Spherical cutoff method

P. J. Steinbach and B. R. Brooks, New Spherical-Cutoff Methods for Long-Range Forces in Macromolecular Simulation, J. Comput. Chem., 15 (1994), 667-683. 

Steinbach, P.J., Brooks, B.R. New spherical-cutoff methods for long-range forces in macromolecu-lar simulation, J. Comp. Chem. 1994,15(7), 667-83. 

The reported results for equilibrium properties were obtained by means of the standard Monte Carlo (MC), molecular dynamics (MD), and Gibbs ensemble (GE) simulation methods [23, 24], For the trial systems of a finite range the simple spherical cutoff was used, whereas in simulations of the full systems either the Ewald summation or the reaction field method were used. For further technical details we refer the reader to the original papers. 

The motion equations have been solved by the Verlet Leap-frog algorithm subject to periodic boundary conditions in a cubic simulation cell and a time step of 2 fs. The simulations have been performed in the NVT ensemble with the Nose-Hoover thermostat [62]. The SHAKE constraints scheme [65] was used. The spherical cutoff radius comprises 1.2 nm. The Ewald sum method was used to treat long-range electrostatic interactions. 

The Spherical Cutoff and Minimum Image Methods. Two commonly used and easily implemented approximations are the so-called spherical cutoff (SC) and minimum image (MI) methods. Both involve a simple... 

Evidently the MI method is less objectionable than cutoff from the standpoint of studying fluid structure. If one wants to add a tail correction to the results, however, a spherical cutoff truncation is much more convenient. One way around this difficulty might be to carry out a Markov chain based on the MI method, but simultaneously to keep track of the energy contributions from pairs within some cutoff distance, and finally to add a tail correction to the latter. A very attractive approach to approximating long-range effects in fluids is the reaction field (RF) method described above, which seems to be free of some of the faults of the Ewald method. It is, however, most convenient to use a spherical cavity in the continuum. If this is done one evidently does not entirely escape these problems associated with a spherical cutoff, however, and some way around them should be sought. 

Ewald summation presented above calls for the calculation of AP terms for each of the periodic boxes, a computationally demanding requirement for large biomolecular systems. Recently, Darden et al. proposed an N log N method, called particle mesh Ewald (PME), which incorporates a spherical cutoff R. This method uses lookup tables to calculate the direa space sum and its derivatives. The reciprocal sum is implemented by means of multidimensional piecewise interpolation methods, which permit the calculation of this sum and its first derivative at predefined grids with fast Fourier transform methods. The overhead for this calculation in comparison to Coulomb interactions ranges from 16 to 84% of computer time, depending on the reciprocal sum grid size and the order of polynomial used in calculating this sum. 

Several methods exist for computation of the dielectric constant e of a liquid. In the simplest formulation for a system with a spherical cutoff effectively imbedded in a vacuum, e is related to the average of the square of the total dipole moment, M, for the molecules in the sphere... 

Hollow-fiber ultrafiltration can be a very usefiil tool in obtaining size-fractionated samples for chemical and physical analyses of humic and fulvic acids in surface and groundwaters. The technique is very reproducible because it is strictly a physical separation, it minimizes the potential artifacts associated with the more classical chemical separation techniques. The separation method assumes a spherical structure for the cutoffs and therefore is an empirical approach. Nevertheless, hollow fibers can be used much as 0.45 im filters are used to separate particulate versus dissolved material with a great deal of success. 

Extensive molecular dynamics simulations of dilute and semidilute polyelectrolyte solutions of chains with degree of polymerization N ranging from 16 up to 300 were recently performed by Stevens and Kremer [146-148] and by Liao et al, [149], In these simulations the long-range electrostatic interactions were taken into account by the Ewald summation method, including interactions with all periodic images of the system, Stevens and Kremer [146-148] have used a spherical approximation of Adams and Dubey [150] for the Ewald sum while Liao et al, [149] have applied the PME method [110], In addition to Coulombic interactions, all particles, including monomers and counterions, interacted via a shifted Lennard-Jones potential with cutoff rcui = 2 a... 

Having dealt with methods to perform simulations of confined fluid-solid transitions, we turn to the determination of whether a transition has occurred, as well as measures of the structure of the confined system. The global order of the system indicates whether a system is fluidlike or solidlike and is characterized by the bond order parameters introduced by Steinhaidt et al. (1983). Here a bond is not a chemical bond but a vector joining two neighboring atoms whose distance is less than a cutoff radius. In order to calculate bond order parameters, a spherical harmonic associated with every bond in the system needs to be determined. The quantity is defined as ... 

Any of the methods used in classical Monte Carlo and molecular dynamics simulations may be borrowed in the combined QM/MM approach. However, the use of a finite system in condensed phase simulations is always a severe approximation, even when appropriate periodic or stochastic boundary conditions are employed. A further complication is the use of potential function truncation schemes, particular in ionic aqueous solutions where the long-range Coulombic interactions are significant beyond the cutoff distance.Thus, it is alluring to embed a continuum reaction field model in the quantum mechanical calculations in addition to the explicit solute—solvent interaaions to include the dielectric effect beyond the cutoff distance. - uch an onion shell arrangement has been used in spherical systems, whereas Lee and Warshel introduced an innovative local reaction field method for evaluation of long-... 

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