Unphysical forces

The perspective exploited by transition path sampling, namely, a statistical description of pathways with endpoints located in certain phase-space regions, was hrst introduced by Pratt [27], who described stochastic pathways as chains of states, linked by appropriate transition probabilities. Others have explored similar ideas and have constructed ensembles of pathways using ad hoc probability functionals [28-35]. Pathways found by these methods are reactive, but they are not consistent with the true dynamics of the system, so that their utility for studying transition dynamics is limited. Trajectories in the transition path ensemble from Eq. (1.2), on the other hand, are true dynamical trajectories, free of any bias by unphysical forces or constraints. Indeed, transition path sampling selects reactive trajectories from the set of all trajectories produced by the system s intrinsic dynamics, rather than generating them according to an artificial bias. This important feature of the method allows the calculation of dynamical properties such as rate constants. 

In order to obtain a steady state from Eqs. 38 dissipative heat must be removed from the system. This is achieved by the last (thermostatting) terms of the last two equations in Eqs. 38. In this respect it is essential to observe that accurate values for Uj and A are needed. Any deviations from the assumed streaming and angular velocity profiles (biased profiles) will exert unphysical forces and torques which in turn will affect the shear-induced translational and rotational ordering in the system [209,211,212]. The values for the multipliers and depend on the particular choice of the thermostat. A common choice, also adopted in the work of McWhirter and Patey, is a Gaussian isokinetic thermostat [209] which insures that the kinetic and rotational energies (calculated from the thermal velocities p" and thermal angular velocities ot) - A ) and therefore the temperature are conserved. Other possible choices are the Hoover-Nose or Nose-Hoover-chain thermostats [213-216]. 

Similarly, in studies of lamellar interfaces the calculations using the central-force potentials predict correctly the order of energies for different interfaces but their ratios cannot be determined since the energy of the ordered twin is unphysically low, similarly as that of the SISF. Notwithstcinding, the situation is more complex in the case of interfaces. It has been demonstrated that the atomic structure of an ordered twin with APB type displacement is not predicted correctly in the framework of central-forces and that it is the formation of strong Ti-Ti covalent bonds across the interface which dominates the structure. This character of bonding in TiAl is likely to be even more important in more complex interfaces and it cannot be excluded that it affects directly dislocation cores. 

Although this model seems to reflect well some experimental observations of contact and separation [6,7] the assumptions made in its formulation are in fact unphysical. They assume that the solids do not interact outside the contact region, whereas in reality electrostatic and van der Waals forces are nonzero at separations of several nanometers. The assumptions made by JKR lead to infinite values of stress around the perimeter of the connecting neck between sphere and plane. 

At short distances, approximately equal to the excluded volume diameter, effective pair forces obtained from force matching exhibit unphysically large fluctuations. This is largely due to inadequate sampling of configurations at short distances in... 

Here, is the distance between atoms i andj, C(/ is a dispersion coefficient for atoms i andj, which can be calculated directly from tabulated properties of the individual atoms, and /dampF y) is a damping function to avoid unphysical behavior of the dispersion term for small distances. The only empirical parameter in this expression is S, a scaling factor that is applied uniformly to all pairs of atoms. In applications of DFT-D, this scaling factor has been estimated separately for each functional of interest by optimizing its value with respect to collections of molecular complexes in which dispersion interactions are important. There are no fundamental barriers to applying the ideas of DFT-D within plane-wave DFT calculations. In the work by Neumann and Perrin mentioned above, they showed that adding dispersion corrections to forces... 

Notably, the advantage of applying MM energy functions in AFE simulations lies in the fact that force evaluations are not sensitive to large structural distortions, i.e., force calculations for structures in unphysical state (A / 0 or A /1) do not pose numerical problems. 

In this model, we have seen how careless introduction of a memory function may lead to unphysical behaviour. In the present simple case, we have seen how the physical situation forces us to work with a parameter choice that removes the difficulty. However, as we do not, in general, have a criterion for acceptable memory functions, an ad hoc modification of the equations may easily lead to unexpected and absurd results. Approximative memory functions are hazardous, [Barnett 2001],... 

Unphysical quenching rate is not the only limitation of MD. Since the potential used enable computations of only central forces, it is suitable for simulations of glasses, which are significantly ionic. It is also successful for the simulations of metallic glasses where use is made of optimised pseudo potentials obtained from first-principle calculations. But in largely covalent materials, MD cannot be of much use imless suitable effective potential functions are developed which take care of non-central nature of the forces as well. In the next section we discuss further advances in MD simulations based on the use of quantum mechanical calculations, which optimise the local geometries and therefore provide more accurate simulations of structure. 

As mentioned above, a set of experimental data does not necessarily correspond to a unique molecular structure. Moreover, even unphysical structures may be consistent with a set of experimental data. It is therefore necessary to carefully choose a set of constraints to limit the number of possible structures. The uniqueness theorem of statistical mechanics [30, 31] provides a guide to the number and type of constraints that should be appfied in the RMC method in order to get a unique structure [32]. For systems in which only two- and three-body forces are important, the uniqueness theorem states that a given set of pair correlation function and three-body correlation function determines aU the higher correlation functions. In other words, assuming that only two- and three-body forces are important, the RMC method must be implemented along with constraints that describe the three-body correlations [27]. 

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