Lowest-order average Hamiltonian

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. 

The lowest-order average Hamiltonian in the rotating frame... 

The cycle for the chosen example is the rotation period. Hence, to calculate the lowest-order average Hamiltonian one has to average over one rotation period. The time-dependent terms in Eq. (36) are exp(i(m n)uv t) the average of these over a rotation period is given by... 

The lowest-order average Hamiltonian for C7 (Eq. (135)) is identical with that for DQ-HORROR, except that the effective dipolar scaling factor... 

Briefly, it happens that for the case of an anisotropic chemical shift the lowest order average Hamiltonian and all hi er order corrections to it vanish identically. Thus, the linewldths in the spectra, such as Fig. 1, should be limited by Inhomogeneous broadening not removed by the sample rotation and by homogeneous broadening, arising from incomplete decoupling, molecular motion, etc. 

For qualitative insight based on perturbation theory, the two lowest order energy eorreetions and the first-order wavefunetion eorreetions are undoubtedly the most usetlil. The first-order energy eorresponds to averaging the eflfeets of the perturbation over the approximate wavefunetion Xq, and ean usually be evaluated without diflfieulty. The sum of aJ, Wd ds preeisely equal to tlie expeetation value of the Hamiltonian over... 

Taking the MREV-8 as a basic C element (duration = xr/5), one can generate a y -encoded [194] version of MREV-8 by concatenating the five C elements with an rf phase set equal to k / 2n/5, where k = 0-4 [107]. The averaged Hamiltonian calculated to the lowest order for this so-called T-MREV method has the following form ... 

Finally, in order to ensure an homogeneous treatment of all excited states at the variational level, the MCSCF calculation should be averaged on the states under investigation. The lowest eigenfunetions of the MCSCF Hamiltonian will provide the zeroth-order wavefunetions to build the perturbation on. 

Figure 2. Plot, as a function of the order i of the perturbation procedure, of the average arithmetic error between the energies of HCN CNH obtained for the ab initio surface of Refs. 7 and 8 and the perturbative Hamiltonian of Eq. (17). The lowest 101 rotationless states of the system are taken into account (see Tables VI and VII of Ref. 7). These states have up to 18 quanta of...

The relativistic IHFSCC approach was used in 2001 to calculate the spectra of neutral Xe and Rn atoms, obtaining unprecedented and still unsurpassed accuracy, with an average error of 0.6% for the lowest excitation energies (about 20 per atom) [74]. This accuracy allowed predictions, e.g., for the unobserved states of Rn. It was demonstrated that only the combination of using large active spaces with all-order treatment of dynamic correlation in the framework of a high-quality relativistic Hamiltonian can yield such high level of accuracy. 

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