The Projector Augmented Waveinvented by P.E. Blochl, 1994IBM 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? • Expand |Ψ> in partial waves |Ψ> = ∑i |φi> ci • 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 = <pi|φ~i> • Then |Ψ> = |ψ~> + ∑i ( |φi> - |φ~i> ) <pi|φ~i> • T = 1 + ∑i ( |φi> - |φ~i> ) <pi| • In practice, |φi> 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!
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.