2.1 Computational Methods
43
molecular materials. Unfortunately, due to the complexity of the non-local correlation functionals, they become increasingly expensive with the size of the unit cell
[36].
A second issue surrounding the intrinsic localised nature of DFT correlation is
that DFT functionals tend to over delocalise electrons due to their intrinsic selfinteraction [5]. This introduces problems with calculation of electronic band gaps,
where typical GGA functionals tend to grossly underestimate these gaps [37]. While
there is some improvement with the introduction of HF exchange in the hybrid
functionals, there remain problematic effects with long-range HF exchange, although
cancellation of errors in hybrid methods can solve many of these problems [38].
Development of so-called screened hybrid DFT functionals has therefore appeared.
These methods separate the Coulomb operator within the HF exchange into short—
and long-range effects. The definition of the range (and thus the length-scale of
HF exchange) is varied with an empirically fit parameter. The HSE06 [39] (Heyd,
Scuseria and Ernzerhof) functional is one such screened hybrid functional, and has
proved particularly promising for the calculation of electronic band gaps. It was
therefore selected for this purpose in this thesis.
2.1.5 Basis Sets
Solution of the Schrödinger equation, Eq. 2.2, requires two components, the Hamiltonian (i.e. how to evaluate the energy) and the wavefunction (i.e. what to evaluate
the energy of ). The former is considered within the theories of HF and DFT, while
the latter depends on the so-called basis set [5]. The basis set is a set of functions
whose linear combination represents a wavefunction. Generally, the larger the basis
set, the more accurately the wavefunction will be described. While it is therefore
tempting to use large basis sets, a larger number of functions greatly increases the
computational cost. Compromises must therefore be made.
There are two main types of basis sets: localised and delocalised [8]. The former is
typically used in the study of isolated molecules, while the latter is more commonly
used for periodic (e.g. crystalline) materials. Both have been utilised in this thesis.
2.1.5.1 Localised Basis Set—Isolated Molecules
Modelling the electronic structure of an isolated molecule is typically done using
localised functions that are centred on the individual atoms. These functions form
the atomic orbitals, and their linear combination forms the linear combination of
atomic orbitals, molecular orbital method. Early methods employed Slater-type functions [40] (STOs) as they have a similar form to the hydrogen atom eigenfunctions.
However, these functions could not be handled efficiently by algorithms, and instead
Gaussian-type orbitals (GTO) became popular. While Gaussian functions are more
easily handled, they do not have the necessary cusp at the nucleus, and they decay
43
molecular materials. Unfortunately, due to the complexity of the non-local correlation functionals, they become increasingly expensive with the size of the unit cell
[36].
A second issue surrounding the intrinsic localised nature of DFT correlation is
that DFT functionals tend to over delocalise electrons due to their intrinsic selfinteraction [5]. This introduces problems with calculation of electronic band gaps,
where typical GGA functionals tend to grossly underestimate these gaps [37]. While
there is some improvement with the introduction of HF exchange in the hybrid
functionals, there remain problematic effects with long-range HF exchange, although
cancellation of errors in hybrid methods can solve many of these problems [38].
Development of so-called screened hybrid DFT functionals has therefore appeared.
These methods separate the Coulomb operator within the HF exchange into short—
and long-range effects. The definition of the range (and thus the length-scale of
HF exchange) is varied with an empirically fit parameter. The HSE06 [39] (Heyd,
Scuseria and Ernzerhof) functional is one such screened hybrid functional, and has
proved particularly promising for the calculation of electronic band gaps. It was
therefore selected for this purpose in this thesis.
2.1.5 Basis Sets
Solution of the Schrödinger equation, Eq. 2.2, requires two components, the Hamiltonian (i.e. how to evaluate the energy) and the wavefunction (i.e. what to evaluate
the energy of ). The former is considered within the theories of HF and DFT, while
the latter depends on the so-called basis set [5]. The basis set is a set of functions
whose linear combination represents a wavefunction. Generally, the larger the basis
set, the more accurately the wavefunction will be described. While it is therefore
tempting to use large basis sets, a larger number of functions greatly increases the
computational cost. Compromises must therefore be made.
There are two main types of basis sets: localised and delocalised [8]. The former is
typically used in the study of isolated molecules, while the latter is more commonly
used for periodic (e.g. crystalline) materials. Both have been utilised in this thesis.
2.1.5.1 Localised Basis Set—Isolated Molecules
Modelling the electronic structure of an isolated molecule is typically done using
localised functions that are centred on the individual atoms. These functions form
the atomic orbitals, and their linear combination forms the linear combination of
atomic orbitals, molecular orbital method. Early methods employed Slater-type functions [40] (STOs) as they have a similar form to the hydrogen atom eigenfunctions.
However, these functions could not be handled efficiently by algorithms, and instead
Gaussian-type orbitals (GTO) became popular. While Gaussian functions are more
easily handled, they do not have the necessary cusp at the nucleus, and they decay