Spherical truncation

Microscopic examination of tapioca starch granules reveals smooth, spherical granules 4-35 pm in diameter, commonly with one or more spherical truncations. 

Some insight into (3.36) and its application to the computer situation can be obtained by considering a closely related situation mentioned in Section II.A. This is an infinite system, but one in which the dipolar interactions are spherically truncated. For such a system, (3.36) is an exact result. ... 

Fig. 5. Values of h °(r) for dipolar hard spheres at p —0.4 and —2.75. The big dots are MC results for Af—256, Rf. —4.2d. The solid and dashed curves are the QHNC and LHNC approximations, respectively, for a spherically truncated potential. The dotted curve is the QHNC result for an infinite system with an untruncated potential. (Results from Ref. 58.)...

The other approximations described in Section III.C have not been solved for a spherically truncated potential, but an estimate of their accuracy can be obtained by comparing with the LHNC or QHNC theories for an infinite system. We are of course assuming that the LHNC and QHNC approximations remain accurate for the full (untruncated) dipolar interaction and lie close to the true infinite system result. The MSA, LIN, L3, and LHNC theories for a dense dipolar hard-sphere system are compared in... 

Fig. 13. Values of h"°(r) for hard spheres with dipoles and quadrupoles at p = 0.8 and H = Q = 1.0. The dots are MC results (N=256, R(. =3.4rf), and the solid, dashed, and dash-dot curves represent the QHNC, LHNC, and MSA, respectively, for a spherically truncated potential. (Results from Ref. 59.)...

Hard Spheres with Dipoles and Quadrupoles. The LHNC, QHNC, and mean spherical approximations have been solved for fluids of hard spheres with both dipole and quadrupole moments. Theoretical results for spherically truncated potentials have been compared with Monte Carlo (SC)... 

D. Wolf, P. Keblinski, S. R. Phillpot, and J. E ebrecht, /. Chem. Phys., 110, 8254 (1999). Exact Method for the Simulation of Coulombic Systems by Spherically Truncated, Pairwise... 

This is a spherically truncated and shifted potential, known as STS, which is the potential corresponding to the truncated force. The fact that MC and MD are, in fact, carried out with different potentials is the origin of the observed differences between the density profiles. The STS potential is shallower than the ST potential, and this gives rise to a lower density in the centre of the film. MD can be carried out with an ST potential, if the following modified force scheme is used. 

Saiz-Urra, L., Gonzalez-Diaz, H.> 8c Uriarte, E. (2005). Proteins Markovian 3D-QSAR with spherically-truncated average... 

Molecular dynamics can be used to simulate the properties of bulk materials. This is done by using periodic boundary conditions to eliminate the system-vacuum interface. This complicated subject is treated elsewhere in this encyclopedia. Suffice it to say that for systems with charges one should not truncate the interactions but should rather use Ewald boundary conditions. For systems with only short-range interactions it is usually a good approximation to use minimum image boundary conditions or spherical truncation. Of course, the properties of small clusters can be simulated without the use of boundary conditions but then one risks losing some of the molecules in the system due to evaporation, and the cluster will eventually disappear. In such circumstances it is useful to invent a potential that binds the particles weakly to the center of the cluster, It is also possible to simulate systems with one or more interfaces. 

For the non-periodic systems it is more common to use a list of non-bonded neighbors and spherical truncation of the non-bonded interactions at some cut off distance in the range 10-15 A, often together with a smoothing of the interaction to zero at the cut off. In one case a fast multipole expansion of the long-range Coulomb interactions was used. For periodic systems there is also the possibility to use Ewald summation, and the simulations of the restriction endonucleases were made with the fast particle mesh Ewald (PME) summation algorithm. 

---