Plane Waves and Pseudopotentials

The electron orbitals used to express the single particle density in Equations 2.5 may be expanded in terms of any converged basis set. In practice a plane wave basis set is used as this confers a number of benefits.

1. A plane wave basis set is unbiased, it does not assume any preconceptions of the form of the problem.

2. Due to Bloch's theorem plane waves are the natural choice for the representation of electron orbitals in a periodic system.

3. The kinetic energy operator is diagonal in a plane wave representation. Similarly the potential is diagonal in real space. The use of Fast Fourier Transforms in changing between these representations provides a large saving in computational cost (See Section 2.5).

4. As a plane wave basis set is non-local no Pulay forces will arise when calculating the forces on the ions in the system. Hence these ionic forces may be calculated with greater efficiency. (See Section 2.6)

The principle disadvantage of a plane wave basis set is its inefficiency. The number of basis functions needed to describe atomic wavefunctions accurately near to a nucleus would be prohibitive. This difficulty is overcome by the use of Pseudopotentials to represent the potential of the ionic cores. This approximation makes the assumption that only the valence electrons determine the physical properties of the system. The pseudopotential represents the potential of the nucleus and the core electrons subject to the following conditions.

1. The valence wavefunction remains unchanged outside the core region.

2. The pseudowavefunction within the core matches correctly at the boundary.

3. The phase shift caused by the core is unchanged. (This means that in general the pseudopotential must be different for each angular momentum component.)

4. The norm of the valence wavefunction in the core is unchanged.

An example of a pseudopotential and pseudowavefunction can be seen in Figure 2.1.

  
Figure 2.1: A schematic illustration of all-electron (solid lines) and pseudoelectron (dashed lines) potentials and their corresponding wavefunctions. The radius at which all-electron and pseudoelectron values match is . Source [22].