Path integral formulation of Roothaan's equations

The generating functional Z(N) of the molecular orbital theory of N electrons has been projected into a subspace of atomic orbitals. By application of the saddle-Point approximataions to the corresponding effective action, a set of equations of motion results. These equations are the classical Roothaan's equations of quantum chemistry.     

Chemical forces in terms of the electron density

A previous study of the small deformations produced by the molecular environment in the atomic densities has concluded that these deformations reflect and support all the concepts of the empirical structural chemistry. In the present work the study is extended to the chemical forces. Four types are distinguished: Van der Waals, Pauli, bonding and nuclear unscreening forces. For each type, the origin, the associated density deformations, the force generated by these deformations and the main role played by this force are investigated. It is stressed that Van der Waals complexes, usual molecules, conformers and chemical processes can be described in terms of these four forces, and are determined by the balance between them. It is proved that simple models for the density are often sufficient for semiquantitative predictions of forces, and can be very helpful for rationalizing the chemical properties and behavior. Developments are illustrated and supported with specific examples. Contribution to the Serafin Fraga Memorial Issue. Financial support: Dirección General de Investigación Científica y Técnica (CTQ2004-06615/BQU).     

On dispersion force correction terms in perturbed equations of state

The first-order correction term for the contribution of dispersion forces in a perturbation expansion has been analyzed using computer simulation and perturbation theory results. It turns out that simple approximations used to make an analytic evaluation of the correction integral possible may result, ironically, in more complicated (and erroneous) behavior in comparison with the exact result.     

Effective potential in density matrix functional theory

In the previous paper it was shown that in the ground state the diagonal of the spin independent second-order density matrix n can be determined by solving a single auxiliary equation of a two-particle problem. Thus the problem of an arbitrary system with even electrons can be reduced to a two-particle problem. The effective potential of the two-particle equation contains a term v(p) of completely kinetic origin. Virial theorem and hierarchy of equations are derived for v(p) and simple approximations are proposed. A relationship between the effective potential u(p) of the shape function equation and the potential v(p) is established.     

Kinetic energy density in terms of electron density for closed-shell atoms in a bare Coulomb field

A long-term aim in density functional theory is to obtain the kinetic energy density t(r) in terms of the ground-state electron density ρ(r). Here, t(r) is written explicitly in terms of ρ(r) for an arbitrary number 𝒩 of closed shells in a bare Coulomb field. In the limit as 𝒩→∞, closed results for t(r) follow from the earlier analysis of ρ(r) by Heilmann and Lieb. [Phys. Rev. A 52 , 3628 (1995)]. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66 : 281–283, 1998     

The sulfur reversion process in natural rubber in terms of crosslink density and crosslink density distribution

Unfilled natural rubber compounds composed of conventional (CV), semi-efficient (SEV), efficient (EV) and sulfur donor (SD) vulcanization systems were heat aged to promote sulfur reversion. Rheometry, hardness, strain-strain characteristics including Mooney-Rivlin analysis, equilibrium solvent swell and Double Quantum (DQ) Nuclear Magnetic Resonance (NMR) were used to monitor crosslink density changes. A loss of crosslink density was observed by rheometry, C₁, equilibrium swelling and by DQ NMR as a function of cure extent. No chain scission reactions were operating in the time/temperature conditions used. All crosslink distributions were unimodal and the network homogeneity followed the order of EV > SD > SEV > CV. The crosslink distribution narrowed during the curing process for the CV and SEV systems. Non-oxidative maturation reactions were advantageous in promoting a more random distribution of crosslinks in the polymer matrix.     

Extracting atoms from molecular electron densities via integral equations

The observation that a molecular electron density is close to the superposition of its constituent atoms leads naturally to the idea of modeling a density by a sum of nuclear-centered, spherically symmetric functions. The functions that are optimal in a least-squares sense are known as Stewart atoms. Previous attempts to construct Stewart atoms by expanding them in an auxiliary basis have been thwarted by slow convergence with respect to the size of the auxiliary basis used. We present a method for constructing Stewart atoms via convolution integrals which bypasses the need for an auxiliary basis, and is able to produce highly accurate approximations to Stewart atoms.     

Effective local potentials for orbital-dependent density functionals

Practicality of the Kohn-Sham density functional scheme for orbital-dependent functionals hinges on the availability of an efficient procedure for constructing local exchange-correlation potentials in finite basis sets. We have shown recently that the optimized effective potential (OEP) method, commonly used for this purpose, is not free from difficulties. Here we propose a robust alternative to OEPs, termed effective local potentials (ELPs), based on minimizing the variance of the difference between a given nonlocal potential and its desired local counterpart. The ELP method is applied to the exact-exchange-only problem and shown to be promising for overcoming troubles with OEPs.     

The two-electron integral transformation and two-body density matrix transformation

Ann ⁵ algorithm for the transformation of quantum-mechanical four centre functions is presented in a form best suited for computers having a virtual memory capability. Part of the work to be submitted for the degree of Ph. D. in the University of Newcastle-upon-Tyne.     

Hierarchy equations for reduced density matrices in different representations

We derive exact relationships for the reduced density matrices in representations where the transformation matrix is a product of one-body transformation matrices. We specialize to the momentum and onebody energy representations. By decoupling the equations we are able to write the Hartree-Fock equation in terms of the first-order density matrix in an arbitrary representation. Applications to reduced local energy and the correlation problem are discussed.     

Hierarchy of equations in the generalized density functional theory

Functional relations and equations of hierarchy in the generalized density functional theory (DFT) are derived from coordinate scaling and adiabatic connection. Local and nonlocal solutions for the noninteracting kinetic energy, exchange energy, correlation energy, and the kinetic energy correction functionals are presented. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

The accuracy of equations approximating the temperature integral part I

Results of an evaluation of the approximation accuracy of the temperature integral obtained by means of different approximation equations are presented. The usability for approximation purposes of the Schloemilch, asymptotic, Bernoullie and Vallet series, depending on the number of expansion terms used in calculations, and the Doyle logarithmic, Zsakó, Coats-Redfern and Turner-Schnitzer-Gorbachev equations has been evaluated. Boundary values ofz, above which the approximation accuracy of a given equation is higher than or equal to the assumed one, are given.

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Zusammenfassung Es werden die Auswertungsergebnisse der Annäherungsgenauigkeit des mittels verschiedener Annäherungsgleichungen erhaltenen Temperaturintegrals gegenübergestellt. Die Anwendbarkeit für Näherungszwecke der Schloemilch-schen, der asymptotischen, der Bernoulli-schen und der Vallet-schen Serien wird in Abhängigkeit von der Zahl der bei den Berechnungen gebrauchten Expansionsausdrücke und von der logaritmischen Gleichung von Doyle, sowie von den Gleichungen von Zsakó, Coats-Redfern und Turner-Schnitzer-Gorbatschev bewertet. Grenzwerte von z, über welchen die Näherungsgenauigkeit einer gegebenen Gleichung höher oder gleich der angenommenen ist, werden gegeben.

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Résumé Dans cette publication, on a cherché à évaluer l'exactitude des équations permettant d'atteindre par différentes approximations l'intégrale de température. On a étudié en particulier la possibilité d'utiliser à des fins d'approximation, les méthodes de Schloemilch, Bernoulli, Vallet, Doyle, Zsakó, Coats-Redfern et Turner-Schnitzer-Gorbachev. On donne les valeurs aux limites de z, au-dessus desquelles l'exactitude de l'approximation d'une équation donnée est supérieure ou égale à celle qui est supposée.

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, . , , , , , — —— . «z», .

     

Fluid-phase diagrams of binary mixtures from constant pressure integral equations

A new algorithm for solving integral equations of the theory of liquids at fixed pressure is introduced. Combining this technique with the Lee's star function approximation for the chemical potentials, we obtain an efficient method to investigate fluid-phase diagrams of binary mixtures. We have tested the capabilities of such technique to study symmetric and asymmetric phase diagrams in nonadditive hard spheres and Lennard-Jones mixtures. We find that the integral equation theories, although approximate, can provide a flexible tool to determine the fluid-phase diagrams whose accuracy is critically dependent on the quality of the closure and of the resulting chemical potentials.     

The study of structure formation in a polymer-containing ionic liquid in terms of the integral equation theory

The structural properties of a polymer-containing ionic liquid under the conditions of good solubility of a flexible polymer are studied theoretically. Two systems are discussed: In one, polymer solubility is due to the presence of specific interaction between polymer chains and solvent cations; in the other, polymer solubility is due to the presence of specific interactions between the polymer and solvent anions. The dependences of the structural characteristics of a solution on the polymer concentration and the energy of attraction between polymer chains and solvent ions are calculated. In a semidilute polymer solution, long-range correlations of polymer chains with a power dependence of the characteristic scale of ordering on the polymer density appear. The conditions under which, along with the intermediate ordering typical of a pure ionic liquid, the long-range ordering of the solvent cations and anions occur after addition of a polymer to the ionic liquid are studied.     

Diffusion current on a nonuniform electrode: Calculation with integral equations

Identity of mathematical problems concerning calculation of the distribution of reactants’ concentrations and the current near the surface of a nonuniform (strip) electrode and distribution of displacements and forces in the case of an elastic layer “antiplane” deformation caused by the punch action. Formulas for calculating the current at a strip electrode are derived for various ratios between the electrode width and the diffusion layer thickness by means of asymptotic methods designed for calculating problems of mechanical contact interactions. It is noted that calculations of the diffusion current for involved activity distributions at the electrode surface may benefit from asymptotic methods of mechanics of contact interactions.     

The accuracy of equations approximating the temperature integral. Part II

The present paper shows the results of an analysis of the approximation accuracy of the temperature integral through the logarithmic equation elaborated by MacCallum and Tanner. The accuracy of this equation is higher than that of the similar logarithmic equation presented by Doyle, although substantially lower than the accuracy of other approximative equations appearing in the literature. Due to its simple form, this equation may be used for initial and quick elaborations of results of kinetic measurements made under conditions of linear temperature increase.     

Curl equations as substratum for the derivation of the electronic nonadiabatic coupling terms

The idea to derive the nonadiabatic coupling terms by solving the Curl equations (Avery, J.; Baer, M.; Billing, G. D. Mol Phys 2002, 100, 1011) is extended to a three‐state system where the first and second states form one conical intersection, i.e., τ₁₂ and the second and the third states form another conical intersection, i.e., τ₂₃. Whereas the two‐state Curl equations form a set of linear differential equations, the extension to a three‐state system not only increases the number of equations but also leads to nonlinear terms. In the present study, we developed a perturbative scheme, which guarantees convergence if the overlap between the two interacting conical intersections is not too strong. Among other things, we also revealed that the nonadiabatic coupling term between the first and third states, i.e., τ₁₃ (such interactions do not originate from conical intersection) is formed due to the interaction between τ₁₂ and τ₂₃. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Effective atomic number and electron density of marble concrete

The effective atomic numbers (Z _eff) and effective electron density (N _e) of different type concrete have been measured and the results were compared with the calculation obtained using the mass attenuation coefficients (μ/ρ) obtained via XCOM in the photon energy range of 1 keV–100 GeV. Six different concrete in where marble has been used in the rate of 0, 5, 10, 15, 20, 25 %, has been used in the study.     

The solution of adsorption integral equations by means of the regularization method

An overall adsorption quantity for a heterogeneous solid is usually expressed by an integral equation, which contains a distribution function that describes heterogeneous properties of this solid. The calculation of this distribution function is an ill-posed problem. The current article shows that the difficulties arising from the ill-posed nature of an adsorption equation can be overcome with the regularization method. This work presents general principles of regularization for solving the ill-posed problems without detailed mathematical considerations. The application of the regularization method to calculate a distribution function from any overall adsorption functions is illustrated with both simulated and experimental adsorption isotherms.     

Electric double layers with electrolyte mixtures: integral equations theories and simulations

A study of a planar electric double layer (EDL) in the presence of mixtures of electrolyte is presented. In particular, results from the Hyper-Netted-Chain/Mean-Spherical-Approximation (HNC/MSA) theory are compared with Monte Carlo (MC) simulations. In this way, the charge inversion induced by mixtures of multivalent and monovalent counterions is probed. Since overcharging phenomena in nature emerge under such conditions, the role of ion-ion correlations in the EDL appears as a crucial point in this kind of study. Unlike previous related works, a realistic hydrated ion size is used in the HNC/MSA calculations and simulations. In this way, a qualitative agreement between the results obtained from the theory and MC simulations is found. However, some discrepancies arise when the charge inversion is expected to be more noticeable, namely at high surface charges and/or elevated concentrations of multivalent electrolytes. Such differences are explained in terms of an overestimation of the charge inversion by the integral equation (IE) formalism.