On topological and geometrical distance matrices

Differences between topological and geometrical distance matrices are examined. Some examples of geometrical distances when graphs are embedded in spaces of different dimensions are given. Relations of topological distance matrices to other graph matrices are shown. The topological distance matrices are defined in the Hilbert space and their elements are distances through the graph lattices.     

Approximately N-representable density functional density matrices

For the first time, we obtain practical density matrices approximately N-representable by correlated-determinant wave functions, which are functionals of the electron density and entirely defined by information obtainable from the X-ray coherent diffraction experiment. © 1994 John Wiley & Sons, Inc.     

Exact exchange—correlation potential from low-order density matrices

The local energy equation derived by Dawson and March (1984) following the study of Cohen and Frishberg (1976) involves the low-order density matrices. Using this equation, a formally exact route is laid down by means of which the exchange-correlation potential of density functional theory can be derived in terms of the third- and lower-order density matrices. An integral equation for an approximate exchange potential is obtained, which may be solved iteratively within the Slater-Kohn-Sham scheme. © 1997 John Wiley & Sons, Inc.     

The ‘Temperature Integral’ — Its use and abuse

This paper first reviews the history of the temperature dependence of reaction rate in reaction kinetics. The various equations which are in use today for expressing this dependence were delineated by van't Hoff almost one hundred years ago. Since an exponential form best describes this dependence for most thermal analysis reactions and, due to the fact that the simple Arrhenius equation (with a temperature-independent preexponential factor) has traditionally been used for this purpose, the mathematically intractable temperature integral often has become a necessary evil in the analysis of thermal analysis kinetics. Methods which avoid the temperature integral in kinetics analysis are discussed. The merits of various evaluations and approximations for the temperature integral are described and assessed in this paper.     

Inequalities for fermion density matrices

A partial trace over the occupation numbers of all but k states in the density matrix of an ensemble with an arbitrary number of single-particle states is defined as the (reduced) k-state density matrix. This matrix is used to obtain a complete, practical solution to the problem of determining the representability of the diagonal elements of the one- and two-particle (reduced) density matrices. This solution is expressed as a series of linear inequalities involving the density-matrix elements; the inequalities are identical with those derived previously by Davidson and McCrae by a different method. In addition, our method is used to obtain nonlinear, matrix inequalities on the off-diagonal elements of the density matrices.     

Traces of spin-adapted reduced density matrices

The traces of the p-order reduced density matrices (p-RDM) split into independent contributions associated to the subsets of p-electron eigenstates of the Ŝ² and Ŝ_z operators. Here, we report the partial traces for the blocks of the low-order RDMs corresponding to pure spin states of an N-electron system. A systematic method for calculating those of higher order RDMs is described and some useful relations are also given. All these relations which must be fulfilled independently by a RDM can be considered as N- and S-representability conditions © 1997 John Wiley & Sons, Inc.     

Permutational symmetry of reduced density matrices

The problems of permutational symmetry of the density matrices in reduction are studied. Some necessary and sufficient conditions for N, [λ₁], [λ₂]-derivability problem are given.     

Photopolymerization of hydroxyethylmethacrylate in the formation of organic–inorganic hybrid sol–gel matrices

Hydroxyethylmethacrylate (HEMA) monomer in different contents was used in the preparation of transparent organic-inorganic hybrid sol-gel matrices. The matrices were studied by analytical techniques before and after irradiation. An increase in the homogeneity of the composite material after the irradiation of HEMA, and a chemical linkage of the monomer with the surface has been observed. Photopolymerization processes were studied by differential scanning calorimetry (DSC). Details of the irradiation times, polymerization rates, and DSC related parameters of HEMA in the sol-gel matrices are reported. © 1996 John Wiley & Sons, Inc.     

Nonspherical model density matrices for Rung 3.5 density functionals

"Rung 3.5" exchange-correlation functionals for Kohn-Sham density functional theory depend linearly on the nonlocal one-particle density matrix of the noninteracting Kohn-Sham reference system. Rung 3.5 functionals also require a semilocal model for the one-particle density matrix. This work presents new model density matrices for Rung 3.5 functionals. The resulting functionals give reasonable predictions for total energies, molecular thermochemistry and kinetics, odd-electron bonds, and conjugated polymer bandgaps. Global-hybrid-like combinations of semilocal and Rung 3.5 exchange, and empirical density matrix models, also show promise.     

Role of electron–electron coalescence density in density functional theory

An expression for the evaluation of electron–electron coalescence density as a functional of the density for any electron system is proposed. The formula, clarifies previously advanced upper bounds for this quantity and provides a method to independently estimate the system‐averaged on‐top exchange–correlation hole. The relationship with the on‐top pair density shows that producing the true electron–electron coalescense should be considered as a leading physical requirement for trial wave functions in any energy minimization scheme. © 2002 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

An optimized preparation of bismaleimide–diamine co‐polymer matrices

The main aim of this study was to develop an improved method for the preparation of a bismaleimide–diamine (BMI/DDM) polymer matrix, achieving shorter curing time, longer processing time (pot life), and good thermal mechanical properties. A matrix of BMI/DDM thermoset was prepared at optimal conditions and formulation, containing BMI and DDM in a 2:1 mol ratio with 0.1 wt% of dicumyl peroxide (DCP) as the curing accelerator. An optimal temperature of 150°C was selected for both melt‐mixing and curing processes. The mechanism of matrix preparation was also investigated using differential scanning calorimetry and quantitative Fourier transformed infrared analysis. DCP at the optimal concentration was found to accelerate cross‐linking reactions between BMI and DDM without inhibiting the chain‐extension reaction of BMI. The specified formulation exhibited longer gel time (208 s/g) and shorter post‐curing time (2 h) compared to other formulations. In addition, thermomechanical behavior and thermal stability were analyzed by dynamic mechanical analysis and thermomechanical analysis, and thermogravimetric analysis, respectively. The storage modulus (E′), glass transition temperature (T_g), and decomposition temperature (T_d) of the BMI/DDM thermosets increased with the BMI content of the formulations, while the coefficient of thermal expansion and damping behavior (tan δ) decreased in a similar manner, primarily because of an increase in the degree of cross‐linking. Copyright © 2014 John Wiley & Sons, Ltd.     

Scattering properties of core–shell particles in plastic matrices

Blending submicron rubber particles with plastics can enhance the mechanical strength of the composite material. However, the difference in refractive index between the particle and matrix scatters light, making the material more opaque. We consider the possibility of reducing a particle's scattering cross section by adding coatings. We find that adding coatings can reduce the amount of scattering by changing the effective dielectric contrast between the particle and the matrix. We also found that, when the refractive index of the particle is very close to that of the matrix the order of the layers can have significant effects on the transmitted light. Such effects may be useful for engineering the optical properties of particle‐doped plastics. Resonant effects akin to those found in antireflection coatings on planar surfaces are difficult to obtain and rarely provide a significant reduction in scattering. We discuss theoretical models that can qualitatively explain some of our results. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 3534–3548, 2005     

Suggested calculations using transition density matrices

It is suggested that certain transition density matrices, N-representable in a limit, be used in a variational calculation. It is noted that such trial matrices should yield reasonable values for the ground state energies of small atoms or molecules provided a set of overlap integrals is maximised.

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Zusammenfassung Es wird vorgeschlagen, bestimmte Übergangsdichtematrizen, die im Limit N-darstellbar sind, in einer Variationsrechnung zu benutzen. Es wird festgestellt, daß solche Näherungsmatrizen gute Werte für die Energie des Grundzustandes kleiner Atome oder Moleküle geben sollten, falls im Satz von Überlappungsintegralen maximiert wird.

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Résumé On propose d'utiliser dans un calcul variationnel certaines matrices densité de transition, N représentables à la limite. Ces matrices d'essai devraient fournir des valeurs raissonnables pour l'énergie de l'état fondamental dans les petits atomes et les petites molécules à condition de maximiser un ensemble d'intégrales de recouvrement.

     

Electrostatic potential–density relationships in molecules

In this work, we present Hartree–Fock and density functional theory nuclear electrostatic potential–density relationships for several homonuclear diatomic molecules. The results of our calculations are encouraging with relatively low errors in the energies. Implications for the application of this work to novel molecular energy equations as well as the need for further improvement of these newly proposed relationships in molecules have been addressed. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Algebraic structure of fermion density matrices. II

The properties of the algebraic structure of fermion density matrices are studied. The algebraic structure of a density matrix leads to a more varied and detailed classification scheme than that offered by the usual shell structure approach. Investigation of the algebraic structure of fermion density matrices by the methods of algebraic topology leads to a classification scheme based on Betti numbers.     

Algebraic structure of fermion density matrices. I

In this paper we define the algebraic structure of a reduced fermion density matrix. We relate the algebraic structure to certain symmetry properties of the reduced density matrix.     

Tracy–Widom statistic for the largest eigenvalue of autoscaled real matrices

Eigenanalysis is common practice in biostatistics, and the largest eigenvalue of a data set contains valuable information about the data. However, to make inferences about the size of the largest eigenvalue, its distribution must be known. Johnstone's theorem states that the largest eigenvalues l₁ of real random covariance matrices are distributed according to the Tracy–Widom distribution of order 1 when properly normalized to , where η_np and ξ_np are functions of the data matrix dimensions n and p. Very often, data are expressed in terms of correlations (autoscaling) for which case Johnstone's theorem does not work because the normalizing parameters η_np and ξ_np are not theoretically known. In this paper we propose a semi‐empirical method based on test‐equating theory to numerically approximate the normalization parameters in the case of autoscaled matrices. This opens the way of making inferences regarding the largest eigenvalue of an autoscaled data set. The method is illustrated by means of application to two real‐life data sets. Copyright © 2011 John Wiley & Sons, Ltd.     

Thermal characterization of polyester–melamine coating matrices prepared under nonisothermal conditions

Thermal analysis methods (differential scanning calorimetry, thermogravimetric analysis, and dynamic mechanical thermal analysis) were used to characterize the nature of polyester–melamine coating matrices prepared under nonisothermal, high‐temperature, rapid‐cure conditions. The results were interpreted in terms of the formation of two interpenetrating networks with different glass‐transition temperatures (a cocondensed polyester–melamine network and a self‐condensed melamine–melamine network), a phenomenon not generally seen in chemically similar, isothermally cured matrices. The self‐condensed network manifested at high melamine levels, but the relative concentrations of the two networks were critically dependent on the cure conditions. The optimal cure (defined in terms of the attainment of a peak metal temperature) was achieved at different oven temperatures and different oven dwell times, and so the actual energy absorbed varied over a wide range. Careful control of the energy absorption, by the selection of appropriate cure conditions, controlled the relative concentrations of the two networks and, therefore, the flexibility and hardness of the resultant coatings. © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 41: 1603–1621, 2003