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.)

     

The conformational preferences of γ-lactam and its role in constraining peptide structure

Summary The conformational constraints imposed by -lactams in peptides have been studied using valence force field energy calculations and flexible geometry maps. It has been found that while cyclisation restrains the of the lactam, non-bonded interactions contribute to the constraints on of the lactam. The -lactam also affects the (,) of the residue after it in a peptide sequence. For an l-lactam, the ring geometry restricts to about-120°, and has two minima, the lowest energy around-140° and a higher minimum (5 kcal/mol higher) at 60°, making an l--lactam more favourably accommodated in a near extended conformation than in position 2 of a type II -turn. The energy of the +60° minimum can be lowered substantially until it is more favoured than the-140° minimum by progressive substitution of bulkier groups on the amide N of the l--lactam. The (,) maps of the residue succeeding a -lactam show subtle differences from those of standard N-methylated residues. The dependence of the constraints on the chirality of -lactams and N-substituted -lactams, in terms of the formation of secondary structures like -turns is discussed and the comparison of the theoretical conformations with experimental results is highlighted.     

Analytic evaluation of multicenter integrals for gaussian-type orbitals

This work contains the evaluation of multicenter integrals with Cartesian Gaussian functions occurring in ||||² These integrals have to be used if it is necessary to calculate the lower bounds for eigenvalues with the method of the minimization of the variance [1], Considering the varianceF() = ||H||² - (H!| )², the integrals from (HY, Y) are well known in contrast to those for ||H ||².     

Solution of the perturbed eigenvalue equation by the low-rank perturbation method

In its simplest form, the low-rank perturbation (LRP) method solves the perturbed matrix eigenvalue equationsA (B + V) =, whereA, B andV are nth-order Hermitian matrices, and where the eigenstates and the eigenvalues of the unperturbed matrixB are known. The method can be applied to arbitrary perturbations V, but it is numerically most efficient if the rank ofV is small. A special case of low-rank perturbations are localized perturbations (e.g. replacement of one atom with another, creation and destruction of a chemical bond, local interaction of large molecules, etc.). In the case of local perturbations with a fixed localizability I, the operation count for the calculation of a single eigenvalue and/or a single eigenstate isO(l ² n). In the more general case of a delocalized perturbation with a fixed rank p, the operation count for the derivation of all eigenvalues and/or all eigenstates is O( ² n ²). For largen, the performance of the LRP method is hence at least one order of magnitude better than the performance of other methods. The obtained numerical results demonstrate that the LRP method is numerically reliable, and that the performance of the method is in accord with predicted operation counts.Research supported by the Robert A. Welch Foundation (Houston, Texas), and by the Yugoslav Ministry for Development (Grant P-339).     

Hypervirial theorems and symmetry

It is pointed out that to ensure that an optimal variational wave function having a certain symmetry satisfies the hypervirial theorem forW, it is sufficient thatiSW, whereS is the projector onto the symmetry type in question, be a possible variation of . Application is made to the tensor hypervirial theorem for atoms.     

Crystallization of glasses in the system SiO2-Li2O-TiO2-Al2O3, investigated in situ at high temperature by XRD and DTA methods

The processes of crystallization of fibres (diameter 10–15m) and coarse powders (grain size 500–1000m) with four compositions in the system SiO₂-Li₂O-TiO₂-Al₂O₃ were studied by conventional and in situ high-temperature XRD, DTA, SEM and optical microscopy. Activation energies of crystallization and morphological indices were deduced from the kinetic curves obtained by recording the high-temperature XRD peak intensity as a function of time. The glass-ceramic fibres drawn from compositions which exhibit glass-inglass phase separation show prevailing not-oriented bulk crystallization, whereas prevailing surface crystallization was found for single-phase glass fibres. Homogeneously-dispersed crystallization was obtained on heating fibres of these compositions. The partially cocrystallized glass fibres of eutectic composition between Li-metasilicate and-spodumene gave rise to a very fine and homogeneously-dispersed sub-microstructure.

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Zusammenfassung Die Prozesse der Kristallisation von Fasern (Durchmesser 10–15m) und groben Pulvern (Korngröße 500–1000m) von SiO₂-LiO₂-TiO₂-Al₂O₃ in vier verschiedenen Zusammensetzungen wurden mittels herkömmlicher und in situ XRD, DTA, SEM und optischer Mikroskopie untersucht. Aktivierungsenergien der Kristallisation und morphologische Indices wurden aus den durch Registrierung der Intensität der Hochtemperatur-XRT-Peaks in Abhängigkeit von der Zeit erhaltenen kinetischen Kurven ermittelt. Die aus Gemischen einer die Glas-in-Glas-Phasentrennung bedingenden Zusammensetzung gezogenen Glas-Keramik-Fasern zeigen vorwiegend eine nicht-orientierte Bulk-Kristallisation, während vorherrschend Oberflächenkristallisation bei Einphasen-Glasfasern festgestellt wurde. Eine homogen verteilte Kristallisation wurde beim Erhitzen der Fasern dieser Zusammensetzungen erhalten. Bei partiell ko-kristallisierten Glasfasern einer eutektischen Mischung von Li-Metasilikat und (-Spodumen liegt eine sehr feine und homogen-disperse Submikrostruktur vor.

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( 10–15 ) ( 500–1000 ) SiO₂-Li₂O-iO₂AL₂O₃ , , . , - . , --, ë, - . - . — - - - .

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This work was supported by the Italian C. N. R., Progetto Finalizzato Chimica Fine e Secondaria — Sottoprogetto Metodologie.

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We are indebted to Dr. B. Locardi of Stazione Sperimentale del Vetro, Murano (Venice) for furnishing the initial glasses and fibres.

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We are also indebted to the G. Giacomello Institute (C. N. R., Rome) for furnishing the computer programme for the unit cell parameter refinement. Assistance by M. Pistolato and F. Venuda in the experimental work is gratefully acknowledged.     

Relativistic virial theorem for diatomic molecules. Application to H 2 +

Summary The virial theorem for a molecule in the relativistic clamped-nuclei approximation is derived. The individual energy contributionsA (momentum energy),B (mass energy),T=A+B (kinetic energy) andV (potential energy) are expressed in terms ofE, E/R (derivate w.r.t. the nuclear coordinates) and the relativistic correction E/² (derivative w.r.t. Sommerfield's fine-structure constant ). IfE and E/R are known as functions of , then all individual energy terms are also known as functions of . As an example, numerical results for H ₂ ⁺ are presented. The relativistic and nonrelativistic potential energy curves and the paradoxical behavior of their different contributions are analyzed and interpreted in both the largeR and shortR ranges.Dedicated to Professor W. Kutzelnigg on the occasion of his 60th birthday     

Interaction of a Single State with a Known Infinite System Containing One-Parameter Eigenvalue Band

Interaction of a quantum system S ₁ ^a containing a single state |> with a known infinite-dimensional quantum system S ^b containing an eigenvalue band [ _a , _b ] is considered. A new approach for the treatment of the combined system S =S ₁ ^a S ^b is developed. This system contains embedded eigenstates |()> with continuous eigenvalues [ _a , _b ], and, in addition, it may contain isolated eigenstates | _I > with discrete eigenvalues _I [ _a , _b ]. Exact expressions for the solution of the combined system are derived. In particular, due to the interaction with the system S ^b , eigenvalue E of the state |> shifts and, in addition, if E[ _a , _b ] this shifted eigenvalue broadens. Exact expressions for the eigenvalue shift and for the eigenvalue distribution of the state |> are derived. In the case of the weak coupling this eigenvalue distribution reduces to the standard resonance curve. Also, exact expressions for the time evolution of the state |(t)> that is initially prepared in the state |(0)>|> are obtained. Here again in the case of the weak coupling this time evolution reduces to the familiar exponential decay. The suggested method is exact and it applies to each coupling of the system S ₁ ^a with the system S ^b , however strong. It also presents a relatively good approximation for the interaction of a nondegenerate eigenstate | _s > of an arbitrary system S ^a with an infinite system S ^b containing a single eigenvalue band, provided this eigenstate is relatively well separated from other eigenstates of S ^a and provided the interaction between the systems S ^a and S ^b is not excessively strong.     

Generalized quantum mechanical two-centre problems. IV. On the accuracy of simple LCAO approximations for H 2 + states and the Eckart criterion

If x denotes an exact solution of the quantum mechanical two centre Coulomb problem, we optimize a normalized LCAO approximation by making the overlap S = (x¦) a maximum. In this context we study how a weight factor (r _a r _b )^–1 in the definition of the inner product changes the approximation and the expectation value of electronic energy. Finally we compare the lower bound given by the Eckart criterion with the exact overlap. Results are reported for H ₂ ⁺ states 1s_g and 2p_u.Dedicated to Professor Hermann Hartmann on occasion of his 70th birthday on May 4th, 1984     

Activity coefficients for ternary systems: VI. The system HCl + MgCl2 + H2O at different temperatures; application of Pitzer's equations

Electromotive-force measurements have been made on HCl–MgCl₂–H₂O mixtures at 5, 15, 25, 35 and 45°C at eleven different ionic strengths from 0.1–5.0 mol-kg ^–1 . The results are interpreted in terms of the simple Harned's equations, as well as the more complicated Pitzer ion-component treatment of multicomponent electrolyte mixtures. Activity coefficients for HCl in the salt mixtures obey Harned's rule up to and including I=5.0. For the salt in the acid mixtures, Harned's rule holds true up to and including I=0.5. The contribution of higher-order electrostatic terms (E and E') in the Pitzer equations is important for accurate evaluations of unlike cation-cation interactions (_H,Mg), and cation-anion-cation interactions (_H,Mg,Cl). The values of^S_H,Mg and _H,Mg,Cl (determined with E and E'), _H,Mg and _H,Mg,Cl (determined without E and E'), as well as the trace activity coefficients of HCl, _tr ^A , in solutions of MgCl₂ (where ionic strength fraction of the salt,y _B = 1) at all the experimental temperatures and ionic strengths, are reported. Results of this study are compared with those for similar systems. At I=0.1 and 25°C, the results of the Brönsted-Guggenheim specific interaction theory are discussed briefly.     

Single-Ion Contributions to Activity Coefficient Derivatives, Second Moment Coefficients, and the Liquid Junction Potential

We discuss several interrelated single-ion thermodynamic properties required to calculate the liquid junction potential between two solutions of the same binary electrolyte. According to a previously reported molecular theory of nonuniform electrolyte solutions in nonequilibrium, is determined by the transport numbers of the ions, and by the second moment coefficients H ⁽²⁾ of the charge densities around the ions. The latter may be viewed as the single-ion contributors to the second moment condition of Stillinger and Lovett. For a solution of a single binary electrolyte, we relate the H ⁽²⁾ (R) to the derivatives of the single-ion activity coefficients with respect to the ionic strength. In the light of these results, we examine, in some detail, the role played by the specific short-range interionic interactions in determining . We investigate this matter by means of integral equation calculations for realistic models of LiCl and NaCl aqueous solutions in the 0–1 mol-dm^–3 range. In addition to the hypernetted-chain (HNC) relation, we perform calculations under a new integral equation closure that is a hybrid between the HNC and Percus–Yevick closures. Like the HNC approximation, the new closure satisfies the Stillinger and Lovett condition. However, for the models considered in this study, the two closures predict different dependence of the H ⁽²⁾ and of on the specific part of the interionic interactions.     

Band structure of even and ions of odd polyenes

The analysis of experimental data for singlet transitions (E _n) of even polyenes (I), cations (II) and anions (III) of odd polyenes show that for infinite chains E (I)/E (II)=E (I)/E (III) = 2:1. It is shown that the energy gap is equal for the three systems. In cases (II) and (III) there is a level (NBMO) in the gap which is vacant in (II) and occupied in (III). That is why the first optical transition in (II) and (III) depends on the semiwidth of the gap.     

On the Interaction of Two Finite-Dimensional Quantum Systems

Interaction of quantum system S ^a described by the generalised × eigenvalue equation A| _s =E _s S ^a | _s (s=1,...,) with quantum system S ^b described by the generalised n×n eigenvalue equation B| _i = _i S ^b | _i (i=1,...,n) is considered. With the system S ^a is associated -dimensional space X ^a and with the system S ^b is associated an n-dimensional space X _n ^b that is orthogonal to X ^a . Combined system S is described by the generalised (+n)×(+n) eigenvalue equation [A+B+V]| _k = _k [S ^a +S ^b +P]| _k (k=1,...,n+) where operators V and P represent interaction between those two systems. All operators are Hermitian, while operators S ^a ,S ^b and S=S ^a +S ^b +P are, in addition, positive definite. It is shown that each eigenvalue _{k i} of the combined system is the eigenvalue of the × eigenvalue equation . Operator in this equation is expressed in terms of the eigenvalues _i of the system S ^b and in terms of matrix elements _s |V| _i and _s |P| _i where vectors | _s form a base in X ^a . Eigenstate | _k ^a of this equation is the projection of the eigenstate | _k of the combined system on the space X ^a . Projection | _k ^b of | _k on the space X _n ^b is given by | _k ^b =( _k S ^b –B)^–1(V– _k P})| _k ^a where ( _k S ^b –B)^–1 is inverse of ( _k S ^b –B) in X _n ^b . Hence, if the solution to the system S ^b is known, one can obtain all eigenvalues _{k i} } and all the corresponding eigenstates | _k of the combined system as a solution of the above × eigenvalue equation that refers to the system S ^a alone. Slightly more complicated expressions are obtained for the eigenvalues _{k i} } and the corresponding eigenstates, provided such eigenvalues and eigenstates exist.     

Thermal membrane potential across cation-exchange membranes for various halide solutions

Thermal membrane potential across cation-exchange membranes was measured for various halide solutions. Linear relationships between thermal membrane potential () and temperature difference (T) were observed, and the temperature coefficient of thermal membrane potential (/T) decreases with increase in the molality of the external solution as predicted by a theory reported. When counterions of membranes are hydrogen ions, the sign of the coefficients turns into minus near 0.01 mol/kg of the external solution for all membranes, although the coefficients are always positive for all the other forms. The dependence of the coefficients on the molality of the external solutions sometimes deviated slightly from the theory. This deviation is attributed to some change in water content of the membranes. A reproducible method of measuring water content of membranes was applied to discuss the change in the state of the membranes.     

Studies of the energy spectrum of α, ω- substituted polymethine chains. II. Polymethine chains with odd number of methine groups

The analysis of the experimental data for the energy of the longest wavelength optical transitions _n,opt of substituted polymethines X (CH)_2n+1 X shows that in the asymptotic case (n) _,opt does not tend to zero, as it follows from the empirically established correlations, but has a finite, non-zero value. It is shown that the energy gap of odd polymethines is the same as that of the even polymethines - the polyenes (E 2 eV). The substituents (X N, O, B) are responsible for the appearance of levels in the gap. These, depending on the substituent character, are vacant (X B) or occupied (X N, O). The transition from or to such a level determines the longest wavelength optical transition energy of polymethines.     

A new improved expression for gamma-ray detection efficiency of Ge(Li) detectors

In the present paper, a new improved expression for -ray detection efficiency of Ge(Li) detectors, ⁰ , is given. It is represented as a continuous function of x (viz. E ^–1) with a maximum and decreases very rapidly to a small positive value as -ray energy, E, drops to 40 keV or lower, but slowly as E rises to 1.7 MeV or higher. Since it can well represent the whole physical process of the -ray detection, this expression may be one of the simplest and most precise representations, for ⁰ at the present time.     

Vibrational isotope effect by the low rank perturbation method

Mathematical formalism of the low rank perturbation method (LRP) is applied to the vibrational isotope effect in the harmonic approximation. A pair of two n-atom isotopic molecules A and B which are identical except for isotopic substitutions at atomic sites is considered. Relations which express vibrational frequencies _k and normal modes _k of the perturbed isotopic molecule B in terms of the vibrational frequencies _i and normal modes _i of the unperturbed molecule A are derived. In these relations complete specification of the unperturbed normal modes _i is not required. Only amplitudes | _i of normal modes _i at sites affected by the isotopic substitution are needed.     

Solution of the generalized eigenvalue equation perturbed by a generalized low rank perturbation

Summary The problem of finding eigenvalues and eigenstates of the generalized perturbed eigenvalue equation = g3(+) is considered. The eigenvalues and the eigenstates of the unperturbed eigenvalue equation = are assumed to be known. Matrices , and can be arbitrary, except for the requirement that be nonsingular and that the eigenstates of the unperturbed equation be complete. It is shown that the eigenvalues and the eigenstates of the perturbed equation can be easily obtained if the rank of the generalized perturbation , is small. A special case of low rank perturbations are piecewise local perturbations which are common in physics and chemistry. If the perturbation is piecewise local with fixed localizability, the operation count for the derivation of a single eigenvalue and/or a single eigenstate is (n). If the perturbation has a fixed rank, the operation count for the derivation of all eigenvalues and/or all eigenstates is (n ²).Research supported by the Welch Foundation of Houston, Texas, and by the Yugoslav Ministry for Development (Grant P-96)     

Kinetics and mechanism of the hydrolysis of sodium carboxymethylcellulose (Na-CMC) by a cellulase complex

TheSomogyi-Nelson colorimetric method is applied in a new manner more suitable for evaluating the kinetics of the enzyme hydrolysis of sodium carboxymethylcellulose (Na-CMC) catalyzed by the cellulase complex. By means of selective inhibition of a chosen enzyme from the cellulase complex it became possible to trace the effect of the other enzymes included in its composition.

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Kinetik und Mechanismus der Hydrolyse von Natriumcarboxymethylcellulose (Na-CMC) durch einen Cellulase-Komplex

Zusammenfassung Die kolorimetrische Methode nachSomogyi undNelson wird nach einem neuen Verfahren zur Verfolgung der Kinetik der hydrolytischen Spaltung von Natriumcarboxymethylcellulose (Na-CMC), katalysiert durch den Cellulase-Komplex, angewandt. Durch selektive Inhibierung eines bestimmten Enzyms des Cellulase-Komplexes kann man die Wirkung der anderen zu seiner gesamten Zusammensetzung gehörenden Enzyme verfolgen.

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Symbols Used E enzyme (E—cellulase;E—exo-cellobiohydrolase;E—-glucosidase) - [E] _w weight concentration of enzymeE - S substrate (Na-CMC—sodium carboxymethylcellulose) - [S]₀ weight concentration of substrateS - I inhibitor (I—lactose;I—calcium chloride;I—condurrite-B-epoxide) - P product (P—oligosaccharides;P—cellobiose;P—D-glucose) - P end product (K , K , K ) - DP degree of polymerization - DS degree of substitution - ES enzyme-substrate complex (E S, E S, E S) - EP enzyme-product complex (E P, E P) - EI enzyme-inhibitor complex (E I, E I, E I) - M _s molecular mass of substrateS - K _s substrate constant (K _s , K _s , K _s ) - K _I inhibitor constant (K _I , K _I , K _I ) - K _m Michaelis-Menten constant - k ₊₁,k ₊₂ (k ₊₂ ,k ₊₂ ,k ₊₂ ) forward rate constants - k _–1 reverse rate constant - ₀ initial rate of reaction - V maximal reaction rate - A change in absorbance - molar absorption coefficient - wavelength Herrn Prof. Dr.Hans Tuppy zum 60. Geburtstag herzlichst gewidmet.     

Augmentation of the Generalised n×n Eigenvalue Equation to a Generalised (n+1)×(n+1) Eigenvalue Equation

Generalised n×n eigenvalue equation B| _i = _i S ^b | _i (i=1,...,n) where B and S ^b are n×n Hermitian matrices while S ^b is in addition positive definite is considered. This equation is augmented to a generalised (n+1)(n+1) eigenvalue equation H| _k = _k S| _k (k=1,...,n+1) where Hermitian matrices H and S represent matrices B and S ^b , respectively, augmented by one additional row and one additional column. It is shown how the eigenvalues _k and the eigenvectors | _k of the augmented eigenvalue equation can be expressed in terms of the eigenvalues _i and the eigenvectors | _i of the original eigenvalue equation. Operation count to obtain by this method all augmented eigenvalues and eigenvectors is of the order O(n ²). Unless matrices involved are of some special kind such as sparse matrices or alike, this operation count is one order of magnitude smaller than operation count required by other presently known methods. In many practical cases operation count to obtain a single selected eigenvalue and/or eigenvector by this method is of the order O(n). In the case of the generalised eigenvalue equation, all other methods usually require again O(n ³) operations, even if only a single eigenvalue and/or eigenvector is required. Thus in many cases of interest operation count to obtain a selected eigenvalue and/or eigenvector by this method is two orders of magnitude smaller than operation count required by other methods.