Finally, after quite some years of work I have successfully defended my PhD thesis
with the title **Development of a modular quantum-chemistry framework
for the investigation of novel basis functions** on 22nd May.
An electronic version of the text has now been published online.

To make the topic of my work accessible both for chemists as well as mathematicians
the thesis introduces the numerical modelling of chemical systems
from a rather mathematical perspective.
For example the first chapters are devoted to **spectral theory** as well
as **Ritz-Galerkin** discretisation techniques and
iterative **diagonalisation algorithms**.
In the following chapters the main equations underlying quantum chemical modelling,
i.e. the **Schrödinger equation** as well as the **Hartree-Fock approximation**,
are presented stating known results about their mathematical properties.
Afterwards the focus is shifted to numerical techniques for **solving Hartree-Fock**
and further to developing a unified algorithmic approach for dealing
with **self-consistent field** problems irrespective of the basis function type employed.
This leads to a discussion of **contraction-based methods**,
**lazy matrices** as well as the `molsturm`

package.
Last but not least convergence results for Coulomb-Sturmian-based quantum chemistry
are presented.

For more details see the full abstract given below.

# Published thesis versions

My thesis is made available in three forms.
Firstly the officially published documents can be accessed using the
DOI 10.11588/heidok.00024519.
Please use this DOI as well as
this bibtex entry for citations.

Secondly the LaTeX source code
is made available on github.
I will keep amending this repository as errata and typos are discovered.
Note, that I deliberately do *not* publish this code as free software yet,
since there are a few legal issues I want to clearify first.
Please let me know, however, if you wish to use some of the LaTeX and CMake code
and I am sure we can work something out.

Thirdly, from time to time I plan on making new "releases" out of the
amended LaTeX code on github. The link
https:/michael-herbst.com/publications/2018.05_phd_corrected.pdf
will be used to point to the most recent version.

# Full thesis abstract

State-of-the-art methods
for the calculation of electronic structures of molecules
predominantly use Gaussian basis functions.
The algorithms employed inside existing code packages
are consequently often highly optimised keeping
only their numerical requirements in mind.
For the investigation of novel approaches,
utilising other basis functions, this is an obstacle,
since requirements might differ.
In contrast, this thesis develops
the highly flexible program package `molsturm`

,
which is designed in order to facilitate rapid design,
implementation and assessment
of methods employing different basis function types.
A key component of `molsturm`

is
a Hartree-Fock (HF) self-consistent field (SCF) scheme,
which is suitable to be combined
with any basis function type.

First
the mathematical background of quantum mechanics
as well as some numerical techniques are reviewed.
Care is taken to emphasise the often overlooked subtleties when
discretising an infinite-dimensional spectral problem
in order to obtain a finite-dimensional eigenproblem.
Common quantum-chemical methods such as full configuration interaction
and HF are discussed providing insight
into their mathematical properties.
Different formulations of HF are contrasted
and appropriate (SCF) solution schemes formulated.

Next discretisation approaches based on four different types of basis functions
are compared
both with respect to the computational challenges as well as
their ability to describe the physical features of the wave function.
Besides (1) Slater-type orbitals and (2) Gaussian-type orbitals,
the discussion considers (3) finite elements,
which are piecewise polynomials on a grid,
as well as (4) Coulomb-Sturmians,
which are the analytical solutions to a Schrödinger-like equation.
A novel algorithmic approach based on matrix-vector contraction
expressions is developed,
which is able to adapt to the numerical requirements of all basis functions considered.
It is shown that this ansatz not only allows to formulate
SCF algorithms in a basis-function independent way,
but furthermore improves the theoretically achievable
computational scaling for finite-element-based discretisations
as well as performance improvements for Coulomb-Sturmian-based discretisations.
The adequacy of standard SCF algorithms with respect to a contraction-based
setting is investigated
and for the example of the optimal damping algorithm
an approximate modification to achieve such a setting is presented.

With respect to recent trends in the development of modern computer hardware
the potentials and drawbacks of contraction-based approaches are evaluated.
One drawback, namely the typically more involved and harder-to-read code,
is identified and a data structure named lazy matrix is introduced
to overcome this.
Lazy matrices are a generalisation of the usual matrix concept,
suitable for encapsulating contraction expressions.
Such objects still look like matrices from the user perspective,
including the possibility to perform operations like matrix sums and products.
As a result programming contraction-based algorithms
becomes similarly convenient as working with normal matrices.
An implementation of lazy matrices in the `lazyten`

linear algebra library
is developed in the course of the thesis, followed by an example
demonstrating the applicability in the context of the HF problem.

Building on top of the aforementioned concepts the design of `molsturm`

is outlined.
It is shown how a combination of lazy matrices and a contraction-based SCF scheme
separates the code describing the SCF procedure
from the code dealing with the basis function type.
It is discussed how this allows to add a new basis function type
to `molsturm`

by only making code changes in a single integral interface library.
On top of that, we demonstrate by the means of examples
how the readily scriptable interface of `molsturm`

can be employed to implement and assess novel quantum-chemical methods
or to combine the features of `molsturm`

with existing third-party packages.

Finally, the thesis discusses an application of `molsturm`

towards the investigation of the convergence properties
of Coulomb-Sturmian-based quantum-chemical calculations.
Results for the convergence
of the ground-state energies at HF level
are reported for atoms of the second and the third period
of the periodic table.
Particular emphasis is put on a discussion about the required
maximal angular momentum quantum numbers
in order to achieve convergence
of the discretisation of the angular part of the wave function.
Some modifications required for a treatment at correlated level are suggested,
followed by a discussion of the effect of the Coulomb-Sturmian exponent.
An algorithm for obtaining an optimal exponent is devised
and some optimal exponents for the atoms of the
second and the third period of the periodic table at HF level are given.
Furthermore, the first results of a Coulomb-Sturmian-based excited
states calculation based on the algebraic-diagrammatic construction
scheme for the polarisation propagator are presented.

# Acknowledgements

No doubt the research leading up to a PhD thesis cannot be successfully achieved
without the help and support from many people, both scientifically
as well as on a personal level. In the past years I have been very lucky
to be surrounded by people, who created an environment
for fruitful discussion. I am happy for everyone from related fields
of science and technology, who broadened my own perspective by getting me in touch
with new and unusual ideas and who helped me escape the box of my own narrow thinking.
Many people should be named in this respect,
most of them are already given credit in the acknowledgement section of my
PhD thesis,
so let me be rather brief here and simply say thanks!

# Download thesis

To download the most recent version
including **corrections** for all known errors and typos
use this link.
To **cite the thesis** please use
this bibtex entry.