Error in forces

Figure 7.5. Comparison between average force-distance profiles in the absence and in the presence of 1.9 DNA on average between adjacent particles. Circles, Fbg, squares, F,> triangles, Fi = Fir - Fftg Dashed lines correspond to the error in force-distance profiles, calculated from the standard deviation of the measured profiles. (Adopted from [26].)...

Figure 8 Comparison of mean absolute errors in force field calculations on conformational energy differences for organic molecules of different structural classes. (Compiled from the results reported in Refs. 25 and 27.)...

Consider two charges qi and 2 placed at random positions Ui and U2 in the unit cell. The mean square error in force due to charge assignment is... 

Mackay, M. E., Halley, P. J. Angular comphance error in force rebalance torque transducers. /. Rheol. 

This situation, despite the fact that reliability is increasing, is very undesirable. A considerable effort will be needed to revise the shape of the potential functions such that transferability is greatly enhanced and the number of atom types can be reduced. After all, there is only one type of carbon it has mass 12 and charge 6 and that is all that matters. What is obviously most needed is to incorporate essential many-body interactions in a proper way. In all present non-polarisable force fields many-body interactions are incorporated in an average way into pair-additive terms. In general, errors in one term are compensated by parameter adjustments in other terms, and the resulting force field is only valid for a limited range of environments. 

The force function /(r) differs from that in ref. [65] by a factor r, yielding simpler expressions. Some errors in that reference have been corrected. 

The two sources of stochasticity are conceptually and computationally quite distinct. In (A) we do not know the exact equations of motion and we solve instead phenomenological equations. There is no systematic way in which we can approach the exact equations of motion. For example, rarely in the Langevin approach the friction and the random force are extracted from a microscopic model. This makes it necessary to use a rather arbitrary selection of parameters, such as the amplitude of the random force or the friction coefficient. On the other hand, the equations in (B) are based on atomic information and it is the solution that is approximate. For ejcample, to compute a trajectory we make the ad-hoc assumption of a Gaussian distribution of numerical errors. In the present article we also argue that because of practical reasons it is not possible to ignore the numerical errors, even in approach (A). 

Table 1. Maximum error in the energy using the semi-implicit method with the energy conserving method (6) for the strong forces.

We apply the semi-implicit algorithm to handle the weak potentials Vi, and the energy conserving method (16) for the stiff forces. The maximal error in the total energy, i.e. 

Table 7.1 presents us with something of a dilemma. We would obviously desire to explore i much of the phase space as possible but this may be compromised by the need for a sma time step. One possible approach is to use a multiple time step method. The underlyir rationale is that certain interactions evolve more rapidly with rime than other interaction The twin-range method (Section 6.7.1) is a crude type of multiple time step approach, i that interactions involving atoms between the lower and upper cutoff distance remai constant and change only when the neighbour list is updated. However, this approac can lead to an accumulation of numerical errors in calculated properties. A more soph sticated approach is to approximate the forces due to these atoms using a Taylor seri< expansion [Streett et al. 1978] ... 

It is important to verify that the simulation describes the chemical system correctly. Any given property of the system should show a normal (Gaussian) distribution around the average value. If a normal distribution is not obtained, then a systematic error in the calculation is indicated. Comparing computed values to the experimental results will indicate the reasonableness of the force field, number of solvent molecules, and other aspects of the model system. 

As for the dielectric constant, when explicit solvent molecules are included in the calculations, a value of 1, as in vacuum, should be used because the solvent molecules themselves will perform the charge screening. The omission of explicit solvent molecules can be partially accounted for by the use of an / -dependent dielectric, where the dielectric constant increases as the distance between the atoms, increases (e.g., at a separation of 1 A the dielectric constant equals 1 at a 3 A separation the dielectric equals 3 and so on). Alternatives include sigmoidal dielectrics [80] however, their use has not been widespread. In any case, it is important that the dielectric constant used for a computation correspond to that for which the force field being used was designed use of alternative dielectric constants will lead to improper weighting of the different electrostatic interactions, which may lead to significant errors in the computations. 

One way to do this is afforded by the predictor-corrector method. We ignore terms higher than those shown explicitly, and calculate the predicted terms starting with bP(t). However, this procedure will not give the correct trajectory because we have not included the force law. This is done at the corrector step. We calculate from the new position rP the force at time t + St and hence the correct acceleration a (t -f 5t). This can be compared with the predicted acceleration aP(f -I- St) to estimate the size of the error in the prediction step... 

For comparison with experimental frequencies (which necessarily are anharmonic), there is normally little point in improving the theoretical level beyond MP2 with a TZ(2df,2pd) type basis set unless anharmonicity constants are calculated explicitly. Although anharmonicity can be approximately accounted for by scaling the harmonic frequencies by 0.97, the remaining errors in the harmonic force constants at this level are normally smaller than the corresponding errors due to variations in anharmonicity. 

Basically, there may be three reasons for the inconsistency between the theoretical and experimental friction factors (1) discrepancy between the actual conditions of a given experiment and the assumptions used in deriving the theoretical value, (2) error in measurements, and (3) effects due to decreasing the characteristic scale of the problem, which leads to changing correlation between the mass and surface forces (Ho and Tai 1998). 

This is equivalent to assuming that the standard error in the i1 1 measurement of the response variable is proportional to its value, again a rather "safe" assumption as it forces least squares to pay equal attention to all data points. 

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