Normalization and Fermi–Coulomb and Coulomb hole sum rules for approximate wave functions

For approximate wave functions, we prove the theorem that there is a one‐to‐one correspondence between the constraints of normalization and of the Fermi–Coulomb and Coulomb hole charge sum rules at each electron position. This correspondence is surprising in light of the fact that normalization depends on the probability of finding an electron at some position. In contrast, the Fermi–Coulomb hole sum rule depends on the probability of two electrons staying apart because of correlations due to the Pauli exclusion principle and Coulomb repulsion, while the Coulomb hole sum rule depends on Coulomb repulsion. We demonstrate the theorem for the ground state of the He atom by the use of two different approximate wave functions that are functionals rather than functions. The first of these wave function functionals is constructed to satisfy the constraint of normalization, and the second that of the Coulomb hole sum rule for each electron position. Each is then shown to satisfy the other corresponding sum rule. The significance of the theorem for the construction of approximate “exchange‐correlation” and “correlation” energy functionals of density functional theory is also discussed. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Instabilities of the symmetry-adapted restricted-hartree–fock ground state in infinite polyenes. I. Singlet instabilities

The singlet instabilities of the RHF ground state in infinite polyenes have been studied in the framework of a semiempirical PPP Hamiltonian, accounting for long-range Coulomb interactions until convergence of the ground-state energy per electron value. The symmetry-adapted RHF solution (SAS ) has been shown to be unstable to the formation of bond-order alternation waves (BAW 's) and charge-density waves (CDW 's). The CDW solutions have been shown to be higher in energy than the corresponding BAW solutions and to represent saddle points of the energy hypersurface, unstable to the formation of BAW 's for physically realistic range of variation of the semiempirical parameters. Analytical formulas for the SAS ground-state energy per electron have been derived in case of a Coulomb law and a Mataga–Nishimoto formula for the two-center Coulomb integrals.     

Study of correlation holes. II. CI calculations on model polyatomic systems

The shape of correlation holes in many-electron systems is at present scarcely known, even where correlated wave functions are available. We investigate here the kind of electron correlation brought about by configuration interaction (CI ), within a given basis set, in the wavefunction of a polyatomic system. The model ring system H₆ (in two different bonding circumstances) and H₁₄ have been chosen for a detailed study, because of their paradigmatic importance. We set out the equal-spin and different-spin correlation holes as obtained from complete CI calculations in H6 and partial ct in H₁₄, both within a minimal basis set. We basically find the spinless correlation as being short range, while the spin-dependent correlation holes show long-range oscillations of antiferromagnetic character. We also present a natural spin-geminal analysis of the two-body reduced density matrices in these systems; we find a peculiarity possibly related to the long-range correlation discussed above. Finally, we compare the electron correlation as given from our CI wavefunction to other pictures of electron correlation, as obtained essentially from alternant molecular orbital wave functions and from the electron–gas literature.     

SCF and CI Studies of Hund's rules for the effects of electronic correlation and delocalization. I

In connection with the reinterpretation of Hund's multiplicity rules for molecules, a detailed study has been made of the energy differences in the total energy and its components for the triplet and singlet Π_u states of the hydrogen molecule and the analogous states of the four- and six-membered hydrogen atom rings. For the hydrogen molecule, both SCF and CI studies indicated that the outer electron is considerably more contracted in the triplet than in the singlet state. In both approximations, the energy difference is dominated for all bond distances of chemical and physical significance by the electron-nuclear attraction component and not by the electron repulsion component as predicted by simple first-order perturbation theory. Although the correlation energy for each of the states is of the same magnitude as the energy differences considered here, the difference of the correlation energies is much smaller. It had little effect on the qualitative differences between these states of the hydrogen molecule. For the four- and six-membered rings, SCF studies were made on the lowest singlet and triplet states where one electron was promoted from the σ_g to a Π_u orbital. Even though the coupled electrons were more delocalized in these cases, the electron repulsion became relatively more important. However in all cases, the lower state had the highest electron repulsion energy and lower electron-nuclear attraction. The triplet state continued to have the more contracted outer open-shell orbital.     

Modification of PSRK mixing rules and results for vapor–liquid equilibria, enthalpy of mixing and activity coefficients at infinite dilution

The predictive Soave–Redlich–Kwong (PSRK) equation of state (EOS) is a well-established method for the prediction of thermodynamic properties required in process simulation. But there are still some problems to be solved, e.g. the reliability for strong asymmetric mixtures of components which are very different in size. The following modifications are introduced in the PSRK mixing rules: the Flory–Huggins term in the mixing rule for the EOS parameter a, and the combinatorial part in the UNIFAC model are skipped simultaneously; a nonlinear mixing rule for the EOS parameterb, instead of the linear mixing rule, is proposed. With these two modifications better results are obtained for vapor–liquid equilibria and activity coefficients at infinite dilution for alkane–alkane systems, specially for asymmetric systems. In order to obtain better results for enthalpy of mixing, temperature-dependent parameters are used. Group interaction parameters have been fitted for several groups, and the results are compared with the Modified UNIFAC (Dortmund), and the PSRK methods.     

An instability condition for the Hartree–Fock solution of the infinite one‐dimensional system with two‐crossing bands. I. Singlet‐instability check of metallic carbon nanotube

An instability condition is derived for the Hartree–Fock solution so that it can be applied to the system in which the highest occupied and the lowest unoccupied bands cross at the in‐between point in the Brillouin zone. The instability check developed here is further applied to a metallic single‐walled carbon nanotube having the two‐crossing bands toward prediction of its instability. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 574–582, 2000     

Electronic-phase transition associated with instability of the Hartree–Fock solution of infinite one-dimensional system

Electronic-phase transition associated with the singlet instability problem of the Hartree-Fock solution of the extended system is studied employing the metallic trans-polyacetylene. Concerning the eigenstates of the stability matrix set up for the crystal orbitals at the Fermi level, classification of the electronic phases is attempted and the origin of their emergence in relation to the interelectronic interaction is discussed.     

Left–right and dynamic correlation

Within density functional theory, it is natural to separate the correlation energy into two parts: left–right correlation and dynamic correlation. Left–right correlation arises from the exchange part of functionals, and dynamic correlation arises from the correlation part of functionals. We examine the nature of these correlation energies as molecules are distorted. We observe that such a natural separation is not possible using the methods of quantum chemistry. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Cluster CPA calculations for infinite and half–infinite chains and applications to coupled polymer chains

It is shown how the cluster CPA can be used to obtain the local density of states p(E) in a half-infinite or infinite random polymer chain. Calculations were performed for p(E) in a system composed of two coupled random chains with nearest neighbors interaction.     

The computational application of the Cahn–Ingold–Prelog rules 1 and 2

Cahn–Ingold–Prelog (C? I? P) rules 1 and 2 (material differences) have been expressed in the formal logic of a high-level computer language (FORTRAN IV) on a laboratory minicomputer (PDP 11/40) by the program CHIRAL. An oversight in the C? I? P formalism in the treatment of rings was treated on an ad hoc basis; the problem has been expressed simply in terms of elementary graph theory in a way consistent with and similar to chemical notation. An example of the operation of CHIRAL is provided and the internal procedures discussed.     

Properties of ethylene–propylene–vinyl chloride graft copolymers. I. Viscoelasticity of ethylene–propylene copolymers

The viscoelastic behavior of two different ethylene–propylene copolymers was studied as a function of the molar ratios of the components and the distribution of the lengths of the ethylene and propylene sequences. The glass transition temperatures T_g agree with the values calculated from relations between T_g and component ratio established by other authors. The copolymer with longer ethylene and propylene sequences was found to exhibit a relaxation spectrum with a slope less steep than ?0.5. This broadening is explained by the broader distribution of friction factors of the statistical segments in this copolymer and by differences in crystallike nearest-neighbor packing.     

Potentials in density functional theory and the importance of sum rules

In an ab initio approach to density functional theory one needs to know the electronic pair-density averaged over the coupling strength of the pair-interaction. As this pair-function is not available without having solved the associated N-electron problem, one has to resort to universal properties of the pair-density that are independent of specific features of the ground-state wavefunction. By exploiting these universal properties and so-called sum rules for the pair-correlation factors we derive very simple approximate spin-dependent expressions for the exchange-correlation energy per particle and for the associated potential in the Kohn-Sham equations. There is some similarity of the resulting density functionals with those obtained from the widely applied local spin density approximation (LSDA) based on electron gas theory. As the application of the latter to exceedingly inhomogeneous gases in realistic systems is very debatable, the manifest similarity seems to suggest that LSDA can consistently be justified only via the above pair-density analysis, but the justification of certain electron gas refinements may remain questionable. We shortly review similar attempts made by other authors and particularly focus on the issue of self-interaction and the “overbinding problem”. We demonstrate for the 3d- and 4d-metals that our approximation yields density of states (DOS), magnetic moments and Stoner parameters that are practically identical with respective data obtained from up-to-date LSDA- or gradient corrected (GGA-)potentials. There is also an excellent agreement of the DOS for the insulators (semi-conductors) C, Si, Ge, and GaAs. We show that our approach yields cohesive energies for these materials that are very close to the GGA-values indicating a distinct improvement over the standard LSDA-values. The calculations have been performed with the aid of the WIEN 97 computer code based on the Full Potential Linear Augmented Plane Wave (FLAPW-) method.     

Carboxylated core–shell particles: I. A system showing hindered swelling behavior

We study the swelling behavior of carboxylated core–shell particles. It is well-known that these particles swell with increasing pH due to the electrostatic repulsion between carboxylate groups. Our results reveal that the swelling behavior is affected by the preparation method. We find that the swelling is promoted in those particles which were initially in a highly swollen state (pH  10). However, the swelling is hindered for those particles which were not previously in this trigger pH. In the hindered systems, a compact conformation of the polymer shell is induced by hydrophobic attractions between the non-charged segments which compete against the swelling driving force. In addition, an interesting hysteresis behavior emerges when promoted systems are subjected to a heating–cooling cycle; a new stable system appears with a less extended polymer shell conformation. Furthermore, salt-induced swelling experiments corroborate not only polymer restructuring but also assembly among carboxylate groups which affects their ionization grade.     

Crystallization of branched polymers. I. Ethylene–vinyl acetate and ethylene–acrylic acid copolymers

The following quantities were measured on a number of ethylene–vinyl acetate (EVA) and ethylene–acrylic acid (EAA) copolymers: (1) the small-angle x-ray scattering invariant, (2) the overall density, and (3) the crystallinity. Assuming a two-phase structure, the separate values of the densities of the crystalline and amorphous regions can be calculated from these data. Of these, the crystalline density is compared with the value obtained from the lattice constants. A systematic difference is observed which is ascribed to the presence of comonomeric side groups in the crystalline regions. For the EVA and EAA samples, their concentration is at least 0.3 and 0.5 times the overall concentration, respectively. The amorphous densities are found to be higher than the values calculated from completely amorphous copolymers by extrapolation procedures.     

Kinetics of polymerization initiated by Mn3+–substrate redox system. I

The kinetics and mechanism of polymerization of acrylamide (AM) initiated by manganese (III) acetate (MTA)–diglycolic acid (DGA) redox system in aqueous sulfuric acid were studied in the temperature range 20-35°C. The overall rates of polymerization and the disappearance of Mn³⁺ and the kinetic chain lengths of polyacrylamide were determined. The polymerization reaction is initiated by the organic free radical arising from the Mn³⁺–diglycolic acid reaction and the termination is by the metal ions. The rate of polymerization of acrylamide was found to be proportional to the first power of monomer and diglycolic acid and independent of manganese(III) acetate. The various rate parameters were evaluated.     

Polyurethane cationomers. I. Structure–properties relationships

Polyurethane (PU) cationomers have been synthesized by quaternizing tertiary amine-containing linear polyurethanes using different quaternizers containing acid groups. The effect of chemical structure of PU cationomers on the physical properties was studied. The mechanical properties of PU cationomers were improved with decreasing molecular weight of poly(caprolactone) glycol, and increasing concentration of quaternary ammonium. Decreasing the carbon number in the alkyl group of the N-alkyl diethanol-amine chain-extenders, and using rigid symmetrical diisocyanates, the mechanical properties of the PU cationomers were increased. The effects of these factors on the glass transition temperature of PU cationomers were also examined. The mechanical properties of the PU cationomers decreased by immersion in water and recovered after removal of the water.     

Kinetics and mechanism of the cationic polymerization of tetrahydrofuran in solution. I. THF–CCI4 system

Polymerization of THF in CCl₄ solvent was initiated with 1,3-dioxolan-2-ylium eations with AsF₆^?, PF₆^?, and SbF₆^? anions as well as with esters of fluorosulfonic and trifluoromethanesulfonic acids. With these esters polymerization proceeds with a marked acceleration period, due to slow initiation. The corresponding rate constants of initiation and their dependence on the polarity of the THF/CCl₄ mixture were determined. The rate constant of propagation on the macroion-pairs (k_p^±) of the polytetrahydrofurylium cation with AsF₆^?, PF₆^?, and SbF₆^? and CF₃SO₃^?, anions was found to be independent in CCl₄ solvent on the anion structure and given by the expression: k_p^± = 2.93 × 10^?2 exp {?4.7 × 10³/T} at [THF]₀ = 8.0M. This constant depends on the polarity of the polymerization mixture, and at 25°C for the THF-CCl₄ system, k_p^± = 1.78 × 10^?2 exp {?4.9/D}; thus, in CCl₄ at [THF]₀ = 8.0M, and at 25° k_p^± = 4.0 × 10^?21/mole-sec. In the polymerization with derivatives of CF₃SO₃H (able to form the corresponding macroester) the overall polymerization rate is much lower than that with complex anions because of the reversible conversion of the macroion-pairs into the macroester (internal return). The macroester is much less reactive than the macroionpair (10²–10³ times) in the monomer addition reaction. At [THF]₀ = 8.0M and at 25°C, 96.5% of the growing species exists in the macroester form. Polymerization of THF initiated with derivatives of CF₃SO₃H is a subject of a strong special salt-effect. At a sufficiently high ratio of [AgSbF₆] to [I]₀, where the initiator I is C₂H₅OSO₂CF₃, the overall polymerization rate is equal to that observed for the polymerization of THF on the macroion-pairs, since the internal return within the triflate ion-pair (the macroester formation) is eliminated and polymerization proceeds on the macroion-pairs with SbF₆- anions exclusively.