The projector augmented wave method (PAW) is a technique used in ab initio electronic structure calculations. It is a generalization of the pseudopotential and linear augmented-plane-wave methods, and allows for density functional theory calculations to be performed with greater computational efficiency.[1]
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.
l{T}
|\tilde{\Psi}\rangle
|\Psi\rangle
|\Psi\rangle=l{T}|\tilde{\Psi}\rangle
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
|\tilde{\Psi}\rangle
|\Psi\rangle
l{T}=1+\sumR\hat{l{T}}R
where
\hat{l{T}}R
\OmegaR
R
Around each atom, it is useful to expand the pseudo wavefunction into pseudo partial waves:
|\tilde{\Psi}\rangle=\sumi|\tilde{\phi}i\rangleci
\OmegaR
Because the operator
l{T}
ci
|pi\rangle
ci=\langlepi|\tilde{\Psi}\rangle
where
\langlepi|\tilde{\phi}j\rangle=\deltaij
|\phii\rangle=l{T}|\tilde{\phi}i\rangle
l{T}
|\phii\rangle
|\tilde{\phi}i\rangle
|pi\rangle
and we can explicitly write it down as
l{T}=1+\sumi\left(|\phii\rangle-|\tilde{\phi}i\rangle\right)\langlepi|
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.[2] [3] [4]
The PAW transformation allows all-electron observables to be calculated using the pseudo-wavefunction from a pseudopotential calculation, conveniently avoiding having to ever represent the all-electron wavefunction explicitly in memory. This is particularly important for the calculation of properties such as NMR,[5] which strongly depend on the form of the wavefunction near the nucleus. Starting with the definition of the expectation value of an operator:
ai=\langle\Psi|\hat{A}|\Psi\rangle
where you can substitute in the pseudo wavefunction as you know
|\Psi\rangle=l{T}|\tilde{\Psi}\rangle
ai=\langle\tilde{\Psi}|l{T}\dagger\hat{A}l{T}|\tilde{\Psi}\rangle
from which you can define the pseudo operator, indicated by a tilde:
\tilde{A}=l{T}\dagger\hat{A}l{T}
If the operator
\hat{A}
l{T}
\tilde{A}=\hat{A}+\sumi,j|pi\rangle\left(\langle\phii|\hat{A}|\phij\rangle-\langle\tilde{\phi}i|\hat{A}|\tilde{\phi}j\rangle\right)\langlepj|
where the indices
i,j
In the original paper, Blöchl notes that there is a degree of freedom in this equation for an arbitrary operator
\hat{B}
\hat{B}-\sumi,j|pi\rangle\langle\tilde{\phi}i|\hat{B}|\tilde{\phi}j\rangle\langlepj|
which can be seen as the basis for implementation of pseudopotentials within PAW, as the nuclear coulomb potential can now be substituted with a smoother one.