The internal coordinate path Hamiltonian; application to methanol and malonaldehyde

The internal coordinate path Hamiltonian is introduced for the study of the vibrations of molecules which have one large amplitude motion. The Hamiltonian is represented in terms of a one path coordinate and 3N—7 normal coordinates. The variational method is used to solve the Schrödinger equation. The molecules studied are methanol and malonaldehyde. For methanol the internal coordinate is a dihedral angle, for malonaldehyde it is the difference in the distances between the migrating hydrogen and the neighbouring oxygen atoms. For methanol there is little coupling between the path and the normal coordinates and so no complications were encountered in the calculations which used harmonic surfaces generated by density functional and M?ller—Plesset theory. Fundamental frequencies were predicted. Malonaldehyde is a different story. There is substantial coupling between the path coordinate and several of the normal coordinates. This introduces many complications: an anharmonic surface is essential and large variational configuration interaction calculations are essential for convergence. Furthermore, because the Coriolis terms require the evaluation of derivatives of both the nuclear coordinates and the normal coordinate eigenvectors along the path, great care must be taken with these numerical procedures. B3LYP predicts too low a transition state which overemphasizes the large Coriolis terms near the transition state. This may be one of the reasons why our fundamental vibrations are in poor agreement with observation. It is most encouraging that the tunnelling splitting is 58 cm^?1 (obs. 21.56 cm^?1), obtained with our quartic density functional surface.     

Measurement of longitudinal relaxation times in crowded 1H NMR spectra using one- and two-dimensional maximum quantum (MAXY) NMR spectroscopy

Methods for measuring longitudinal relaxation times of protons in heavily overlapped ¹H NMR spectra are introduced and exemplified using a solution of cholesteryl acetate. The methods are based on 1-dimensional and 2-dimensional maximum quantum NMR spectroscopy (MAXY), which makes possible the selective detection of CH, CH₂ and CH₃ ¹H NMR resonances. A modification of the BIRD pulse sequence to achieve selective inversion of protons bonded to either ¹²C or ¹³C is given. The approach should find application in studies of molecular dynamics where isotopic enrichment is not possible and the level of available sample dictates the use of ¹H NMR spectroscopy.     

Two studies in process modelling: injection moulding and resin film infusion

This paper discusses two problems from important areas of processmodelling, namely injection moulding and resin film infusion.The first comes from the food-processing sector, and involvesthe filling of tubs with liquid foodstuffs. The approach takenhere is computational, with the foodstuffs modelled as incompressibleviscous (Newtonian or non-Newtonian) fluids. In practice, theliquids have multiple free surfaces; the correct treatment ofthese surfaces is a crucial element of the modelling herein. Resin film infusion is one of the preferred processes for themanufacture of composite materials. In essence it involves squeezinga viscous incompressible fluid into a woven carbon-fibre matrix.With the matrix treated as a porous medium, and with exothermicreactions (and consequent thermal effects) neglected, simpleanalytical results are obtainable which can be verified by straightforwardengineering experiments.     

Residues in toric varieties

We study residues on a complete toric variety X, which are defined in terms of the homogeneous coordinate ring of X.We first prove a global transformation law for toric residues. When the fan of the toric variety has a simplicial cone of maximal dimension, we can produce an element with toric residue equal to 1. We also show that in certain situations, the toric residue is an isomorphism on an appropriate graded piece of the quotient ring. When X is simplicial, we prove that the toric residue is a sum of local residues. In the case of equal degrees, we also show how to represent X as a quotient (Y\{0})/C* such that the toric residue becomes the local residue at 0 in Y.     

The numerical solution of an inverse scattering problem for acoustic waves

In a previous paper, the authors presented a dual space methodfor the numerical solution of the two-dimensional inverse scatteringproblem for acoustic waves in an inhomogeneous medium. Here,by making major modifications to the dual space method, a dramaticimprovement in the numerical performance of this method is achievedfor solving the inverse scattering problem.     

A Sequential Method for Discrete Optimization Problems and its Application to the Assignment, Travelling Salesman, and Three Machine Scheduling Problems

A discrete optimization problem is defined in which a numberof items are to be arranged in a sequence of positions in orderto obtain a minimum cost configuration. A simple sequentialmethod is proposed for the solution of this problem. The methodis composed of two stages. The first stage builds up an initialarrangement by introducing the items into the positions individually.The second stage adjusts these positions until no further improvementcan be made by a simple alteration of the arrangement. The aimof the method is to provide near optimum results with relativelylittle calculation. The scheme is shown to be successful ona number of well known problems in operations research.