The Projector Augmented Wave
invented by P.E. Blochl, 1994
IBM Research Division, Zürich Research Laboratory
Electronic Structure Course, UC Davis
by Ryan Snow
Gruezi!
Pseudopotentials
Computationally efficient
Soft pseudopotentials
Nodeless w.f.
Frozen Core Approximation
Molecular Dynamics
No Pulay Forces
Now fully ab initio
Norm conservation within a core
radius
Haman, Schluter, Chiang, PRL 1971
A Problem with Pseudopotentials
Some Elements have
numerically “hard” wave
functions
transition elements
first row elements
B,C,N,O,F
requires large basis
Computational cost is order
N3, where N is the size of
basis set.
Vanderbilt, PRB 41, 7892 (1990)
Two solutions to the pseudopotential problem
Vanderbilt's Ultrasoft Pseudopotentials (USPP)
Relaxes the norm conservation condition
fully nonlocal pseudopotential is generated directly
Blochl's Projector Augmented Waves (PAW)
also relaxes the norm conservation condition
Keeps the full wave functions while working with
soft, pseudo- wave functions
combines LAPW and pseudopotential methods
accuracy, simplicity, and MD
implemented in vasp, abinit, abpaw, pwpaw, socorro, etc.
PAW overview
Features:
An All-Electron wave function |Ψ>
A soft, pseudo- wave function |ψ~>
A linear transformation between these:
|Ψ> = T |ψ~>
Operators, including the total energy, can be evaluated in
either the transformed, all-electron space of |Ψ>, or in a
Heisenberg picture with transformed operators and |ψ~>
= after transforming |Ψ> = T |ψ~>
= where A~ = T~ A T
PAW—How does it work?
1) Expand |Ψ> in partial waves |Ψ> = ∑i |φi> ci
2) Expand |ψ~> in partial waves |ψ~> = ∑i |φ~i> ci
One |φ~> for each |φ>
Let |Ψ> = T |ψ~>,
The ci are functionals of the |ψ~>: ci =
3) Then |Ψ> = |ψ~> + ∑i ( |φi> - |φ~i> )
T = 1 + ∑i ( |φi> - |φ~i> ) are evaluated numerically on a radial grid;
|φ~i> and |pi> are expanded in planewaves
Early tests of paw method
Kresse, PRB 59, 1758 (1999)
60 meV/μB error for USPP magnetic energies
A more stringent test of paw method
hcp-bcc-hcp-fcc-hcp pattern across transition element rows
4d
Structural phase stability possibly governed by Zd
Delocalized s and p band energies rise in energy faster than d band
energies with the application of pressure
Continuous sp -> d promotion with pressure
as Zd increases, will Mo transition bcc->hcp ??
Much qualitative and quantitative disagreement in theory and experiment!
direct fcc transition at 620 GPa
direct fcc transition at 650 GPa
Summary
We predict the direct bcc->fcc transition at 610 (HGH PP,LDA),
620 (APW+lo,LDA), and 650 Gpa (APW+lo, GGA)
Other predictions: also bcc->fcc
Belonoshko et.al., PAW/vasp 720GPa
Boettgar 660 Gpa
Christensen etal., 600 Gpa
Other predictions: bcc --> hcp, and then hcp-->fcc
Moriarty, LMTO 420 and 620 Gpa
Jona & Marcus PAW/vasp 620 and 770 Gpa
Soderlind etal. 520, 740, and fcc-->bcc at 34,000 GPa
Sikka, >490 Gpa
Smirnova etal. FP-LMTO 620 Gpa
Smirnova etal. LMTO-GF-CPA 730 GPa
Experiment
DAC has shown no phase transition in bcc Molybdenum
from 0 to 560 GPa.
Shock data is controversial, with some claiming a transition
at 210 GPa, others not.