Fast particle methods

The detectors used with UPLC systems have to be able to handle very fast scanning methods because peak half-height widths of around 1 s are typically obtained with columns packed with 1.7-p.m particles. In order to accurately and reproducibly integrate an analyte peak, the detector sampling rate must be high enough to capture enough data points across the peak. Conceptually, the sensitivity increase for UPLC... 

For enhanced throughput, fast RPLC methods using monolithic silica columns [238], small size particles (3 pm) packed columns [173, 195, 226, 235-237, 239, 240], ultra high pressure liquid chromatography (UHPLC) columns packed with sub-2 pm particles [145,196, 227, 245, 247] and 2.6 pm core-shell particles HPLC columns [246] have been proposed for the high-throughput separation and quantification of tamoxifen/metabolites. 

Considering long-range MD, a variety of approximate methods have been developed to overcome the bottleneck that characterizes the forces treatment these include particle mesh algorithms, hierarchical methods, and fast multipole methods. One of the most promising developments is the cell multipole method, which scales linearly with N, requires only modest memory, and is well suited to highly parallel and vector computers. 

There is a growing number of approaches to treat the essentially infinite reach of charge-charge interactions. To mention just a few of the more traditional numerical ones which are well adapted to the requirements of MD, we have charge group cut-off [63], Ewald [72] summation, smooth particle Ewald [66] summation and particle-particle-particle-mesh (P M) [73]. There are also several variations of hierarchical methods [74] a few examples are the method of Bames and Hut (BH) [75], the fast multipole method (EMM), with [76] and without [77] multipoles, and the cell multipole method [78]. 

Exclusion effects are also possible with PRP-1 column depending on the size of the solute molecule. This was evident In the retention behavior of the macrocycllc heptaene antibiotic, amphotericin B, on a PKP-1 column. The reverse-phase fast LC method recently developed In the author s laboratory using a 5-cm long Cig column (3 micron particle size) gave a retention time of 5.5 minutes for this compound. This method also provided excellent resolution of amphotericin B from the cofermented X component (25), another heptaene closely related to It. The mobile phase consisted of... 

The description of the mDC method in the present work is supplemented with mathematical details that we Have used to introduce multipolar densities efficiently into the model. In particular, we describe the mathematics needed to construct atomic multipole expansions from atomic orbitals (AOs) and interact the expansions with point-multipole and Gaussian-multipole functions. With that goal, we present the key elements required to use the spherical tensor gradient operator (STGO) and the real-valued solid harmonics perform multipole translations for use in the Fast Multipole Method (FMM) electrostatically interact point-multipole expansions interact Gaussian-multipoles in a manner suitable for real-space Particle Mesh Ewald (PME) corrections and we list the relevant real-valued spherical harmonic Gaunt coefficients for the expansion of AO product densities into atom-centered multipoles. 

Giese, T. J., and York, D. M. (2008). Extension of adaptive tree code and fast multipole methods to high angular momentum particle charge densities,/. Comput Chem. 29(12), 1895-1904. 

Kurzak,)., and Pettitt, B. M. (2006). Fast multipole methods for particle dynamics. Mol SimuL 32, pp. 775-790. 

A variety of methods are available for computing electrostatic energies (and forces), including the Fast Multipole Method [38], the Particle-Particle-Particle-Mesh Method [179], and methods based on the technique of Ewald Summation [125, 367] we only discuss a particular variant of the latter approach here and not... 

Shaw, D. A fast, scalable method for the parallel evaluation of distance-limited pairwise particle interactions. J. Comput. Chem. 26, 1318—1328 (2005). doi 10.1002/jcc.20267... 

Fu, Y, Klimkowski, K. J., Rodin, G. J., Berger, E., and et al., A fast solution method for three-dimensional many-particle problems of linear elasticity. International Journal for Numerical Methods in Engineering, 42,1215-1229 (1998). 

FIGURE 26.7 The schemahc of MD Fast Multipole Method idea. In the lowest, parricle-parhcle level, the interactions on particle I are computed from all the particles which are located in the same and neighboring cells, while the interactions from the larger cells are approximated. 

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