Singer polymals. I. Evaluation of matrix elements

Hamiltonian matrix elements between pairs of explicitly correlated Singer polymals are derived. A new method for 1/r₁₂ matrix elements is found which avoids the 6 by 6 matrix inversion required at each quadrature point in the old method. The new formula requires about half the computer time of the old one. The electron repulsion and nuclear attraction matrix elements are shown to be related to incomplete elliptic integrals. Simplified formulas are derived for the special cases of cylindrical symmetry, spherical symmetry, and single-center expansions. An upper bound approximation formula for the purpose of neglecting matrix elements is derived. These bounds may be useful for single-center expansions but are found to be too high for the general case.     

Spin-free approach for evaluation of electronic matrix elements using character operators of ℒN

A spin-free method is presented for evaluating electronic matrix elements over a spin-independent many-electron Hamiltonian. The spin-adapted basis of configuration state functions is obtained using a nonorthogonal spin basis consisting of projected spin eigenfunctions. The general expressions for the matrix elements are given explicitly, and it is demonstrated how the matrix elements may be obtained simply from the knowledge of the irreducible characters of the permutation group ℒ_N. The presented formulas are very general and may be applied in connection with both spin-coupled valence bond studies and in conventional configuration interaction (CI) methods based on an orthonormal orbital basis. © 1996 John Wiley & Sons, Inc.     

Spin-independent three-body effective valence-shell operators: Application to molecular oxygen

A mathematical construction is presented that uniquely defines a set of spin-independent effective valence-shell Hamiltonian (H^v) three-body matrix elements. These spin-independent H^v matrix elements separate direct and exchange portions of the three-body H^v matrix elements and therefore provide the most natural form for comparisons with parameterization schemes of semiempirical electronic structure methods in which the three-body matrix elements are incorporated into semiempirical one- and two-body Hamiltonian matrix elements in an averaged manner. Ab initio H^v three-body matrix elements of O₂ are computed through third order of quasidegenerate perturbation theory and are analyzed as a function of internuclear distance and atomic orbital overlap to aid in understanding how these three-body matrix elements may be averaged into semiempirical one- and two-body matrix elements. © 1992 John Wiley & Sons, Inc.     

The preparation and homogeneity study of synthetic multi-element standards (SMELS) for QC/QA of k0-NAA

Three new synthetic multi-element standards (SMELS) were developed for testing the performance of the k ₀-standardization neutron activation analysis (k ₀-NAA) method when implemented in a laboratory. SMELS consist of a phenol-formaldehyde polymer matrix spiked with different groups of elements according to the half-lives of the formed radionuclides. They are classified as Type I for short-, Type II for medium- and Type III for long-lived radionuclides. This article presents the preparation of SMELS including different aspects such as choice of the matrix and spiking elements, chemical compounds, target concentrations, purity of the matrix and spiking procedure and the homogeneity study for all the spiked elements in order to demonstrate the quality of the produced materials.     

Spin-Orbit coupling in aromatic hydrocarbons: A semiempirical evaluation of the radiative lifetimes of naphtalene and anthracene

Singlet-triplet spin-orbit matrix elements, which govern the lowest ³B_1u ← ¹A_g transition in typical aromatic molecules like naphtalene and anthracene, are calculated with INDO molecular orbitals and the conventional spin-orbit one-electron hamiltonian. The correct order of magnitude of the triplet radiative lifetimes is obtained for the two molecules, when INDO MO coefficients are referred to a symmetrically orthogonalized basis. The possibility of using the semiempirical Hamiltonian is explored using ab initio wave-functions for a few test cases. Reasonably accurate doublet-doublet and singlet-triplet matrix elements have been computed.     

Investigations of the carbide phase and the matrix of non-ledeburitic high-speed steel with Ti and Nb by atomic absorption spectrometry and energy dispersive X-ray spectroscopy methods

This paper presents a computing procedure for the indirect assessment of the matrix composition by AAS of two-stage prepared carbide isolates. Partitioning of carbon and metallic elements between matrix and carbides has been determined. Austenitizing has caused considerable increase in Mo and Cr contents in the matrix related to dissolution of carbides of the M₆C type. Tempering resulted in a decrease in contents of elements forming M₂₃C₆ secondary carbides. Changes to the Ti and Nb contents in the matrices were relatively small in specimens given various heat treatments. The matrix of annealed steel has been shown to be carbonfree, whereas the carbon content in the tempered state is high in comparison to conventional high-speed steels.     

Detection limits in XPS for more than 6000 binary systems using Al and Mg Kα X‐rays

A simple approach to estimating the detection limits of X‐ray photoelectron spectroscopy (XPS) for any element in any elemental matrix is presented, using the intensity of the background at the expected position for the photoelectron peak to be detected. The approach has been extended to estimate the detection limit for all elements from lithium to bismuth in a similar range of elemental matrices. Using a number of assumptions, it is possible to obtain a reasonable estimate of the background intensity at any electron kinetic energy in the XPS spectrum of an element. Therefore, a detection limit for an arbitrary element homogeneously distributed in that matrix can be estimated. The results show that, although most elements are detectable at about the 1 at.% to 0.1 at.% level, for heavy elements in a light element matrix, the detection limit can be better than 0.01 at.%, whereas for light elements in a heavy element matrix, detection limits above 10 at.% are not uncommon. Two charts detailing the detection limits for all combinations of trace and matrix elements from lithium (Z = 3) to bismuth (Z = 83) are provided for Al Kα and Mg Kα X‐ray sources using a typical hemispherical analyser instrument which provides 10⁶ counts eV for the Ag 3d_5/2 peak from pure silver. These detection limits can be scaled to estimate the detection limits for any given instrument and operating conditions if the intensity of the Ag 3d_5/2 peak from pure silver under those conditions is known. © 2014 Crown Copyright. Surface and Interface Analysis © 2014 John Wiley & Sons Ltd.     

Anreicherung von elementspuren aus hochreinem gallium durch partielles lösen der matrix : Analyse des spurenkonzentrats durch flammen-atomabsorptions-spektrometrie mit der “injektionsmethode”

Trace elements can be preconcentrated from high-purity gallium by simple dissolution of the sample in HCl—HNO₃, until a residue of less than 5 mg remains. Traces of metals which are nobler electrochemically than the matrix (Ag, Au, Bi, Co, Cu, Fe, Hg, Ni, Pb, Pd and Sn) are enriched in the residue (recovery > 95 %), a concentration factor of about 10³ being obtained. Traces of Cd, Co, Fe, Tl and Zn can be extracted from gallium without dissolving the matrix by treatment with a mixture of HI and I₂. Most of the preconcentrated elements are determined by flame atomic absorption spectrometry with the “injection method”. The limit of detection (3σ) for different elements was found to be between 0.06 and 0.004 p.p.m. The relative standard deviation is generally about 5 %.     

SEM and EDXS analysis of the reaction products of compounds AxM6X8 (A = Tl,K; M = V,Ti; X = S,Se) with I2/CH3CN and H2O

The ternary transition metal chalcogenides A_xM₆X₈ (A=Tl, K; M=V, Ti; X=S, Se) build up a three dimensional framework with large hexagonal channels running parallel to the crystallographic c-axis. The electropositive elements thallium or potassium are confined within these channels. It is possible to remove the Tl or K atoms via a chemical redox reaction with an I₂/CH₃CN solution or with H₂O. Using SEM it is demonstrated that the host matrix “V₆S₈” is only slightly affected by the redox agent. In contrast the host matrix “Ti₆Se₈” of Tl_xTi₆Se₈ reacts with the I₂/CH₃CN solution. The results of the EDXS analyses clearly show that the removal of the electropositive elements proceeds only along the large channels and not through the host matrix.     

Application of a Matrix of Piezosorption Sensors to Analysis of Ethanol-Containing Gas Mixtures

General relationships were established for operation of piezosorption sensors when exposed to an ethanol-based multicomponent gas mixture. The operation conditions were optimized for the sensors; a matrix comprising six piezosorption elements was designed for analyzing gas mixtures containing low-boiling C₁-C₅ alcohols and ethyl acetate. The sensor matrix is suitable for foodstuff analysis.     

Separation/preconcentration of trace heavy metals in urine, sediment and dialysis concentrates by coprecipitation with samarium hydroxide for atomic absorption spectrometry

Multi-element determination of trace elements in urine and dialysis solutions by atomic absorption spectrometry has been investigated. Coprecipitation with samarium hydroxide was used for preconcentration of trace elements and elimination of matrix elements. To 10 ml of each sample was added 500 μl of 2 mg ml⁻¹ samarium solutions; the pH was then adjusted to 12.2 in order to collect trace heavy metals on samarium hydroxide. The precipitate was separated by centrifugation and dissolved in 1 ml of 1 mol l⁻¹ HNO₃. Coprecipitation parameters and matrix effects are discussed. The precision, based on replicate analysis, is around 5% for the analytes, and recovery is quantitative, based on analysis of spiked samples and solutions including matrix components. The time required for the coprecipitation and determination was about 30 min.     

Error matrices in gas-electron diffraction: II. Influence of weight matrix

The properties of three different forms of error matrices in electron diffraction are investigated, assuming the presence of stationary, Gaussian, Markovian noise in the primary data. The error matrices studied are M_x^p based on the optimum weight matrix P the bona fide error matrix M_x^w based on the nonoptimum weight matrix W, and the false error matrix M_x^o commonly calculated by diffractionists using the formula for the optimum error matrix while incorporating a nonoptimum weighting. Simple formulae relating the elements of the various matrices are derived in the case where W is the best diagonal weight matrix and where geometric constraints are not imposed on parameters. The influence of geometric constraints is tested. Calculations indicate that diagonal weight matrices in ordinary circumstances give results imperceptibly inferior to the results obtained with the best nondiagonal weight matrices. Elements of M_x^w closely approach those of M_x^p whereas elements of the false error matrix, taken alone, may be very misleading.     

Improved treatment for matrix elements of spin-dependent operators

In this paper a general method for the evaluation of the matrix elements of spin-dependent operators is proposed to improve the treatment primitively suggesteed by Cooper and Musher. This approach is largely based on the recent results which the present authors have achieved in the representation theory for the inner- and outer-product reduction of the symmetric group. It is shown that the so-called outer-product coupling coefficients (OPCC ) can be used to generalize the method for constructing the irreducible tensor operators of group S_n. Together with the use of inner-product coupling coefficients (IPCC ), an expression for the matrix elements of spin-dependent operators is presented as the product of a Racah coefficient for S_n and a reduced matrix element which can be expressed in terms of IPCC, OPCC , and the related integrals. The treatment for one- and two-electron spin-dependent operators is discussed in detail.     

Spin-Dependent operators in the spin-free quantum chemistry

The two-particle spatial density matrix components introduced by McWeeny are expressed in terms of the Fock coordinate wave function, which is constructed from an arbitrary function of N spatial coordinates. The integral relations for these components are verified. The necessary matrix elements of a standard representation of the S_N group are calculated.     

Improvements in calibration procedures for the quantitative determination of trace elements in carbonate material (mussel shells) by laser ablation ICP-MS

A better repeatability and accuracy in the quantitative determination of trace elements in mussel shells or carbonate-based materials by LA-ICP-MS was achieved by using a series of multielement calibration standards prepared by co-precipitation of twelve elements into a CaCO₃ matrix in order to improve the homogeneity of the resulting powder samples. Pressed powder discs of good mechanical stability could be obtained at a pressure of 50 MPa, without the addition of a binder. An UV laser (modified Nd:YAG, 266 nm) was used in the Q-switched mode at a repetition rate of 10 Hz and an energy level of 3.5 mJ. Correlation coefficients (R) for the linear calibration graphs (concentration range: 1.5–400 μg/g) for Cr, Mn, Co, Cu, Zn, As, Cd, Sn, Ba, and Pb are generally better than 0.997. The detection limits for all elements investigated are in the sub-μg/g range. Incorporation of elements into the matrix by co-precipitation has shown as a superior method for producing calibration standards than the simple mixture of the analytes (in carbonate or oxide form) with the matrix (CaCO₃) or addition of standard solutions to a carbonate powder base. Two examples of the quantitative determination of toxic elements in mussel shells will be presented. Received: 10 January 2000 / Revised: 16 March 2000 / Accepted: 26 March 2000     

Use of Markov Chains in Treating Depropagation in Copolymerization. II. Distribution of Monomer Units in Copolymers

A regular Markov chain is set equivalent to n-component copolymerization (n ≥ 2) with depropagation. It is shown that the elements of the matrix of the average times for first approach M (m_ij), represent the average number of units m_k (k = 1, 2, …, n; k ± j), occupying places between the units m_i and m_j in the macromolecules. These are the average lengths of the blocks, beginning with any monomer unit m_i and ending with any unit m_j, which however do not meet further in the block. For n = 2, the composition of these blocks is fully determined, because they are composed only of one type of units. When, however, n ≥ 3, two or more types of monomer units are included in these blocks. By means of the absorbing Markov chains, expressions for the composition of these blocks when n ≥ 3 are also obtained. In binary copolymerization the diagonal elements of the matrix M (m_ij) give the average lengths of the homoblocks ～ m_i-m_i-…-m_i ～ in the macromolecules. For the n-component copolymerizatior. (n ≥ 3), the elements m_ii do not give this information. Expressions for these lengths in this case may be obtained by additional probability analysis.     

Acid effects in ICP-AES

Summary The influence of concentration and nature of the acid, type and tuning of RF-generator, method of solution introduction (flow injection or two-nebulizer method) are investigated to understand the nature of acid matrix interferences in ICP-AES. Sixteen elements and nine different spectrometers wer used to study the matrix interferences caused by HCl, HNO₃, HClO₄, H₂SO₄, H₃PO₄. There are two effects connected with acid: depression of the analyte emission signals and irregular signal changes for different elements. The values of these effects depend on the RF-generator type. The obvious results of introducing acid in the plasma are a decrease in plasma temperature and departures from LTE.

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Einfluß von Säuren in der ICP-AES

     

Spectral Interference in Antimony Analysis with High Temperature Furnace Atomic Absorption

Abstract

Background correction in atomic absorption spectrophotometry using a high temperature furnace has a close relation to the chemical treatment and the detection limit. Conventionally used for this correction is the D₂ lamp method, but spectral interference is inevitable in this method because the spectral bandwidth of the light from the D₂ lamp after passing through a monochromator is as wide as 0.1 to 5 nm. In the analysis of Sb in lead alloy or in steel, there is spectral interference due to high concentration of elements such as Pb, Cu or Fe, in the matrix, so accurate measurement is impossible unless these elements are removed at the pretreatment stage.

In polarized analyte Zeeman atomic absorption spectrophotometry developed by the authors, the wavelength of the measuring beam is the same as that of the reference and they are identical in spectral line profile as well as in geometrical shape. They differ only in the direction of polarization. For these reasons, we found that there is no spectral interference like that recognized in the D₂ lamp correction technique. As a result, a direct analysis was done simply by dissolving gunmetal or steel in acid without removing the matrix elements Pb, Cu or Fe.     

Systematic investigations of the multi-element preconcentration from copper by precipitation of the matrix as copper(I) oxide and copper(I) thiocyanate

Two methods for multi-element preconcentration from copper by reductive matrix precipitation are presented. In systematic investigations on the coprecipitation behaviour of Ag, Al, Au, Bi, Cd, Co, Cr, Fe, Ga, In, Mn, Mo, NJ, Pb, Sb, Se, Sn, Te and Zn during precipitation of the copper matrix as Cu₂O or CuSCN, the separation parameters were optimized. By combination with a hexamethyleneammonium hexamethylenedithiocarbamate collector precipitation, a concentration of 8 elements (Cu₂O precipitation) or 13 elements (CuSCN precipitation) in a small volume was achieved. The limits of detection of the procedures are, depending on the element, 0.1–5 μg g^?1 for flame atomic absorption spectrometry (AAS) and 0.01–0.1 μg g^?1 for graphite furnace AAS. The relative standard deviations are about 3%. The analytical performance of the procedures is compared with that of an electrolytic copper separation.