Cutoff function technique

One may use the cutoff function technique (see Section 4.3) and change t/r fr) near the points r = R to regularize it. The modified function belongs to the domain of definition of the Laplacian operator (an alternative method is mollification, see Sect. 2.17, 2.18 in [22]) and has approximately the same mean energy. Then simple estimates (see e.g. Section V.5.3 in [25]) and relations of Equation (2.4) demonstrate that all of the values R(r), (r) are uniformly bounded, for some constant Co and large enough R, and the following simple estimates hold for any r ... 

Abstract. A smooth empirical potential is constructed for use in off-lattice protein folding studies. Our potential is a function of the amino acid labels and of the distances between the Ca atoms of a protein. The potential is a sum of smooth surface potential terms that model solvent interactions and of pair potentials that are functions of a distance, with a smooth cutoff at 12 Angstrom. Techniques include the use of a fully automatic and reliable estimator for smooth densities, of cluster analysis to group together amino acid pairs with similar distance distributions, and of quadratic progrmnming to find appropriate weights with which the various terms enter the total potential. For nine small test proteins, the new potential has local minima within 1.3-4.7A of the PDB geometry, with one exception that has an error of S.SA. 

The use of the periodic boundary conditions in the two directions perpendicular to the interface normal (X and Y) implies that the system has infinite extent in these directions. To make the computational cost reasonable, one must truncate the number of interactions that each molecule experiences. The simplest possible technique is to include, for each molecule i, the interaction with all the other molecules that are within a sphere of radius which is smaller than half the shortest box axis. One selects, from among the infinite possible images of each molecule, the one that is the closest to the molecule i under consideration. This is called the minimum image convention, and more details about its implementation can be found elsewhere [2]. To arrive at the correct bulk properties, any ensemble average calculated by this technique must be corrected for the contribution of the interactions beyond the cutoff distance. The fixed analytical corrections are calculated by assuming some simple form of the statistical mechanics distribution function for distances greater then R. ... 

The thermal functions for the five alkali metal monatomic gases are calculated by the same procedure. Oberved and estimated atomic energy levels are included in the partition function calculation, using an Ionization potential lowering (IP-kT) technique as the cutoff procedure in the energy level summation (16). 

Although valence band information could be acquired by conventional X-ray sources, analysis of the valence band region is not as simple as the core region, since all the components in the sample contribute in this narrow region (with E of 30 eV or less). Due to the broad line width of conventional X-ray sources and the low ionization cross section. X-ray-excited valence band spectroscopy is less commonly used for surface analysis. Instead, ultraviolet sources (e.g.. He I and He II) are adopted to acquire the valence band spectra, a surface technique called ultraviolet photoelectron spectroscopy (UPS). He I and He n resonance lines have inherently narrow widths of only a few meVs and high ionization cross sections in the valence band. This technique is widely used in the study of adsorption phenomena and valence band structure of metals, alloys, and semiconductors. Work functions can be derived from the Fermi level and the secondary electron (SE) cutoff of the UPS spectrum. 

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