48
2 Experimental and Computational Methods
The choice of pseudopotential is not unique, and many approaches exist. However,
all pseudopotentials must obey the simple criteria including:
1. The core charge of the pseudo-wavefunction must be identical to that of the
atomic wavefunction.
2. The eigenvalues of the pseudo-electrons must be the same as in the atomic
wavefunction.
3. The pseudo-wavefunction and its first and second derivatives must be continuous
at r c .
A variety of pseudopotential types have been developed that satisfy these conditions. The most commonly employed pseudopotentials are the ultra-soft pseudopotentials (USPP). These were introduced by Vanderbilt [52] to allow the lowest
possible cut-off energies for plane-wave basis sets.
In addition to the above criteria, norm-conserving pseudopotentials [53] (NCPP)
can be generated such that the pseudo- and all-electron wavefunctions yield the same
charge density [5]. This is done by generating a pseudopotential that maintains
r c
0
ϕ
∗
AE (r)ϕ AE (r)dr =
r c
0
ϕ
∗
ps (r)ϕ ps (r)dr
(2.33)
where ϕ AE (r ) and ϕ ps (r ) are the all-electron and pseudopotential wavefunctions,
respectively. This ensures equality of electronic charge both inside and outside the
core region. Because of this, NCPPs ensure accurate reproduction of the scattering
properties of ions and are most easily developed into DFT (this is particularly true
for density functional perturbation theory, DFPT) codes. The work presented in this
thesis therefore employs norm-conserving pseudopotentials throughout all lattice
dynamics calculations. All pseudopotentials were taken from databases available
within the quantum chemical software: CASTEP (00PBE_OP for C, H, N and O
atoms).
2.1.6 Phonon Calculations
There are two approaches to calculating vibrational properties within DFT: the linear
response (also known as density functional perturbation theory, DFPT) and the finite
differences approach. Only DFPT is used in this thesis and will therefore be discussed
briefly here. An extensive review on the subject can be found in Reference [54] and
an excellent introduction to the field in Reference [55]. The use of DFPT over finite
differences methods in this thesis is due to the fact that the latter requires use of
supercells to calculate the frequencies at wave vectors away from zone centre. This
very quickly becomes computationally intractable for larger, low symmetry systems
such as those used in this work.
2 Experimental and Computational Methods
The choice of pseudopotential is not unique, and many approaches exist. However,
all pseudopotentials must obey the simple criteria including:
1. The core charge of the pseudo-wavefunction must be identical to that of the
atomic wavefunction.
2. The eigenvalues of the pseudo-electrons must be the same as in the atomic
wavefunction.
3. The pseudo-wavefunction and its first and second derivatives must be continuous
at r c .
A variety of pseudopotential types have been developed that satisfy these conditions. The most commonly employed pseudopotentials are the ultra-soft pseudopotentials (USPP). These were introduced by Vanderbilt [52] to allow the lowest
possible cut-off energies for plane-wave basis sets.
In addition to the above criteria, norm-conserving pseudopotentials [53] (NCPP)
can be generated such that the pseudo- and all-electron wavefunctions yield the same
charge density [5]. This is done by generating a pseudopotential that maintains
r c
0
ϕ
∗
AE (r)ϕ AE (r)dr =
r c
0
ϕ
∗
ps (r)ϕ ps (r)dr
(2.33)
where ϕ AE (r ) and ϕ ps (r ) are the all-electron and pseudopotential wavefunctions,
respectively. This ensures equality of electronic charge both inside and outside the
core region. Because of this, NCPPs ensure accurate reproduction of the scattering
properties of ions and are most easily developed into DFT (this is particularly true
for density functional perturbation theory, DFPT) codes. The work presented in this
thesis therefore employs norm-conserving pseudopotentials throughout all lattice
dynamics calculations. All pseudopotentials were taken from databases available
within the quantum chemical software: CASTEP (00PBE_OP for C, H, N and O
atoms).
2.1.6 Phonon Calculations
There are two approaches to calculating vibrational properties within DFT: the linear
response (also known as density functional perturbation theory, DFPT) and the finite
differences approach. Only DFPT is used in this thesis and will therefore be discussed
briefly here. An extensive review on the subject can be found in Reference [54] and
an excellent introduction to the field in Reference [55]. The use of DFPT over finite
differences methods in this thesis is due to the fact that the latter requires use of
supercells to calculate the frequencies at wave vectors away from zone centre. This
very quickly becomes computationally intractable for larger, low symmetry systems
such as those used in this work.