Relativistic virial theorem for atoms

A relativistic virial theorem is derived for atoms in a general manner. The virial ratio consists of the usual V/T term and a correction term W/T, where T, V, and W are the kinetic energy, the potential energy, and correction terms, respectively. Explicit forms of W are presented for four specific nuclear potential models. Numerical calculations for a uniform nuclear charge model show that the magnitude of the correction term W/T increases with increasing atomic numbers and that it modifies the ratio V/T considerably for atoms with large atomic numbers in particular. Received: 21 November 2000 / Accepted: 8 January 2001 / Published online: 3 April 2001     

Electron correlations in Kohn–Sham exchange‐only theory

We provide an interpretation for the “exchange” energy and potential of Kohn–Sham exchange‐only theory, or equivalently that of the optimized potential method (OPM), which shows that in addition to contribution due to the Pauli exclusion principle, there is a kinetic component to these properties. The interpretation is in terms of a conservative field R ^OPM( r ), which is a sum of two fields, one representative of Pauli electron correlations and the other of kinetic effects. The OPM exchange potential is derived via the differential virial theorem to be the work done to move an electron in the field R ^OPM( r ). The OPM exchange energy is then expressed via the integral virial theorem in terms of this field. A similar interpretation for the energy and potential may also be derived directly from the OPM integral equation. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 71:473–480, 1999     

Nonadiabatic core polarization and penetration correction in alkali-like atoms: Model computations on excited states of helium

Correlation between the motion of a highly excited outer electron and that of the remaining ionic “core” of an atom is generally treated in an adiabatic approximation, in which it is assumed that the outer electron affects the core in the same way as a stationary point charge. An alternative approach to this correlation problem which avoids the adiabatic approximation is tested here on the 1s2p, 1s3d, and 1s4f states of helium. The results provide the first accurate test of the adiabatic approximation and of a simple correction for the nonzero velocity of the outer electron. The approach used here is based on neglect, in the “correlation” part of the wave function, of the possibility that the outer electron comes closer to the nucleus than any core electron (“penetration”). A correction for this neglect is derived and tested on a version of the adiabatic approximation that likewise neglects penetration.     

Ionic dehydration reactions: The effect of location of the activated complex on kinetic energy release

A broad range of kinetic energy release has been found for the 1,4 elimination of water in a series of ortho substituted benzyl alcohols and benzoic acids. It is suggested that the trend reflects, in part, the position of the activated complex on the reaction coordinate. More specifically, reactions which proceed via an ‘early’ transition state release only small quantities of energy, whereas those processes occurring ‘later’ release a larger fraction of the available energy. Consequently, the o- methyl derivatives give large kinetic energy release compared to o- amino or hydroxy substituted compounds. Father verification is obtained from the small kinetic energy release observed for the 1,4 ionic dehydration in simple,acyclic alcohols.     

Asymptotic behavior of the primitive function of different “symmetry-adapted” perturbation schemes for the H ground state

The leading terms in the asymptotic 1/R expansion of the wave functions and energies of various “symmetry-adapted” perturbation schemes for intermolecular forces, as well as for the “polarization approximation” (PA ) are derived for the H ground state, both exactly (i.e., to infinite order in λ) and in the first two orders of the λ expansion. It is pointed out that only in the PA and the Hirschfelder–Silbey scheme is the formal primitive function Φ genuinely primitive, whereas Φ in the other schemes is asymmetric in a rather strange way. In this lack of genuine primitivity lies the reason why in these schemes the leading term of the 1/R expansion is only recovered to infinite order in λ and requires knowledge of the R^?4 term of the wave function, provided that one uses the “internal” energy expression. Four different energy expressions are compared that behave differently in the different schemes. The results obtained here are basis independent, but the implications of the basis completeness problem are discussed as well.     

Note on the atomic correlation energy

In the introductory section, we compare the total, kinetic, nuclear-electron, Coulomb, exchange, and correlation energies of ground-state atoms. From the analyses of the data, one can conclude that the Hartree-Fock (HF) model is notably good and might require only a small perturbation to become essentially an “accurate” model. For this reason and considering past literature, we present a semiempirical extension of the HF model. We start with a calibration of three independent models, each one with an effective Hamiltonian, which introduces a small perturbation on the kinetic, the nuclear-electron, or the Coulomb HF operators. The perturbations are expressed as very simple functions of products of orbital probability density. The three perturbations yield very equivalent results and the computed ground-state energies are reasonably near to the accurate nonrelativistic energies recently provided by E. Davidson and his collaborators for the 2–18 electron systems and the estimates by Clementi and his collaborators for the 19–54 electron systems. The first ionization potentials from He to Cs, the second ionization potentials from Li to Zn, and excitation energies for npⁿ, 3dⁿ, and 4s¹3dⁿ configurations are used as additional verification and validation. The above three effective Hamiltonians are then combined in order to redistribute the correlation energy correction in a way which exactly satisfies the virial theorem and maintains the HF energy ratios between kinetic, nuclear-electron, and electron-electron interaction energies; the resulting effective Hamiltonian, named “virial constrained,” yields good quality data comparable to those obtained from the three independent effective operators. Concerning excitation energies, these effective Hamiltonians yield values only in modest agreement with experimental data, even if definitively superior to HF computations. To further improve the computed excitation energies, we applied an empirical scaling in the vector coupling coefficient; this correction yields very reasonable excitations for all the configurations that we have considered. We conclude that the use of effective potentials to introduce small perturbations density-dependent onto the HF model constitutes a broad class of practical and reliable semiempirical solutions to atomic many-electron problems, can provide an alternative to popular proposals from density functional theory, and should prepare the ground for “generalized HF models.” © 1997 John Wiley & Sons, Inc. Int J Quant Chem 62: 571–591, 1997     

Effect of pressure on free radical copolymerization kinetics. II. Novel methods of measuring monomer reactivity ratios under high-pressure conditions

Two new techniques for the determination of monomer reactivity ratios in copolymerization under high-pressure conditions have been developed, viz., the “sandwich” and the “quenching” method. Both methods are based on repeated quantitative gas chromatographic analysis of the reaction mixture during the low-pressure stages preceding and succeeding the high-pressure stage, of which the kinetics is under investigation. Application of the “sandwich” method implies the occurrence of reaction during both low-pressure stages and consequently the low-pressure kinetic data are required to obtain the transition points of low to high pressure and vice versa. These points constitute the initial and final conditions of the relevant high-pressure reaction. On the contrary, in the “quenching” method no reaction occurs during the low-pressure stages, owing to the lower temperature and the high activation energy of the initiator decomposition. As a consequence, the initial and final conditions of the high-pressure stage can be determined by a simple averaging procedure. Both methods have been tested for the ethylene—vinyl acetate copolymerization at 62°C and 600 kg/cm² with tert-butyl alcohol as solvent, and appear to lead to almost identical monomer reactivity ratios, although the “quenching” method is slightly preferred in case of copolymerization reactions. Both methods are particularly valuable when one of the reactants is gaseous or the reaction produces a gas. Further merits and drawbacks of both methods are discussed.     

Kinetics of the thermal decomposition of CaCo3 in Co2 and some observations on the kinetic compensation effect

The kinetics of the thermal decomposition of CaCO₃ in CO₂ were investigated using both dynamic and isothermal techniques. Values of apparent activation energies range from 200–1000 kcal mol^?1 depending upon sample size and heating rate. It is concluded that thermal transport rather than mass transport or chemical processes is rate determining. The results are compared with earlier work in O₂ and discussed in terms of “the kinetic compensation effect”, i.e., the reported linear relationship between the logarithm of the preexponential term and the activation energy derived from the Arrhenius equation.     

Reactivity Tracers of the Hydrolysis of Fe(II)‐Bis(salicylidene) Complexes: Initial–Transition States Analysis and Enhanced Impact of Tensides on the Kinetic Trends

The kinetics of the acid hydrolysis reaction of Fe(II)‐bis(salicylidene) complexes were followed under pseudo–first‐order conditions ([H⁺] >> [complex]) at 298 K. The ligands of the studied azomethine complexes were derived from the condensation of salicylaldehyde with different five α‐amino acids. The hydrolysis reactions were studied in acidic medium at different ratios (v/v) of aqua–organic mixtures. The decrease in the dielectric constant values of the reaction mixture enhances the reactivity of the reaction. The transfer chemical potentials of the initial and transition states (IS–TS) from water into mixed solvents were determined from the solubility measurements combined with the kinetic data. Nonlinear plots of logk_obs versus 1/D (the reciprocal of the dielectric constant) suggest the influence of the solvation of IS–TS on the reaction reactivity. Furthermore, the acid hydrolysis reactions were screened in the presence of different concentrations of cationic and anionic tensides. The addition of surfactants to the reaction mixture accelerates the reaction reactivity. The obtained kinetic data were used to determine the values of δ_mΔG^# (the change in the activation barrier) for the studied complexes when transferred from “water to various ratios (v/v) of water–co‐organic binary mixtures” and from “water to water containing different [surfactant].” It was found that the reactivity of the acid hydrolysis reaction was controlled by the hydrophobicity of the studied chelates.     

Square‐Wave Voltammetry of the Molybdenum‐1,10 Phenanthroline‐Fulvic Acids Complex: Redox Kinetics Measurements

The reduction process of molybdenum in the presence of fulvic acids and phenanthroline was investigated by square-wave voltammetry (SWV). The mixed-ligand complex of molybdenum exhibits a pronounced tendency to adsorb onto the mercury electrode surface. The electrode reaction proceeds as a surface process in which both components of the redox couple are firmly confined to the electrode surface. The kinetics of the electrode reaction is studied utilizing the properties of “split SW peaks” and “quasireversible maximum”. The kinetic parameters obtained with two different square-wave voltammetric methods are in good agreement. In 0.5 mol/L NaCl solution with pH 2.5 the kinetic parameters are: standard rate constant k_s=8±2 s⁻¹, cathodic electron transfer coefficient α=0.41±0.05, and number of exchanged electrons n=2. The SW kinetic measurements are confirmed by cyclic voltammetric method.     

Prediction of artifact peak intensity in linked scans for dissociations occurring in the first field-free region of sector mass spectrometers

Linked scans are commonly used on double-focusing mass spectrometers to obtain tandem mass spectrometry (MS/MS) spectra. The appearance of artifact peaks in linked scan MS/MS spectra from dissociations occurring in the first field-free region are a result of poor parent ion resolution, and they often can complicate the interpretation of the MS/MS spectra. The kinetic energy release associated with dissociation of ions of similar m/z to the “selected” parent ion is the main factor in determining the intensity of artifact peaks. A means of predicting the intensities of these artifact peaks in product ion and constant neutral loss scans is presented here. The method requires straightforward calculations based on Lacey-Macdonaldion intensity diagrams. The exact calculations require knowledge of the kinetic energy release of a particular dissociation, the kinetic energy spread of the main beam, and the parent ion and product ion mass-to-charge ratios. Adequate predictions, however, can be made by assuming a general kinetic energy release for any given reaction and a typical instrument energy resolution. Theoretical predictions are in good agreement with experimental data obtained from the product ion scans of unlabeled and isotopically labeled tirilazad and unlabeled and labeled leucine enkephalin methyl ester. There is also excellent agreement between experiment and theory in the constant neutral loss scans of rubidium bromide clusters.     

The Chemistry of Isolated Cations

Organic ions decompose in the field-free regions of a mass spectrometer after a relatively well-defined lifetime (t ≈ 10^?5s). This lifetime is very long compared with the time-scale of molecular bond vibrations and consequently the excess internal energy (“non fixed” energy) present in the transition state for dissociation is comparable with that found in solution experiments. In general, only the energetically most favorable decay routes are able to compete. Moreover, information concerning kinetic energy release (T) accompanying unimolecular decomposition may be obtained from a consideration of the metastable peak for the process; these extremely valuable data are inaccessible to solution experiments because of the effects of collisions. The chemistry of ions may therefore be investigated conveniently by generating the ions of interest in a mass spectrometer and analyzing the reactions which occur in metastable transitions.     

Cure Characterization of a New Bismaleimide Resin using Differential Scanning Calorimetry

The cure characterization of a new BMI resin was investigated to ascertain a suitable cure model for the material. A series of isothermal differential scanning calorimetry (DSC) runs from 170°C to 220°C provided information about the cure characterization. All “kinetic triplet” (i. e., Arrhenius parameters A, activation energy E, and the reaction model, f(α)) of this new BMI system were calculated. It was found that the cure characterization of this new BMI could be expressed by nth‐order cure reaction.     

Kinetic Screening for the Acid‐Catalyzed Hydrolysis of Some Hydrophobic Fe(II) Schiff Base Amino Acid Chelates and Reactivity Trends in the Presence of Alkali Halide and Surfactant

Kinetics of acid‐catalyzed hydrolysis of some high‐spin Fe(II) Schiff base amino acid complexes were followed spectrophotometrically at 298 K under pseudo–first‐order conditions. The studied ligands were derived from the condensation of 5‐bromosalicylaldehyde with different four amino acids (phenylalanine, aspartic acid, histidine, and arginine). The acid hydrolysis reaction was studied in aqueous media and in the presence of different concentrations of the alkali halide (KBr) and cationic surfactant (cetyl‐trimethyl ammonium bromide, CTAB). The general rate equation was suggested to be rate = k_obs[complex], where k_obs = k₂[H⁺]. The increase in [KBr] enhances the reactivity of the reaction, and the addition of CTAB to the reaction mixture accelerates the reaction reactivity. The obtained kinetic data were used to determine the values of δ_mΔG^# (the change in the activation barrier) for the studied complexes when transferred from “water to water containing different [KBr]” and from “water to water containing altered [CTAB].”     

Thomas–Fermi term as the simplest correction to the Weizsacker term

The nature of the correlation function G(1,2) appearing in the definition of the reduced first order density operator γ′(1,2) = ρ(1)^1/2ρ(2)^1/2G(1,2) is analyzed. It is shown that when G(1,2) is expanded in terms of plane waves in the context of a single-determinant approximation to the wave function, the correction to the Weizsacker term in the kinetic energy density expression is the Thomas–Fermi term.     

Recent results in insect pheromone chemistry

Pheromones are substances which are secreted to the outside by an animal, especially an insect, and received by a second individual of the same species, in which they release a specific reaction, for example, a definite behavior or developmental process. The name is derived from the Greek pherein (to bear, transmit) and hormon (impel, excite). Pheromones include, inter alia, sex attractants, warning substances, and “aggregation pheromones”. Many of these substances are effective at very low concentration. The insect pheromones are predominantly derivatives of long-chain, slightly branched hydrocarbons.     

Vacuum and oxidative pyrolysis of poly-p-xylylene. II. Chromatographic analysis and kinetics of reaction products

Reaction products of vacuum and oxidative degradation of poly-p-xylylene have been quantitatively determined by chromatographic analysis as function of time, temperature and oxygen pressure. Respective Arrhenius parameters were also ascertained for some of the reaction products and for the sums of all products. The energies of activation for the sums agree quite satisfactorily with the energies of activation obtained previously by uninterrupted experiments in quartz-spoon reaction vessels. The results found here can be described in terms of mechanisms previously postulated on the basis of the total loss in weight (or volatile production) data. Scission of “weak” links (due to abnormal structures) takes place followed by formation of various products. The whole process is governed by the initial chain scission reaction; however, the energies of activation for each of the products do not need to be identical with that of the chain scission reaction. Each product is formed by a reaction which has its own characteristic number average kinetic chain lengths; the latter have their specific energy of activation values. Oxidative degradation produces the same organic compounds as vacuum degradation and in addition CO, CO₂, and H₂O. Oxidized intermediate compounds are apparently fairly rapidly decarboxylated and decarbonylated. Oxidative chain scission is appreciably faster than that in vacuum. Almost simultaneous “weak” link and “normal” chain scission are taking place initiating the formation of a number of products.     

Kinetic energy release during CO loss by rearrangement of α-benzoylcarbenium ions

Primary α-benzoylcarbenium ions (a) and tertiary α-benzoyldimethylcarbenium ions (b) are obtained by chemical ionization of ω-hydroxyacetophenone and its dimethyl derivative, respectively. Both α-acylcarbenium ions decompose by a rearrangement reaction and subsequent loss of a CO molecule. The kinetic energy released during this process by metastable ions a and b has been determined under different experimental conditions. The kinetic energy released during the CO elimination from the tertiary α-benzoyldimethylcarbenium ions b is independent of the experimental conditions and gives rise to dish-topped peaks with T₅₀=440±20 meV in the MIKE spectra of b. In contrast to this the kinetic energy releases and the peak-shapes in the MIKE spectra of the primary α-benzoylcarbenium ion a varies with the experimental conditions. The mechanism of this rearrangement reaction is discussed, and it is shown by a MNDO calculation of the heats of formation of the relevant ions that the different characteristics of the kinetic energy release during the fragmentation of primary and tertiary carbenium ions can be attributed to different types of reaction energy profiles.     

Energy disposal in gas-phase nucleophilic displacement reactions

The partitioning of reaction exothermicity into relative translational energy of the products of gas-phase S_N2 (F^? + CH₃Cl) and nucleophilic aromatic substitution (F^? + C₆H₅Cl) reactions has been investigated using kinetic energy release Fourier transform ion cyclotron resonance spectroscopy. The chloride product ion is observed to be highly translationally excited for the S_N2 reaction, indicating a cold internal energy distribution for the products. For the chlorobenzene reaction the products are not generated with large translational energies. The results are compared with a statistical model. Ion-intensity profiles for the CH₃Cl reaction deviate significantly from the statistical model whereas the chlorobenzene results are consistent with this model. The kinetic energy release for the CH₃C1 reaction is compared with energy-disposal results for the photodissociation and dissociative electron-attachment processes of halomethanes. In all three cases a node in the molecular orbital between the carbon atom and the departing halogen results in a repulsive energy release. Ion-retention curves for the nucleophilic aromatic substitution reaction are consistent with the existence of a long-lived ion-dipole complex on the exit channel for this reaction.     

Time Programmable Locking/Unlocking of the Calix[4]arene Scaffold by Means of Chemical Fuels

In this work, we report that 2-cyano-2-phenylpropanoic acid and its p-Cl, p-CH₃ and p-OCH₃ derivatives can be used as chemical fuels to control the geometry of the calix[4]arene scaffold in its cone conformation. It is shown that, under the action of the fuel, the cone calix[4]arene platform assumes a “locked” shape with two opposite aromatic rings strongly convergent and the other two strongly divergent (“pinched cone” conformation). Only when the fuel is exhausted, the cone calix[4]arene scaffold returns to its resting, “unlocked” shape. Remarkably, the duration of the “locked” state can be controlled at will by varying the fuel structure or amount. A kinetic study of the process shows that the consume of the fuel is catalyzed by the “unlocked” calixarene that behaves as an autocatalyst for its own production. A mechanism is proposed for the reaction of fuel consumption.