Environment potential in Hartree-Fock-Slater formalism

A new formulation of the environment potential approach (EPA) is developed within the Hartree-Fock-Slater framework. As the results of illustrative applications to C₅ cluster which is the smallest unit of diamond lattice, EPA is shown to annul efficiently the fictitious-surface effects which are inevitably brought into a finite-size-cluster model.     

Conformation of linear polyethylene in 1-chloronaphthalene: Light scattering characterization of polymers. IV

Light scattering measurements were carried out on a linear polyethylene sample NBS 1475 in 1-chloronaphthalene at 135 and 115°C to determine the weight-average molecular weight, the second virial coefficient A₂, and the z-average mean-square radius of gyration. By use of these results, the system is analyzed in terms of the interpenetration function Ψ for A₂. Observed values of A₂ are rather large but the excluded volume is nevertheless relatively small. Such behavior seem to be similar to that of semiflexible polymers. The characteristic ratio C_n,LS as determined by light scattering is found to be almost twice the literature value of 6.7, which was obtained from viscosity measurements. This discrepancy is explained by comparing the theoretical value of the Flory viscosity parameter Φ at the nondraining limit with values calculated from the light scattering results.     

Nitrogen-15 nuclear magnetic resonance spectroscopy. Natural abundance nitrogen-15 spectra of the cis and trans isomers of secondary alkylformamides

The ¹⁵NMR chemical shifts were measured of a number of N-substituted formamides and acetamides at the natural abundance level. The ratios of the cis and trans isomers for several N-alkylformamides were also determined. Substituent effects on the ¹⁵N chemical shifts of formamides are compared with those of some other nitrogen-containing compounds. There is a consistent pattern of behavior of the one bond spin–spin coupling constants [¹J(¹⁵NH)] wherein the trans isomers of N-alkylformamides are larger than those of the cis isomers.     

Further results on graph equations for line graphs and n-th power graphs

We present solutions of seven graph equations involving the line graph, complement and n-th power operations. One such equation $\text{L(G)}{}^{\mathrm{n}}\text{=}\text{G}\text{?}$ generalizes a result of M. Aigner. In addition, some comments are made about graphs satisfying $\text{G}{}^{\mathrm{n}}\text{=}\text{G}\text{?}$.     

Principle of inclusion-exclusion on semilattices

We introduce an enumeration theorem under lattice action. Let L be a finite semilattice and Ω be a nonempty set. Let $\text{f: L → P(Ω)}$ be a map satisfying f(x ? y) ? f(x) ∩ f(y), where ? and $\text{P}$(Ω) mean “join” and the power set of Ω, respectively. Then

$\text{m}\text{∪}\text{x?L}\text{?(x)}\text{=}\text{Σ}\text{c?C}\text{(?1)}{}^{\mathrm{l(c)}}\text{m}\text{∩}\text{x?c}\text{?(x)}$

, where C is the set of all chains in L and l(c) denotes the length of a chain c. Also the theorem can be dualized. Furthermore, we describe two applications of the theorem to a Boolean lattice of sets and a partition lattice of a set.