2.1 Computational Methods
45
ψ j,k (r) = u j,k (r)e
i k.r
(2.28)
In this equation, u j (r) describes a function with the periodicity of the potential,
such that u j (r + g) = u j (r), where g is the translational length of the crystal. The
term j describes the band index, and k a wave vector within the first Brillouin zone. In
a typical calculation, k is sampled at a discrete number of points. This is often done
using a Monkhorst-Pack grid [45], which is an unbiased means of selecting a subset
of k points to sample based on a rectangular grid of points, spaced evenly throughout
the Brillouin zone. Appropriate dimensions for this grid require convergence testing
on a system-by-system basis [46], and depends on complexities in the underlying
electronic structure.
As u j of Eq. 2.28 is periodic, it can be expressed as a Fourier series,
u j,k (r) =
G
c j,G ex p(i G · r)
(2.29)
where G is a reciprocal lattice vector and c j,G are expansion coefficients for each
plane wave. It follows that the electron wavefunctions can be constructed as a linear
combination of these plane waves,
u j,k (r) =
G
c j, k+G ex p(i(k + G) · r
(2.30)
Similar to the GTO, a DFT calculation therefore aims to minimize the energy of
this function by optimizing the expansion coefficients through self-consistent field
cycles.
Electronic wavefunctions represented by plane waves are fast to compute, and
further, the set of plane waves is universal and does not depend on the position or
type of atoms in the unit cell [5]. Hence, in contrast to localised basis sets, all systems
can be treated with the same set of functions. In principle, the plane wave series is
infinite. However, this is clearly not computationally tractable, and a truncation must
be made. This is done by setting an upper energy limit for the plane wave kinetic
energies (and thus electron kinetic energies) to be incorporated in the wavefunction,
E cut ,
E cut =
2
2m
|k + G|
2
(2.31)
In contrast to GTOs, plane wave basis sets are therefore readily improved, simply
by the addition of more plane waves (i.e. increasing E cut ). An appropriate E cut must
be optimised for each system studied, and depends intimately on the pseudopotential
(described below) and type of atom being investigated. The ability of plane waves to
correctly model atomic structure is shown in Fig. 2.2.
It is also possible (albeit less common) to build Bloch functions of Eq. 2.28
based on GTOs. For an N-electron system, this is done by expanding the unknown
45
ψ j,k (r) = u j,k (r)e
i k.r
(2.28)
In this equation, u j (r) describes a function with the periodicity of the potential,
such that u j (r + g) = u j (r), where g is the translational length of the crystal. The
term j describes the band index, and k a wave vector within the first Brillouin zone. In
a typical calculation, k is sampled at a discrete number of points. This is often done
using a Monkhorst-Pack grid [45], which is an unbiased means of selecting a subset
of k points to sample based on a rectangular grid of points, spaced evenly throughout
the Brillouin zone. Appropriate dimensions for this grid require convergence testing
on a system-by-system basis [46], and depends on complexities in the underlying
electronic structure.
As u j of Eq. 2.28 is periodic, it can be expressed as a Fourier series,
u j,k (r) =
G
c j,G ex p(i G · r)
(2.29)
where G is a reciprocal lattice vector and c j,G are expansion coefficients for each
plane wave. It follows that the electron wavefunctions can be constructed as a linear
combination of these plane waves,
u j,k (r) =
G
c j, k+G ex p(i(k + G) · r
(2.30)
Similar to the GTO, a DFT calculation therefore aims to minimize the energy of
this function by optimizing the expansion coefficients through self-consistent field
cycles.
Electronic wavefunctions represented by plane waves are fast to compute, and
further, the set of plane waves is universal and does not depend on the position or
type of atoms in the unit cell [5]. Hence, in contrast to localised basis sets, all systems
can be treated with the same set of functions. In principle, the plane wave series is
infinite. However, this is clearly not computationally tractable, and a truncation must
be made. This is done by setting an upper energy limit for the plane wave kinetic
energies (and thus electron kinetic energies) to be incorporated in the wavefunction,
E cut ,
E cut =
2
2m
|k + G|
2
(2.31)
In contrast to GTOs, plane wave basis sets are therefore readily improved, simply
by the addition of more plane waves (i.e. increasing E cut ). An appropriate E cut must
be optimised for each system studied, and depends intimately on the pseudopotential
(described below) and type of atom being investigated. The ability of plane waves to
correctly model atomic structure is shown in Fig. 2.2.
It is also possible (albeit less common) to build Bloch functions of Eq. 2.28
based on GTOs. For an N-electron system, this is done by expanding the unknown