Projector augmented wave method

Valence wavefunctions tend to have rapid oscillations near ion cores due to the requirement that they be orthogonal to core states; this situation is problematic because it requires many Fourier components (or in the case of grid-based methods, a very fine mesh) to describe the wavefunctions accurately. The PAW approach addresses this issue by transforming these rapidly oscillating wavefunctions into smooth wavefunctions which are more computationally convenient, and provides a way to calculate all-electron properties from these smooth wavefunctions. This approach is somewhat reminiscent of a change from the Schrödinger picture to the Heisenberg picture.

Transforming the wavefunction

The linear transformation ${\displaystyle \mathcal{T}}$ transforms the fictitious pseudo wavefunction ${\displaystyle |\tilde{\mathrm{\Psi }}⟩}$ to the all-electron wavefunction ${\displaystyle |\mathrm{\Psi }⟩}$:

${\displaystyle |\mathrm{\Psi }⟩=\mathcal{T}|\tilde{\mathrm{\Psi }}⟩}$

Note that the "all-electron" wavefunction is a Kohn–Sham single particle wavefunction, and should not be confused with the many-body wavefunction. In order to have ${\displaystyle |\tilde{\mathrm{\Psi }}⟩}$ and ${\displaystyle |\mathrm{\Psi }⟩}$ differ only in the regions near the ion cores, we write

${\displaystyle \mathcal{T}=1+\sum _R{\hat{\mathcal{T}}}_R}$,

where ${\displaystyle {\hat{\mathcal{T}}}_R}$ is non-zero only within some spherical augmentation region ${\displaystyle {\mathrm{\Omega }}_R}$ enclosing atom ${\displaystyle R}$.

Around each atom, it is useful to expand the pseudo wavefunction into pseudo partial waves:

${\displaystyle |\tilde{\mathrm{\Psi }}⟩=\sum _i|{\tilde{\varphi }}_i⟩c_i}$ within ${\displaystyle {\mathrm{\Omega }}_R}$.

Because the operator ${\displaystyle \mathcal{T}}$ is linear, the coefficients ${\displaystyle c_i}$ can be written as an inner product with a set of so-called projector functions, ${\displaystyle |p_i⟩}$:

${\displaystyle c_i=⟨p_i|\tilde{\mathrm{\Psi }}⟩}$

where ${\displaystyle ⟨p_i|{\tilde{\varphi }}_j⟩={\delta }_{ij}}$. The all-electron partial waves, ${\displaystyle |{\varphi }_i⟩=\mathcal{T}|{\tilde{\varphi }}_i⟩}$, are typically chosen to be solutions to the Kohn–Sham Schrödinger equation for an isolated atom. The transformation ${\displaystyle \mathcal{T}}$ is thus specified by three quantities:

1. a set of all-electron partial waves ${\displaystyle |{\varphi }_i⟩}$
2. a set of pseudo partial waves ${\displaystyle |{\tilde{\varphi }}_i⟩}$
3. a set of projector functions ${\displaystyle |p_i⟩}$

and we can explicitly write it down as

${\displaystyle \mathcal{T}=1+\sum _i\left(|{\varphi }_i⟩-|{\tilde{\varphi }}_i⟩\right)⟨p_i|}$

Outside the augmentation regions, the pseudo partial waves are equal to the all-electron partial waves. Inside the spheres, they can be any smooth continuation, such as a linear combination of polynomials or Bessel functions.

The PAW method is typically combined with the frozen core approximation, in which the core states are assumed to be unaffected by the ion's environment. There are several online repositories of pre-computed atomic PAW data.