The Projector Augmented Waveinvented 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~>

<A> = <Ψ|A|Ψ> after transforming |Ψ> = T |ψ~> <A> = <ψ~|A~|ψ~> where A~ = T~ A T

PAW—How does it work?

1) Expand |Ψ> in partial waves |Ψ> = ∑i |φ

i> c

i

2) Expand |ψ~> in partial waves |ψ~> = ∑i |φ~

i> c

i

One |φ~> for each |φ> Let |Ψ> = T |ψ~>,

The ci are functionals of the |ψ~>: c

i = <p

i|φ~

i>

3) Then |Ψ> = |ψ~> + ∑i ( |φ

i> - |φ~

i> ) <p

i|φ~

i>

T = 1 + ∑i ( |φ

i> - |φ~

i> ) <p

i|

4) In practice, |φi> are evaluated numerically on a radial grid;

|φ~i> and |p

i> are expanded in planewaves

Early tests of paw method

Kresse, PRB 59, 1758 (1999)60 meV/μ

B 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.