Continuous Symmetry Measures of Density Maps

Pinsky, Mark; Danovich, David; Avnir, David

doi:10.1021/jp1021505

Cited by 8 publications

(5 citation statements)

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“…In particular these results suggest that in concerted reactions one can find a minimal symmetry measure value that is not necessarily zero (i.e., ideally symmetric), with respect to the original expected symmetry, and that higher reactivity in such reactions is associated with lower CSM values, that is, with higher quantitative symmetry values. Finally, we draw attention to a recent study by Casanova et al, who suggested a symmetry measure of the electron density , that could be suitable to test the influence of substitution on the orbital symmetry of concerted reactions; extension of our nuclei geometry analysis to orbitals analysis by this and by related methods , is in progress.…”

Section: Discussionmentioning

confidence: 98%

Quantifying Asymmetry in Concerted Reactions: Solvents Effect on a Diels–Alder Cycloaddition

Tuvi-Arad

Avnir

2011

J. Org. Chem.

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We propose the notion that if asymmetry characterizes a concerted reaction, a quantitative treatment in terms of continuous symmetry can bridge the gap between the Woodward-Hoffmann (WH) rules, originally formulated for symmetry-idealized unsubstituted reactants, and the fact that these rules hold for a much wider scope of reactions. Instead of focusing on symmetry conservation along the minimum energy path, we suggest that the distortion with respect to the original expected symmetry must attain a certain minimal value, not necessarily zero. To demonstrate this approach we studied the effect of solvents on the symmetry and reactivity of the classical [4 + 2] Diels-Alder cycloaddition of (E,E)-1,4-dimethoxy-1,3-butadiene with tetracyanoethylene, revealing the predictive value of this approach. Calculations of the enthalpy of activation and the charge separation at the transition state (TS) predict increased reactivity with the polarity of the solvent. The symmetry measure is in excellent correlation with the enthalpy of activation and the charge separation at the TS, indicating the higher reactivity of the more symmetric case, thus quantifying the main teaching of the WH rules. The advantages of using a global structural parameter that takes into account all geometrical parameters, i.e., the symmetry measure, over specific ones (e.g., asynchronicity) are discussed.

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Section: Discussionmentioning

confidence: 98%

Quantifying Asymmetry in Concerted Reactions: Solvents Effect on a Diels–Alder Cycloaddition

Tuvi-Arad

Avnir

2011

J. Org. Chem.

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“…(2). Here, we adopt the approach of Pinsky et al52 and we further develop it as follows. (1) Apply the symmetry operation once, using the symmetry element found in the Section “Finding an Initial Position for the Symmetry Element.” Two sets of points are thus obtained: the original set, X, and the set obtained by the operation, Y.…”

Section: The Methodologymentioning

confidence: 99%

“…The continuous symmetry measure (CSM) methodology33–54 will be used in this article, because it is simple and versatile. The CSM is the normalized distance between the original structure and the closest symmetrical object.…”

Section: Introductionmentioning

confidence: 99%

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Quantitative symmetry and chirality—A fast computational algorithm for large structures: Proteins, macromolecules, nanotubes, and unit cells

2011

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Symmetry is one of the most fundamental properties of nature and is used to understand and investigate physical properties. Classically, symmetry is treated as a binary qualitative property, although other physical properties are quantitative. Using the continuous symmetry measure (CSM) methodology one can quantify symmetry and correlate it quantitatively to physical, chemical, and biological properties. The exact analytical procedures for calculating the CSM are computationally expensive and the calculation time grows rapidly as the structure contains more atoms. In this article, we present a new method for calculating the CSM and the related continuous chirality measure (CCM) for large systems. The new method is much faster than the full analytical procedures and it reduces the calculation time dependency from N! to N(2), where N is the number of atoms in the structure. We evaluate the cost of the applied approximations, estimate the error of the method, and show that deviations from the analytical solutions are within an error of 2%, and in many cases even less. The method is applicable at the moment for the cyclic symmetry point groups- C(i), C(s), C(n), and S(n), and therefore it can be used also for chirality measures, which are the minimal of the S(n) measures. We demonstrate the application of the method for large structures across chemistry: proteins, macromolecules, nanotubes, and large unit cells of crystals.

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“…The holographic electron density theorem has important role in local shape analysis, in drug design, , in toxicological risk assessment, and in the prediction of “latent properties”, for example, excited state properties not exhibited in the ground electronic ground state, , and in a wide range of additional applications by various authors. − ,− …”

Section: Applications Of the Holographic Electron Density Theorem To ...mentioning

confidence: 99%

Fuzzy Electron Density Fragments in Macromolecular Quantum Chemistry, Combinatorial Quantum Chemistry, Functional Group Analysis, and Shape–Activity Relations

Mezey

2014

Acc. Chem. Res.

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Conspectus Just as complete molecules have no boundaries and have "fuzzy" electron density clouds approaching zero density exponentially at large distances from the nearest nucleus, a physically justified choice for electron density fragments exhibits similar behavior. Whereas fuzzy electron densities, just as any fuzzy object, such as a thicker cloud on a foggy day, do not lend themselves to easy visualization, one may partially overcome this by using isocontours. Whereas a faithful representation of the complete fuzzy density would need infinitely many such isocontours, nevertheless, by choosing a selected few, one can still obtain a limited pictorial representation. Clearly, such images are of limited value, and one better relies on more complete mathematical representations, using, for example, density matrices of fuzzy fragment densities. A fuzzy density fragmentation can be obtained in an exactly additive way, using the output from any of the common quantum chemical computational techniques, such as Hartree-Fock, MP2, and various density functional approaches. Such "fuzzy" electron density fragments properly represented have proven to be useful in a rather wide range of applications, for example, (a) using them as additive building blocks leading to efficient linear scaling macromolecular quantum chemistry computational techniques, (b) the study of quantum chemical functional groups, (c) using approximate fuzzy fragment information as allowed by the holographic electron density theorem, (d) the study of correlations between local shape and activity, including through-bond and through-space components of interactions between parts of molecules and relations between local molecular shape and substituent effects, (e) using them as tools of density matrix extrapolation in conformational changes, (f) physically valid averaging and statistical distribution of several local electron densities of common stoichiometry, useful in electron density databank mining, for example, in medicinal drug design, and (g) tools for combinatorial quantum chemistry approaches using fuzzy fragment databanks and rapid construction of a large number of approximate electron densities for large sets of related molecules, relevant in theoretical molecular and nanostructure design.

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Continuous Symmetry Measures of Density Maps

Cited by 8 publications

References 25 publications

Quantifying Asymmetry in Concerted Reactions: Solvents Effect on a Diels–Alder Cycloaddition

Quantifying Asymmetry in Concerted Reactions: Solvents Effect on a Diels–Alder Cycloaddition

Quantitative symmetry and chirality—A fast computational algorithm for large structures: Proteins, macromolecules, nanotubes, and unit cells

Fuzzy Electron Density Fragments in Macromolecular Quantum Chemistry, Combinatorial Quantum Chemistry, Functional Group Analysis, and Shape–Activity Relations

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