11
2.4 Implementation of DFT in the GPAW(Gridbased
Projector Augmented Wave) code
The first step in a practical implementation of DFT is choosing a basis for
the expansion of the wavefunctions. There are wide variety of bases and one
is preferred over the other depending on the kind of the calculations. In the
current version GPAW has plane wave, grid and linear combination of atomic
orbitals (LCAO) as basis sets 19, 20. In principle, one can solve the all electron
problem without making any approximation for the core electrons, but
that is not usually the case. Since for most of the applications the valence electrons
govern the behavior of materials, its desirable to make approximations
for the core electrons in order to make the calculations computationally less
demanding. Many codes use pseudopotential in which the core electrons act
as mere spectators and provide an effective potential to the valence electrons
21. One of the drawbacks of the pseudopotential method is that one completely
looses the information of the core electrons which might be required in
few cases. In order to circumvent this issue with the pseudopotentials, Blöchl
proposed the projector augmented wave (PAW) method 22.
2.4.1 A Brief Introduction to the PAW Method
The oscillatory behavior of the wavefunctions in the core regions requires large
number of basis functions for the expansion, therefore, making the calculations
computationally demanding. In the Blöchl formalism a linear transformation
is applied to an auxilliary smooth wavefunction in order to obtain the full all
electron Kohn-Sham (KS) wavefunction. The operation can be written as 23:
| ni = T | ˜ ni, (2.15)
where | ni and | ˜ ni are the true and auxilliary wavefunctions respectively.
One of the properties required by the transformation operator is that it should
not affect the wavefunction outside a given cutoff radius. The above requirement
is due the similar nature of the true wavefunction and the auxilliary
wavefunction outside the cutoff radius. Therefore T can be written as:
T = I +
X
a
T a, (2.16)
a denotes the atom index and with the expression above the T a does not have
any effect outside the cutoff radius. The true wavefunction inside the augmentation
sphere can be expanded in terms of the partial waves and the partial