Canonical representation

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). 

The development of localized-orbital aspects of molecular orbital theory can be regarded as a successful attempt to deal with the two kinds of comparisons from a unified theoretical standpoint. It is based on a characteristic flexibility of the molecular orbital wavefunction as regards the choice of the molecular orbitals themselves the same many-electron Slater determinant can be expressed in terms of various sets of molecular orbitals. In the classical spectroscopic approach one particular set, the canonical set, is used. On the other hand, for the same wavefunction an alternative set can be found which is especially suited for comparing corresponding states of structurally related molecules. This is the set of localized molecular orbitals. Thus, it is possible to cast one many-electron molecular-orbital wavefunction into several forms, which are adapted for use in different comparisons fora comparison of the ground state of a molecule with its excited states the canonical representation is most effective for a comparison of a particular state of a molecule with corresponding states in related molecules, the localized representation is most effective. In this way the molecular orbital theory provides a unified approach to both types of problems. 

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

In the canonical representation the total energy correction of a given order n is denoted by he localized representation consists of two terms The... 

In order to have an insight into the three-body effect,we continue the study of the He-clusters. Fortunately, there are published examples for several He-clusters, as cited above. All of these studies, however, were performed in the canonical representation. The use of the localized representation allows us to separate the dispersion and the charge transfer components of the interaction energy for the three-body effects as it was similarly done for the two-body effects. The calculation of the interaction energy in the SMO-LMBPT fiumework has been discussed in detail in several papers [8-10] The formulae given at the correlated level, however, were restricted to the two-body interaction. 

The definition of aromaticity conceived by Hiickel strictly applies to monocyclic ring systems, but indole, constructed from the fusion of benzene and pyrrole, behaves as an aromatic compound, like quinoline and isoquinoline. The ring fusion, however, affects the properties of both components. This is reflected in the valence bond description of indole, shown in Scheme 7.1, where one canonical representation shows electron density shared between N-1 and C-3 in the pyrrole unit (implying enamine character). Note that although other canonical forms can be drawn, where the lone-pair electrons are delocalized into the benzenoid ring, their energy content is relatively high and they are of limited importance. 

Let us first discuss the results obtained in canonical representation. The energy quantities calculated by using eight different basis sets for a water monomer are given in Table I. 

The various canonical representations of thermodynamics are an issue of concern. Tarsitant and Vicentini (67) analyzed a number of textbooks on thermodynamics, and pointed out divergent attitudes not only towards the definition of fundamental thermodynamic concepts, but also towards the epistemological status of the subject. According to these authors, these attitudes... 

In the initial representation this state corresponds to a statistical mixture. Now two cases are possible. If we deal with the ground state, the statistical mixture Eqs. (58) and (59) may be reduced to the canonical representation... 

Using this ordering, we can construct a canonical representation of a given tree by reordering the children of each vertex according to the order defined above. See Figure 6. From this canonical representation, we can construct a linear code which represents the tree uniquely. There are many possible ways to do this one way is defined by the following rules. 

AOS = variation of the oxidation state of the metal, ANEV = variation of the number of valence electrons, ACN = variation of coordination number, AQ = variation of the global charge. c Two different representations of the same reaction. d This is not a reaction but a canonical representation. 

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). 

Grochowski P, Lesyng B (2003) Extended Hellmann-Feynman Forces, Canonical Representations, and Exponential Propagators in the Mixed Quantum-Classical Molecular Dynamics. J. Chem. Phys. 119 11541-11555... 

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. 

The tautomers of cytosine have also been examined with continuum solvation methods. Both Miertus-Scrocco-Tomasi (MST) and PCM computations predict that 13a is the lowest energy conformer in aqueous solution. This is consistent with the explicit water computations. Aqueous solvation, therefore, substantially alters the relative energies of the tautomers of cytosine. The canonical representation of cytosine 13a, the tautomer invoked in Watson-Crick base pairing, is in fact the most favorable tautomer in solution, but not in the gas phase. 

In the description of nature afforded by quantum mechanics, one classifies and characterizes the state of a total system in terms of the eigenvalues of a set of commuting observables acting on an element of the Hilbert space, the state vector. Molecular orbital theory in its canonical representation as originally... 

In addition to the canonical representation, where the independent variables are N and v(r), the convexity of the functional E N, v] allows the existence of other representations.8 The grand potential,... 

Derivatives from two representations are not independent they can be related by a reduction of derivatives scheme, analogous with the thermodynamical procedure.9 For the previous representations, one finds that the second derivatives of the grand canonical representation can be written in the following form,... 

Figure 2. Canonical representations of the MLCT excited state in (a) nbn and (b) npbn ruthenium complexes, showing how torsion between the phenylene rings can act to diminish electronic coupling...

The embedding of encoded watermark bits from u, in the jth structure Mj is considered. A block diagram of the embedding scheme is depicted in Fig. 3. First, a canonic representation of the structure is obtained as described above. Next, the host data vector Xj is extracted (see also Section 4.7). Here, it is assumed that L j elements are extracted from the structure Mj, and the elements of are scaled such that the watermark w,- can be embedded with a variance = l. 

The upper part of the block diagram in Fig. 4 depicts the watermark detection scheme for one stmcture Mj. First, the data is transformed into its canonical representation. Next, the received vector rj is extracted. The extraction method must be identical to the host vector extraction used for watermark embedding. Thus, the length of rj is also Lx J. Second, tlie 64-bit hash of Mj is derived and the pseudo-random vectors t j. kj and ij are computed dependent on the copyright holders key K. After applying tlie spread transform, the demodulated soft watennark letters yj are derived from and kj as described in Section 2. The probability p dn,j = 1) of receiving a watennark letter dnj = 1 from the nth clement of is given by... 

Two-dimensional representations alternative to the molecular graph are the linear notation systems, for example, Wiswesser Line Notation system (WLN) [Smith and Baker, 1975], SMILES [Weininger, 1988, 1990, 2003 Weininger, Weininger et al., 1989 Convard, Dubost et al., 1994 Hinze and Welz, 1996], and SMARTS (SMART - Daylight Chemical Information Systems, 2004). CAST (CAnonical representation of STereochemistry) is a method that gives a linear notation that canonically represents stereochemistry around a specific site in a molecule [Satoh, Koshino et al, 2000, 2001, 2002],... 

Similar to —> MNA descriptors, signature descriptors are holographic vectors of count numbers of all the atomic signatures present in the data set molecules, the signature of an atom being a canonical representation of the atom environment up to a predefined depth, that is, a circular substructure descriptor [Visco, Pophale et al, 2002 Faulon, Visco et al, 2003]. 

AIC Akaike Information Content CAST CAnonical representation of... 

Because SMIRKS is a combination of SMILES and SMARTS and because there is no canonical representation of SMARTS, there is no canonical representation of SMIRKS. SMARTS can be considered as a set of instructions on how to match substructures of SMILES. SMIRKS can similarly be considered as a set of instructions on how to identify reactive atoms and combine or alter them in order to carry out a specific transformation of a set of SMILES. 

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