1. 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!

2. 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

3. 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)

4. 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.

5. 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

6. 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

7. Early tests of paw method Kresse, PRB 59, 1758 (1999) 60 meV/μ B error for USPP magnetic energies

8. 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!

11. direct fcc transition at 620 GPa

12. direct fcc transition at 650 GPa

14. 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

15. 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.