Smooth truncation

Hyperbolic Secant Inversion Pulses and Wideband Uniform Rate Smooth Truncation... 

Siegel et al. showed that enhancement of the CT can also be obtained using hyperbolic secant (HS) pulses to invert selectively the STs [74], Unlike the DFS waveform, whose frequency sweep is generated by a constant rf-pulse phase while modulating the amplitude, the HS pulse utilizes both amplitude and phase modulation, yielding an enhancement exceeding that obtained by DFS or RAPT [61, 74, 75]. Most recently, the pulse sequence called wideband uniform-rate smooth truncation (WURST) [76] was introduced to achieve selective adiabatic inversion using a lower power of the rf-field than that required for the HS pulses [77,78]. One of its applications involved more efficient detection of insensitive nuclei, such as 33S [79]. 

From equation (8) three modes of adiabatic spin inversion using RF pulses may be defined as (a) amplitude modulated pulses, e.g. I-BURP [13], G3[16], I-SNOB [17], (b) frequency modulated pulses, e.g. chirp [18,20], tangential sweep [20,21] and (c) both amplitude and frequency modulated pulses, e.g. the hyperbolic secant [22] or WURST (Wide band Uniform Rate Smooth Truncation) [23] pulse. 

The interactions between pairs of gas molecules were estimated using the LJ potential =As alr) - Glr), where r is the site-site distance). Two sets of LJ parameters for gas molecules (i.e., and cr) were chosen. One was slkB = 149 K (fe is the Boltzmann constant) and G = 0.378 nm, and the other was dkB 231 K and a = 0.405 nm. The former corresponded to the parameter set for a spherically-approximated CH4 molecule, and the latter for a Xe molecule. The interactions between H2O and gas molecules were also estimated using the LJ potential, and the s and a values for the H20-gas LJ potential were determined using the Lorentz-Berthelot rules. Long-range interactions were smoothly truncated at an intermolecular distance of 1 nm using a switching function. 

Fig. 8. Density variation of the inherent structure pressure for a fluid with a smoothly truncated Lennard-Jones potential (Sastry etal., 1997b). Regions A, B, and C identify distinguishing density intervals for the inherent structures discussed in the text.

Smoothly truncated Lennard-Jones fluid (Sastry et al., 1997b). 

Figure 9.10. Schematic excitation profiles for (a) a low-power rectangular pulse and (b) a smoothly truncated shaped pulse.

WURST Wideband, uniform rate and smooth truncation (broadband decoupling sequence) 9.2... 

Rose and Benjamin [194, 187] studied the adsorption of Na+ and CF on the Pt(lOO) surface and concluded that the structure of the solvation complex around Na and Cl is very similar to the structure in the bulk . In this study, the same model for the water-metal interactions was used as in the preceding study of Li+ and F hydration [189]. The ion-surface interactions, however, were described by a modified image charge potential which was smoothly truncated. The authors discussed in some detail the anisotropy of the hydration complex that is induced by the density inhomogeneities at the interface. In a study of Fe + and Fe + hydration [197],... 

Two major ways exist for calculating interactions between atoms in such periodic systems. The first involves truncating the interactions (for example, beyond a certain distance), and the second is to explicitly or effectively calculate the full set of interactions within the periodic system. Truncation effectively means modifying the Hamiltonian for the periodic system to exclude some, typically more distant, interactions. Some adverse effects of truncation within a region can be reduced by shifting and/or smoothly truncating the potentials. This... 

Levitt has noted the importance of smooth truncation of atomic forces in leading to stable trajectories of proteins (/) and methods for the smooth truncation of atomic forces are discussed in detail by Steinbach and Brooks (2) (see also Allen Tildesley (5)). Although there are other approaches to more accurately represent all of the electrostatic interactions, the Ewald (4) approach is one of the most accurate ways... 

Very recently, the WURST-QCPMG sequence was developed to achieve uniform excitation of quadrupolar nuclei across very wide bandwidth using frequency-sweep chirped pulses [86]. The WURST (wideband uniform-rate smooth truncation) pulses provide broadband excitation. Integrating the WURST pulses with QCPMG results in the WURST-QCPMG method, which has proven to be a highly efficient approach for acquiring ultra-wideHne NMR spectra [74]. 

Shifting functions- 77 provide an alternative to smoothly truncated potential energies. As the name suggests, the true potential is shifted so as to make it zero at the cutoff distance. One form of a shifted potential is ... 

A flexible, three-center model labeled F3C has been compared with other water models within the context of biomolecular simulations. This model contains a smooth truncation function acting on all interactions, so dielectric properties may not be well represented. This model stabilizes the hydrogen bond network, as evident from the oxygen-oxygen radial distribution functions, suggesting that the correct local ordering of liquid water is reproduced. A version of this model with a different truncation scheme has also been evaluated. " ... 

---