Sobolev Spaces with only Trivial Isometries

We will give some conditions for Sobolev spaces on bounded Lipschitz domains to admit only trivial isometries.     

The convergence properties of direct methods of energy minimization with respect to linear coefficients in the LCAO-MO-SCF approach

It is shown that in the LCAO-MO-SCF problem, if the molecular orbital orthonormality constraints are introduced in the manner first suggested by Fletcher, then the Hessian of the problem is singular. It is suggested that this singularity may well account for the slow convergence observed using direct energy minimization methods to solve the SCF problem. Ways of avoiding the consequences of this singularity are discussed.     

Molecular structure calculations without clamping the nuclei

A number of recent papers have considered ways in which molecular structure may be calculated when both the electrons and the nuclei are treated from the outset as quantum particles. This is in contrast to the conventional approach in which the nuclei initially have their positions fixed and so merely provide a potential for electronic motion. The usual approach is generally assumed to be justified by the 1927 work of Born and Oppenheimer. In this paper we discuss what precisely might be anticipated in the way of molecular structure from a mathematical consideration of the spectral properties of the full Coulomb Hamiltonian, to what extent the very idea of molecular structure might be dependent upon treating the nuclei simply as providing a potential and the extent to which the work of Born and Oppenheimer can be used to support this position.     

Non-empirical LCAO-MO-SCF-CI calculations on organic molecules with Gaussian type functions

This paper (Part I) describes the theoretical and computational bases of some non-empirical calculations on small organic molecules to be reported in later papers (Parts II et seq). Approximate solutions for the usual fixed nucleus electronic Hamiltonian, in the term of wave functions composed of Slater determinants, are discussed, with particular emphasis on their computational utility. Possible choices of basis functions, from which to form the determinants are examined, and the advantages of Gaussian type functions (GTF) centered on the component atoms are pointed out. Some of the properties of molecules which can be calculated using such approximate wave functions are outlined. Finally an attempt is made to discuss the current limitations of non-empirical calculations of the type described here, and some guesses are made about their future. Brief outlines as a set of appendices are given of the mathematical formalism and computational details of the calculations.     

Darboux transformations for higher-rank Kadomtsev-Petviashvili and Krichever-Novikov equations

It is shown that the action of a special rank 2 or rank 3 Darboux transformation, calledtransference, upon a pair of commuting ordinary differential operators of orders 4 and 6 implements the Abelian sum on their elliptic joint spectrum. A consequence of this is that, under the deformation of these commuting operators by the KP flow, every rank 2 KP solution corresponds to a solution of the Krichever-Novikov (KN) equation, and vice versa, with the transference process providing the correspondence between (2+1) and (1+1) dimensions. For a singular joint spectrum, it is further shown that transference at the singular point produces a correspondence between solutions of the singular KN equation and those of the KdV equation. These correspondences are illustrated by considering examples of a nondecaying rational KdV or Boussinesq solutions and the corresponding rational, singular-KN and rational KP solutions which the transference process generates.     

Disordered ground states for classical discrete-state problems in one dimension

It is known that one-dimensional lattice problems with a discrete, finite set of states per site generically have periodic ground states (GSs). We consider slightly less generic cases, in which the Hamiltonian is constrained by either spin (S) or spatial (I) inversion symmetry (or both). We show that such constraints give rise to the possibility ofdisordered GSs over a finite fraction of the coupling-parameter space—that is, without invoking any nongeneric fine tuning of coupling constants, beyond that arising from symmetry. We find that such disordered GSs can arise for many values of the number of statesk at each site and the ranger of the interaction. The Ising (k=2) case is the least prone to disorder:I symmetry allows for disordered GSs (without fine tuning) only forr5, whileS symmetry never gives rise to disordered GSs.     

A configuration interaction study of phosphine using bonded functions

A computational study is made of the effect of basis set upon the energy, properties and inversion barrier of the phosphine molecule. The calculations are performed at both the SCF and CI level. The flexibility of the double zeta basis is discussed in the light of the results.     

Ultraviolet protection factors for clothing: an intercomparison of measurement systems

In recent years the need to standardize measurement protocols for quantifying the degree of ultraviolet radiation (UVR) protection provided by clothing has led to the introduction of a number of standards around the world. To date, these standards have specified spectral measurements of UVR transmission by clothing and fabrics. Development of a standard test method has become an important part of the testing process, and this article presents results from an intercomparison involving 10 independent testing laboratories and 11 different UVR transmission measurement instruments. In addition to comparing the measured ultraviolet protection factors (UPF), this intercomparison also incorporates detailed scan results from all 10 laboratories and highlights differences in performance of the various instruments in different wavelength regions. Careful examination of these differences can indicate where changes to the systems could be made to allow improvements both in equipment performance and in agreement of the final results. The variability in the measurements of UPF in this study suggest that the protection categories in standards may need to be broadened.     

An experimental and theoretical study of competitive adsorption at the n-heptane/water interface

A model to calculate the interfacial concentration of competing surface active species in a two-phase oil/water system was developed. To enable the calculation of the surface excess of 2-hydroxy-5-nonylacetophenone oxime (HNAPO, active ingredient of LIX 84) in the presence of surfactants competing for interfacial area, an interfacial adsorption competition model was derived for noninteracting surface active species in a n-heptane/aqueous system, assuming ideal enthalpy and entropy of mixing. The model was found to be valid for HNAPO in the presence of sodium dodecyl sulfate (SDS) or dodecyldimethyl(3-sulfopropyl)ammonium (DDSA). In the case of dodecyltrimethylammonium chloride (DTAC) or octa(ethylene glycol) mono-n-dodecyl ether (C12E8) as the competing surfactants with HNAPO, the predicted surface excess values from the model fit less favorably. The difference was shown to not be due to nonideal entropy of mixing.