Projector-augmented-wave approach

The partitioning of the density introduced in (14) is borrowed from the projector augmented-wave approach [5]. Its special form separates the smooth contributions n, characteristic of the interatomic regions, from the quickly varying terms close to the atoms n.4. ua is a smooth local term which compensates for the overlap of the soft and the hard densities in the atomic region, so that the integrals can still be expanded over all space. In our current implementation the densities h r),nA r), and h-A(r) are expanded in plane-waves and products of primitive Gaussians centered on atom A, respectively... 

The projector augmented-wave (PAW) DFT method was invented by Blochl to generalize both the pseudopotential and the LAPW DFT teclmiques [M]- PAW, however, provides all-electron one-particle wavefiinctions not accessible with the pseudopotential approach. The central idea of the PAW is to express the all-electron quantities in tenns of a pseudo-wavefiinction (easily expanded in plane waves) tenn that describes mterstitial contributions well, and one-centre corrections expanded in tenns of atom-centred fiinctions, that allow for the recovery of the all-electron quantities. The LAPW method is a special case of the PAW method and the pseudopotential fonnalism is obtained by an approximation. Comparisons of the PAW method to other all-electron methods show an accuracy similar to the FLAPW results and an efficiency comparable to plane wave pseudopotential calculations [, ]. PAW is also fonnulated to carry out DFT dynamics, where the forces on nuclei and wavefiinctions are calculated from the PAW wavefiinctions. (Another all-electron DFT molecular dynamics teclmique using a mixed-basis approach is applied in [84].)... 

Most plane wave calculations use ultrasoft pseudopotentials (USPP),15,28 which describe the core electrons of atoms in a mathematically efficient form that greatly reduces the computational cost associated with heavy atoms. An increasing number of calculations used the projector augmented wave (PAW) approach instead.28 In most circumstances where both approaches can be used, the differences between USPP and PAW calculations are minor. Some exceptions to this observation include transition metals with large magnetic moments (e.g., Fe) and alkali metals.28... 

At first sight, the pseudopotential approach and the different partial-wave methods do not seem to have very much in common. In the first approach, the inner, atom-like wave functions are discarded altogether and replaced by a much weaker potential. In the second group, the outer wave function augments exactly these atom-like partial waves. The projector-augmented wave (PAW) method by Blochl [237], however, combines the two ideas into a unified electronic-structure method. Without going into detail, the PAW method can be looked upon as a pseudopotential method in which the pseudopotential instantaneously adapts to the electronic environment. This is because the PAW method is, in fact, a complete all-electron method, and its internal pseu-... 

One way to reduce the computational cost of DFT (or WFT) calculations is to recognize that the core electrons of an atom have only an indirect influence on the atom chemistry. It thus makes sense to look for ways to precompute the atomic cores, essentially factoring them out of the larger electronic structure problem. The simplest way to do this is to freeze the core electrons, or to not allow their density to vary from that of a reference atom. This frozen core approach is generally more computationally efficient. One class of frozen core methods is the pseudopotential (PP) approach. The pseudopotential replaces the core electrons with an effective atom-centered potential that represents their influence on valence electrons and allows relativistic effects important to the core electrons to be incorporated. The advent of ultrasoft pseudopotentials (US-PPs) [18] enabled the explosion in supercell DFT calculations we have seen over the last 15 years. The projector-augmented wave (PAW) [19] is a less empirical and more accurate and transferable approach to partitioning the relativistic core and valence electrons and is also widely used today. Both the PP and PAW approaches require careful parameterizations of each atom type. 

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