Theory of projected probabilities on non-orthogonal states: Application to electronic populations in molecules

Often it is important to consider the expansion of a quantum state ) in terms of physically meaningful basis states. For example, molecular orbitals can be expressed as linear combinations of atomic orbitals, or vibrational states can be expressed as super positions of local or normal mode eigenstates. In such expansions, it then becomes desirable to determine how much character a quantum state has in one of these basis states. One way of accimplishing this task is to calculate the projected probability of |) on basis state |j). In this paper, we consider this general quantum mechanical problem. If the basis states are orthonormal, then the projected probability of|) on |j) is of course |<|j>|². However, if the basis states are not orthogonal, then this result is no longer valid and one must develop a more general theory to calculate these projected probabilities. An earlier paper used one-dimensional projection operators to initiate this theory and gave closed form results for the case of two non-orthogonal basis states [1]. One- and many-dimensional projection operators, together with linear algebraic techniques, are used to extend this theory to the n non-orthogonal basis state case. Explicit closed form results are given for the two- and three-state cases, and a general algorithm is developed for the case of four or more basis states. Application of the theory is made to atomic populations in three- to six-atom molecules, and comparisons are made to the related work of Mulliken.     

The kinetic energy components of LCAO-molecular orbitals: A comment on the “Electron-in-a-Box”-“LCAO-MO-Model” analogy

In many textbooks attention is drawn to the close analogy that seems to exist between the Electron-in-a-Box-wave functions _n and their LCAO-MO counterparts _J (J = n) for the movement of an electron in a -system. It is often implied that the wave lengths of _n and of _J (J = n) which satisfy to a high degree the relation =, have the same physical meaning. It is shown that this is not the case. for a linear system (e.g. a one-dimensional Electron-in-a-Box-model) is directly connected with the momentum of the electron and therefore with its kinetic energy according to the deBroglie relation. However, there is no such simple relationship between A and the corresponding kinetic energy component in LCAO-MO's _J . (The necessary two-center kinetic energy integrals have been computed for 1s-type atomic orbitals.)

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Zusammenfassung In vielen elementaren Textbüchern wird die Aufmerksamkeit auf die scheinbar enge Verwandtschaft hingelenkt, die zwischen den Wellenfunktionen _n für ein Electron-in-a-Box-Modell und den entsprechenden LCAO-MOs _J (J=n) für die Bewegung eines Elektrons in einem -System besteht. Unter anderem wird oft implizit angenommen, daß die Wellenlängen der Funktion _n und von _J (J=n), die weitgehend der Bedingung = genügen, die gleiche physikalische Bedeutung haben. In dieser Arbeit wird gezeigt, daß dies nicht der Fall ist. Für ein lineares System (z. B. ein eindimensionales Electron-in-a-Box-Modell) ist über die deBroglie'sche Beziehung direkt mit dem Impuls und damit mit der kinetischen Energie des Elektrons verknüpft. Im Gegensatz dazu existiert keine einfache Beziehung zwischen und der entsprechenden Komponenten der kinetischen Energie in einem LCAO-MO _J . (Die notwendigen Zweizentrenintegrale der kinetischen Energie wurden für Atomorbitale vom 1s-Typus berechnet.)

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Résumé Dans les textes élémentaires de chimie théorique on attire souvent l'attention sur l'analogie qui semble exister entre les fonctions d'onde _n pour un modèle «Electron-in-a-Box» et les fonctions correspondantes LCAO-MO _J (J=n) décrivant le mouvement d'un électron dans un système . En particulier cette comparaison implique que les «longueurs d'onde» de _n et de _J (J=n), qui satisfont pratiquement la relation =, ont la même signification physique. Dans ce travail on montre, que ceci n'est pas le cas. Pour un système linéaire (c.à.d. un modèle linéaire du type «Electron-in-a-Box») est reliée directement à la quantité de mouvement et par là à l'énergie cinétique, par la relation de deBroglie. Par contre on ne trouve pas une dépendance analogue entre et la composante correspondante de l'énergie cinétique dans une orbitale moléculaire LCAO _J. (Les intégrales bicentriques pour les composantes d'énergie cinétique nécessaires à ce calcul ont été déterminées pour des orbitales atomiques du type 1s.)

     

Effect of the crystalline environment upon the rotational conformation about the N-C and C-C′ bonds (Φ andΨ) in amides and peptides

SCFab initio and PCILO computations indicate the intrinsic preference of the torsion angle for 60 °. In the crystal structure of simple methylamides and peptides the observed values for this torsion angle lie between 0 °–30 °. Different procedures for computing lattice energies and total crystallographic conformational energies (lattice + torsional) utilized by other authors, failed to account for this situation. We show that the procedure developed recently in our laboratory for computing lattice energies indicates that the minima of these energies for in acetamide and N-methylacetamide correspond well to 0 °–30 °. Because of the low value of the barrier of the torsional potential for this angle, the total crystallographic conformational energy corresponds also to = 0 °–30 ° in agreement with the experimental data.     

Theory of bridge-assisted electron transfer in systems with an intermediate link at high temperatures

From solution of the time-dependent wave equation by specifying the (t) function in the form of a linear combination a_i(t)_i+a_d(t)_d+a_j(t)_j (_i, _d, _j are the wave functions corresponding to localization of the electron on the donor, the intermediate link, and the acceptor), we obtain an expression for the electron transfer probability in a system consisting of six components: one direct transfer and five interference components. We have studied the effect of electronic structure and vibrational motion of the components of the system on the probability components. This has allowed us to find the dependence of the electron transfer probability on the ionization potential or the electron affinity of the intermediate link, playing the role of a catalyst or inhibitor of the process.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 25, No. 2, pp. 134–142, March–April, 1989.     

Leitfähigkeitsmessungen in Phosphoroxychlorid, 3. Mitt. Triäthylamin

Zusammenfassung Am Beispiel des Triäthylamins wird das Leitfähigkeitsverhalten eines gekoppelten Dissoziationsgleichgewichtes mit nichtleitender Zwischenstufe (ABC⁺+D^–) untersucht.Mit 3 AbbildungenZugleich 14. Mitteilung der Reihe: Das Solvosystem Phosphoroxychlorid.2. Mitt:M. Baaz undV. Gutmann, Mh. Chem.90, 256 (1959).     

Komplexbildung im System Ni2+—o-Methylbenzamidoxim

Zusammenfassung Die Komplexe des Ni²⁺ mit o-Methylbenzamidoxim wurden in neutraler und in alkalischer Lösung spektrophotometrisch untersucht. Die Bildungskonstanten sindK ₁=40 für 11 undK ₂=1,7·10² für 12 in neutraler Lösung und ₁ = =(3,92 ±0,2) · 10⁴für 11 und lgK = lg ₁ + lg ₂ = 3,45 ± ±0,15 für 12 bei 25° und =1 in alkalischer Lösung.

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Complex formation in the systemeNi ²⁺—o-methylbenzamide oxime

The complexes of Ni²⁺ with o-methylbenzamide oxime were investigated spectrophotometrically in neutral as well as in alkaline solution. The formation constants areK ₁=40 for 11 andK ₂=1.7·10² for 12 in the neutral solution and ₁ = =(3.92 ±0.2) · 10⁴ for 11 and lgK = lg ₁ + lg ₂ = 3.45 ± ±0.15 for 12 at 25° and =1 for the alkaline solution. *** DIRECT SUPPORT *** A3615139 00007

     

The role of excited states in the photolysis of n-butyraldehyde, II. Intersystem crossing quantum yields

Singlet triplet intersystem crossing quantum yields were determined for n-butyraldehyde, using the photosensitized cis-trans isomerization of piperylene to monitor the process. At room temperature and 313 nm, 0.58±0.05 and 0.72±0.05 were obtained for the triplet yield in the vapor phase and in isooctane, respectively.

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- - - . 0,58±0,05 0,72±0,05 313 .

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For Part I, see Ref. /1/     

Comparative selectivities of two copper-based catalysts: Mixed Cu?Cr and Cu?Al oxides for allylic alcohol reactions in the presence of molecular hydrogen

In the presence of mixed copper-chromium (aluminium) oxide, allylic alcohols react with molecular hydrogen and lead to several primary products. This is due to the simultaneous presence of two active sites in the mixed oxides. Copper species (Cu⁺) are responsible for hydrogenation (HYD) and the chromium (Cr³⁺) (aluminium [Al³⁺]) species for the isomerization (I) and the hydrodeoxygenation (HDO) reactions. However, the stronger acidic character of Al³⁺, compared with Cr³⁺, entails some differences evidenced by the HYD/(I+HDO) and HDO/I ratios.

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- () , . . (Cu⁺ (), (Cr³⁺) ( Al³⁺) (), (). , Al³⁺ Cr³⁺ , /(+) /.

     

Critical discussion of simple adsorption methods used to evaluate the micropore size distribution

The well-known simple adsorption methods used to evaluate the micropore size distribution from low pressure adsorption isotherms were examined by employing model isotherms for slit-like graphite micropores obtained from nonlocal density functional theory. It was shown that in the range of pore sizes from about 0.4 to 0.9 nm, the Horvath Kawazoe (HK) method satisfactorily reproduces the shape of the micropore size distribution, but the pore sizes are underestimated. In the case of micropores wider than 0.9 nm, the method fails as the formation of the monolayer on the pore walls produces a peak corresponding to 0.6 nm micropores on the HK pore size distribution. Therefore, the HK method indicates the presence of microporosity even for nonporous samples. The Dubinin-Astakhov adsorption isotherms were also examined and it was shown that their application to represent local adsorption isotherms for homogeneous pores is questionable. However, the adsorption potential distributions seem to be promising for micropore analysis.Nomenclature A Adsorption potential kJ/mol - C ₁ Constant in Eq. 3 and 4 kJ * nm/mol - C ₂ Constant in Eq. 3 and 4 nm³ - C ₃ Constant in Eq. 3 and 4 nm⁹ - C ₄ Constant in Eq. 3 and 4 - d Adsorbate molecule diameter nm - d _A Adsorbent atom diameter nm - G Change in the Gibbs free energy kJ/mol - J Pore size distribution cm³/(g*nm) - R The universal gas constant = 8.31431 J/(mol * K) - T Absolute temperature K - V Amount adsorbed expressed in cm³ of liquid adsorbate per 1 g of the adsorbent = 0.0015468 * amount adsorbed expressed in cm³ STP/g cm³/g - x Pore width nm - X Differential adsorption potential distribution cm³ * mol/(g*kJ) - Constant defined as nm - p Pressure Pa - p ₀ Saturated pressure = 760 torr = 101325 Pa Pa - P _c Condensation pressure Pa - Degree of pore filling - S _BET BET specific surface area m²/g - S _ex External surface area obtained fromt-plot method m²/g - V _mi Micropore volume obtained fromt-plot method cm³/g - V ₁ Total pore volume cm³/g - E Characteristic energy in the Dubinin-Astakhov equation kJ/mol - n Exponent in the Dubinin-Astakhov equation      

Thermal and some other properties of the complexes crystallizing from the system Ni(II)-en-[Ni(CN)4]2−-H2O

Seven complex compounds exhibiting the compositions Ni(en)₃Ni(CN)₄·H₂O (I), Ni(en)₃Ni(CN)₄ (II),-Ni(en)₂Ni(CN)₄ (III), Ni(en)Ni(CN)₄·2H₂O (IV), Ni(en)Ni(CN)₄ (V), Ni(en)₂Ni(CN)₄ · 2.5H₂O (VI) and-Ni(en)₂Ni(CN)₄ (VII) were prepared from the system Ni-en-[Ni(CN)₄]^2–-H₂O. These compounds were examined by the methods of infrared spectroscopy, X-ray powder diffractometry, UV-VIS reflectance spectroscopy, and also by the measurement of magnetic moments. The thermal stability, the stoichiometry of thermal decomposition and the mutual transformations were investigated with a derivatograph. The reactions proceeding according to the following schemes were observed if the system was heated to appropriate temperature: (I)(II)(III)(V)(IV) and (VI)(VII)(III)(V)(IV) Process (VII)(III) represents isomerization. The reversibility of the process (V)(IV) is due to the high hygroscopicity of the anhydrous complex. The changes in structure in the course of the individual processes are discussed.

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Zusammenfassung Aus einem System Ni-en-[Ni(CN)₄]^2–-H₂O wurden sieben Komplexe der Formeln Ni(en)₃Ni(CN)₄·H₂O (I), Ni(en)₃Ni(CN)₄ (II),-Ni(en)₂Ni(CN)₄ (III), Ni(en)Ni(CN)₄·2H₂O (IV), Ni(en)Ni(CN)₄ (V), Ni(en)₂Ni(CN)₄ · 2.5H₂O (VI) und-Ni(en)₂Ni(CN)₄ (VII) hergestellt. Diese Verbindungen wurden mittels IR-Spektroskopie, Röntgenpulverdiffraktometrie, UV-Reflexionsspektroskopie und durch Messungen des magnetischen Momentes untersucht. Die Wärmestabilität, die Stöchiometrie des thermischen Zerfalles und die gegenseitigen Umwandlungen wurden mittels eines Derivatographen untersucht. Wird das System auf geeignete Temperaturen erhitzt, kann der Reaktionsverlauf durch folgendes Schema dargestellt werden: (I)(II)(III)(V)(IV) und (VI)(VII)(III)(V)(IV).Der Prozeß (VII)(III) verkörpert eine Isomerisierung. Die Umkehrbarkeit von Prozeß (V)(IV) ist auf die ausgeprägten Hygroskopieeigenschaften des wasserfreien Komplexes zurückzuführen. Es werden die im Ablaufe der einzelnen Prozesse vorgehenden Strukturveränderungen besprochen.

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Ni- -[No(N)]² -₂ Ni(en)₃Ni(CN)₄ · ₂ (I), Ni(en)₃Ni(CN)₄ (II),-Ni(en)₂Ni(CN)₄ (III), Ni(en)Ni(CN)₄-2H₂O (IV), Ni(en)Ni(CN)₄ (V), Ni(en)₂Ni(CN)₄ · 2,5H₂O (VI) -Ni(en)₂Ni(CN)₄ (VII). , , - , . , . (I)(II)(III)(V)(IV) (VI)(VII)(III)(V)(IV). (VII)(III) . (V)(IV) . .

     

IR investigation of NH4Y zeolite samples poisoned with sodium vapor

The interaction between sodium atoms (obtained by thermal decomposition of NaN₃) and acidic OH groups of the zeolite framework has been investigated by IR spectroscopy. Reduction in band intensities could be observed, its extent depending on the amount of NaN₃ and temperature. Impregnated samples showed displacement of the N=NN IR band characteristic of salt occlusion.

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, NaN₃, OH . e, NaN₃ . N=NN^–, .

     

Stabilized zeolites of Y-type. Their adsorption and catalytic properties

The stabilized decationized form of a Y-type zeolite has been prepared by the self-steaming calcination method. The temperature of the stabilization process influenced the nature of the interaction with ethylene.

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Y . .

     

Generation of model reactions leading to limit cycle behavior

The system of differential equations by Feistel and Ebeling has been generalized. Some new formal kinetic reactions with two internal components, which may exhibit limit cycle behavior have been studied. Based upon the numerical integration of the deterministic models of these reactions the oscillatory character of the systems has been confirmed.

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. , . .

     

Glass transition and crystallization effects demonstrated by thermal analysis of liquid-quenched ternary glasses Te80Ge20−xA x V (AV=Sb,Bi)

In DSC studies of liquid-quenched ternary chalcogenide glasses Te₈₀Ge_20–xA _x ^V (A^V=Sb, Bi), the characteristic temperatures (glass transition and crystallization temperatures) were determined. Changes in the thermal stabilities of these glasses, depending on the element A (Sb, Bi) from group V of the periodic table and on its content in the alloy were evaluated. Moreover, the effect of changes in the glass composition on the glass formation ability expressed by the parameter K_g1 was determined.

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Zusammenfassung In DSC-Untersuchungen abgeschreckter ternärer Chalkogenidgläser des Typs Te₈₀Ge_20–xA _x ^v (A^x=Sb, Bi) wurden die charakteristischen Temperaturen (die GlasÜbergangs- und Kristallisationstemperaturen) bestimmt. Die Änderungen der Thermostabilität dieser Gläser wurden in Abhängigkeit von dem Element A (=Sb, Bi) aus der V Gruppe der Periodensystems und von seinem Gehalt in der Legierung ausgewertet. Ausserdem wurde der Einfluß der Änderungen in der Glaszusammensetzung auf die Glasbildungsfähigkeit, ausgedrückt durch den Parameter K_g1, bestimmt.

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Résumé On a déterminé par analyse calorimétrique différentielle (DSC) les températures caractéristiques (températures de transition vitreuse et de cristallisation), des verres ternaires à chalcogénures formés par trempe à partir du liquide, du type Te₈₀Ge_20–xA _x ^v (A^v=Sb, Bi). On a évalué la variation de la stabilité thermique de ces verres en fonction de l'élément A (=Sb, Bi) du Vème groupe du tableau périodique et de sa teneur dans l'alliage. De plus, on a déterminé l'effet des variations de la composition du verre sur la capacité de formation du verre qui s'exprime par le paramètreK _g1.

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Te₈₀Ge_20–xA _x ^v (A^v=Sb, Bi), , . A(=Sb,Bi) . , K _gl, .

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Research supported by the U. S. National Science Foundation under Grant No. GF 421 76     

Sintering of supported palladium

Sintering results of Pd supported on carbon black at 723–973 K, in water vapor, in H₂ and under high vacuum are reported. They are compared with sintering data of Pd on other supports. The observed sequences are: SiO₂C>sepioliteAl₂O₃>AlPO₄ and H₂O>high vacuum>H₂.

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Pd 723–973 , , H₂ . Pd . : SiO₂Al₂O₃>AlPO₄ H₂O> >H₂.

     

On the dioximine complexes of transition metals

Twelve new complexes NH₄[Co(DH)₂(SO₃)(amine)]·t H₂O (DH ₂=dimethylglyoxime) have been synthesized and characterized. Their i.r. spectra show the SO₃ to be co-ordinated to the Co atom through the S atom. The thermal decompositions of a series of derivatives of this type have been investigated with a derivatograph. The first process is an endothermic dehydration reaction, occurring in a single stage or in two successive ones. The loss of the crystallization water is followed by another endothermic reaction, without a clear stoichiometry, which is referred to as deamination. At higher temperatures, exothermic pyrolysis processes occur. From the TG curves, kinetic parameters have been derived for the dehydration and deamination reactions.

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Zusammenfassung Zwölf neue Komplexe der allgemeinen Formel NH₄[Co(DH)₂(SO₃)(amin)]·t H₂O (DH=Dimethylglyoxim) wurden synthetisiert und charakterisiert. Die IR-Spektren zeigen, daß SO₃ über das S-Atom koordinativ an das Co-Atom gebunden ist. Die thermische Zersetzung einer Reihe von Derivaten dieses Typs wurde mittels eines Derivatographen untersucht. Der erste Prozeß ist die in einem einzigen oder in zwei aufeinander folgenden Schritten verlaufende endotherme Dehydratisierung. Auf die Abgabe des Kristallwassers folgt eine andere endotherme Reaktion unklarer Stöchiometrie, die als Deaminierung bezeichnet wird. Bei höheren Temperaturen verlaufen exotherme Pyrolyseprozesse. Aus den TG-Kurven wurden kinetische Parameter für die Dehydratisierung und die Deaminierung bestimmt.

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NH₄Co(DH)₂(SO₃)(a)]·tH₂O, DH ₂ — . , . . , . «». . .

     

Thermal analysis of thermosetting resins

Applications of thermoanalytical methods for the study of the resinification and curing of phenolic resins are reviewed. Examples are given to illustrate the use of DSC for practical industrial systems such as paper impregnants and saturated papers.

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Zusammenfassung Es wird eine Übersicht über die Anwendungen thermoanalytischer Methoden zur Untersuchung der Harzbildung und der Aushrtung von phenolischen Harzen gegeben. Anhand von Beispielen wird die Verwendung der DSC für praktische industrielle Systeme wie Papierfüllstoffe und gefüllte Papiere aufgezeigt.

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, . , .

     

Thermal properties of 5-(subst)-amino-1,2,3,4-thiatriazoles

The thermal properties of 5-(subst)-amino-1,2,3,4-thiatriazoles were studied using TG and DTA. 5-amino-1,2,3,4-thiatriazole was actively decomposed at the melting temperature. The active thermal decomposition proceeds at a very fast rate, being accompanied by exothermic reactions of recombination. In comparison with this, the different-5-(subst)-amino-1,2,3,4-thiatriazoles are decomposed at slower rates in several degradative steps. The effects of some substituents on the total course of thermal degradation are discussed.

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Zusammenfassung Die thermischen Eigenschaften von 5-(subst)-Amino-1,2,3,4,-Thiatriazolen wurden mittels der Methoden TG und DTA untersucht. Das 5-Amino-1,2,3,4-Thiatriazol wurde bei der Schmelztemperatur aktiv zersetzt. Die aktive thermische Zersetzung verläuft mit einer sehr großen Geschwindigkeit und wird von den exothermen Reaktionen der Rekombination begleitet. Im Vergleich hierzu werden die verschiedenen 5-(subst)-Amino-1,2,3,4-Thiatriazole mit einer geringeren Geschwindigkeit, in mehreren Zersetzungsstufen abgebaut. Die Wirkung einiger Substituenten auf den Gesamtablauf der thermischen Zersetzung wird erörtert.

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Résumé On a étudié par TG et ATD les propriétés thermiques des 5-(subst)-amino-1,2,3,4-thiatriazoles. Le 5-amino-1,2,3,4-thiatriazole se décompose activement à la température de fusion. La décomposition thermique active s'effectue avec une très grande vitesse et s'accompagne de réactions exothermiques de recombinaison. Par contre, les divers 5-(subst)-amino-1,2,3,4-thiatriazoles se décomposent avec une vitesse plus faible et en plusieurs étapes. On discute l'effet de quelques substituants sur le processus complet de la dégradation thermique.

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, , 5-()--1,2,3,4-. 5--1,2,3,4- . , . , 5-()--1,2,3,4- . .