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Plane Waves as a Basis

When studying the electronic structure of condensed matter systems, one is investigating the behaviour of a number of electrons in the order of $ \sim 10^{28}$ per mole of atoms. Many extended systems are periodic in structure, corresponding to one of the Bravais lattices, so one can define an infinite periodic system and perform calculations for only the electrons in the periodic cell.

Bloch's theorem shows that the wavefunction, $ \psi_n$ , of an electron in band $ n$ , for a periodic system can be expressed as a combination of a plane wave part and a periodic cell part [6]:

$\displaystyle \psi_{n}\ensuremath{(\ensuremath{\bm{r}})}= u_{n}\ensuremath{(\ensuremath{\bm{r}})}e^{i\ensuremath{\bm{k}}\cdot\ensuremath{\bm{r}}},$ (1.24)

where the plane wave part has wave vector $ \ensuremath{\bm{k}}$ , which is confined to the first Brillouin zone. The periodic part has the same periodicity as the lattice, i.e. $ u_n(\ensuremath{\bm{r}}+\bm{R})=u_n\ensuremath{(\ensuremath{\bm{r}})}$ , where $ \bm{R}$ is one of the lattice vectors.

This leads us to choose a plane wave basis set to describe the wavefunction within the periodic cell. The periodic part of the wavefunction can then be written as:

$\displaystyle u_n\ensuremath{(\ensuremath{\bm{r}})}= \sum_{\ensuremath{\bm{G}}} c_{n,\ensuremath{\bm{G}}}e^{i\ensuremath{\bm{G}}\cdot\ensuremath{\bm{r}}},$ (1.25)

where we have plane wave coefficients $ c_{n,\ensuremath{\bm{G}}}$ and $ \ensuremath{\bm{G}}$ are the reciprocal lattice vectors that satisfy the relation $ \ensuremath{\bm{G}}\cdot\bm{R} = 2\pi m$ , where $ m$ is an integer. If we combine equations 1.24 and 1.25, the Kohn-Sham orbitals can therefore be written as an infinite sum of plane waves:

$\displaystyle \psi_n\ensuremath{(\ensuremath{\bm{r}})}= \sum_{\ensuremath{\bm{G...
...{G}})}e^{i(\ensuremath{\bm{k}}+ \ensuremath{\bm{G}})\cdot \ensuremath{\bm{r}}},$ (1.26)

where $ c_{n,\ensuremath{\bm{k}}+\ensuremath{\bm{G}}}$ are the coefficients of the plane waves describing the wavefunction.

Bloch's theorem allows us to take an infinite system but only calculate a finite number of electronic wavefunctions. However, this leaves an infinite number of k-points as each electron occupies a definite $ \ensuremath{\bm{k}}$ . In practice, we need only choose a sample of k-points as the wavefunction varies slowly over small regions of k-space. The electronic wavefunctions at k-points that are close will be nearly identical. Therefore a region of k-space can be represented by the wavefunction at a single k-point. Efficient k-point sampling schemes have been developed, such as the one given by Monkhorst and Pack [7]. The symmetry of the lattice can be used to reduce the number of k-points required. The Brillouin zone can be made irreducible by applying the point group symmetries of the lattice, leaving no k-points related by symmetry.

The sum over $ \ensuremath{\bm{G}}$ vectors in equation 1.26 is infinite in order to fully describe the wavefunction, i.e. for the plane wave basis set to be complete. When devising a computational implementation one must choose a finite end to the sum. For most realistic wavefunctions, there will be a scale below which the wavefunction can be described as smoothly varying. This means that the coefficients $ c_{n,\ensuremath{\bm{k}}+\ensuremath{\bm{G}}}$ will become small for large $ \vert\ensuremath{\bm{k}}+\ensuremath{\bm{G}}\vert$ . The cutoff point is referred to as the plane wave kinetic energy cutoff:

$\displaystyle E_{cut} \ge \ensuremath{\frac{1}{2}}\vert\ensuremath{\bm{k}}+\ensuremath{\bm{G}}\vert^2,$ (1.27)

i.e. it is greater than or equal to the highest kinetic energy of the plane waves used. This corresponds to a sphere in reciprocal space within which all the used $ \vert\ensuremath{\bm{k}}+\ensuremath{\bm{G}}\vert$ vectors lie.

When performing calculations one must always be careful to select an appropriate sampling of k-points and plane wave cutoff energy. This is done by performing calculations at successively higher cutoff energies and finer grids of k-points until the quantities of interest no longer change - a test of convergence. An example of this for the total energy of silicon in the diamond structure is shown in Figure 1.1.
Figure 1.1: Graphs for the total energy per atom in silicon against plane wave cutoff energy and k-point grid size.
Image convergence

The use of plane waves as a basis set is advantageous in a number of ways. In terms of the accuracy required for the system in question, one can always improve the accuracy by increasing the plane wave cutoff energy and therefore tending towards the complete basis set. Real space quantities, such as potentials, can be easily transformed to reciprocal space using standard numerical techniques, in order to obtain the plane wave coefficients. Derivatives in real space become multiplications in reciprocal space, so quantities such as the kinetic energy of the Kohn-Sham orbitals can be easily evaluated. The use of plane waves treats all regions of space equally, so can be applied generally, even for non-periodic systems, if an appropriate periodic supercell is used. We make use of this in our investigation of surfaces in Chapter 3. A simple schematic of the supercell approximation is shown in Figure 1.2. However, this includes regions of vacuum, so there is an added memory and computational cost in such cases. A plane wave basis set also lends well to distribution of data and processing in a parallel computing environment. This allows larger and more complicated systems to be simulated with higher accuracy.
Figure 1.2: A schematic of the supercell approximation. Vacuum padding of the cell reduces the effect of the molecule ``seeing'' its own image in the next repeating unit. Here we have increased the size of the right hand cell by five times over the left hand cell. The same principle can be applied to defects in solids and to surfaces.
Image supercell

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Next: The Pseudopotential Approximation Up: Plane Wave Pseudopotential Method Previous: Plane Wave Pseudopotential Method   Contents
Stewart Clark 2012-08-09