Eigenvalues of saturated hydrocarbons

A simplified Hückel-type molecular-orbital (MO) model for the valence electrons of saturated hydrocarbons is proposed and the consequent eigenvalue spectrum considered. A first foundational result is obtained, which every chemist “knows”, namely that: alkanes are stable, with half their (Hückel-type MO) eigenvalues positive and half negative.     

Successive optical resolution of two compounds by one enantiopure compound

By using (1R,2R)-1,2-diphenylethylenediamine as a single enantiopure compound, we achieved a novel successive optical resolution of more than one kind of racemic compound through supramolecular crystallization.     

Evaluation of Successive Fractions for Optimum Quantification of Bergenin and Gallic Acid in Three Industrially Important Bergenia species by High-Performance Thin-Layer Chromatography

JPC – Journal of Planar Chromatography – Modern TLC -     

Eigenvalues of the real generalized eigenvalue equation perturbed by a low-rank perturbation

The low-rank perturbation (LRP) method solves the perturbed eigenvalue equation (B +V) _k = _k (C +P) _k , where the eigenvalues and the eigenstates of the related unperturbed eigenvalue equationB _i = _i C _i are known. The method is designed for arbitraryn-by-n matricesB, V, C, andP, with the only restriction that the eigenstates _i of the unperturbed equation should form a complete set. We consider here a real LRP problem where all matrices are Hermitian, and where in addition matricesC and (C +P) are positive definite. These conditions guarantee reality of the eigenvalues _k and _i . In the original formulation of the LRP method, each eigenvalue _k is obtained iteratively, starting from some approximate eigenvalue _k . If this approximate eigenvalue is not well chosen, the iteration may sometimes diverge. It is shown that in the case of a real LRP problem, this danger can be completely eliminated. If the rank of the generalized perturbation {V, P} is small with respect ton, then one can easily bracket and hence locate to any desirable accuracy the eigenvalues _k (k = 1, ...,n) of the perturbed equation. The calculation of alln eigenvalues requiresO(² n ²) operations. In addition, if the perturbation (V, P) is local with the localizabilityl p, then onlyO(² n) operations are required for a derivation of a single eigenvalue.     

Eigenvalues of propagation through liquid crystals

Analytical expressions are given for the propagation constant through an homogeneous liquid crystal, following the 4 × 4 matrix formulation of Berreman. Also propagation through polarizers can be calculated analytically; it is a special case of a situation described by Teitler and Henvis.     

Lower Bounds to Ground-State Eigenvalues

A new lower bound method is presented for ground-state eigenvalues which relies on the Eckart inequality. The method bears similarity to the variational method and the Temple lower bound formula. Restrictions are that an exactly soluble base Hamiltonian must be available with a positive-semidefinite perturbation to the Hamiltonian of interest. A sample calculation shows that our method is able to sometimes best the Temple bound and also perform rigorously when the Temple bound does not.     

Eigenvalues of a statistical mechanics matrix

We consider the problem of finding the spectrum of an n × n matrix which arises in the study of a certain model of long-range interactions in a one-dimensional statistical mechanics system. Our analysis exhibits a curious resemblance of the suitably normalized distribution of eigenvalues to the Marčenko–Pastur law in the limit n → ∞.     

Eigenvalues of propagation through liquid crystals

Abstract

Analytical expressions are given for the propagation constant through an homogeneous liquid crystal, following the 4 × 4 matrix formulation of Berreman. Also propagation through polarizers can be calculated analytically; it is a special case of a situation described by Teitler and Henvis.     

Eigenvalues of graphs with threefold symmetry

In many cases, the spectrum of one graph contains the entire spectrum of a second, smaller graph. The larger (composite) graph and the smaller (component) graph are said to be subspectral. Rules are given for constructing two subspectral components of a composite graph which has threefold symmetry such that the eigenvalues of one component and the eigenvalues of the second component taken twice comprise the complete spectrum of the composite graph. The mathematical basis for these rules is shown to be a unitary transformation upon the eigenvalue equation of the adjacency matrix of the composite graph by the matrix which represents threefold rotation.     

Compressing Double [7]Helicene by Successive Charging with Electrons

Chemical reduction of a benzo‐fused double [7]helicene ( 1 ) with two alkali metals, K and Rb, provided access to three different reduced states of 1 . The doubly‐reduced helicene 1 ^2? has been characterized by single‐crystal X‐ray diffraction as a solvent‐separated ion triplet with two potassium counterions. The triply‐ and tetra‐reduced helicenes, 1 ^3? and 1 ^4?, have been crystallized together in an equimolar ratio and both form the contact‐ion complexes with two Rb⁺ ions each, leaving three remaining Rb⁺ ions wrapped by crown ether and THF molecules. As structural consequence of the stepwise reduction of 1 , the central axis of helicene becomes more compressed upon electron addition (1.42 Å in 1 ^4? vs. 2.09 Å in 1 ). This is accompanied by an extra core twist, as the peripheral dihedral angle increases from 16.5° in 1 to 20.7° in 1 ^4?. Theoretical calculations provided the pattern of negative charge build‐up and distribution over the contorted helicene framework upon each electron addition, and the results are consistent with the X‐ray crystallographic and NMR spectroscopic data.     

Lower Bounds to Ground-State Eigenvalues II

An alternative to the Temple method for calculating a lower bound to the ground-state eigenvalue of an operator is presented. The method presented is an improvement and generalization upon a similar method [M.G. Marmorino, J. Math. Chem. 32 (2002) 19–29]. For the system tested the lower bound of the current method is significantly superior to the Temple method and the previous method. Furthermore, the current method (like its predecessor) is able to generate a lower bound when the Temple method fails.     

A Tetracyclic Octaphosphane by Successive Addition,Inversion, and Condensation Reactions

An example of an octaphosphane of type R₂P₈ (R=(DDP)Ga) was isolated by treatment of cage compound (DDP)GaP₄ ( 2 , DDP=(2,6-diisopropylphenyl)(4-((2,6-diisopropylphenyl)imino)pent-2-en-2-yl)amide) with (C₆F₅)₂PBr. The initially formed endo-exo butterfly shaped pentaphosphane 7 rapidly rearranges to the more stable exo–exo isomer 8 , which undergoes dimerization to decaphosphane 11 . Compound 11 unexpectedly eliminates tetraaryldiphosphane 13 to give tetracyclo[3.3.0.0^2,7.0^4,6]octaphosphane [(DDP)GaBr]₂P₈ ( 12 ). The reaction steps were confirmed by crystal structure analysis of the key intermediates and supported by kinetic studies using NMR techniques.     

A Tetracyclic Octaphosphane by Successive Addition,Inversion, and Condensation Reactions

An example of an octaphosphane of type R₂P₈ (R=(DDP)Ga) was isolated by treatment of cage compound (DDP)GaP₄ ( 2 , DDP=(2,6‐diisopropylphenyl)(4‐((2,6‐diisopropylphenyl)imino)pent‐2‐en‐2‐yl)amide) with (C₆F₅)₂PBr. The initially formed endo ‐exo butterfly shaped pentaphosphane 7 rapidly rearranges to the more stable exo –exo isomer 8 , which undergoes dimerization to decaphosphane 11 . Compound 11 unexpectedly eliminates tetraaryldiphosphane 13 to give tetracyclo[3.3.0.0^2,7.0^4,6]octaphosphane [(DDP)GaBr]₂P₈ ( 12 ). The reaction steps were confirmed by crystal structure analysis of the key intermediates and supported by kinetic studies using NMR techniques.     

Successive uranium and thorium adsorption from Egyptian monazite by solvent impregnated foam

The present work deals with uranium and thorium recovery from the Egyptian monazite sulfate leach liquor using the extraction chromatography technique (solvent impregnated material), where tributylamine (TBA) and di-n-octylamine (DOA) solvents were impregnated onto foam uranium and thorium separate recovery. The calculated theoretical capacities of the latter solvents were about 1.4 gU/g foam and 1.6 gTh/g foam, respectively. The attained uranium and thorium adsorption efficiencies (using ion-exchange columnar technique) were about 75 and 70% of its theoretical capacities, respectively. Using 1 M NaCl–0.1 M H₂SO₄ and 2 M H₂SO₄ as eluent solutions for uranium and thorium from the loaded solvents impregnated foam gave 95.8 and 98.7% elution efficiencies, respectively.     

Eigenvalues of single-cycle class-sums in the symmetric group

Explicit expressions for the eigenvalues of the class sums [(p)(1)^n?p]_n, p = 2, 3,…,14, of the symmetric group S_n are presented. Partial results are given for the eigenvalues corresponding to arbitrary p.     

Successive chelatometric titration of calcium and magnesium

A simple procedure for the successive chelatometric titration of calcium and magnesium in natural waters, with 3',3'-bis(2-hydroxy-3-carboxy-naphthaleneazo)phenolphthalein as indicator, is described.     

Inner-shell Eigenvalues from valence orbital only calculations

Methods are examined for calculating Inner-shell Eigenvalues from molecular orbital models which do not explicitly include core basis functions. If the “valence-orbital only” calculation is a good one, a rather straightforward method can be used to obtain core eigenvalues with a rms error of ±0.01 a.u. compared with ab-initio values. Even simpler methods can be used to reproduce trends among core eigenvalues. The AAMOM valence technique, and to a lesser extent, the INDO model, can be used to yield core eigenvalues for orbitals centered on carbon and nitrogen: for oxygen the agreement is poorer. Extended Hückel or Iterative Extended Hückel methods cannot be used for this purpose with any degree of confidence. Eigenvalues from ab-initio studies or from AAMOM and INDO can be used in assigning ionization processes (XPS) from orbitals localized on carbon and nitrogen: ionization processes from oxygen orbitals are not well treated. An attempt is made to explain this behaviour.     

Variations of radon in soils induced by external factors

Experiments were designed to evaluate, in a controlled way, some of the variations that can be expected in radon concentration in soil due mainly to meteorological factors and to close spacing monitoring in the field. Electronic continuous radon monitoring and the track-etch method were used to record the radon -decay. The radon emanation in daily measurements and in long term surveys showed a general effect modulated by temperature and pressure variations. Radon monitoring in soil for nearby stations showed that a 10% range of variation can be expected with the track-etch method and as a result of non-homogeneous emanation pattern of the soil.