Local bond order parameter

Lechner W, DeUago C (2008) Accurate determination of crystal stmctures based on averaged local bond order parameters.. J Chem Phys 129... 

The local bond-order parameters are a measure of the local structure around a particle and are constructed as follows. First we define a (2/ -i-1) dimensional complex vector with the components... 

The orientation of the unit vector ry is determined by the polar and azimuthal angles 0y and 4 ij The rotationally invariant local bond-order parameters are then defined as follows ... 

Fig. 4. Distribution functions of the local bond-order parameters q, q, 64 and tog from Monte Carlo simulations in a hard-sphere system. Here the cutoff radius rq for the local environment of a particle is chosen to be 1.4

This expression has the same form as the modified embedded atom model. Taylor represented the local atomic density by bond-order parameters and different radial functions as discussed in Sect. 2.3.1 in Chap. 2. By choosing appropriate radial functions, he obtained the original modified embedded-atom formula, but systematic improvement of the formula is also possible in his framework. 

Strong layering of the confined system was observed due to the interaction with the attractive pore walls. The structure of the confined mixture was investigated by calculating for each layer i the 2D bond-order parameters (n = 4 and 6 for a square and triangular structure, respectively). On./ was determined as the average value of the local order parameter nX ), which measures the bond order at a position r of a particle in the layer i [17] ... 

Before we can calculate Nn in a Monte Carlo simulation we need to have a numerical technique that enables us to distinguish between particles in a liquid and solid environment. To this end, we use local bond-order analysis introduced by Steinhardt et al. [11] and applied to study nucleation by Frenkel and coworkers [8, 12, 13]. The advantage of this analysis is that it is only sensitive to the overall degree of crystallinity in the system, but independent of any specific crystal structure. This requirement is important as otherwise we would apply an external biasing potential, that could force the system to crystallize in a specific structure. A second advantage is that these bond-order parameters can be constructed so as to be independent of the reference frame. 

Fig. 27. Structure analysis of two independent crystal nuclei of size n = 100 and 200 from the simulations with parameters = 8 and k = 10. The figure shows the results for the fit parameters for the local bond-order analysis as a function of the distance from the center of mass of the nuclei. The core of the cluster of size n = 100 has a clear bcc signature, where the cluster of size n = 200 shows a clear fee structure...

The distribution of the intermolecular vector is also of value in distinguishing between smectic A and smectic B phases with the latter having long range bond orientational order [23, 24]. At the local level we can define a bond orientational order parameter, PeCn) for molecule i at position q by [25]... 

Formally, S2 represents a decrease in the autocorrelation function caused by the motion S2=0 corresponds to completely unrestricted motion of a bond (N-H in this case), while S2=0 is expected if the bond reorientations are frozen. It was shown recently that the order parameter may be related to the statistical mechanical properties of a protein molecule [33-35] hence, changes in the NMR-derived order parameters can indicate localized contributions to overall molecular entropy. 

A linear correlation between 13C chemical shifts and local n electron densities has been reported for monocyclic (4n + 2) n electron systems such as benzene and nonbenzenoid aromatic ions [76] (Section 3.1.3, Fig. 3.2). In contrast to theoretical predictions (86.7 ppm per n electron [75]), the experimental slope is 160 ppm per it electron (Fig. 3.2), so that additional parameters such as o electron density and bond order have to be taken into account [381]. Another semiempirical approach based on perturbational MO theory predicts alkyl-induced 13C chemical shifts in aromatic hydrocarbons by means of a two-parameter equation parameters are the atom-atom polarizability nijt obtained from HMO calculations, and an empirically determined substituent constant [382]. 

As in the MD method, PES for KMC can be derived from first-principles methods or using empirical energy functionals described above. However, the KMC method requires the accurate evaluation of the PES not only near the local minima, but also for transition regions between them. The corresponding empirical potentials are called reactive, since they can be used to calculate parameters of chemical reactions. The development of reactive potentials is quite a difficult problem, since chemical reactions usually include the breaking or formation of new bonds and a reconfiguration of the electronic structure. At present, a few types of reactive empirical potentials can semi-quantitatively reproduce the results of first-principles calculations these are EAM and MEAM potentials for metals and bond-order potentials (Tersoff and Brenner) for covalent semiconductors and organics. 

In order to include curvature-dependence in both the covalent and non-bonding interactions, we used the adaptive intermolecular reactive bond-order (AIREBO) potential,24 with modified van der Waals interactions. This potential uses the same bonding interactions as Brenner s REBO potential,25,26 both of which correctly account for local curvature dependence in the covalent bonding interactions. Chemisorption is thus treated accurately, but there is no explicit or implicit curvature dependence in the Lennard-Jones (L-J) parameters used to describe the non-bonded van der Waals interactions (physisorption). Consequently, we modified the Lennard-Jones parameters to make them explicitly dependent on the curvature of the nanotube. 

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