Shifted force potential

Figure 10 Potentials and forces calculated for the shifted-force potential and taper-7 potential A = 0).

Computing the interatomic forces is the most time-consuming part in an MD simulation. The use of a cutoff radius is a standard trick of the trade that reduces computational cost by neglecting interactions between atoms separated by a distance larger than the specified cutoff. As described earlier, this truncation results in a discontinuity of both the potential and the force at the cutoff distance, but the drawback thus entailed can be avoided by implementation of either the shifted-force potential or a taper function. 

As a first illustration we consider the model discussed in Section 1.3.3, namely a fluid of simple molecules confined between chemically striped solid surfaces (see Fig. 5.2). As before in Section 5.4 we treat the confined fluid as a thermodynamically open system. Hence, equilibrium states correspond to minima of the grand potential 11 introduced in Eqs. (1.66) and (1.67). The fluid fluid interaction is described by the intermolecular potential ug (r) introduced in Eq. (5.38) where the associated shifted-force potential is introduced in Eq. (5.39). The fluid substrate interaction is described by 1 1 (x, z) in the continuum representation [see Eq. (5.68)], where x replaces x because of the misaligmncnt of the sul)stratcs relative to each other [see Eq. (5.103)]. 

The Shifted Force Potential. When two sites interact through a spherically symmetric potential function, they exert upon each other equal and opposite forces that act along the vector f between them. (A wavy underline indicates a vector quantity.) The magnitude of the force is given by F=-(du/dr). 

The modified force and potential energy are plotted in Figure Kb). Both potential energy and force now go smoothly to zero at r=r. The modification to the force is a constant equal to the magnitude of the original discontinuity at r, hence the name shifted force potential. The modification to the poten-... 

The small changes to the force and potential energy, resulting from the change to a shifted force potential function, provide improved accuracy eind stcibility in computer simulations, with negligible changes to structure and time correlation functions, at short to moderate times, calculated from the usual truncated potential. Corrections to calculated thermodynamic properties to account for modifications to the potential can be calculated by a perturbation method similar to that used for the long tail corrections. This matter will be discussed in detail in a separate paper. 

Figure 7 Rate constant versus driving force for the back-ET process at electrostatically sensitized Sn02 interface for a series of complexes spanning a range of redox potentials. The filled square symbols ( ) correspond to data collected at pH = 9. Also included in the figure are the results of variable pH studies for Os(5-Cl-phen)l+ (0) and Os(3,4,7,8-CH3-phen)2(m-im)l+ (O). The pH variations serve to shift the potential of the conduction band edge. (Adapted from Ref. 36.)...

At equilibrium, it vanishes. Let us now assume that an external force K P ) has shifted the interface by an amount 3, and that the / particles were not able to follow this shift. The potential field f ( ) is now perturbed relative to the cloud of / particles as indicated in Figure 10-15. From Eqn. (10.54), one calculates the restoring... 

In the shifted force method, both the potential and the force are smoothed out to zero at a certain cutoff distance Rq... 

Figure 9 Electrostatic interaction energy between two hydrogen atoms represented by the shifted-force Coulomb and shielded Coulomb potentials.

Figure 9 shows the Coulomb potential, the shifted-force Coulomb potential (Eq. [31]), the shielded Coulomb potential (Eq. [32]), and the shielded shifted-force Coulomb potential (Eq. [33]) calculated for interaction between two hydrogen atoms with a charge = + e. ... 

Just as in the shifted-force method, the taper method replaces a strict cutoff radius by a cutoff region in which the interactions are smoothly tapered to zero. This is done by multiplying the Coulomb potential from r = Ator = B... 

Finally, it should be noted that both the shifted-force method and the tapered method can be made more computer efficient by the construction of lookup tables for the potentials and their derivatives. 

The great expense in calculation time due to the inevitably large particle numbers in single-file systems calls for the application of simplified potentials. Figure 1 shows the results obtained for spherical molecules diffusing in an unstructured tube [22]. Particle-particle and particle-wall interactions have been simulated by a shifted-force Lennard-Jones potential [26] and an... 

The effect of screening is illustrated in Fig. 1.8. Further modifications substitute potential energies that smoothly go to a constant value (normally zero) at a finite distance, as, e.g. the damped, shifted force model [134]. 

Each simulation extended to about 15000 time-steps after a pre-equilibration of approximately the same length. Ewald summation in tabulated form was used for the Coulomb interactions, and the shifted-force method (e.g., Allen and Tildesley 1987) was used for the non-Coulomb parts of the BJH potential. All technical details of the simulations are described in detail elsewhere (Kalinichev and Heinzinger 1992, 1995 Kalinichev 1993). 

When the electromotive force Ec - Ea) is supplied, the corrosion cell is formed with the current flowing between the anode and the cathode. The cathodic electrode potential is shifted to the less noble direction. The shifting of potentials... 

The limitation for MD simulation is the number of molecules N that can realistically be modeled in a computer. Since the computation for any particular molecule requires the consideration of all other molecules as potential collision partners, the amount of computation required is proportional to N. Some saving in computer time is achieved by cutting off the weak tail of the potential at say = 2.5a and shifting the potential by a linear term in r so that force goes smoothly to zero at the cut off. As a result, only nearby molecules are treated as potential collision partners and the computation time for N molecules no longer scale with N. ... 

It should be noted that the force described by equation (8) includes solute/solvent boundary force contributions from the dielectric and ion pressures (second and third terms) in addition to the familiar direct force on a charge in an electrostatic field (first term). With expressions for the potential distribution, the electrostatic free energy, and the electrostatic force, the PB treatment of solvent effects can be applied to a variety of macromolecular properties, including pATa shifts, redox potential shifts, binding energies, association rates, folding and stabilization. 

Figure 1. Potential energy u (solid lines) and force F (dashed lines) as a function of site-site distance r for truncated (a) and shifted-force (h) potential functions. The modified force in (h) is formed by shifting the force-distance curve in (a) upward a distance H, equal to the magnitude of the discontinuity at r = r,.. The modified potential energy in (h) is then defined by Equations 1 and 2.

Imprcssed-current cathodic protection entails the use of an external power source in combination with a stable anode. The potential of the specimen being protected is forced to negative values and its rate of corrosion is consequently reduced. The result of impressing a cathodic current on the structure is shown in Figure 18.13 For the parameters used to draw this figure we obtain jeon 48.8 fxA cm and Fcorr = —0.554 V vs. SHE. Applying a cathodic current density of 72 (xA cm shifts the potential to —0.58 V and the anodic current density (which represents the rate of corrosion of the cathodically protected surface) is reduced to 10 iA cm", as shown by line 1. 

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