The Projector Augmented Wave invented by P.E. Blochl, 1994 IBM Research Division, Zürich Research Laboratory Electronic Structure Course, UC Davis by Ryan.

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Presentation transcript:

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 N 3, 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 |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 c i are functionals of the |ψ ~ >: c i = 3)Then |Ψ> = |ψ ~ > + ∑ i ( |φ i > - |φ ~ i > )  T = 1 + ∑ i ( |φ i > - |φ ~ i > ) <p i | 4)In practice, |φ i > 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 Z d 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 Z d 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.