Unit Cell and Plane Wave Basis

The unit cell of a periodically repeated system is defined by the lattice vectors ${\bf a}_1$ , ${\bf a}_2$ , and ${\bf a}_3$ . The lattice vectors can be combined into a three by three matrix ${\bf h} = [ {\bf a}_1, {\bf a}_2, {\bf a}_3 ]$ . The volume $\Omega$ of the cell is calculated as the determinant of $\bf h$

$$\Omega = {\rm det} {\bf h} \enspace .$$ (52)

Further, scaled coordinates ${\bf s}$ are introduced that are related to ${\bf r}$ via ${\bf h}$

$${\bf r} = {\bf h} {\bf s} \enspace .$$ (53)

Periodic boundary conditions can be enforced by using

$${\bf r}_{\rm pbc} = {\bf r} - {\bf h} \! \left[ {\bf h}^{-1} {\bf r} \right]_{\rm NINT} \enspace ,$$ (54)

where $[ \cdots ]_{\rm NINT}$ denotes the nearest integer value. The coordinates ${\bf r}_{\rm pbc}$ will be always within the box centered around the origin of the coordinate system. Reciprocal lattice vectors ${\bf b}_{i}$ are defined as

$${\bf b}_{i} \cdot {\bf a}_{j} = 2 \pi \delta_{ij} \enspace$$ (55)

and can also be arranged to a three by three matrix

$$[{\bf b}_1, {\bf b}_2, {\bf b}_3] = 2 \pi ({\bf h}^{\rm t})^{-1} \enspace .$$ (56)

Plane waves build a complete and orthonormal basis with the above periodicity

$$f^{\rm PW}_{\bf G}({\bf r}) = {1 \over \sqrt{\Omega}} \exp \! \l... ...mega}} \exp \! \left[{2 \pi i {\bf g} \cdot {\bf s}} \right] \enspace ,$$ (57)

with the reciprocal space vectors

$${\bf G} = 2 \pi ( {\bf h}^{\rm t} )^{-1}{\bf g} \enspace ,$$ (58)

where ${\bf g} = [i,j,k]$ is a triple of integer values. A periodic function can be expanded in this basis

$$\psi({\bf r}) = \psi({\bf r} + {\bf L}) = {1 \over \sqrt{\Omega}}... ... G} \psi({\bf G}) \exp \! \left[{i {\bf G} \cdot {\bf r}}\right] \enspace ,$$ (59)

where $\psi({\bf r})$ and $\psi({\bf G})$ are related by a three-dimensional Fourier transform. The direct lattice vectors ${\bf L}$ connect equivalent points in different cells.

The Kohn-Sham potential of a periodic system exhibits the same periodicity as the direct lattice

$$V^{\rm KS}({\bf r}) = V^{\rm KS}({\bf r} + {\bf L}) \enspace ,$$ (60)

and the Kohn-Sham orbitals can be written in Bloch form (see e.g. Ref. [52])

$$\Phi ({\bf r}) = \Phi_{i} ({\bf r},{\bf k}) = \exp \! \left[{i {\bf k} \cdot {\bf r}} \right] \; u_{i} ({\bf r},{\bf k}) \enspace ,$$ (61)

where ${\bf k}$ is a vector in the first Brillouin zone. The functions $u_{i} ({\bf r},{\bf k})$ have the periodicity of the direct lattice

$$u_{i} ({\bf r},{\bf k}) = u_{i} ({\bf r} + {\bf L},{\bf k}) \enspace .$$ (62)

The index $i$ runs over all states and the states have an occupation $f_{i}({\bf k})$ associated with them. The periodic functions $u_{i} ({\bf r},{\bf k})$ are now expanded in the plane wave basis

$$u_{i} ({\bf r},{\bf k}) = {1 \over \sqrt{\Omega}} \sum_{\bf G} c_{i}({\bf G},{\bf k}) \exp \! \left[ {i {\bf G} \cdot {\bf r}} \right] \enspace ,$$ (63)

and the Kohn-Sham orbitals are

$$\Phi_{i} ({\bf r},{\bf k}) = {1 \over \sqrt{\Omega}} \sum_{\bf G}... ...{\bf k}) \exp \! \left[{i ({\bf G} + {\bf k}) \cdot {\bf r}}\right] \enspace ,$$ (64)

where $c_{i}({\bf G},{\bf k})$ are complex numbers. With this expansion the density can also be expanded into a plane wave basis

$$n ({\bf r}) = {1 \over \Omega} \sum_{i} \int \! d{\bf k} \; f_{i}({\bf k}) \sum... ...{i}({\bf G},{\bf k}) \exp \! \left[{i ({\bf G} + {\bf k}) \cdot {\bf r}}\right]$$ (65)

$$= \sum_{\bf G} n({\bf G}) \exp \! \left[{i {\bf G} \cdot {\bf r}}\right] \enspace ,$$ (66)

where the sum over ${\bf G}$ vectors in Eq. (66) expands over double the range given by the wavefunction expansion. This is one of the main advantages of the plane wave basis. Whereas for atomic orbital basis sets the number of functions needed to describe the density grows quadratically with the size of the system, there is only a linear dependence for plane waves.

In actual calculations the infinite sums over ${\bf G}$ vectors and cells has to be truncated. Furthermore, we have to approximate the integral over the Brillouin zone by a finite sum over special ${\bf k}$ -points

$$\int \! d{\bf k} \rightarrow \sum_{\bf k} w_{\rm k} \enspace ,$$ (67)

where $w_k$ are the weights of the integration points. From now on we will assume that the Brillouin zone integration can be done efficiently by a single point at $k = 0$ , the so called $\Gamma$ -point.

The truncation of the plane wave basis rests on the fact that the Kohn-Sham potential $V^{\rm KS} ({\bf G})$ converges rapidly with increasing modulus of ${\bf G}$ . For this reason only ${\bf G}$ vectors with a kinetic energy lower than a given maximum cutoff

$${1\over 2} \; \vert{\bf G}\vert^2 \leq E_{\rm cut} \enspace$$ (68)

are included in the basis. With this choice of the basis the precision of the calculation within the approximations of density functional theory is controlled by one parameter $E_{\rm cut}$ only.

The number of plane waves for a given cutoff depends on the unit cell. A good estimate for the size of the basis is

$$N_{\rm PW} = {1 \over 2 \pi^2} \; \Omega E_{\rm cut}^{3/2} \enspace ,$$ (69)

where $E_{\rm cut}$ is in Hartree units. The basis set needed to describe the density calculated from the Kohn-Sham orbitals has a corresponding cutoff that is four times the cutoff of the orbitals. The number of plane waves needed at a given density cutoff is therefore eight times the number of plane waves needed for the orbitals.