Essential spectrum and -solutions of one-dimensional Schrödinger operators

In 1949, Hartman and Wintner showed that if the eigenvalue equations of a one-dimensional Schrödinger operator possess square integrable solutions, then the essential spectrum is nowhere dense. Furthermore, they conjectured that this statement could be improved and that under this condition the essential spectrum might always be void. This is shown to be false. It is proved that, on the contrary, every closed, nowhere dense set does occur as the essential spectrum of Schrödinger operators which satisfy the condition of existence of -solutions. The proof of this theorem is based on inverse spectral theory.

     

Salting coefficients for gases in seawater from scaled-particle theory

Experimental values of the salting coefficients for He, Ne, Ar, O₂, and N₂ in seawater are compared with values calculated from scaled-particle theory. The agreement is reasonably good; the average difference between calculated and observed values at 25°C is 0.007. Scaled-particle theory predicts correctly thatk _s should decrease as the temperature increases and that this effect should be most pronounced at low temperatures. However, the predicted magnitude ofdK _s/dt is only about half of that observed.     

Swelling of physically cross-linked copolymers in water vapour

A general thermodynamic equation for the swelling of a cross-linked polymer system in the vapour of a swelling agent has been derived under isothermal and isobaric conditions. The equation is used to describe the equilibrium uptake of water by elastomers Arnitel^® (DSM), which are hard-soft-segment block copolymers in which the hard segments poly(butylene terephthalate) crystallise and are responsible for a physical cross-linking. The gravimetrically determined degrees of swelling of different elastomers in water vapour of various partial pressures are analysed and discussed by the application of an extended semi-empirical swelling equation of Flory-Huggins-Staverman-van Santen.     

On the mathematical formulation and interpretation of LACO MO theory

The mathematical basis of LCAO MO theory is studied, both within the Hartree-Fock approximation and in more exact formulations. The basic LCAO expansion for molecular orbitals ¦> in terms of atomic orbitals ¦x> is conveniently written ¦> = ¦x> S ^–1 B where S is the overlap matrix for atomic orbitals and B is the matrix of atomic orbital-molecular orbital overlaps. It is suggested that matrices P and Q, defined by P=B B and Q=BnB where n is the matrix of molecular orbital occupation numbers, are appropriate to the interpretation of molecular calculations in terms of atomic orbital components, electronic populations and the degree of bonding. Implications for Hartree-Fock calculations are investigated.     

Effects of Strong Ion‐Pairing on the Electrochemical Reduction of Cyclooctatetraene in Tetrahydrofuran in the Presence of Lithium Ion. Peak Coalescence Does Not Imply Potential Inversion

The first study of the voltammetric reduction of cyclooctatetraene (COT) in tetrahydrofuran (THF) in the presence of lithium ion is reported. A single wave is observed at ?2.23 V vs. Ag/0.1 M AgNO₃. Density functional calculations have been carried out on a variety of COT/Li/THF species in order to clarify the nature and role of ion pairing in this system. The dominant species in solution are the COT/Li/(THF)₂ anion radical and the COT/Li₂/(THF)₄ dianion. Computer simulations have been carried out to further understand the effects of ion pairing on the reduction. The simulations show that coalescence of two waves into one can occur in the presence of strong ion pairing even when the second reduction potential is negative of the first.     

The Mathematical Theory of Ito Diffusions on Hypersurfaces, with Applications to NMR Relaxation Problems

A mathematical framework for translational Brownian motion on hypersurfaces is presented, using an imbedding of the surface and Ito diffusions in the ambient space. This includes a survey of Ito calculus and differential geometry. Computational methods for time correlation functions relevant to spin relaxation studies on curved interfaces are given, and explicit calculations of time correlation functions and order parameters for a Rippled surface are presented.     

Direct calculation of ionization potentials of closed-shell atoms and molecules

Vertical ionization potentials, electron affinities and information about quasi-particles can be obtained by using the technique of the single-particle propagator. The expansion of the self-energy part up to third order perturbation theory can be evaluated numerically, but does not lead, in most cases, to satisfying results. A theoretical and numerical analysis of the diagrammatic expansion of the self-energy part requires the introduction of a renormalized interaction and renormalized hole and particle lines.     

Three-dimensional aromaticity in deltahedral boranes and carboranes

Methods derived from topology and graph theory indicate that the deltahedral boranes B _n H _n ^2– and the corresponding carboranes C₂B_n–2H _n (6 n 12) may be regarded as three-dimensional delocalized aromatic systems in which surface bonding and core bonding correspond to -bonding and -bonding, respectively, in planar polygonal two-dimensional hydrocarbons CinnH _n ^(n–6)+ (n=5/7). The two extreme types of topologies which may be used to model core bonding in deltahedral boranes and carboranes are the deltahedral (D _n ) topology based on the skeleton of the underlying deltahedron and the complete (K _n ) topology based on the corresponding complete graph. Analyses of the Hoffmann-Lipscomb LCAO extended Hückel computations, the Armstrong—Perkins—Stewart self-consistent molecular orbital computations, and SCF MOab initio GAUSSIAN-82 computations on B₆H₆ ^2– indicate that the approximation of the atomic orbitals by the sum of the molecular orbitals, as is typical in modernab initio computations, leads to significantly weaker apparent core bonding approximated more closely by deltahedral (D _n ) topology than by complete (K _n ) topology.This work was presented at the Workshop The Modern Problems of Heteroorganic Chemistry sponsored by the A. N. Nesmeyanov Institute of Organoelement Compounds of the Russian Academy of Sciences (May 8–13, 1993).Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 8, pp. 1353–1360, August, 1993.     

Systematics of f-element crystal-field interaction

Systematic behaviors of free-ion and crystal-field interactions are elucidated as a function of N, the number of f electrons in a lanthanide or actinide ion. Experimentally determined values of the free-ion interaction parameters are compared with those calculated based on Hartree-Fock theory. Comparison is also made between the lanthanide series in 4f^N configurations and the actinide series in 5f^N configurations. Variation in intra-ionic electrostatic interaction, spin-orbit coupling, and ion-ligand interaction is analyzed in comparison between the iso-f-electron lanthanide and actinide ions. Based on an exchange-charge model of crystal-field theory, crystal-field parameters of the f-element ions in various crystals are summarized in terms of point charge contribution and covalence effect. A systematic correlation is found between the free-ion parameters and the crystal-field strength. Increase of the crystal-field interaction results in a reduction in the free-ion parameters.