One Dimensional Chemistry and the Generalised Local Density Approximation

Abstract

In this thesis we explore two distinct topics: a unique model of one dimensional chemistry and the development of the generalised local density approximation.

The specific one dimensional system we study is one in which three dimensional particles, both electrons and nuclei, have been strictly confined to move on a line. This means we retain the Coulomb interaction of the three dimensional particles. This is problematic, the singularity of this interaction is exceptionally strong in one dimension and requires special techniques be used to circumvent it.

In our study we employ the Dirichlet boundary conditions, which require the wavefunction to vanish whenever two particles touch. This brings some severe consequences including, most controversially, that the nuclei are impenetrable to the electrons. However it does permit finite binding energies of electrons to nuclei and has a number of unique features which make it an interesting system to study for insight into electron behaviour.

Here we explore the mechanics of chemistry within this model. We construct an unusual periodic table for one dimensional elements and explore the mechanisms by which they bind into molecules. Ultimately, we are able to develop a set of simple rules with which one can easily predict the outcome of a reaction in this unique model.

The generalised local density approximation is a new method for constructing density functional approximations. The local density approximation, the most simplistic of all density functionals, is built off the properties of the infinite uniform electron gas and as result replicates them exactly. It is also possible to construct finite uniform electron gases and, contrary to expectations, the regular local density approximation is unable to model these finite gases correctly.

The generalised local density approximation incorporates these finite gases into its construction. We account for their differences from the infinite gases by introducing a new parameter which measures the proximity of electrons at a point in space. We can then construct a correlation functional for one dimensional systems using a method which closely resembles that used for previous local density approximations. Because of the definition of the new parameter, one would formally consider this functional to be a meta-GGA functional. However it retains the exceptionally simple form of a local density approximation.

This new functional is observed to have greatly improved accuracy compared to the standard local density approximation when tested against a variety of systems, including reactions of the one dimensional molecules identified earlier. Although we only construct a one dimensional functional here, the method is easily generalised to other dimensions. Show more