Algorithm, CANON

Internally IGOR uses since 1980 canonical representation of chemical reactions that is based on a linear combination B + kE (k is an integer the maximum entry of any B or E, e.g k = 10). The latter is subjected to a canonical indexing procedure by an extended version of the algorithm CANON (ref. 21,22). 

Accordingly an EMB(A) is representable by up to n distinct but equivalent fee-matrices [31]. One of these fee-matrices of the EMB(A) may be chosen as the canonical representation, e.g. by the algorithm CANON [32]. Note that any constitutional symmetries in EMB(A) are also detected by CANON. 

Such an index may be automatically assigned by a canonical ordering procedure such as the algorithm CANON [32]. 

HOC algorithms hierarchically ordered extended connectivities algorithms - canonical numbering... 

A special extension of SMILES is USMILES (sometimes described as Broad SMILES) [23-25]. This Unique SMILES of Daylight is a canonical representation of a structure. This means that the coding is independent of the internal atomic numbering and results always in the same canonical, unambiguous, and unique description of the compound, granted by an algorithm (see Section 2.5.2). 

Various methods have been developed for a unique and unambiguous numbering of the atoms of a molecule and thus for deriving a canonical code for this molecule [76]. Besides eigenvalues of adjacency matrices [77], it is mainly the Morgan Algorithm that is used [79]. 

It is clear that Eq. (85) is numerically reliable provided is sufficiently small. However, a detailed investigation in Ref. 69 reveals that can be as large as some ten percent of the diameter of a fluid molecule. Likewise, rj should not be smaller than, say, the distance at which the radial pair correlation function has its first minimum (corresponding to the nearest-neighbor shell). Under these conditions, and if combined with a neighbor list technique, savings in computer time of up to 40% over conventional implementations are measured for the first (canonical) step of the algorithm detailed in Sec. IIIB. These are achieved because, for pairwise interactions, only 1+ 2 contributions need to be computed here before i is moved U and F2), and only contributions need to be evaluated after i is displaced... 

To test the results of the chemical potential evaluation, the grand canonical ensemble Monte Carlo simulation of the bulk associating fluid has also been performed. The algorithm of this simulation was identical to that described in Ref. 172. All the calculations have been performed for states far from the liquid-gas coexistence curve [173]. 

MD runs for polymers typically exceed the stability Umits of a micro-canonical simulation, so using the fluctuation-dissipation theorem one can define a canonical ensemble and stabilize the runs. For the noise term one can use equally distributed random numbers which have the mean value and the second moment required by Eq. (13). In most cases the equations of motion are then solved using a third- or fifth-order predictor-corrector or Verlet s algorithms. 

It should be appreciated that canonical correlation analysis, as the name implies, is about correlation not about variance. The first step in the algorithm is to move from the original data matrices X and Y, to their singular vectors, Ux and Uy, respectively. The singular values, or the variances of the PCs of X and Y, play no role. 

For Hamiltonian dynamics with a canonical or microcanonical distribution of initial conditions the acceptance probability for pathways generated with the shifting algorithm is particularly simple. Provided forward and backward shifting moves are carried out with the same probability the acceptance probability from (7.23) reduces to... 

This section is used to introduce the momentum-enhanced hybrid Monte Carlo (MEHMC) method that in principle converges to the canonical distribution. This ad hoc method uses averaged momenta to bias the initial choice of momenta at each step in a hybrid Monte Carlo (HMC) procedure. Because these average momenta are associated with essential degrees of freedom, conformation space is sampled effectively. The relationship of the method to other enhanced sampling algorithms is discussed. 

Fig. 10.12. Vapor-liquid phase behavior for the Lennard-Jones fluid. Solid triangles and hollow squares indicate the results of the particle addition/deletion and volume scaling variants of the flat-histogram simulation using the Wang-Landau algorithm. Crosses are from a histogram reweighting study based on grand-canonical measurements at seven state points. The solid line is from Lotfi, et al. [76], Reprinted figure with permission from [75]. 2002 by the American Physical Society...

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