by means of average Hamiltonian theory.This is accomplished by transforming TYq to the Zeeman interaction frame and then applying the spherical tensor rotation properties to the spin elements 72,The resulting quadrupolar Hamiltonian TTq in the rotating frame is given by ... 

Mixing sequences for total through-bond correlation spectroscopy in solids (TOBSY) have been developed for fast MAS experiments. Possible sequences with the desired Hamiltonian (the homonuclear isotropic J interaction) have been identified using lowest order average Hamiltonian theory combined with numerical simulations as a function of the MAS frequency. An experimental TOBSY spectrum of a uniformly C-labelled decapeptide at 20 kHz MAS has been obtained using one of the new sequences. The spectrum allows to assign the resonances to the respective spin systems. 

MQCs are not excited uniformly and the efficiency with which the various orders of MQC are excited depends specifically on the parameters of the spin system (dipolar couplings, scalar couplings, quadrupolar couplings, chemical shifts) in the spin system and the choice of the preparation time t. Many researchers have co-added spectra acquired with different preparation times to ensure that all transitions are observed with reasonable intensity. A number of broadband excitation techniques have been developed,13-15 where the value of t in the preparation sequence has been varied either in a pseudo-random or systematic fashion to achieve a more uniform excitation in the multiple quantum domain. An experimental search method has been used to optimise the delays in the preparation period of the MQ excitation sequence16 and Wimperis17 used average Hamiltonian theory to propose... 

The most efficient way to speed up spin diffusion is the so-called r.f.-driven spin-diffusion experiment [15, 19] where the chemical-shift differences are removed by r.f. irradiation. For small chemical-shift differences, r.f.-driven spin diffusion can be implemented by applying a continuous-wave r.f. field to the S-spins which can theoretically be described by a transformation into a tilted rotating frame (see Appendix B). To zeroth-order average Hamiltonian theory the chemical-shift differences are removed (fl, — fty = 0 for all spins i and j) and the dipolar-coupling frequencies are scaled by a factor s = -1/2. The scaled-down (or ideally vanishing) chemical-shift difference allows one to keep the zero-quantum line narrow by decoupling the protons. This results in fast spin-diffusion rates. Furthermore, the rate constants are now determined by the S-spin coupling network, and the proton spins need not be considered for the data analysis. 

The performance of these multiple-pulse sequences can be calculated using average Hamiltonian theory [23]. It has been shown that sequences with all pulses along one axis (with positive and negative amplitudes), like WALTZ... 

J.S. Waugh, Average Hamiltonian Theory, in The Encyclopedia of Nuclear Magnetic Resonance, Wiley, 1996, and references therein. 

The SRTS sequence consists of a preparatory pulse and an arbitrary long train of the phase-coherent RF pulses of the same flip angle applied with a constant short-repetition time. As was noted above, the "short time" in this case should be interpreted as the pulse spacing T within the sequence that meets the condition T T2 Hd. The state that is established in the spin system after the time, T2, is traditionally defined as the "steady-state free precession" (SSFP), ° and includes two other states (or sub-states) quasi-stationary, that exists at times T2effective relaxation time) and stationary, that is established after the time " 3Tie after the start of the sequence.The SSFP is a very particular state which requires a specific mechanism for its description. This mechanism was devised in articles on the basis of the effective field concept and canonical transformations. Later approaches on the basis of the average-Hamiltonian theory were developed. ... 

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