Proof of the Fukui conjecture via resolution of singularities and related methods. V

The present article is a direct continuation of part IV of this series. The Local Analyticity Proposition (LAP1), which admits a proof via resolution of singularities is a major key to proving the Fukui conjecture via resolution of singularities and related methods. By LAP1, the essential part of the mechanism of the “asymptotic linearity phenomena” is extracted and is elucidated by using tools from the theory of algebraic and analytic curves. Here in the present article, we complete the proof of the LAP1 by using fundamental tools developed in parts III and IV of this series, thus completing the proof of the Fukui conjecture via resolution of singularities and related methods. This series of articles I-V establishes, for the first time, a new linkage between (i) the mathematical field of resolution of singularities and (ii) the chemical field of additivity problems tackled and solved in a unifying manner via the repeat space theory (RST), which is the central theory in the First and Second Generation Fukui Project. A new development called the Matrix Art Program in the Second Generation Fukui Project has also been expounded with a graphical representation of energy band curves of a carbon nanotube.     

Proof of the Fukui conjecture via resolution of singularities and related methods: III

The present article is a direct continuation of the second part of this series. In conjunction with the analysis of the energy band curves of carbon nanotubes, we develop here fundamental theoretical tools, which are essential to prove the Local Analyticity Proposition (LAP). The LAP enables one to prove the Fukui conjecture (the guiding conjecture for developing the repeat space theory) in a new and powerful context of the theory of algebraic curves and resolution of singularities. The present fundamental tools also serve as modular tools for the repeat space theory, by which one can solve a variety of additivity and molecular network problems in a unifying manner.     

Proof of the Fukui conjecture via resolution of singularities and related methods. II $$^{\star}$$

The present article is a direct continuation of the first part of this series. We reduce a proof of the Fukui conjecture (concerning the additivity problem of the zero-point vibrational energies of hydrocarbons) to that of a proposition related to the theory of algebraic curves, so that we can focus on the key mechanism of the additivity phenomena. Namely, by establishing what is called the Basic Piecewise Monotone Theorem (BPMT), we reduce a proof of the Fukui conjecture to that of a proposition, called the Local Analyticity Proposition, Version 1 (LAP1), which admits a proof via resolution of singularities. By LAP1, the essential part of the mechanism of the asymptotic linearity phenomena is extracted and is elucidated by using tools from the mathematical theory of algebraic curves, whose language is of vital importance in analyzing the crux of the additivity mechanism. Dedicated to the memory of Prof. Kenichi Fukui (1918–1998).     

New Proof of the Fukui Conjecture by the Functional Asymptotic Linearity Theorem

The present article provides a new proof of the Fukui conjecture concerning the additivity problem of the zero-point vibrational energies of hydrocarbons. This conjecture played a prominent role in the initial development of the repeat space theory (RST), and continues to be of vital significance in the recent development of the theory of the generalized repeat space X _r (q,d). The new proof of the Fukui conjecture has been given here by establishing the functional version of the Asymptotic Linearity Theorem (ALT), the Functional ALT. This enhanced version of the ALT directly implies the validity of the Fukui conjecture; it easily unifies, in a broad perspective, a variety of additivity phenomena in physico-chemical network systems having many identical moieties, and efficiently solves some interpretational problems of the empirical additivity formulae from experimental chemistry. The proof of the functional version of the ALT is based on a new method transferable to the extended theoretical framework of the generalized repeat space X _r (q,d).     

Proof of the Fukui conjecture via resolution of singularities and related methods. I

The present article is the preliminary part of a series devoted to extending the foundation of the Asymptotic Linearity Theorems (ALTs), which prove the Fukui conjecture concerning the additivity problem of the zero-point vibrational energies of hydrocarbons. In this article, we establish a theorem, referred to as the Boundedness Theorem, through which one can easily form a chain of logical implications that reduces a proof of the Fukui conjecture to that of the Piecewise Monotone Lemma (PML). This chain of logical implications serves as a basis throughout this series of articles. The PML, which has been indispensable for demonstrating any version of the ALTs and has required for its proof a mathematical language not generally known to chemists, is directly related to the theory of algebraic curves. Proofs of the original and enhanced versions of the PML are obtainable via resolution of singularities and related methods.Dedicated to the memory of Prof. Kenichi Fukui (1918–1998).     

Repeat Space Theory Applied to Carbon Nanotubes and Related Molecular Networks. I

The present article is the first part of a series devoted to extending the Repeat Space Theory (RST) to apply to carbon nanotubes and related molecular networks. Four key problems are formulated whose affirmative solutions imply the formation of the initial investigative bridge between the research field of nanotubes and that of the additivity and other network problems studied and solved by using the RST. All of these four problems are solved affirmatively by using tools from the RST. The Piecewise Monotone Lemmas (PMLs) are cornerstones of the proof of the Fukui conjecture concerning the additivity problems of hydrocarbons. The solution of the fourth problem gives a generalized analytical formula of the pi-electron energy band curves of nanotube (a, b), with two new complex parameters c and d. These two parameters bring forth a broad class of analytic curves to which the PMLs and associated theoretical devices apply. Based on the above affirmative solutions of the problems, a central theorem in the RST, called the asymptotic linearity theorem (ALT) has been applied to nanotubes and monocyclic polyenes. Analytical formulae derived in this application of the ALT illuminate in a new global context (i) the conductivity of nanotubes and (ii) the aromaticity of monocyclic polyenes; moreover an analytical formula obtained by using the ALT provides a fresh insight into Hückel’s (4n+2) rule. The present article forms a foundation of the forthcoming articles in this series. The present series of articles is closely associated with the series of articles entitled ‘Proof of the Fukui conjecture via resolution of singularities and related methods’ published in the JOMC.     

The Functional Delta Existence Theorem and the Reduction of a Proof of the Fukui Conjecture to That of the Special Functional Asymptotic Linearity Theorem

This article establishes a fundamental existence theorem, called the Functional Delta Existence Theorem (DET), which is significant for a new development in the repeat space theory (RST) and also for elucidating an empirical asymptotic principle from experimental chemistry. By using the Functional DET, we reduce a proof of the Fukui conjecture to that of a special and simpler version of the Asymptotic Linearity Theorem (ALT). This reduction provides a basis for the forthcoming series of articles entitled “Proof of the Fukui conjecture via resolution of singularities and related methods”. A proof of the Functional DET is given here in a unifying manner so that an investigative link is formed among: (i) fundamental methodology in the RST, which is referred to as the approach via the aspect of form and general topology, (ii) frontier electron theory of reactivity indices, and (iii) the Shingu–Fujimoto empirical asymptotic principle for long chain molecules.     

Multidimensional Magic Mountains and Matrix Art for the generalized repeat space theory

The Asymptotic Linearity Theorem (ALT), which proves the Fukui conjecture in a broader context, plays a significant role in the repeat space theory (RST), which is the central unifying theory in the First and the Second Generation Fukui Project. Proving the Asymptotic Linearity Theorem Extension Conjecture (ALTEC) is a fundamental problem in the repeat space theory. The present paper constructs a class of functions MagicMtθ, which serves as a powerful tool for proving the Asymptotic Linearity Theorem Extension Conjecture and related propositions. The d-dimensional generalization?μ _d,n,θ of MagicMt _θ , which is given in the present paper and is called a ‘d-dimensional Magic Mountain’, provides inwardly repeating fractals in multidimensional spaces useful for interdisciplinary research that uses the generalized repeat space theory.     

Performance of density functional theory methods to describe intramolecular hydrogen shifts

The performance of three exchange and correlation density functionals, LDA, BLYP and B3LYP, with four basis sets is tested in three intramolecular hydrogen shift reactions. The best reaction and activation energies come from the hybrid functional B3LYP with triple-ζ basis sets, when they are compared with high-level post-Hartree-Fock results from the literature. For a fixed molecular geometry, the electrophilic Fukui function is computed from a finite difference approximation. Fukui function shows a small dependence with both the exchange and correlation functional and the basis set. Evolution of the Fukui function along the reaction path describes important changes in the basic sites of the corresponding molecules. These results are in agreement with the chemical behavior of those species.     

Quantifying Electronic Effects in QM and QM/MM Biomolecular Modeling with the Fukui Function

Multi-scale quantum-mechanical/molecular-mechanical(QM/MM) and large-scale QM simulation provide valuable insight into enzyme mechanism and structure-property relationships. Analysis of the electron density afforded through these methods can enhance our understanding of how the enzyme environment modulates reactivity at the enzyme active site. From this perspective, tools from conceptual density functional theory to interrogate electron densities can provide added insight into enzyme function. We recently introduced the highly parallelizable Fukui shift analysis(FSA) method, which identifies how frontier states of an active site are altered by the presence of an additional QM residue to identify when QM treatment of a residue is essential as a result of quantum-mechanically affecting the behavior of the active site. We now demonstrate and analyze distance and residue dependence of Fukui function shifts in pairs of residues representing different non-covalent interactions. We also show how the interpretation of the Fukui function as a measure of relative nucleophilicity provides insight into enzymes that carry out S_N2 methyl transfer. The FSA method represents a promising approach for the systematic, unbiased determination of quantum mechanical effects in enzymes and for other complex systems that necessitate multi-scale modeling.     

Chemical reactivity and selectivity using Fukui functions: basis set and population scheme dependence in the framework of B3LYP theory

The use of Fukui functions for the site selectivity of the formaldehyde molecule for nucleophilic, electrophilic and radical attacks has been made with special emphasis to the dependence of Fukui values on the basis sets as well as population schemes in the framework of B3LYP theory. Out of the five population schemes selected viz., Mulliken population analysis, natural population analysis, CHELP, CHELPG and atoms in molecules (AIM), it is found that the CHELPG and AIM schemes predict precise reactive site with less dependency on the basis sets. Charges derived from Hirshfeld partitioning, calculated using the BLYP/dnd method (implemented in the DMOL³ package), provide non-negative Fukui values for all the molecular systems considered in this study. Supporting results have been obtained for acetaldehyde and acetone molecules at the 6-31+G** basis set level. These results support the fact that high Fukui values correspond to soft–soft interaction sites. On the other hand, the correlation of the low Fukui value to the hard–hard interaction site merits further investigation. Received: 10 November 2001 / Accepted: 6 March 2002 / Published online: 13 June 2002     

Bond fukui indices: Comparison of frozen molecular orbital and finite differences through mulliken populations

Bond Fukui functions and matrices are introduced for ab initio levels of theory using a Mulliken atoms in molecules model. It is shown how these indices may be obtained from first‐order density matrix derivatives without need for going to second‐order density matrices as in a previous work. The importance of taking into account the nonorthogonality of the basis in ab initio calculations is shown, contrasting the present results with previous work based on Hückel theory. It is shown how the extension of Fukui functions to Fukui matrices allows getting more insight into the nature of bond Fukui functions. All presently introduced indices respect the necessary normalization conditions and include the classical single atom condensed Fukui functions. © 2013 Wiley Periodicals, Inc.     

Polarization justified Fukui functions: the theory and applications for molecules

The Fukui functions based on the computable local polarizability vector have been presented for a group of simple molecules. The necessary approximation for the density functional theory softness kernel has been supported by a theoretical analysis unifying and generalizing early concepts produced by the several authors. The exact relation between local polarizability vector and the derivative of the nonlocal part of the electronic potential over the electric field has been demonstrated. The resulting Fukui functions are unique and represent a reasonable refinement when compared to the classical ones that are calculated as the finite difference of the density in molecular ions. The new Fukui functions are strongly validated by their direct link to electron dipole polarizabilities that are reported experimentally and by other computational methods.     

Comprehensive quantum chemical calculations and molecular docking analysis of uracil mustard by first principle

Uracil mustard belongs to the nitrogen mustard family and is primarily used in anticancer drugs. The research that follows, investigates many quantum chemical features such as the computation of global minimum energies with no negative wavenumber values using the Density Functional Theory (DFT) with Becke three functional and 6-311G (d, p)/6–311++G (d, p) basis sets. All the vibrational modes have been calibrated and justified in comparison to their experimental counterparts. Mustard's polarizability and hyperpolarizability components, Natural Bond Analysis (NBO), electronic properties, Fukui function analysis, various global parameters, Quantum Theory of Atoms In Molecule (QTAIM) analysis, ADMET analysis, and docking analysis have all been investigated using the same theory and basis sets, indicating its biochemical significance. The biological activity of the molecule is reported by using PASS software. The Full fitness score and binding affinity parameters are utilized to determine the binding strength with 6cq3 protein. The acidity of the title molecule is calculated in water solvent by polarizable continuum model (PCM) solvent effects (estimated in water). The HOMO, LUMO, and MESP plots are used to explore the nature of binding and surfaces. The Fukui functions are computed using Mulliken atomic charges for neutral atoms, cations, and anions. The Ultraviolet–visible (UV–vis) of the molecule is computed employing the TD-DFT method.     

Modeling temporary anions in density functional theory: calculation of the Fukui function

Two approaches are investigated for modeling electron densities of temporary anions in density functional theory (DFT). Both rely on an artificial binding of the excess electron, in one case by a compact basis set and in the other by a potential wall. The key feature of the calculations is that the degree of binding is controlled in both cases by knowledge of the negative electron affinity of the corresponding neutral, approximated in terms of DFT local functional frontier orbital eigenvalues and vertical ionization potential, A=-(epsilon(LUMO)+epsilon(HOMO))-I. To illustrate the two approaches, Fukui functions for nucleophilic attack are determined in four molecules with increasingly negative electron affinities. They yield very similar results, which are notably different to those determined without artificial electron binding. The use of a potential wall has the attractive feature that large, diffuse basis sets can be used, avoiding the need for a compact basis, tailored to a particular molecule.     

2,4-二氯苯酚的电化学氧化降解反应研究

采用循环伏安法和原位红外光谱技术研究了2,4-二氯苯酚在Pt电极上的电化学氧化降解反应,结合Fukui函数值预测了2,4-二氯苯酚在电化学氧化过程中的反应位点. 结果表明,Pt电极对2,4-二氯苯酚有良好的电催化活性,2,4-二氯苯酚在电极表面反应主要有3个途径：直接通过电化学反应脱去氯离子,生成苯酚;在·OH的进攻下,C—Cl键断裂,4位Cl较2位Cl先脱去,生成苯二酚,并可进一步氧化生成苯醌以及不饱和羧酸;在·OH的进攻下发生苯环开环反应,生成含氯不饱和羧酸. 在1700 mV左右,2,4-二氯苯酚可经电化学氧化生成CO₂.