Canonical indexing

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). 

V. Kvasnicka, J. Pospichal, Canonical Indexing and Constructive Enumeration of Molecular Graphs, J.Chem.Inf.Comput.Sci., 30 (1990) 99-105. 

An out-of-space string Kp can only contribute if it is produced by K m)h and if E j transforms it back into an allowed string (Jp). But if the orbitals are numbered consecutively within each RAS space, and if equation (139) is used to define the canonical index (ij), such contributions are impossible. For example, consider the case where a maximum of two electrons are allowed in RAS III. If string Ip already contains two electrons in RAS III, then a single replacement E (Ip)) which promotes a third electron to RAS III will yield an out-of-space string. The shift operator E could produce an allowed string again if it moves any of the three electrons in RAS III back down into RAS I or RAS II. However, if the summation over orbitals is restricted to (ij) > (kl), this is impossible, since k > i. 

Kvasnicka, V. and Pospichal, J. (1990) Canonical indexing and constructive enumeration of molecular graphs. J. Chem. Inf. Comput. ScL,... 

The permuted labels will not, in general, be in standard order order orders them enablng the transformed canonical index to be computed. The signs can now be discarded, having been used. / it = abs(it)-, jt = absljty, if it < jt)... 

We first form the canonical index ((a6)(cd)) from the atomic centre labels. Then apply the permutations induced by the symmetry operations (summarised in the above table) and form the canonical index ((a 6 )(c d )) for each permutation there are three possibilities ... 

We now form the canonical index of an integral involving a basis function on each of the centres let ((ij)(k()) be such an index, it is only necessary that one of 4>i, 4>j, k and (j>i be on each of the centres a,b,c and d, not that they be on the respective centres. Now apply the transformations contained in the coded linear transformation array to this set of basis-function labels there are, this time, four possibilities ... 

At last, a permuted set of labels worth saving, put them in the sub-list and keep their canonical index /... 

A precondition for an efficient manipulation of BE-matrices in the computer is a canonical indexing of the atoms in a molecule. In order to generate a unique numbering we use the connectivity matrix of the molecular graph and the labels already assigned to its vertices, i.e. the chemical symbols. 

The constructive enumeration proposed by Kvasni ka and Pospfchal based on the GLAC generates only non-redundant canonically indexed graphs. This method does not need to check whether the produced graphs have already been constructed in the previous steps. 

Particles may be indexed by a canonical code, defined to be the smallest binary number representing an intermediate state of a given paiticle. Empirically derived properties of particles with canonical codes with widths tv < 16 evolving under a radius-r PFA rule, are summarized in table 3.4 [park86]. 

With applications to protein solution thermodynamics in mind, we now present an alternative derivation of the potential distribution theorem. Consider a macroscopic solution consisting of the solute of interest and the solvent. We describe a macroscopic subsystem of this solution based on the grand canonical ensemble of statistical thermodynamics, accordingly specified by a temperature, a volume, and chemical potentials for all solution species including the solute of interest, which is identified with a subscript index 1. The average number of solute molecules in this subsystem is... 

One of the most popular such techniques is parallel tempering in the canonical ensemble, for which the index parameter is temperature [13-15], While parallel tempering (or replica exchange) strategies had been independently proposed on multiple occasions in various scientific areas, perhaps the earliest seed of the idea can be found in early work by Swendsen and Wang [37]. 

Diagonal matrix elements of the P3 self-energy approximation may be expressed in terms of canonical Hartree-Fock orbital energies and electron repulsion integrals in this basis. For ionization energies, where the index p pertains to an occupied spinorbital in the Hartree-Fock determinant,... 

Such a unique indexing of the atoms affords unique connectivity lists, or matrices, respectively, which may be termed canonical representation b> of chemical constitution. 

Unfortunately, Maxwell s equations can be solved analytically for only a few simple canonical resonator structures, such as spheres (Stratton, 1997) and infinitely long cylinders of circular cross-sections (Jones, 1964). For arbitrary-shape microresonators, numerical solution is required, even in the 2-D formulation. Most 2-D methods and algorithms for the simulation of microresonator properties rely on the Effective Index (El) method to account for the planar microresonator finite thickness (Chin, 1994). The El method enables reducing the original 3-D problem to a pair of 2-D problems for transverse-electric and transverse-magnetic polarized modes and perform numerical calculations in the plane of the resonator. Here, the effective... 

As one increases the indexing, the disparity between the number of Cartesian functions and the number of canonical functions increases. Thus, with f-type GTOs (indices summing to 3) there are 10 Cartesian functions and 7 canonical functions, with g-type 15 and 10, etc. GTOs can be taken arbitrarily high in angular momentum. 

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Here, the index e stands for electronic, and the index k for either translational, rotational or vibrational degrees of freedom, the quantities 0 are the modules of canonical distributions [see (4) and (5)], and e are the energy levels. The function/ x(e ) is an expression of the fact that only certain electronic levels of individual chemical components — the long-lived, chemically interacting ones - in the system are considered. 

This Hamiltonian describes a set of coupled oscillators. Finally, applying the canonical transformation from Aqi to new variables xq (q is now the index of independent modes)... 

The basic building blocks of the theory are Heisenberg operators (x) which create and destroy respectively, particles of type m at the space-time point x = x, (x. For the purposes of chemistry we can take the index nzs>e for electrons and a for nuclei only. Of course when energies are much larger than chemical energies, nuclei appear to be composite particles, and we must then introduce fields for their constituents (quarks, rishons). We shall not make any explicit reference to the spins carried by these fields beyond noting that odd-integral spins require fermi statistics, so that for fermi fields we have canonical anticommutation relations (CARS)... 

Both the bond character and the free valence index (but not the energy) depend in principle on the choice of the canonical configurations. In general, however, it is easy to make a proper choice on the basis of chemical intuition. In naphthalene the results for both quantities are as given below (see also Table 24 and p. 210). o.I0... 

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