Basis Set (chemistry) - Plane-wave Basis Sets

Plane-wave Basis Sets

In addition to localized basis sets, plane-wave basis sets can also be used in quantum-chemical simulations. Typically, a finite number of plane-wave functions are used, below a specific cutoff energy which is chosen for a certain calculation. These basis sets are popular in calculations involving periodic boundary conditions. Certain integrals and operations are much easier to code and carry out with plane-wave basis functions than with their localized counterparts.

In practice, plane-wave basis sets are often used in combination with an 'effective core potential' or pseudopotential, so that the plane waves are only used to describe the valence charge density. This is because core electrons tend to be concentrated very close to the atomic nuclei, resulting in large wavefunction and density gradients near the nuclei which are not easily described by a plane-wave basis set unless a very high energy cutoff, and therefore small wavelength, is used. This combined method of a plane-wave basis set with a core pseudopotential is often abbreviated as a PSPW calculation.

Furthermore, as all functions in the basis are mutually orthogonal and are not associated with any particular atom, plane-wave basis sets do not exhibit basis-set superposition error. However, they are less well suited to gas-phase calculations. Using Fast Fourier Transforms, one can work with plane-wave basis sets in reciprocal space in which not only the aforementioned integrals, such as the kinetic energy, but also derivatives are computationally less demanding to be carried out. Another important advantage of a plane-wave basis is that it is guaranteed to converge in a smooth, monotonic manner to the target wavefunction, while there is only a guarantee of monotonic convergence for all Gaussian-type basis sets when used in variational calculations. (An exception to the latter point is the correlation consistent basis sets.) The properties of the Fourier Transform allow a vector representing the gradient of the total energy with respect to the plane-wave coefficients to be calculated with a computational effort that scales as NPW*ln(NPW) where NPW is the number of plane-waves. When this property is combined with separable pseudopotentials of the Kleinman-Bylander type and pre-conditioned conjugate gradient solution techniques, the dynamic simulation of periodic problems containing hundreds of atoms becomes possible.