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).     

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.     

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).     

Structural analysis of certain linear operators representing chemical network systems via the existence and uniqueness theorems of spectral resolution VII

The present Part VII of this series of articles is a direct extension of Part VI, where (1) fundamental methodology in the repeat space theory (RST) and (2) frontier electron theory of reactivity indices were theoretically linked. This part presents an estimate of the size of the regular index set, which was a central notion in Part VI, and two new theorems that are simpler and more powerful than the main theorem in Part VI. The main theorem in this part enables one to globally contextualize the Generalized Alpha Existence Theorem (a theorem essential in the RST and proved in Part V) and the μ Existence Theorem (derived from the main theorem in Part VI) into the star algebra structure of the generalized repeat space ??_r(q, d). © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

Static contact angle hysteresis on smooth, homogeneous solid substrates

A theory of contact angle hysteresis on smooth, homogeneous solid substrates is developed in terms of shape of disjoining/conjoining pressure isotherm and quasi-equilibrium phenomena. It is shown that all contact angles, θ, in the range θ _r?<?θ?<?θ _a, which are different from the unique equilibrium contact angle θ?≠?θ _e, correspond to the state of slow “microscopic” advancing or receding motion of the liquid if θ _e ?<?θ?<?θ _a or θ _r?<?θ?<?θ _e, respectively. This “microscopic” motion almost abruptly becomes fast “macroscopic” advancing or receding motion after the contact angle reaches the critical values θ?=?θ _a or θ _r?=?θ, correspondingly. The values of the static receding, θ _r, and static advancing, θ _a, contact angles in cylindrical capillaries were calculated earlier, based on the shape of disjoining pressure isotherm. It is shown that an advancing contact angle of a droplet on a solid substrate depends on the drop volume and is not a unique characteristic of the liquid–solid system. The suggested mechanism of contact angle hysteresis has direct experimental confirmation.     

The Curl–Divergence equations for the electronic non-adiabatic coupling terms: study of the C2H molecule and the H2+H system

This Letter is part of an effort to use the Curl equations to calculate non-adiabatic coupling terms, subject to ab initio boundary conditions. As examples we consider two-state, planar, systems characterized by two coordinates, θ and q and treat the corresponding non-adiabatic coupling terms, namely, τ_θ(q,θ) and τ_q(q,θ). The theory, which yields τ_q(q,θ) once τ_θ(q,θ) is given, is applied to three cases: an analytical model and two ab initio treatments – one for the C₂H molecule and one for the H+H₂ molecular system. In all three cases encouraging agreements were obtained between the theoretical τ_q(q,θ) values and the ab initio ones.     

Motion of a laser-cooled three-level ion in a harmonic trap

The relative stabilities of eight-atom Si clusters were calculated with the semiempirical MO method SINDO1. With the aid of the group theoretical concept of kernels and epikernels, stationary points ofD _{2d, C 2v} andC _s symmetry on the potential energy surface near the unstable cube were found. Apart from these stationary points which adopt epikernel symmetries, aD _4d stationary point not derivable from group theory is possible in configuration space. The cubic array with eight equivalent atoms can be classified as a diradical.     

Visualization of electron correlation in ground states of He and H−

For the ground states of He and H^?, conditional probability densities d(r₂,θ₁₂r₁) for finding two electrons in different relative positions have been calculated for several wavefunctions including simple one-configuration wavefunctions and Hylleraas—Kinoshita type functions. The calculations, including graphing, were all carried out on a desktop computing calculator.     

A formula for angular and hyperangular integration

A formula is derived which allows angular or hyperangular integration to be performed on any function of the coordinates of a D-dimensional space, provided that it is possible to expand the function as a polynomial in the coordinates x₁,x₂,...,x_d. The expansion need not be carried out for the formula to be applied. This revised version was published online in July 2006 with corrections to the Cover Date.     

Structural analysis of certain linear operators representing chemical network systems via the existence and uniqueness theorems of spectral resolution. IV

Part IV of this series consists of two complementary subparts devoted to attain the following two goals: (i) By shifting from the previous setting of the Banach algebra B (ℬ︁)= B (ℬ︁, ℬ︁) to a broader setting of the space B (X, ℬ︁) of all bounded linear operators from a normed space X to a Banach space ℬ︁, we extend our previous theoretical framework to incorporate part of the theory of additive correlation involving the Asymptotic Linearity Theorems, which have been developed for a study of correlation between structure and properties in molecules having many identical moieties, especially in macromolecules having repeating units. (ii) By reverting our focus to the special algebra B (ℋ︁) with ℋ︁ being a Hilbert space, we develop a theorem which is useful for a structural analysis of spectral symmetry of linear operators representing physico-chemical network systems. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67 : 57–69, 1998     

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.     

Synergistic improvement of gas sensing performance by micro-gravimetrically extracted kinetic/thermodynamic parameters

A novel method is explored for comprehensive design/optimization of organophosphorus sensing material, which is loaded on mass-type microcantilever sensor. Conventionally, by directly observing the gas sensing response, it is difficult to build quantitative relationship with the intrinsic structure of the material. To break through this difficulty, resonant cantilever is employed as gravimetric tool to implement molecule adsorption experiment. Based on the sensing data, key kinetic/thermodynamic parameters of the material to the molecule, including adsorption heat −ΔH°, adsorption/desorption rate constants K_a and K_d, active-site number per unit mass N′ and surface coverage θ, can be quantitatively extracted according to physical–chemistry theories. With gaseous DMMP (simulant of organophosphorus agents) as sensing target, the optimization route for three sensing materials is successfully demonstrated. Firstly, a hyper-branched polymer is evaluated. Though suffering low sensitivity due to insufficient N′, the bis(4-hydroxyphenyl)-hexafluoropropane (BHPF) sensing-group exhibits satisfactory reproducibility due to appropriate −ΔH°. To achieve more sensing-sites, KIT-5 mesoporous-silica with higher surface-area is assessed, resulting in good sensitivity but too high −ΔH° that brings poor repeatability. After comprehensive consideration, the confirmed BHPF sensing-group is grafted on the KIT-5 carrier to form an optimized DMMP sensing nanomaterial. Experimental results indicate that, featuring appropriate kinetic/thermodynamic parameters of −ΔH°, K_a, K_d, N′ and θ, the BHPF-functionalized KIT-5 mesoporous silica exhibits synergistic improvement among reproducibility, sensitivity and response/recovery speed. The optimized material shows complete signal recovery, 55% sensitivity improvement than the hyper-branched polymer and 2 ∼ 3 folds faster response/recovery speed than the KIT-5 mesoporous silica.     

Studies with model membranes: Part IX. Evaluation of the thermodynamic effective fixed charge density and permselectivity of parchment supported cobalt tungstate and mercuric chromate membranes

Thermodynamic effective fixed charge densities of cobalt tungstate and mercuric chromate parchment supported membranes were evaluated by a number of methods particularly those of Teorell-Meyer-Sievers [19]. Altug and Hair [23] and the most recent one of Kobatake et al. [24, 27] based on the thermodynamics of irreversible processes. The value of the permselectivity was also obtained for the two membranes based on Kobatake et al.'s procedure. Kobatake et al.' equation was used under two limiting conditions of concentration of electrolyte solution, namely (1) in the concentrated range, and (2) in the dilute range. The charge densities obtained under these conditions namely θ_c and θ_d were found to differ (θ_c=1/2θ_d), but it was interesting to note that θ_c values were more closer to those of the TMS values. The theoretical predictions for membrane potential using Kobatake et al.'s equation are borne out quite satisfactorily by our experimental results for both membranes.     

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.     

Structural,Electronic and Tunable Magnetic Properties of Transition Metal Doped Rh<Subscript>8</Subscript> Cluster from First Principles Calculation

The configurations, electronic and magnetic properties of the Rh₇M (M?=?3d, 4d transition metal) are systematically investigated within the framework of the generalized gradient approximation density-functional theory (DFT-GGA). The results indicated that the ground state structures of Rh₇M (M?=?3d) clusters prefer to a bicapped octahedron configuration, while the Rh₇M (M?=?4d) clusters present a different degree of geometry reconstruction relative to the perfect cubic structure of Rh₈ cluster. In most cases, the doped clusters show relatively higher stability, indicating that impurity atoms could stabilize the pure Rh₈ cluster; the Rh₇M (M?=?3d, 4d) have smaller frontier orbital energy gaps in comparison to the host. The magnetic moments of Rh₇M (M?=?3d, 4d) exhibit a tunable magnetism with range from 3μ_B to 13μ_B and the Fe atom doping enhances the magnetic moment of mixed cluster.