Integrals and derivatives for correlated Gaussian functions using matrix differential calculus

The matrix differential calculus is applied for the first time to a quantum chemical problem via new matrix derivations of integral formulas and gradients for Hamiltonian matrix elements in a basis of correlated Gaussian functions. Requisite mathematical background material on Kronecker products, Hadamard products, the vec and vech operators, linear structures, and matrix differential calculus is presented. New matrix forms for the kinetic and potential energy operators are presented. Integrals for overlap, kinetic energy, and potential energy matrix elements are derived in matrix form using matrix calculus. The gradient of the energy functional with respect to the correlated Gaussian exponent matrices is derived. Burdensome summation notation is entirely replaced with a compact matrix notation that is both theoretically and computationally insightful. © 1996 John Wiley & Sons, Inc.     

Investigation of plasma-related matrix effects in inductively coupled plasma-atomic emission spectrometry caused by matrices with low second ionization potentials—identification of the secondary factor

Plasma-related matrix effects induced by a comprehensive list of matrix elements (a total of fifty-one matrices) in inductively coupled plasma-atomic emission spectrometry were investigated and used to confirm that matrix effects caused by elements with a low second ionization potential are more severe than those from matrix elements having a low first ionization potential. Although the matrix effect is correlated unambiguously with the second ionization potential of a matrix, the correlation is not monotonic, which suggests that at least one other factor is operative. Through study of a large pool of matrix elements, it becomes possible to identify another critical parameter that defines the magnitude of the matrix effect; namely the presence of low-lying energy levels in the doubly charged matrix ion. Penning ionization by Ar excited states is proposed as the dominant mechanism for both analyte ionization/excitation and matrix effects; matrices with a low second ionization potential can effectively quench the population of Ar excited states through successive Penning ionization followed by ion-electron recombination and lead to more severe matrix effects.     

Partitioning of Polymerizing Fluids in Random Microporous Media: Application of the Replica Ornstein-Zernike Equations

We have investigated a model for a polymerizing fluid in which each of the particles has two bonding sites, such that chains can be formed via a chemical association mechanism. The fluid model is considered to be in a random quenched microporous matrix. The matrix species are assumed to be either impermeable to adsorbed fluid particles or permeable, such that the surface of the matrix particles represents a permeable membrane of finite width. We have studied the influence of the matrix species on the formation of chains due to association. The model is investigated by means of the associative replica Ornstein-Zernike equations with the Percus-Yevick closure and the ideal chain approximation. We have observed that the average chain length is longer in the presence of an impermeable matrix than in the case where the matrix is absent. Matrix is therefore conducive to the growth of the polymerizing species in micropores. There is a decrease in the average chain length with increasing permeability of matrix species. This behavior reaffirms the attenuating role of the permeable matrix species as a whole. Copyright 1999 Academic Press.     

Elements of irreducible tensorial matrices generated by finite groups with applications to ligand field Hamiltonians

Using symmetry to determine Hamiltonian matrix elements for quantum systems with finite group symmetry is a special case of obtaining group-generated irreducible tensorial matrices. A group-generated irreducible tensorial matrix transforms irreducibly under the group and is a linear combination of group transformations on a reference matrix. The reference matrix elements may be appropriate integrals or parameters. The methods of normalized irreducible tensorial matrices (NITM) are employed to express elements of the generated matrix in terms of those of the reference matrix without performing the actual transformations. Only NTTM components of the reference matrix with the same transformation properties as the group-generated matrix will contribute to its elements. The elements of invariant symmetry-generated matrices are proportional to simple averages of certain elements of the reference matrix. This relation is substantially more efficient than previous techniques for evaluating matrix elements of octahedral and tetragonal d-type ligand-field Hamiltonians.     

Disposable sodium electrodes

In this paper we describe a disposable sodium sensor in double matrix membrane technology. This sensor is prepared from filter paper with an evaporated silver conducting line on one side. For insulation the sensor is laminated with a pre-perforated heat-sealing film. A defined volume of an ion-sensitive polymer matrix membrane cocktail is filled into one hole. So an ion-sensitive coated film sensor in double matrix membrane technology is produced. The double matrix membrane is formed by the polymer matrix membrane and by the additional filter paper matrix. The response behaviour of the sodium electrode is comparable with conventional macro ion-selective electrodes. By this technology a mass production of low cost sensors is possible.     

Numerical simulation of nanoparticle melting inside a matrix

A theoretical model is proposed for describing the melting of a metal nanoparticle embedded into a solid matrix. The model is based on a thermodynamic approach that takes into account matrix elasticity. The melting process is described for gold nanoparticles embedded in a solid matrix whose elastic modulus is varied in a wide range. Both spherical and ellipsoidal particles are considered. It is shown that particle melting temperature can be both higher and lower than the melting point of a bulk sample depending on the interaction intensity of the solid and liquid particle surfaces with the matrix. An increase in the shear modulus of the matrix causes a rise in the nanoparticle melting temperature, with the effect of the matrix elasticity becoming noticeable at some critical shear modulus. The conditions are revealed at which only a surface layer of a nanoparticle, the thickness of which depends on the particle radius and temperature, is melted.     

荧光光谱解析中矩阵病态对检测结果的影响

在工程数据处理中,经常要进行矩阵变换,矩阵病态性影响计算结果的稳定性,而矩阵的条件数与矩阵病态性密切相关.针对荧光光谱原始信号,通过分析解谱矩阵的条件数,给出通过解谱矩阵进行信号解谱时,解谱矩阵病态性对解谱数据结果的影响以及在提取解谱矩阵时如何减少矩阵病态性的方法.实验结果表明,数据突变会导致矩阵病态性增加,同时带来荧光光谱结果失真,在实际解谱矩阵提取及数据处理过程中,需重视并增强荧光染料谱图的平滑性.     

Effect of matrix bidispersity on the morphology of polymer-grafted nanoparticle-filled polymer nanocomposites

We present a simulation study showing the effect of bidispersity in matrix homopolymer length on the wetting/dewetting of homopolymer-grafted nanoparticles and the morphology of polymer nanocomposites where the graft and matrix polymer chemistries are identical. In a bidisperse matrix with equal number of short and long chains and average matrix length greater than the monodisperse graft length, the densely grafted polymer layer is preferentially wet by the short chains and relatively dewet by the long chains. This is driven by a larger gain in entropy of mixing between graft and matrix for short matrix chains than long matrix chains. Despite the preferential wetting of the short and dewetting of long chains, matrix length bidispersity does not significantly change the overall wetting of the grafted layer. Unlike graft length bidispersity that significantly improves particle dispersion, matrix length bidispersity slightly increases particle aggregation in the polymer matrix. © 2014 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2014 , 52, 1661–1668     

Form of multicomponent Fickian diffusion coefficients matrix

The form of multicomponent Fickian diffusion coefficients matrix in thermodynamically stable mixtures is established based on the form of phenomenological coefficients and thermodynamic factors. While phenomenological coefficients form a symmetric positive definite matrix, the determinant of thermodynamic factors matrix is positive. As a result, the Fickian diffusion coefficients matrix has a positive determinant, but its elements — including diagonal elements — can be negative. Comprehensive survey of reported diffusion coefficients data for ternary and quaternary mixtures, confirms that invariably the determinant of the Fickian diffusion coefficients matrix is positive.     

Surface Green's function calculations: a nonrecursive scheme with an infinite number of principal layers

A novel computational method for a surface Green's function matrix is introduced for the calculation of electrical current in molecular wires. The proposed nonrecursive approach includes an infinite number of principal layers and yields the second-order matrix equation for the transformed Green's function matrix. The solution is found by the direct diagonalization of the auxiliary matrix without any iteration process. As soon as complex roots of the auxiliary matrix (approximately GS) are calculated, the gaps and the bands in the surface electronic structure are found. It is shown that the solution of a second-order matrix equation determines the spectral density matrix, that is, the density of states for noninteracting electrons. Single and double principal layer models are studied both analytically and numerically. The energy interval for nonvanishing spectral matrices is determined. This method is applicable to matrices of any rank.     

The relationship between the eigenvalues and eigenvectors of a similarity matrix and its associated Carbó index matrix

The eigenvalues and eigenvectors of a quantum similarity matrix are also generalized eigenvalues and eigenvectors of the associated matrix of Carbó indices. This establishes bounds on the spectrum of the Carbó index matrix; for example, a quantum similarity matrix is positive semidefinite if and only if the associated Carbó index matrix is also positive semidefinite. The generalized eigenvalue problem for the Carbó index matrix has a diagonal metric matrix on the right-hand-side. Every generalized eigenvalue problem can be written in this diagonal form (i.e., this form is not special to this application). This diagonally structure generalized eigenvalue problem is especially convenient because it can be converted to a conventional eigenvalue problem by a particularly simple partial Löwdin transformation.     

Quasirelativistic theory equivalent to fully relativistic theory

The Dirac operator in a matrix representation in a kinetically balanced basis is transformed to a quasirelativistic Hamiltonian matrix, that has the same electronic eigenstates as the original Dirac matrix. This transformation involves a matrix X, for which an exact identity is derived, and which can be constructed either in a noniterative way or by various iteration schemes, without requiring an expansion parameter. The convergence behavior of five different iteration schemes is studied numerically, with very promising results.     

Observation of a physical matrix effect during cold vapour generation measurement of mercury in emissions samples

The observation of a physical matrix effect during the cold vapour generation–atomic fluorescence measurement of mercury in emissions samples is reported. The effect is as a result of the different efficiencies of liberation of reduced mercury from solution as the matrix of the solution under test varies. The result of this is that peak area to peak height ratios decease as matrix concentration increases, passing through a minimum, before the ratio then increases as matrix concentration further increases. In the test matrices examined – acidified potassium dichromate and sodium chloride solutions – the possible biases caused by differences between the calibration standard matrix and the test sample matrix were as large as 2.8% (relative) representing peak area to peak height ratios for calibration standards and matrix samples of 45 and 43.75, respectively. For the system considered there is a good correlation between the density of the matrix and point of optimum liberation of dissolved mercury for both matrix types. Several methods employing matrix matching and mathematical correction to overcome the bias are presented and their relative merits discussed; the most promising being the use of peak area, rather than peak height, for quantification.     

The use of pencil lead as a matrix and calibrant for matrix-assisted laser desorption/ionisation

Pencil lead is shown to be an effective matrix and calibrant in matrix-assisted laser desorption/ionisation (MALDI) mass spectrometry. Various groups of analytes, including peptides, polymers and actinide metals, can be readily ionised using MALDI when deposited onto a pencil lead matrix. The matrix is seen to have advantages in sample preparation relating to its hydrophobic properties and almost complete suppression of the matrix during analysis. Using pencil lead as a matrix is a quick and convenient method of qualitative analysis and has been shown to be quantitative for the isotope ratio analysis of actinide metals.     

Density matrix purification with rigorous error control

Density matrix purification, although being a powerful tool for linear scaling construction of the density matrix in electronic structure calculations, has been limited by uncontrolled error accumulation. In this article, a strategy for the removal of small matrix elements in density matrix purification is proposed with which the forward error can be rigorously controlled. The total forward error is separated into two parts, the error in eigenvalues and the error in the occupied invariant subspace. We use the concept of canonical angles to measure and control differences between exact and approximate occupied subspaces. We also analyze the conditioning of the density matrix construction problem and propose a method for calculation of interior eigenvalues to be used together with density matrix purification.     

Matrix isolation vibrational spectroscopy as a tool for studying conformational isomerism

The use of matrix isolation vibrational spectroscopy to study conformational isomerism is described. Methods of distinguishing conformational splitting from matrix splitting are discussed. Examples are given of molecules for which the conformational equilibrium existing in the gas phase prior to deposition is trapped out in the matrix, of molecules which exhibit reversible interconversion of conformers at matrix temperatures, and of a molecule for which the conformational distribution trapped out in the matrix is strongly dependent on matrix polarity. Results obtained for molecules which exhibit infrared-induced isomerisation in matrices are discussed.     

基体石墨的孔隙结构及其对单质Ag扩散的影响

高温气冷堆燃料元件的基体石墨是一种多孔复合材料,是燃料元件的主要组成部分,其结构影响燃料元件的性能和裂变产物在燃料元件中的扩散。 本文利用压汞法表征基体石墨的孔隙结构,并讨论了基体石墨制备工艺中最大压制压强与热处理过程对孔隙结构的影响。 结果表明,基体石墨大孔孔径分布为6001900 nm,高温热处理使基体石墨的总孔隙率、中值孔径、大孔孔容均减小;基体石墨热处理样品的大孔孔容随最大压制压强的增加而呈线性减少,热处理过程单质Ag在石墨基体中的扩散速度与大孔孔容变化具有正相关性。     

A Concise Description of an Old Problem: Application of Matrices to Obtain the Balancing Coefficients of Chemical Equations

It is not difficult to balance chemical equations and thus it is hardly given more thoughts. However, there is a mathematical principle for the balancing of chemical equations and this principle may be used for automation. The balancing of chemical equation is carried out, by formulating a reaction matrix and the latter is used in a matrix equation. The matrix equation is then solved to obtain the balancing coefficients. The conventional matrix inverse method cannot always be used and hence the solution is obtained by row reduced operations. These operations and any matrix manipulation are carried out with Matlab. Further this novel method can be used to classify chemical equations as nonfeasible, unique and nonunique.     

MALDI-TOF mass spectrometry of a combinatorial peptide library: effect of matrix composition on signal suppression

The effect of matrix composition on signal suppression caused by a dominant compound under MALDI ionization was studied using the combinatorial TQTXT pentapeptide library as a model system. The peptide library is composed of 19 components with all proteinogenic amino acids except cysteine in position X. From these compounds, only the Arg peptide (TQTRT) was detected with sufficient intensity in the MALDI-TOF mass spectrum under typical MALDI conditions (CCA matrix). The analysis of a set of compounds utilized as different matrix components, additives and a cationizing agent revealed that the composition of the matrix is a critical point in signal suppression. Highly improved ion yields were achieved by using a CCA/DHB mixture as a matrix. The addition of K(+) as a cationizing agent to the CCA matrix resulted in MALDI-TOF mass spectra with relative ion intensities very similar to those obtained by electrospray ionization.     

The effect of matrix structure on the diffusion of fluids in porous media

The effect of matrix structure on the transport properties of adsorbed fluids is studied using computer simulations and percolation theory. The model system consists of a fluid of hard spheres diffusing in a matrix of hard spheres fixed in space. Three different arrangements of the fixed spheres, random, templated, and polymeric, are investigated. For a given matrix volume fraction the diffusion coefficient of the fluid, D, is sensitive to the manner in which the matrix is constructed, with large differences between the three types of matrices. The matrix is mapped onto an effective lattice composed of vertices and bonds using a Voronoi tessellation method where the connectivity of bonds is determined using a geometric criterion, i.e., a bond is connected if a fluid particle can pass directly between the two pores the bond connects, and disconnected otherwise. The percolation threshold is then determined from the connectivity of the bonds. D displays universal scaling behavior in the reduced volume fraction, i.e., D approximately (1-phi(m)phi(c))(gamma), where phi(m) is the matrix volume fraction and phi(c) is the matrix volume fraction at the percolation threshold. We find that gamma approximately 2.2, independent of matrix type, which is different from the result gamma approximately 1.53 for diffusion in lattice models, but similar to that for conduction in Swiss cheese models. Lattice simulations with biased hopping probabilities are consistent with the continuous-space simulations, and this shows that the universal behavior of diffusion is sensitive to details of local dynamics.