Use of the mixed basis method for ab initio SCF MO calculations

A previously described mixed basis method for performing SCF MO calculations has been applied to the benzene molecule and to the allyl cation. A basis set of Slater type orbitals is used to calculate the one-electron and one-centre two-electron integrals whilst the many-centre two-electron integrals are approximated by small gaussian type orbital expansions of the STO basis. Comparison of the results with all-gaussian basis sets and literature values indicates that the mixed basis 2-GTO approximation is inadequate for molecules of this size because of the consistent underestimation of the electron repulsion integrals. The use of gaussian exponents chosen by a least-squares procedure rather than variationally gives better mixed basis results, but the indications are that a 4-GTO expansion is necessary for reliable mixed basis calculations. A method for more accurate integral evaluation by gaussian expansions of orbital products is suggested.

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Zusammenfassung Eine früher beschriebene Methode mit gemischter Basis zur Durchführung von SCF-MO-Berechnungen wurde auf das Benzolmolekül sowie das Allylkation angewendet. Ein Basissatz aus Slaterorbitalen wird verwendet, um die Einelektronen- und die Einzentrenzweielektronenintegrale zu berechnen, während die Mehrzentrenzweielektronenintegrale mit Hilfe von Entwicklungen der Slaterorbitale nach wenigen Gaußfunktionen angenähert werden. Ein Vergleich mit Ergebnissen von Rechnungen mit Gesamtbasissätzen von Gaußfunktionen und mit Literaturwerten zeigt, daß der gemischte Basissatz 2-GTO ungeeignet für Moleküle von dieser Größe ist, da die Elektronenabstoßungsintegrale durchgehend zu klein berechnet werden. Die Verwendung von Exponenten der Gauß-funktion, die mit Hilfe einer Methode der kleinsten Quadrate gewonnen wurde, ergibt bei der gemischten Basis bessere Ergebnisse als diejenigen, die nach der Variationsmethode gewonnen wurden; es zeigt sich jedoch, daß 4-GTO-Entwicklungen für angemessene Berechnungen mit gemischter Basis notwendig werden. Eine Methode für eine genauere Integralberechnung mit Hilfe der Entwicklung von Orbitalprodukten nach Gaußfunktionen wird vorgeschlagen.

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Résumé Application à la molécule de benzène d'une méthode de base mixte précédemment décrite pour effectuer les calculs SCF MO. Une base d'orbitales de Slater est utilisée pour calculer les intégrales monoélectroniques et les intégrales biélectroniques monocentriques; les intégrales biélectroniques polycentriques sont approchées au moyen d'une expression des orbitales de Slater en orbitales gaussiennes. La comparaison des résultats avec ceux obtenus en bases gaussiennes et avec ceux donnés dans la littérature indique que l'approximation 2-GTO n'est pas adaptée aux molécules de cette taille car elle provoque une sous-estimation constante des intégrales de répulsion électronique. L'emploi d'exposants gaussiens choisis par un procédé de moindres carrés plutôt que par une méthode variationnelle donne de meilleurs résultats en bases mixtes mais il s'avère nécessaire d'utiliser un développement du type 4-GTO. On propose une méthode pour l'évaluation plus précise des intégrales par développement gaussien des produits d'orbitales.

     

Formation of multimeric steroid metal adducts and implications for isomer mixture separation by traveling wave ion mobility spectrometry

Steroid analysis is essential to the fields of medicine and forensics, but such analyses can present some complex analytical challenges. While chromatographic methods require long acquisition times and often provide incomplete separation, ion mobility spectrometry (IMS) as coupled to mass spectrometry (MS) has demonstrated significant promise for the separation of steroids, particularly in concert with metal adduction and multimerization. In this study, traveling wave ion mobility spectrometry (TWIMS) was employed to separate multimer steroid metal adducts of isomers in mixtures. The results show the ability to separate steroid isomers with a decrease in resolution compared with single component standards because of the formation of heteromultimers. Additionally, ion‐neutral collision cross sections (CCS) of the species studied were measured in the mixtures and compared with CCSs obtained in single component standards. Good agreement between these values suggests that the CCS may aid in identification of unknowns. Furthermore, a complex mixture composed of five sets of steroid isomers were analyzed, and distinct features for each steroid component were identified. This study further demonstrated the potential of TWIMS‐MS methods for the rapid and isomer‐specific study of steroids in biological samples for use either in tandem with or without chromatographic separation.     

Concomitants and syzygies of spinors in 3-dimensional euclidean space

Summary In 3-dimensional Euclidean space, explicit formulae for the Hermitean concomitants of two irreducible spinsors of type[n+1/2] are obtained, where n is a positive integer. Further, a method, by means of which the syzygies of degree2 in these concomitants can be obtained, is described and is illustrated by considering two irreducible spinors of type[3/2]. Entrata in Redazione il 16 maggio 1972.     

Ranges in silicon of ions with atomic numbers 62 ⩽ Z1 ⩽ 66 at 100 keV

The ranges of ions with atomic numbers 62 ? Z₁ ? 66 at 100 keV in silicon have been measured. Both the projected range and standard deviation are found to vary smoothly with Z₁.     

Symmetric spinors ink-dimensions

Summary In his paper, explicit formulae are given for any irreducible spinor, in any number of dimensions, which is symmetric in its suffixes. The method employed in determining these formulae is immediately applicable to the task of obtaining explicit forms for spinors which are unsymmetric in their suffixes. Entrata in Redazione il 22 maggio 1971.     

Basic spin matrices

Summary In this paper the basic spin matrix, U, corresponding firstly, to an orthogonal transformation in3-dimensional Euclidean space and secondly, to a Lorentz transformation in4-dimensional space-time, is obtained explicitly in terms of the defining set of anticommuting matrices X_i and the elements of the matrix of the transformation. It is found that in3-dimensions, four formulae for U are necessary in order to cover all possible contingencies, whilst in4-dimensions sixteen are necessary. A number of particular cases are also considered both in three and in four dimensions. Entrata in Redazione il 14 settembre 1970.     

RUC-decompositions in symmetric operator spaces

We show that ifX is a Banach space of type 2 andG is a compact Abelian group, then any system of eigenvectors {x }_G (with respect to a strongly continuous representation ofG onX) is an RUC-system. As an application, we exhibit new examples of RUC-bases in certain symmetric spaces of measurable operators.Research supported by the Australian Research Council     

Muon spin relaxation in ferromagneticPdMn

Positive-muon ( ⁺) spin relaxation experiments have been carried out in the dilute ferromagnetic alloy Pd+2 at.% Mn (T _c=5.8 K). In the paramagnetic state the inhomogeneous ⁺ linewidth is proportional to the bulk magnetization. BelowT _c the ⁺ linewidth and the width of the ⁺ local field distribution in zero applied field are both in qualitative accord with the Sherrington-Kirkpatrick theory of disordered magnets.This work was performed under the auspices of the U.S. Department of Energy, and was supported by the U.S. National Science foundation, grant nos. DMR-7909223 and DMR-8115543, and by the Netherlands Stichting voor Fundamenteel Onderzoek der Materie (FOM).