44
2 Experimental and Computational Methods
too rapidly [5, 41]. As such, in practice a contraction of Gaussian primitive functions g j (r), each with an appropriate weighting coefficient (d) is used. As such, a
contracted GTO (CGTO) containing L primitive functions takes the form
χ i (r ) =
L
j=1
d j g j (r )
(2.27)
The larger the number of CGTOs and primitives used to construct a CGTO, the
more accurate will be the model, and the more computationally demanding will be
the calculation. Noting that chemistry typically involves only the valence electrons,
split-valence basis sets are common, where the number of CGTOs used to model the
core orbitals differs from that used for the valence orbitals. The Pople [42] basis set 631G, for example, states that six gaussian primitives are summed to a single CGTO
to model the core shells, while two CGTOs are employed to model the valence
region, one composed of three primitives and one of a single primitive function.
This offers additional flexibility to the valence electrons and permits more accurate
representation of perturbations to these electron orbitals. Additional functions can be
added to further enhance the flexibility of these valence states: polarisation and diffuse
functions. The former describes addition of higher angular momentum functions onto
an atom (e.g. a p-orbital onto an s-orbital), and assists in capturing changes in the
shape of electron density on bonding. Diffuse functions add higher principle quantum
number orbitals (e.g. a larger orbital of the same angular momentum). Polarisation
functions are crucial for accurate capture of anions, polarizable atoms, excited states
and long range interactions. A number of different families of GTOs exist, and differ
mainly in their optimization of the primitive Gaussian exponents. The most common
families [5] are the Dunning [43] and Pople [42] basis sets.
In this thesis, GTOs are used to study bond elongation and excited state potential
energy surfaces of anions. This requires accurate modelling of the atom-atom bonding
interactions, as well as permitting sufficient flexibility to capture the excited state.
Hence in this work both polarization and diffuse functions are incorporated. However,
due to the computational expense of the calculation, only a limited number of these
functions could be used.
2.1.5.2 Condensed Matter, Delocalised Basis Sets and Bloch Theorem
Treating electrons in a solid, which is essentially an infinite array of periodic unit cells,
brings the additional challenge of solving the Schrödinger equation for an infinite
number of electrons. This problem can be solved by considering Bloch’s theorem,
which states that due to the periodicity of a crystalline material, it is necessary to
consider only the electrons that reside within the primitive unit cell. Bloch’s theorem
states that the wavefunction of an electron within a perfectly periodic potential can
be written as [44]
2 Experimental and Computational Methods
too rapidly [5, 41]. As such, in practice a contraction of Gaussian primitive functions g j (r), each with an appropriate weighting coefficient (d) is used. As such, a
contracted GTO (CGTO) containing L primitive functions takes the form
χ i (r ) =
L
j=1
d j g j (r )
(2.27)
The larger the number of CGTOs and primitives used to construct a CGTO, the
more accurate will be the model, and the more computationally demanding will be
the calculation. Noting that chemistry typically involves only the valence electrons,
split-valence basis sets are common, where the number of CGTOs used to model the
core orbitals differs from that used for the valence orbitals. The Pople [42] basis set 631G, for example, states that six gaussian primitives are summed to a single CGTO
to model the core shells, while two CGTOs are employed to model the valence
region, one composed of three primitives and one of a single primitive function.
This offers additional flexibility to the valence electrons and permits more accurate
representation of perturbations to these electron orbitals. Additional functions can be
added to further enhance the flexibility of these valence states: polarisation and diffuse
functions. The former describes addition of higher angular momentum functions onto
an atom (e.g. a p-orbital onto an s-orbital), and assists in capturing changes in the
shape of electron density on bonding. Diffuse functions add higher principle quantum
number orbitals (e.g. a larger orbital of the same angular momentum). Polarisation
functions are crucial for accurate capture of anions, polarizable atoms, excited states
and long range interactions. A number of different families of GTOs exist, and differ
mainly in their optimization of the primitive Gaussian exponents. The most common
families [5] are the Dunning [43] and Pople [42] basis sets.
In this thesis, GTOs are used to study bond elongation and excited state potential
energy surfaces of anions. This requires accurate modelling of the atom-atom bonding
interactions, as well as permitting sufficient flexibility to capture the excited state.
Hence in this work both polarization and diffuse functions are incorporated. However,
due to the computational expense of the calculation, only a limited number of these
functions could be used.
2.1.5.2 Condensed Matter, Delocalised Basis Sets and Bloch Theorem
Treating electrons in a solid, which is essentially an infinite array of periodic unit cells,
brings the additional challenge of solving the Schrödinger equation for an infinite
number of electrons. This problem can be solved by considering Bloch’s theorem,
which states that due to the periodicity of a crystalline material, it is necessary to
consider only the electrons that reside within the primitive unit cell. Bloch’s theorem
states that the wavefunction of an electron within a perfectly periodic potential can
be written as [44]