Force-shifted function

When the cutoff is sharp, discontinuities in the forces and resultant loss of con servation of energy m molecular dynamics calcnla-tionscan result.To minimi/e edge effects of a cu toff, often theciit-off IS implemented with a switching or shifting function to allow the interactions to go smoothly to /ero. 

Here /(R) and pflX) denote the shift and generalized momentum for the molecular vibration of the low frequency a>9 and reduced mass m, at the Rth site of the adsorbate lattice bi+(K) and K) are creation and annihilation operators for the collectivized mode of the adsorbate that is characterized by the squared frequency /2(K) = ml + d>, a,(K)/m , with O / iat(K) representing the Fourier component of the force constant function /jat(R). Shifts i//(R) for all molecules are assumed to be oriented in the same arbitrary direction specified by the unit vector e they are related to the corresponding normal coordinates, ue (K), and secondary quantization operators ... 

The use of both switching and shifting functions modifies the model, since the potential and forces are changed, and therefore affects the results of the simulation. Whether these changes are significant relative to the other approximations in the model depends on the specific system and properties. 

In order to make satisfactory predictions, it was necessary to take a step that prior chapters did not. Rather than using the classical storage and loss moduli G co) and G" co), a novel pair of viscoelastic moduli are here introduced. The ansatz makes predictions for the novel functions. The new functions are actually not very radical indeed, there is a sense in which they are more consistent with prior chapters than are G (co) and G"(co). The previous chapter described the zero-shear viscosity as the low-shear limit t](c, M) of the experimentally accessible viscosity. To describe shear thinning, rj is extended to the shear-dependent For oscillatory motion, one might by analogy have expected rj to become r cd) with complications arising because at nonzero co the displacement and applied force shift in relative phase. The orthodox loss function is G", with... 

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. 

Terms in the energy expression that describe how one motion of the molecule affects another are called cross terms. A cross term commonly used is a stretch-bend term, which describes how equilibrium bond lengths tend to shift as bond angles are changed. Some force fields have no cross terms and may compensate for this by having sophisticated electrostatic functions. The MM4 force field is at the opposite extreme with nine different types of cross terms. 

To initiate a chemical relaxation it is necessary to perturb the system from its initial equilibrium position. This is done by applying a forcing function, which is an appropriate experimental stress to which the system responds with a shift in equilibrium configuration. Forcing functions can be transient (a sudden, essentially discontinuous Jolt ) or periodic (a cyclic stress of constant frequency). 

The RMS displacement as a function of time for the shifted potential simulations are shown in Figure 3. As expected for this functional form, the longer cutoff distances result in a smaller RMS deviation from X-ray. The results for the 100 picosecond analysis section of all of the simulations are summarized in Table IV. For Table IV, the term "rdie" indicates that a distant dependant dielectric was used, cdie indicates that a constant dielectric was used, and eps2 indicates that the electrostatic forces have been scaled by 0.5. 

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