Mathematical Analysis of Quantum
Chemical Models for Small Atoms
Chemical Models for Small Atoms
File(s)
Accepted version
Author(s)
Goddard, BD
Type
Journal Article
Abstract
Quantum chemical methods are of great use in predicting chemical and physical
properties of atoms and molecules. Whilst many of the methods are built upon
a rigorous mathematical foundation, there are very few chemically-specific mathematical
results, and many aspects, such as the choice of optimal basis sets, are
poorly understood.
This thesis centres on the choice of a minimal basis for the second row atoms,
motivated by perturbation theory and physical screening arguments, for which
a full Configuration Interaction (CI) calculation is performed. This calculation
requires no numerical integration and can essentially be performed ‘by hand’.
As well as demonstrating that this method leads to an accurate prediction of
the atomic spectra, a number of computationally useful results concerning the
Hamiltonian matrix and the spectra themselves are rigorously proven.
A rigorous rate of convergence for the CI method applied to the ground state
of the Helium atom, with a basis ordered by angular momentum, is also derived.
This includes a derivation of the leading order constant, enabling extrapolation
of computational results.
Another interesting area of computational chemistry is the prediction of molecular
geometries. This thesis investigates the AH2 trimers, where A is a second
row atom. The relationship between the maximum of the pair density of the central
atom and the bond angle is investigated, using the canonical ground state
wavefunctions derived in the CI calculations. The non-numerical results are independent
of the radial parts of the wavefunctions and inserting the CI wavefunctions
leads to excellent qualitative and reasonably accurate quantitative numerical
predictions.
properties of atoms and molecules. Whilst many of the methods are built upon
a rigorous mathematical foundation, there are very few chemically-specific mathematical
results, and many aspects, such as the choice of optimal basis sets, are
poorly understood.
This thesis centres on the choice of a minimal basis for the second row atoms,
motivated by perturbation theory and physical screening arguments, for which
a full Configuration Interaction (CI) calculation is performed. This calculation
requires no numerical integration and can essentially be performed ‘by hand’.
As well as demonstrating that this method leads to an accurate prediction of
the atomic spectra, a number of computationally useful results concerning the
Hamiltonian matrix and the spectra themselves are rigorously proven.
A rigorous rate of convergence for the CI method applied to the ground state
of the Helium atom, with a basis ordered by angular momentum, is also derived.
This includes a derivation of the leading order constant, enabling extrapolation
of computational results.
Another interesting area of computational chemistry is the prediction of molecular
geometries. This thesis investigates the AH2 trimers, where A is a second
row atom. The relationship between the maximum of the pair density of the central
atom and the bond angle is investigated, using the canonical ground state
wavefunctions derived in the CI calculations. The non-numerical results are independent
of the radial parts of the wavefunctions and inserting the CI wavefunctions
leads to excellent qualitative and reasonably accurate quantitative numerical
predictions.
Version
Accepted version
Date Issued
2011-07-15T12:38:51Z
Copyright Statement
© 2007 The Author
Source
University of Warwick