Switching functions

The function /(r) is a force-switching function that goes smoothly from 1 ar r = 0 to 0 at r = Tc. The long-range part of the field, i.e., what remains from the complete Coulomb field ... 

Another immediate application of r-RESPA is to the case when the force can be subdivided into a short range part and a long range part. One way for effectuating this break up is to introduce a switching function, s x) that is unity at short inter-particle separations and 0 at large inter-particle separations. We introduced this strategy in our earlier non-reversible RESPA paper [15] where we expressed the total force as. 

The Equilibrium version was tested also on a 40 A diameter sphere of water with a switching function transition distance of 8 A to 13 A. There was in this case a slight rise in energy when At was 8 fs. 

These functions allow- the nonbonded potential energy Lo turn off smoothly and systematically, removing artifacts caused by a truncated potential. With an appropriate switching function, the potential function is unaffected except m the region of the switch. 

Example For two atoms having point charges of 0.616 and -0.504 e and a constant dielectric function, the energy curve shows a switching function turned on (Ron) at a nonbonded distance of 10. A and off (R(,rr) al a distance of 14 A. Compare the switched poieniial with the abruptly inincaied poteiiiial. 

An alternative way to eliminate discontinuities in the energy and force equations is to use a switching function. A switching function is a polynomial ftmction of the distance by which the potential energy function is multiplied. Thus the switched potential o (r) is related to the true potential t> r) by v r) = v(r)S(r). Some switching functions are applied to the entire range of the potential up to the cutoff point. One such function is ... 

The switching function has a value of 1 at r = 0 and a value of 0 at r = r., the cutoff distcince. Between these two values it Vciries as shown in Figtire 6.19, which also shows how the potential function is affected. 

Fig. 6.20 A switching function that applies over a narrow range near the cutoff and its effect on the Lennard-Janes potential.

By ensuring that the first derivative is zero at the endpoints the force also approaches zero smoothly. A continuous second derivative is required to ensure that the integration algorithm works properly. If the switch function is assumed to take the following form ... 

When using a switching function in a molecular simulation with a residue-based cutoff it is important that the function has the same value for all pairs of atoms in the two interacting groups. Otherwise, severe fluctuations in the energy can arise when the separation is within the cutoff region. These two contrasting situations can be formally expressed as follows ... 

Na and Nb are the numbers of atoms in the two groups A and B and S is the switching function. With the group-based switching function, it is necessary to define the distance between the two groups (i.e. the two points Ta and Tb). There is no definitive way to do this. As with cutoffs, a special marker atom can be nominated within each residue, or the centre of mass, centre of geometry or centre of charge may be used. 

IS added to the short-range molecule-molecule interaction. Problems with the reaction ethod may arise from discontinuities in the energy and/or force when the number of les j rvithin the cavity of the molecule i changes. These problems can be avoided by dng a switching function for molecules that are near the reaction field boundary. 

Arelatively simple method for alleviating some of the nonphysical behaviors caused by imposing a nonbonded cutoff is to use a potential switching function (equation 14). 

In an attempt to remedy truncation problems, a shifting potential multiplies the nonbonded electrostatic potential by a function that goes to zero. That is, the potential is shifted to zero at the cutoff Roff. Unlike the switching function, the shifted potential does not apply to van der Waals interactions. 

Caution Comparing the shifted constant dielectric to a constant dielectric function without a cutoff shows that the shifted dielectric, unlike a switching function, perturbs the entire electrostatic energy curve, not only the region near the cutoff. 

Note MMh- is derived from the public domain code developed by Dr. Norman Allinger, referred to as MM2(1977), and distributed by the Quantum Chemistry Program Exchange (QCPE). The code for MMh- is not derived from Dr. Allinger s present version of code, which is trademarked MM2 . Specifically, QCMPOlO was used as a starting point for HyperChem MMh- code. The code was extensively modified and extended over several years to include molecular dynamics, switching functions for cubic stretch terms, periodic boundary conditions, superimposed restraints, a default (additional) parameter scheme, and so on. 

Eor larger molecules, such as proteins, use a switching function to dramatically decrease computing time. 

A switched function extends over the range of inner (Ron) to outer (Roff) radius and a shifted function from zero to outer (Roff) radius. Beyond the outer radius, HyperChem does not calculate non-bonded interactions. The suggested outer radius is approximately 14 Angstroms or, in the case of periodic boundary conditions, less than half the smallest box dimension. The inner radius should be approximately 4 Angstroms less than the outer radius. An inner radius less than 2 Angstroms may introduce artifacts to the structure. 

The switching function used by HyperChem, called switch below, alters the nonbonded energy (van der Waals, hydrogen bond, and electrostatic) in the following way ... 

The last letter indicates the extreme value at which the alarm/switch function is trigged ... 

HarM71 Harrison, M. A. Counting theorems and their applications to classification of switching functions. Chap. 4 of Recent Developments in Switching Theory (A. Mukhopadyay, ed.) Academic Press (1971). 

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