Inner and outer radial density functions in many-electron atoms

When any two electrons are considered simultaneously, the radial density function D(r) in many-electron atoms is shown to be rigorously separated into inner D _<(r) and outer D _>(r) radial densities. Accordingly, radial properties such as the electron–nucleus attraction energy V _en and the diamagnetic susceptibility χ _d are the sum of the inner and outer contributions. The electron–electron repulsion energy V _ee has an approximate relation with the minus first moment of the outer density D _>(r). For the 102 atoms He through Lr in their ground states, different characteristics of local maxima in the radial densities D _<(r), D _>(r), and D(r) are reported based on the numerical Hartree-Fock wave functions. Relative contributions of the inner and outer components to V _en and are also discussed for these atoms.     

Direct and exchange contributions to inner and outer radii in many-electron atoms

Within the Hartree-Fock framework, the spinless two-electron density function Γ (r ₁, r ₂) consists of direct Γ_di (r ₁, r ₂) and exchange Γ_ex (r ₁, r ₂) parts. Accordingly, the inner and outer radii in many-electron systems are rigorously separated into the direct and exchange contributions, i.e., and . It is generally shown that and , where is the usual average radius of an electron. Numerical examinations of the direct and exchange contributions for the 102 atoms from He to Lr in their ground states find that the electron exchange works to decrease and increase . However, the exchange parts are very small and the direct parts essentially govern the inner and outer radii.     

Non-alkane behavior of cyclopropane and its derivatives: characterization of unconventional hydrogen bond interactions

Eight cyclopropane derivatives (Δ − R) have been modeled, with R = −H, −CH₃, −NH₂, −C ≡ CH, −C ≡ CCH₃, −OH, −F and −C ≡ N. All geometries have been fully optimized at the MP2/ AUG-cc-pVTZ level of calculations. Natural bond orbital analyses reveal extra p character (sp^λ, λ > 3) in the C-C bonds of the cyclopropyl rings. The banana-like σ _CC bonds in the rings are described in detail. Alkene-like complexes between Δ − R molecules and hydrogen fluoride are identified. These weakly bonded complexes are formed through unconventional hydrogen bond interactions between the hydrogen atom in the HF molecule and the carbon–carbon bonds in the cyclopropane ring. A topological analysis of the electronic charge density and its Laplacian has been used to characterize the interactions. The possible relevance of such complexes in the modeling of substrate–receptor interactions in some anti-AIDS drugs is discussed. Contribution to the Serafin Fraga Memorial Issue.     

Atomic electron-pair distances and subshell radii in position and momentum space

For the 53 neutral atoms from He to Xe in their ground states, the average distances < u> _{n l , n ′ l ′} in position space and < v> _{n l , n ′ l ′} in momentum space between an electron in a subshell nl and another electron in a subshell n ^′ l ^′ are studied, where n and l are the principal and azimuthal quantum numbers of an atomic subshell, respectively. Analysis of 1700 subshell pairs shows that the electron-pair distances < u> _{n l , n ′ l ′} in position space have an empirical but very accurate linear correlation with a one-electron quantity U _{n l , n ′ l ′}≡L _r +S _r ²/(3L _r ), where L _r and S _r are the larger and smaller of subshell radii < r> _{n l} and < r> _{n ′ l ′}, respectively. The correlation coefficients are never smaller than 0.999 for the 66 different combinations of two subshells appearing in the 53 atoms. The same is also true in momentum space, and the electron-pair momentum distances < > _{n l , n ′ l ′} have an accurate linear correlation with a one-electron momentum quantity V _{n l , n ′ l ′}≡L _p +S _p ²/(3L _p ), where L _p and S _p are the larger and smaller of average subshell momenta < p> _{n l} and < p> _{n ′ l ′}, respectively. Trends in the proportionality constants between < u> _{n l , n ′ l ′} and U _{n l , n ′ l ′} and between < > _{n l , n ′ l ′} and V _{n l , n ′ l ′} are discussed based on a hydrogenic model for the subshell radial functions. Received: 8 April 1998 / Accepted: 6 July 1998 / Published online: 18 September 1998     

He photoionization: βN and σN below N = 5 and 6 thresholds

In this paper we use the results of a K-matrix method calculation to evaluate the ionization spectrum of He between the N = 4 and 6 parent ion thresholds. Partial dipole asymmetry parameters β _N below N = 6 threshold have been computed for the first time and compared to the available preliminary experimental findings by Jiang and Püttner [http://www.diss.fu-berlin.de/2006/269]. Dedicated to Prof. Serafin Fraga.     

Interfacial and aggregation properties of aqueous sodium <Emphasis Type="Italic">N</Emphasis>-dodecanoylsarcosinate solutions at different pH

The pH dependence of an anionic surfactant, sodium N-dodecanoylsarcosinate (SLAS), has been studied by measuring interfacial tension, fluorescence, dynamic light scattering, etc., in aqueous solutions with phosphate and borate buffers. The interfacial tension (γ) of SLAS decreases remarkably with a pH decrease and is constant at pH > 7.3. The observed values for the critical micelle concentration (cmc) and the surfactant concentration at which its γ value is reduced by 20 mN/m from that of pure water (C ₂₀) decrease with a pH decrease, while those also become constant at pH > 6.5 and >7.3, respectively. On the other hand, the interfacial excess of SLAS increases at pH < 7.3. These interfacial behaviors have been further investigated by the addition of Tl⁺ which replaces Na⁺ of SLAS. The observed γ values of LAS⁻ with the different counter cations are in the order of H⁺ < Tl⁺ < Na⁺. In order to reveal aggregation properties of SLAS, the aggregation number (N _agg), the micropolarity, the hydrodynamic radius (R _h) of micelle, and the fluorescence anisotropy of Rhodamine B (r) have been evaluated at various pHs. The N _agg value shows a decreasing tendency with a pH increase. The I ₁/I ₃ ratio and the R _h values do not strongly depend on pH. The r value decreases until pH 7 and remains constant at pH > 7.0. These interfacial and micelle properties have been discussed in detail considering the electrostatic interaction and the molecular structures of the hydrophilic headgroup.     

Electronic states of XC<Subscript>3</Subscript>H<Subscript>3</Subscript>Si five-membered rings (X = CH,N, P,and As)

Stable configurations of XC₃H₃Si five-membered rings, 1 _X and 2 _X (X = CH, N, P, and As) in the singlet and triplet states are found at B3LYP/6-311++G** level of theory. All the singlet states of 1 _X and 2 _X have lower potential energy than the triplet state. The ΔG _s−t differences between the singlet and triplet states of 1 _X and 2 _X changes at the B3LYP/6-311++G** level in the order (in kcal/mol): 1 _N (−17.56) > 1 _CH (−15.26) > 1 _P (−4.96) > 1 _As (−3.45) and 2 _CH (−15.26) > 2 _N (−9.21) > 2 _P (−7.39) > 2 _As (−6.15), respectively.     

On the spectral radius of graphs with connectivity at most <Emphasis Type="Italic">k</Emphasis>

In this paper, we study the spectral radius of graphs of order n with κ(G) ≤ k. We show that among those graphs, the maximal spectral radius is obtained uniquely at , which is the graph obtained by joining k edges from k vertices of K _n-1 to an isolated vertex. We also show that the spectral radius of will be very close to n − 2 for a fixed k and a sufficiently large n.     

Structural behavior of hyperbranched polymers in solvents of various qualities: Brownian dynamics simulation

Structural characteristics of third- and fourth-generation dendrimers and irregular hyperbranched polymers of various topologies equal to regular dendrimers in terms of molecular mass in solutions of various topologies are studied via Brownian dynamics simulation. Terminal and inner groups of the studied molecules feature different sensitivities to the quality of a solvent. The mean-square of the radius of gyration, <R _g²>; hydrodynamic radius <R _hyd>; and the radial density distribution functions of monomer units, ρ(r), are calculated. A change in the structural characteristics of molecules induced by worsening in the quality of the solvent is affected by molecular mass, the amount of terminal groups, the ratio between the amount of terminal groups and the total amount of monomer units, and the topology of hyperbranched macromolecules. A comparison of simulation results with the available experimental data makes it possible to use the computersimulation data for determination of the topology of polymers.     

Additive intermolecular potentials from ab initio calculations: trends in Rg2–dihalogen van der Waals trimers

In the present study, the validity of the pairwise additivity of the interactions, derived from the Rg₂ and Rg-dihalogen CCSD(T) potentials, is investigated by means of ab initio electronic structure and quantum-mechanical calculations. The topology of the potential surfaces of three different types of Rg₂–dihalogen vdW complexes is studied and general trends within the Rg₂–dihalogen family are discussed. Calculations of vibrational energies, including all five intermolecular degrees of freedom, are performed on such pairwise-additive potentials. The results are compared with experimental data from high-resolution spectroscopy, and provide further information on the additivity of the intermolecular forces for the He₂-dihalogen trimers. Contribution to the Serafin Fraga Memorial Issue.     

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.     

Steric effects on the singlet-triplet energy gaps of five-membered ring R<Subscript>2</Subscript>C<Subscript>4</Subscript>H<Subscript>2</Subscript>M

Thermal internal energy gaps, ΔE _s−t; enthalpy gaps, ΔH _s−t; Gibbs free energy gaps, ΔD _s−t, between singlet (s) and triplet (t) states of R₂C₄H₂M (M = C, Si, and Ge) were calculated at B3LYP/6-311++G** level of theory. The ΔG _s−t of R₂C₄H₂C was increased in the order (in kcal/mol): R = −CH₃ (−10.51) > −H (−9.59) > i-Pr (−9.51) > t-Bu (−8.98). While, the ΔG _s−t of R₂C₄H₂Si and R₂C₄H₂Ge were increased in the order (in kcal/mol): −CH₃ (17.01) > i-Pr (15.30) > −H (15.26) > t-Bu (14.35) and -H (22.79) > −CH₃ (22.69) > i-Pr (21.66) > t-Bu (21.01), respectively.     

Oxygen desorption from polycrystalline palladium: Thermal desorption of O<Subscript>2</Subscript> from a chemisorbed layer of O<Subscript>ads</Subscript> in the course of the decomposition of PdO surface oxide and in the release of oxygen from the bulk of palladium

The desorption of oxygen from polycrystalline palladium (Pd(poly)) was studied using temperature-programmed desorption (TPD) at 500–1300 K and the amounts of oxygen absorbed by palladium (n) from 0.05 to 50 monolayers. It was found that the desorption of O₂ from Pd(poly), which occurred from a chemisorbed oxygen layer (O_ads), in the release of oxygen from a near-surface metal layer in the course of the decomposition of PdO surface oxide, and in the release of oxygen from the bulk of palladium (O_abs), was governed by repulsive interactions between O_ads atoms and the formation and decomposition of O_ads-Pd*-O_abs structures (Pd* is a surface palladium atom). At θ ≤ 0.5, the repulsive interactions between O_ads atoms (ɛ_aa = 10 kJ/mol) resulted in the desorption of O₂ from Pd(poly) at 650–950 K. At 0.5 ≤ n ≤ 1.0, the release of inserted oxygen from a near-surface palladium layer occurred during TPD in the course of the migration of O_abs atoms to the surface and the formation-decomposition of O_ads-Pd*-O_abs structures. As a result, the desorption of O₂ occurred in accordance with a first-order reaction with a thermal desorption (TD) peak at T _max ∼ 700 K. At 1.0 ≤ n ≤ 2.0, the decomposition of PdO surface oxide occurred at a constant surface cover-age with oxygen during TPD in the course of the formation-decomposition of O_ads-Pd*-O_abs structures. Because of this, the desorption of O₂ occurred in accordance with a zero-order reaction at low temperatures with a TD peak at T _max ∼ 675 K. At 1.0 ≤ n ≤ 50, oxygen atoms diffused from deep palladium layers in the course of TPD and arrived at the surface at high temperatures. As a result, O₂ was desorbed with a high-temperature TD peak at T > 750 K.     

Analytical Hartree-Fock electron densities for singly charged cations and anions

The Hartree-Fock (HF) electron density has an important property that it is identical to the unknown exact density to the first order in the perturbation theory. We generate the spherically averaged HF electron density ρ(r) by using the numerical HF method for the singly charged 53 cations from Li⁺ to Cs⁺ and 43 anions from H⁻ to I⁻ in their ground state. The resultant density is then accurately fitted into an analytical function F(r), which is expressed by a linear combination of basis functions r ⁿⁱ exp(−ζ _i r). The present analytical approximation F(r) has the following properties: (1) F(r) is nonnegative, (2) F(r) is normalized, (3) F(r) reproduces the HF moments <r ^k > (k=−2 to +6), (4) F(0) is equal to ρ(0), (5) F ^′(0) satisfies the cusp condition and (6) F(r) has the correct exponential decay in the long-range asymptotic region. The present results together with our previous ones for neutral atoms provide a compilation of accurate analytical approximations of the HF electron densities for all the neutral and singly charged atoms with the number of electrons N≤54. Received: 11 July 1997 / Accepted: 27 August 1997     

Atomic one- and two-electron subshell moments in position and momentum spaces

For 357 subshells of the 53 neutral atoms He through Xe in their ground states, the two-electron intracule (relative motion) <u ^k > _nl and extracule (center-of-mass motion) <R ^k > _nl subshell moments in position space are examined as well as their counterparts <v ^k > _nl and <P ^k > _nl in momentum space, where n and l are the principal and azimuthal quantum numbers of the atomic subshell, respectively. It is clarified that between the intracule and extracule moments the “2 ^k -rule” is strictly valid, which means <u ^k > _nl = 2 ^k <R ^k > _nl and <v ^k > _nl = 2 ^k <P ^k > _nl for any nl subshell. Theoretical analysis also proves that for a particular case of k = +2, two relations <u ²> _nl = (N _nl −1)<r ²> _nl and <v ²> _nl = (N _nl −1)<p ²> _nl hold exactly, where N _nl (≥2) is the number of electrons in the subshell nl, and <r ^k > _nl and <p ^k > _nl are the familiar one-electron subshell moments in position and momentum spaces, respectively. The latter equality establishes a new and rigorous relation between the second electron-pair moments in momentum space and the total energy of an atom through the virial theorem. For k=+1, −1, and −2, the numerical Hartree-Fock results for the 357 subshells show that there are approximate but accurate linear relations between <u ^k > _nl and <r ^k > _nl and between <v ^k > _nl and <p ^k > _nl , in which the proportionality constant in each space depends on n,l, and k. Received: 27 April 1998 / Accepted: 29 May 1998 / Published online: 28 August 1998     

Adsorption of potential-determining ions on porous glasses of different compositions

The adsorption characteristics are studied for nano- and ultraporous glasses (PGs) produced from sodium borosilicate glasses and a glass containing small amounts of fluoride ions and phosphorus oxide by acid (HCl) leaching and additional alkaline (KOH) and thermal treatment. The surface charges σ₀ of PGs are determined by continuous potentiometric titration in 10⁻³−1 M NaCl, KCl, and (C₂H₅)₄NCl solutions. Only negative surface charges of PGs are observed for all investigated systems. The |σ₀| value is predetermined by the following factors: the composition of PG, the pore radius in the nanometer region (r ≤ 13 nm), the specificity of counterions, the content of secondary silica in the pore space, and the temperature of the additional thermal treatment of the membranes. The introduction of fluoride ions and phosphorus oxide into sodium borosilicate glass, an increase in the pore sizes and the amount of the secondary silica in PGs, and a rise in the specificity of counterions enhance the |σ₀| values, which decrease with a rise in the temperature of the thermal treatment due to the surface dehydration and dehydroxylation. For ultraporous glasses (r > 13 nm), the surface charge is almost independent of the pore radius.     

Simplified accurate approximation for the Kohn-Sham potential using the KLI method

The method of Krieger, Li, and Iafrate (KLI) [Phys. Rev. A46, 5453 (1992) and A47, 165 (1993)] is employed to calculate the Kohn-Sham (KS) potential, V_κσ, for the exchange-only case in which the electron-electron interaction between “core” electrons in the Hartree-Fock exchange energy functional is treated in the local-spin-density (LSD) approximation with and without self-interaction-correction (SIC). The resulting V_κσ(r) maintains the important analytic properties exhibited by the exact KS potential. When the core is taken to include all occupied states except those in the last two occupied subshells of the atom, we find that properties strongly dependent on the valence electron states continue to be accurately approximated. In particular, when the LSDSIC approximation is employed, we find the results of self-consistent calculations of the ionization potential and electron affinity are within 0.3 mRy of the exact KS results and that the energy eigenvalue corresponding to the highest energy occupied orbital and <r²> have an average error of a few tenths of 1% for both atoms and negative ions for Z ≤ 20. Similarly, slightly less accurate results are obtained when the LSD approximation is employed. These results suggest that the KLI method may be accurately and more easily applied to multiatom systems when this additional approximation is made. © 1997 John Wiley & Sons, Inc.     

The effect of stoichiometry on curing and properties of epoxy–clay nanocomposites

Epoxy–clay nanocomposites have been prepared with an organically modified montmorillonite. The epoxy network was based on diglycidyl ether of bisphenol A (DGEBA) cured with diaminodiphenylmethane (DDM). The stoichiometry DGEBA–DDM was varied, the molar ratio of amine hydrogen/epoxy groups, r, ranged from 0.85 to 1.15. The influence of stoichiometry on curing and properties of the nanocomposites was studied using differential scanning calorimetry, dynamic mechanical thermal analysis and X-ray diffraction. All nanocomposites had intercalated clay structures. The clays accelerated the curing reaction whose rate was also increased when increasing r. The heat of reaction, −ΔH (J/g epoxy), increased as r increased, reaching a constant value for r ≥ 1. In the presence of clays −ΔH was lower than in the neat DGEBA–DDM. The glass transition temperature (T _g) of the neat epoxy thermosets reached a maximum at r = 1; however, the nanocomposites showed the T _g maximum at 0.9 < r < 1. The presence of clay lowered the T _g for r > 0.94 and raised T _g for r ≤ 0.85. The elastic modulus of neat epoxy thermosets reached a maximum in the rubber state and a minimum in the glassy state at r = 1. The nanocomposites showed similar behavior, but the maximum and the minimum values of the elastic modulus were reached at stoichiometry r < 1. The comparison of the properties of neat epoxy with those of the nanocomposites varying the stoichiometry indicates that the clay itself induces stoichiometric changes in the system.     

Mean‐Square Radius of Gyration and Degree of Branching of Highly Branched Copolymers Resulting from the Copolymerization of AB2 With AB Monomers

Summary: The evolution of the various structural units incorporated into hyperbranched polymers formed from the copolymerization of AB₂ and AB monomers has been derived by the kinetic scheme. The degree of branching was calculated with a new definition given in this work. The degree of branching monotonously increased with increasing A group conversion (x) and the maximum value could reach 2r/(1 + r)², where r is the initial fraction of AB₂ monomers in the total. Like the average degree of polymerization, the mean‐square radius of gyration of the hyperbranched polymers increased moderately with A group conversion in the range x < 0.9 and displayed an abrupt rise when the copolymerization neared completion. The characteristic ratio of the mean‐square radius of gyration remained constant for the linear polymers. However, the hyperbranched polymers did not possess this character. In comparison with the linear polymerization, the weight average and z‐average degree of polymerization increased due to the addition of the branched monomer units AB₂ and the mean‐square radius of gyration decreased quickly for the products of copolymerization.

     

On the determination of the average molecular weight,radius of gyration,and mass/length ratio of polydisperse solutions of polymerizing rod‐like macromolecular monomers by multi‐angle static light scattering

The angular dependence of the light scattered from polydisperse solutions of rod‐like or worm‐like linear polymers formed by the polymerization of rod‐like macromolecular monomers was studied with the aid of computer simulations. Except at very low conversion degrees, these ensembles are characterized by curved Zimm‐like plots. An appraisal of the use of polynomial fittings for the derivation of the weight‐average molecular weight (M_w) and of the z‐average square radius of gyration (<R_g²>_z) from such plots is presented and discussed. Depending on the average size and shape distribution, the use of polynomial fittings allows the applicability of the Zimm method well beyond the standard condition q²<R_g²>_z<<1, q being the scattering wavevector. In addition, the derivation of the w/z‐average mass/length ratio M_L of the polymers from only partially linear Casassa plots, from which large errors in the derived M_L values can be made, is analyzed. By combining the Casassa method with the complementary Holtzer plots, it is usually possible to assess the reliability of the results and give an estimate of their accuracy. However, it was also found that apparently linear Casassa plots, allowing a good estimate of the M_L value, may arise in particular situations even when the Holtzer plot would indicate otherwise.