Some simple remarks on variational bounds in Faddeev-type formalisms

A bound on the energy obtained from the differential equation forms of Faddeev-type equations is derived. The procedure is based on the standard variation principle for the Schrödinger equation plus the observation that the ordinary Schrödinger wavefunction |ψ(i)> equals the sum of the Faddeev-type “channel wavefunctions” |ψ_λ(i)>.     

Some remarks on the balance as a symbol

Today just as in very early illustrations the balance is regarded a a symbol. It is an attribute of merchants and various trade guilds. In the hands of the Gods of Greece, it indicates victory or death of the hero. In the Egyptian religion, it first appears as an instrument in the death tribunal. In Judaism and for the Romans, it was the Symbol of justice.     

Some remarks on the hydrogen atom

The path integral for the Green's function involving the Coulomb potential in combination with the Kustaanheimo-Stiefel transformation is used to generate the atomic orbitals of the nonrelativistic hydrogen atom as various combinations of the product of one-dimensional isotropic harmonic oscillator wave functions. The use of the transformation is justified, by connecting the homogeneous space with the quotient space in the Feynman quantization formalism.     

Some remarks on the Pariser-Parr-Pople method

Basic assumptions which characterize the Pariser-Parr-Pople method of computing molecular electronic wave functions are critically examined. By restricted variational calculation of the valence state of carbon and nitrogen atoms and ions, it is demonstrated that the usual methods of evaluation of one-centre Coulomb integrals and atomic core energies are rather good. A semi-theoretical means of estimating the core resonance integral is proposed and shown to give fair agreement with the empirical values for C-C, O-O, C-N, and C-O bonds.

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Zusammenfassung Die Grundannahmen der Methode von Pariser-Parr-Pople zur Berechnung molekularer Wellenfunktionen werden kritisch durchleuchtet. Mittels beschränkter Variationsansätze für den Valenzzustand von Kohlenstoff- und Stickstoff-Atomen und -Ionen wird dargelegt, daß die Methoden, wie sie üblicherweise zur Berechnung von Einzentren-Coulombintegralen und atomaren Rumpfenergien angewendet werden, zu recht guten Ergebnissen führen. Ferner wird ein halbtheoretisches Verfahren zur Abschätzung der Rumpfresonanzintegrale vorgeschlagen und nachgewiesen, daß das Resultat gut mit den empirischen Werten für die C-C-, O-O-, C-N- und C-O-Bindung übereinstimmt.

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Résumé Les hypothèses fondamentales de la méthode de Pariser-Parr-Pople pour le calcul des fonctions d'onde électroniques sont examinées critiquement. Un calcul variationel limité des états de valence des atoms et ions de carbone et de nitrogène montre que les méthodes usuelles pour l'évaluation des intégrales de Coulomb monocentriques et des énergies de coeur atomiques sont assez bonnes. Nous proposons un procédé semithéorique pour évaluer l'intégrale de résonance de coeur, qui s'accorde assez bien aux valeurs expérimentales pour les liaisons C-C, O-O, C-N et C-O.

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The research reported in this paper was sponsored in part by the King Gustaf VI Adolf's 70-Years Fund for Swedish Culture, Knut and Alice Wallenberg's Foundation, the Swedish Natural Science Research Council, and in part by the Aeronautical Research Laboratory, OAR, through the European Office, Aerospace Research, United States Air Force, and also in part by the National Aeronautics and Space Administration Research Grant NsG-512.

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On leave of absence from the Department of Chemistry, Faculty of Science, Hokkaido, University, Sapporo, Japan.     

Some remarks on the theory of nitration

Experiments on formation of nitric esters of starch (“O-nitration” of starch) led to the conclusion, that nitration occurs not necessarily by means of NO₂⁺ ions, but NO₃⁻ ions and/or undissociated nitric acid are nitrating agents.

Nitrating mixtures rich in nitric acid may yield products of higher nitrogen content owing to the fact that the reaction occurs in one phase system.     

On the use of explicitly correlated functions in variational computations for small molecules

The explicitly correlated wave functions used in variational molecular calculations are reviewed. Different types of such functions are considered. The state of art and future perspectives are briefly discussed. © 1994 John Wiley & Sons, Inc.     

Some remarks on the maximum bond order

Summary Some remarks are given concerning the recent improvement scheme of calculating Jug's maximum bond border index.     

Some remarks on the dynamics of polymer melts

Considering one long chain (N monomeric units) in a homodisperse melt of chemically identical, but shorter, “solvent” chains (P monomers per chain), we propose some tentative scaling laws for the self-diffusion constant D(N) and the relaxation time T(N) of the solute chain. We also discuss the viscosity increment δ_η due to a small volume fraction Φ of the long chains. We find three regimes of behavior, depending on N and P, and on the distance between entanglement points (assumed smaller than N and P): (A) reptation of the N chain; (B) Stokes–Einstein regime; the solute moves like a usual polymer coil in a viscous fluid of P chains; (C) mixed regime, where D(N) is controlled by reptation, while δ_η is of type B. Contrary to our earlier belief, we find no significant regime where the process of “tube renewal” could be dominant.     

The use of slater orbitals in variational calculations involving one-electron green's functions

In this paper an application of a minimum principle proposed for electronic systems by Hall, Hyslop and Rees, [1] and [2], is considered. It is shown that the evaluation of the required two-centre molecular integrals involving one-electron Green's functions may be facilitated by the introduction of a particular class of trial functions. These functions incorporate the potential energy of the system as a weighting factor and Slater-type orbitals are then used as a basis set. The evaluation of the resulting integrals is discussed and illustrative calculations for the H ₂ ⁺ ion are presented.     

Some remarks on propagators and resolvents

Summary The background of the propagator methods introduced into quantum chemistry by Linderberg and Öhrn is briefly reviewed. Emphasis is put on its connection with the superresolvent associated with the Liouvillians superoperator as shown by Goscinskiet al. The paper stresses the importance of the effectiveness of the resolvent approach in general, and points out that, for the ordinary Hamiltonian, this method is not only conceptually important as the basis for infinite-order perturbation theory and rational approximations but also as the foundation for a large-scale computational effort still to come. Even if it may be premature to ask for a similar effort related to the superoperator, it would certainly be of value to carry out some test calculations based on the Hilbert-Schmidt binary product, in which the Liouvillian superoperator is automatically self-adjoint, and to compare the results with those obtained from the more conventional propagator methods. The importance of the use of approximations based on inner projections in both these approaches is also mentioned.     

Some remarks on the semiempirical one-center electron repulsion integrals

The identification of the stage of ionization for various kinds of one-center electron repulsion integrals, occurring when nonbonding or lone-pair electrons are considered explicitly as well as π-electrons, is discussed for conjugated organic molecules containing heteroatoms N. It is concluded that the value for the negative ions should be used for (π_Cπ_C | π_Cπ_C) in all the states but for (π_Nπ_N | π_Nπ_N) only in the π-π states. In the n-π states, the appropriate value of (π_Nπ_N | π_Nπ_N) is that of the neutral atom if the molecule contains only one N atom. If more than one N atom is involved in the molecule, some weighted mean of the values for the negative ion and for the neutral atom should be used. The value for the neutral atom is most adequate for one-center repulsion integrals other than the (ππ | ππ) type in both the π-π and the n-π states. The method of determining these integrals is also discussed. It is concluded that they are to be determined from the consideration of appropriate electron-transfer reactions except for exchange integrals. The exchange integrals are shown to have to be determined from the Slater–Condon parameters derived from the analysis of the experimental atomic energy levels. Illustrative calculations are given for the lower singlet levels of the formaldehyde, pyrazine, pyridine, and the p-benzoquinone molecule. It is found that the calculated energies of the n-π transitions become much too low unless the (ππ | ππ) values of the heteroatoms in the molecule are chosen differently in the n-π states and in the π-π states as pointed out theoretically in this article.     

Some remarks on substitution effects in sodalites

Some recent examples of substitution effects in the three partial structures of selected sodalites are presented. Framework-cation, and cage-cation substitution effects are discussed for the series |(Eu_xCa_2–x)₄(OH)₈|[(Al_2+xSi_1–x)₄O₂₄]-SOD, cage-anion substitutions for |Ca₈[(WO₄)_x(MoO₄)_1–x]₂|[Al₁₂O₂₄]-SOD. While in a first approximation decomposition into partial structures seems reasonable, it is shown that, however, effects arising from substitutions in one partial structure are not limited to it. This is due to the fact that three main interactions have to be considered, namely, cage-cation – framework, cage-anion – framework, and cage-cation – cage-anion interactions. These can be co-operative or competitive and may lead to structural frustrations.     

Some remarks on the Raman spectra of selected menshutkin complexes

The Raman spectra of the solid complexes naphthalene·2SbCl₃, phenanthrene ·2SbCl₃ and diphenyl ·2SbCl₃ are presented. The differences found in their Raman spectra can be explained by comparing the results with those of NQR and X-ray structure studies given in the literature.