8.12. Projector Augmented-Wave Method (PAW)¶
We can use the Density Functional Theory (DFT) to reduce the many body problem to solve a single particle Schrödinger equation:
The wavefunctions contain cusps (and are oscillatory around each nucleus), also one needs to solve this for all core states.
Next step is to use the known behavior around each atom and take advantage of the known physics somehow. There are two general approaches, either one can incorporate the known physic in the basis (for example the partition of unity in the finite element method), or into the equations. PAW method uses the latter approach.
8.12.1. Projectors, Augmentation Spheres and Smooth Wavefunctions¶
We introduce smooth wavefunctions (we’ll use “~” for smooth functions) by a linear transformation operator :
We construct augmentation spheres around each atom (one can imagine a muffin-tin), where is a cut-off radius, is the atom index, is the atom position.
We write as:
where only acts in the augmentation sphere. We choose a complete basis (also called partial waves) inside the sphere. The smooth partial waves can be obtained using the operator:
Because only acts in the sphere, it follows that
outside the sphere (i.e. for ). We can now expand the smooth wavefunctions using the partial waves basis:
inside the augmentation sphere. Multiplying both sides by :
So both smooth and non-smooth wavefunctions have the same expansion coefficients . We choose smooth projector functions satisfying the following orthogonality and completeness relations inside the augmentation spheres (no restrictions are imposed outside the spheres, so we just define ):
Let’s write using the projectors:
Note that the right hand side is zero outside the augmentation sphere. Thus
In other words, the transformation operator is completely defined using the smooth and non-smooth partial waves and the projector functions. In terms of the wavefunction:
In words, the wavefunction can be decomposed as the sum of the smooth wavefunction and sum over atoms (centers), at each atom we have “1-center all electron” minus “1-center pseudo”.
The projection functions can always be written as
where is any set of linearly independent functions.
Note: the above means all states of interest — either all states, or only the valence states.
8.12.2. Frozen Core Approximation¶
One can either calculate all electrons in the eigenproblem, or only calculate the valence electrons and treat the core states separately. The simplest option is to introduce a frozen core approximation, where
for all core states , here runs over , where is the atom index and are the core states of an atom. This approximation can also be relaxed in various ways.
8.12.3. Expectation Values of Local Operators¶
In the frozen core approximation:
where the tensor is:
The functions are not strictly localized withing the augmentation sphere.
8.12.4. Kohn Sham Equations¶
We multiply the original equations by from the left and introduce the smooth wavefunctions:
The orthogonality of wavefunctions is:
The overlap operator can be written as:
The transformed Hamiltonian is