Theory of Pseudopotentials. Outline of Talk
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1 Theory of Pseudopotentials David Vanderbilt Rutgers University Outline of Talk Introduction Motivation Basic Idea History and Terminology First-Principles Pseudopotentials Construction Scattering Properties Norm Conservation Transferability Tests Relativistic Case Computational Considerations: Softness Ultrasoft Pseudopotentials and PAW Resources Reference list Web resources
2 Motivation Bandstructure of Si: Basic idea of pseudopotentials Pseudo wavefunction All-electron wavefunction Pseudopotential Cutoff radius All-electron potential
3 Pseudopotentials: History Pseudopotentials: Terminology
4 Pseudopotentials: Terminology s, p, and d electrons all feel the same potential Pseudopotentials: Terminology (For molybdenum) s, p, and d electrons feel different potentials
5 Pseudopotentials: Terminology These terms vanish outside r c s, p, and d electrons feel different nonlocal operators First-principles pseudopotentials Philosophy:
6 First-principles pseudopotentials Outline of Talk Introduction Motivation Basic Idea History and Terminology First-Principles Pseudopotentials Construction Scattering Properties Norm Conservation Transferability Tests Relativistic Case Computational Considerations: Softness Ultrasoft Pseudopotentials and PAW Resources Reference list Web resources
7 First-principles PSP construction Same for r > r c First-principles PSP construction Beyond r c : Also! AE =! PS
8 First-principles PSP construction Scattering properties For AE and PS separately: Choose channel l and energy! Find solution of SE that is regular at the origin at this! Compare beyond r c If match! good scattering properties! =! bound! >! bound (Used for construction) (Used for testing)
9 Scattering properties Quantify: Logarithmic derivatives D R (!) D R (!) R R! bound Correct by construction at! bound Norm conservation at! =! bound (Used for construction) r c Charge contained in this region is the same (AE vs. PS)
10 Norm conservation " Scattering properties Fundamental advance of Hamann, Schlüter and Chang, 1979: If norm conservation is imposed, then pseudo D R (!) matches all-electron D R (!) to second order in (!! bound ) R! bound Slope automatically matches First-principles PSP construction
11 Example: Hamann, Schlüter, and Chang (Semilocal PSP), 1979 (Mo) Charge self-consistency in PSP construction
12 Transferability tests PSP was generated in reference configuration, e.g.: [core]s 2 p 2 for Si Now, pick a couple of excited configurations, e.g.: [core]sp 3 [core]s 2 p (+1 ion) For each excited configuration, compare: All-electron calculation Pseudopotential calculation using previously generated PSP Points of comparison: Total energies Energy eigenvalues Logarithmic derivatives Transferability tests Example: HSC pseudopotential for oxygen
13 Relativisitic pseudopotentials Relativisitic pseudopotentials Bachelet, Hamann & Schluter, PRB 26, 4199 (1982)
14 Issues of computational expense The expense is in the target calculation (PSP construction is extremely cheap) First consideration: Compatibility with FFT approach to H"? H "(G) = KE "(G) + (V")(G) FFT FFT "(r) Multiply on Real space mesh (V")(r) Issues of computational expense Cheap, but poor transferability Good transferability, but expensive ~ All modern calculations done this way with one or two projectors
15 Expense vs. accuracy Compare different functional forms: Computational expense Want Semilocal PSP Local PSP Nonlocal (KB) PSP Accuracy Improved softness Now assume nonlocal (KB): Computational expense: N PW Smaller r c Larger r c Smooth cutoff functions Accuracy
16 Softness and plane-wave convergence Usual HSC Vanderbilt, 1985 q cut! 11 au -1 q cut! 7 au -1 Softness and plane-wave convergence Apply maximal smoothness to V ps construction!vanderbilt, 1985 This was only marginally successful in lowering the cutoff needed for the wavefunction Apply maximal smoothness to " ps construction!rappe, Rabe, Kaxiras, Joannopoulos (RRKJ, 1990)!Troullier and Martins (TM, 1991) Much more successful These (especially TM) are standard kind of potentials in use today
17 Softness and plane-wave convergence Plane-wave convergence of energy for free Cu atom (RRKJ, 1990) Outline of Talk Introduction Motivation Basic Idea History and Terminology First-Principles Pseudopotentials Construction Scattering Properties Norm Conservation Transferability Tests Relativistic Case Computational Considerations: Softness Ultrasoft Pseudopotentials and PAW Resources Reference list Web resources
18 Ultrasoft pseudopotentials O 2p wavefunction (nodeless!) AE PS Overshoot required by Norm conservation a.u. Ultrasoft pseudopotentials TM or RRKJ for O, Cu, etc: N PW Smaller r c Larger r c Ultrasoft PP (Vanderbilt, 1990) Accuracy
19 Ultrasoft pseudopotentials O 2p wavefunction (nodeless!) AE Usual PSP Ultrasoft a.u. Ultrasoft pseudopotentials Covergence tests for oxygen USPP, r c = 1.2 au USPP, r c = 1.8 au
20 Ultrasoft pseudopotentials: Formalism Ultrasoft pseudopotentials: Formalism
21 Ultrasoft pseudopotentials: Formalism Ultrasoft pseudopotentials: Formalism Typically, #=(1,2) in each angular momentum channel:
22 Terminology and Comparison In current usage, PSPs are classified as either NCPP = Norm-conserving pseudopotentials USPP = Ultrasoft pseudopotentials However, remember that USPP are norm-conserving in a generalized sense Thus, they retain the good features of NCPP In fact, their accuracy is usually better than NCPP Warning: Extra coding required in solid-state code Not all code packages accept USPP Ultrasoft Pseudopotentials
23 USPP and PAW P.E. Blöchl, Projector Augmented-Wave Method PRB 50, (1994) G. Kresse and D. Joubert, From USPP to PAW PRB 59, 1758 (1999) Outline of Talk Introduction Motivation Basic Idea History and Terminology First-Principles Pseudopotentials Construction Scattering Properties Norm Conservation Transferability Tests Relativistic Case Computational Considerations: Softness Ultrasoft Pseudopotentials and PAW Resources Reference list Web resources
24 Resources: References Resources: Web Sites Jose Luis Martins site for Troullier-Martins potentials: Octopus web interface for pseudopotential generation Vanderbilt Ultrasoft Pseudopotential site:
25 Octopus Web Site Octopus Web Site
26 Octopus Web Site Octopus Web Site
27 Octopus Web Site Ultrasoft Pseudopotential Web Site
28 Ultrasoft Pseudopotential Web Site Summary Introduction Motivation Basic Idea History and Terminology First-Principles Pseudopotentials Construction Scattering Properties Norm Conservation Transferability Tests Relativistic Case Computational Considerations: Softness Ultrasoft Pseudopotentials and PAW Resources Reference list Web resources Talk will be posted on
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