Molecular weight distribution in random crosslinking of polymer chains

The molecular weight distribution (MWD) of crosslinked polymer molecules formed during polymeric network formation is the sum of the fractional MWDs containing 0, 1, 2, 3, … crosslinkages. The MWD for polymer molecules containing ?? crosslinkages is investigated for the random crosslinking of polymer chains whose initial MWD is given by the Schulz-Zimm distribution. For a very narrow initial MWD, each fractional MWD with ?? = 0, 1, 2, … is independent and a multimodal distribution is obtained for the whole distribution. When the initial MWD is uniform, the average crosslinking density within the polymer fraction whose degree of polymerization is r, ρ_r is simply given by ρ_r = ρ_gel,c – 2/r irrespective of the extent of crosslinking reaction where ρ_gel,c is the crosslinking density within gel fraction at the gel point. On the other hand, the MWDs with ?? crosslinkages overlap each other with different ?? values significantly for the broader initial distributions, and ρ_r increases with the progress of crosslinking reactions. The value of ρ_r increases with increasing r but levels off asymptotically at large r. The average crosslinking density of polymer molecules containing ?? crosslinkages ρ_?? is an increasing function of k but soon reaches a plateau; sooner for the broader initial MWDs. For ?? ≥ 1, ρ_?? is always larger than the average crosslinking density of the whole reaction system ρ in the pregelation period, i.e., in terms of the crosslinking density, the difference between polymer molecules with and without crosslinkage is most significant. In general, the average crosslinking density ρ, which is convenient to use in describing the nature of the whole reaction system, cannot be considered as a characteristic degree of crosslinking for polymer molecules containing at least one crosslinkage. Consideration of the bivariate distribution of r and k reveals important aspects of the polymeric network formation that have been obscured in the conventional theories in which the averages including linear polymers are solely considered. © 1995 John Wiley & Sons, Inc.     

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     

Charge densities for singlet and triplet electron pairs

Analytic properties of charge densities associated with singlet and triplet electron pairs, ρ₀( r ) and ρ₁( r ), are presented. In an N‐electron system with total spin S, distributions ρ₀( r ) and ρ₁( r ) are independent of the spin projection quantum number M (spin rotation invariance), as opposed to the usual spin‐up and spin‐down electron densities, ρ_α( r ) and ρ_β( r ). We derive equations showing that in the case of a wave function which is a spin‐eigenfunction, ρ₀( r ) and ρ₁( r ) are linear combinations of the total charge density ρ( r ) and the uncompensated spin density ρ_s( r )=[ρ_α( r )−ρ_β( r )]/2M. For a wave function which is not an eigenfunction of $\mathcal{S}^{2}$, no such relationship exists. In a related discussion, a definition of the high‐spin solution corresponding to a given spin‐unrestricted Hartree–Fock wave function is proposed, and a notion of effectively unpaired electrons is introduced. The distributions ρ₀( r ) and ρ₁( r ) are shown not to be invariant under spin coupling between isolated systems. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 651–660, 2000     

A localized electrons detector for atomic and molecular systems

The local value of the single-particle momentum provides a direct three-dimensional representation of bonding interactions in molecules. It is given exclusively in terms of the electron density and its gradient, and therefore is an ideal localized electrons detector (LED). The results introduced here extend to molecular systems our study of the single-particle local momentum in atomic systems (Bohórquez and Boyd in J Chem Phys 129:024110, 2008; Chem Phys Lett 480:127, 2009). LED is able to clearly identify covalent and hydrogen bonding interactions by depicting distinctive regions around the bond critical points, emerging as a complementary tool in conventional atoms in molecules studies. The local variable we introduce here is an intuitively interpretable 3D electron-pairs locator in atoms and molecules that can be computed either from theoretical or experimentally derived electron densities.     

Basis set quality. II. Information theoretic appraisal of various s- orbitals

The expectation values 〈r^k〉 (?2 ? k ? 4, k = 10), values of the charge density ρ(r) at selected points, and coefficients in the MacLaurin expansion of ρ(r) are used to test the quality of 71 orbital basis sets used for the atomic helium Hartree–Fock problem. These tests in position space are complementary to the momentum space tests previously carried out [Int. J. Quantum Chem. 21 , 419 (1982)]. Information theoretic measures with respect to either or both position and momentum space properties are subsequently defined and the orbitals are ranked accordingly. These measures indicate that, for a given orbital, momentum space properties are more poorly predicted than position space ones. Moreover an improvement in the virial ratio does not necessarily lead to a more balanced orbital with respect to position and momentum space properties. Basis sets containing Slater-type orbitals are again found to be rather accurate. The exponentially damped rational function is confirmed to be the outstanding two-parameter unconventional orbital.     

Slater's nonlocal exchange potential and beyond

The local density approximation (LDA) to the exchange potential V_x( r ), namely the ρ^1/3 electron gas form, was already transcended in Slater's 1951 paper. Here, using Dirac's 1930 form for the exchange energy density ? _x( r ), the Slater (Sl) nonlocal exchange potential V( r ) is defined by 2? _x( r )/ρ( r ). In spherical atomic ions, say the Be or Ne‐like series, this form V( r ) already has the correct behavior in both r → 0 and r → ∞ limits when known properties of the exchange energy density ? _x( r ) and the ground‐state electron density ρ( r ) are invoked. As examples, some emphasis will first be given to the use of the so‐called 1/Z expansion in such spherical atomic ions, for which analytic results can be obtained for both ? _x( r ) and ρ( r ) as the atomic number Z becomes large. The usefulness of the 1/Z expansion is directly demonstrated for the U atomic ion with 18 electrons by comparison with the optimized effective potential prediction. A rather general integral equation for the exchange potential is then proposed. Finally, without appeal to large Z, two‐level systems are considered, with specific reference to the Be atom and to the LiH molecule. In all cases treated, the Slater potential V( r ) is a valuable starting point, even though it needs appreciable quantitative corrections reflecting directly atomic shell structure. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Concerning atomic radii

The identification of the atomic radius with the distance from the nucleus to the position of the minimum in the internuclear electronic density is studied. It is shown that a set of statistically significant radii may be defined for the atoms of a given column of the periodic table bound to any of the atoms of another column. Sets of radii calculated by modified Anderson-Parr relations are presented. The values obtained are consistent with radii obtained using a minimum in electronic density criterion with electron densities calculated from molecular orbital wave functions or approximated by a sum of atomic densities.     

Analytic inhomogeneous electron liquid and its density for model spin-compensated two-electron atomic ions with Coulomb confinement: an exact nonrelativistic Hamiltonian

In earlier work, Howard and March (HM) proposed an analytic ground-state electron density ρ( r ) starting from the s-wave model of the He atom. Subsequently Ancarani has constructed by numerical methods a variational approach for this s-wave model with a lower energy than the HM result. This clearly means that the HM ρ( r ) is not the ground-state electron density of the He s-wave model. Therefore, we derive here an exact nonrelativistic Hamiltonian, with strong radial correlation plus Coulomb confinement, for which the HM ρ( r ) is indeed the ground-state electron density.     

Monotonicity properties of the atomic charge density function

The present knowledge of the monotonicity properties of the spherically averaged electron density ρ(r) and its derivatives, which comes mostly from Roothan-Hartree-Fock calculations, is reviewed and extended to all Hartree-Fock ground-state atoms from hydrogen (Z = 1) to uranium (Z = 92). In looking for electron functions with universal (i.e., valid in the whole periodic table) monotonicity properties, it is found that there exist positive values of α so that the function g_o(r; α) = ρ(r)/r^α is convex, and g₁(r;α) = −ρ′(r)/r^α is not only monotonically decreasing from the origin but also convex. This is, however, not the case for the function g₂(r; α) = ρ′(r)/r^α. Additionally, the conditions which specify values for β such that the function g_n(r; β) = (−1) ′ρ⁽ⁿ⁾(r)/r^β is logarithmically convex are obtained and numerically calculated for n = 0,1 in all neutral atoms below uranium. The last property is used to obtain inequalities of general validity involving three radial expectation values which generalize all the similar ones known to date, as well as other relationships among these quantities and the values of the electron density and its derivatives at the nucleus. © 1996 John Wiley & Sons, Inc.     

Covalent bond orders and atomic anisotropies from iterated stockholder atoms

Iterated stockholder atoms are produced by dividing molecular electron densities into sums of overlapping, near-spherical atomic densities. It is shown that there exists a good correlation between the overlap of the densities of two atoms and the order of the covalent bond between the atoms (as given by simple valence rules). Furthermore, iterated stockholder atoms minimise a functional of the charge density, and this functional can be expressed as a sum of atomic contributions, which are related to the deviation of the atomic densities from spherical symmetry. Since iterated stockholder atoms can be obtained uniquely from the electron density, this work gives an orbital-free method for predicting bond orders and atomic anisotropies from experimental or theoretical charge density data.     

Analytical Schwartz density applied to heavy two-electron ions

An analytical expression of the electron density function p(r) due to Schwartz for two-electron atomic systems is applied to a detailed study of density-dependent properties of relatively heavy two-electron ions. Comparison of the Schwartz results with those from accurate Hartree-Fock and Hylleraas wave functions shows that despite its simple yet analytical form, the Schwartz density has a quantitative applicability in the density study of two-electron atoms within the nonrelativistic framework. © 1997 John Wiley & Sons, Inc.     

Sizes of atoms and ions and covalency of bonding in molecules and crystals

A new system of atomic radii for the elements up to barium inclusive is constructed. Values of the radii are chosen so as the dependence between the dissociation energy of diatomic homonuclear molecules and a depth of atom overlapping is monotonous, and the scatter of data is minimal. The depth of overlapping is calculated as a difference between the sum of atomic radii and an experimental interatomic distance. Conclusions are made that: the radii of free atoms and ions are determined by the value of the electron density equal to 0.01 au; they considerably change in molecules and crystals only as a result of the charge transfer from cation to anion; covalent bonding is well described by the overlapping of free atoms (ions), confined by the surface of the given radius, and its energy depends upon the depth of overlapping of valence electron densities of atoms. A method of overlapping atoms is proposed for the approximate estimation of ionic sizes and charges in bound systems.     

Information entropies of many-electron systems

The Boltzmann–Shannon (BS ) information entropy S_ρ = ∫ ρ(r)log ρ(r)dr measures the spread or extent of the one-electron density ρ(r), which is the basic variable of the density function theory of the many electron systems. This quantity cannot be analytically computed, not even for simple quantum mechanical systems such as, e.g., the harmonic oscillator (HO ) and the hydrogen atom (HA ) in arbitrary excited states. Here, we first review (i) the present knowledge and open problems in the analytical determination of the BS entropies for the HO and HA systems in both position and momentum spaces and (ii) the known rigorous lower and upper bounds to the position and momentum BS entropies of many-electron systems in terms of the radial expectation values in the corresponding space. Then, we find general inequalities which relate the BS entropies and various density functionals. Particular cases of these results are rigorous relationships of the BS entropies and some relevant density functionals (e.g., the Thomas–Fermi kinetic energy, the Dirac–Slater exchange energy, the average electron density) for finite many-electron systems. © 1995 John Wiley & Sons, Inc.     

Exchange energy density and exchange potential via a hartree–fock plus mp2 study of the electron liquid in the ground-state conformer of glycine

Very recent criticisms of existing exchange-correlation functionals by Wanko et al. applied to systems of biological interest have led us to reopen the question of the ground-state conformer of glycine: the simplest amino acid. We immediately show that the global minimum of the Hartree–Fock (HF) ground-state leads to a planar structure of the five non-hydrogenic nuclei, in the non-ionized form NH₂–CH₂–COOH. This is shown to lie lower in energy than the zwitterion structure NHB₃ ⁺–CH₂–COO^?, as required by experiment. Refinement of the nuclear geometry using second-order Møller–Plesset perturbation theory (MP2) is also carried out, and bond lengths are found to accord satisfactorily with experimentally determined values. The ground-state electron density for the MP2 geometry is then redetermined by HF theory and equidensity contours are displayed. The HF first-order density matrix γ( r , r ′) is then used to obtain similar exchange-energy density (ε _x ( r )) contours for the lowest conformer of glycine. At first sight, their shape looks almost the same as for the density ρ( r ), which seems to vindicate the LDA proportional to ρ( r )^3/4. However, by way of an analytically soluble model for an atomic ion, it is shown that this has to be corrected to obtain an accurate HF exchange energy E_x as the volume integral of ε _x ( r ). Finally, recognizing that for larger amino acids, the use of HF plus MP2 perturbation corrections will become prohibitive, we have used the HF information for ε _x ( r ) and ρ( r ) to plot the truly non-local exchange potential proposed by Slater, from the density matrix γ( r , r ′). This latter calculation should be practicable for large amino acids, but there adopting Becke's one-parameter form of ε _x ( r ) correcting LDA exchange. Some future directions are suggested.     

Analytical density-dependent representation of Hartree—Fock atomic potentials

A procedure to represent atomic electron charge densities [L. Fernandez Pacios, J. Phys. Chem., 95 , 10653 (1991); J. Phys. Chem., 96 , 7294 (1992)] is here generalized to obtain simple analytical functions for potential energy contributions. Based upon suitable functions to describe atomic electron densities in a physically meaningful form, the procedure is developed to define density-dependent analytical expressions for the electrostatic (classical) and exchange (quantum) potentials by means of proper approximate functionals. Calculations of correlation energies by using various density-functional approaches are also performed. The whole scheme is used to represent Hartree–Fock limit atomic wave functions by Clementi–Roetti. This way, a set of analytically simple, nonbasis set-dependent functions are defined with the aim to be further implemented in energy decomposition schemes for molecular interactions studies using atomic instead of electronic building blocks. © 1993 John Wiley & Sons, Inc.     

Momentum space properties of atoms

A systematic investigation of momentum densities Π (p), Compton profiles J (q), and internally folded densities B (r) has been made for the ground states of the atoms from H to Kr. The data are presented in reduced form by means of the characteristic momentum p* = q* and length r* = 1/p*, where p* = [J (0)/2πΠ (0)]^½. The results indicate that p* and J (0) contain the gross qualitative behavior of the atomic properties in momentum space. They also show that subtle effects over and above the gross scale and size are mostly periodic.