Pseudopotentials: design, testing, typical errors
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1 Pseudopotentials: design, testing, typical errors Kevin F. Garrity Part 1 National Institute of Standards and Technology (NIST) Uncertainty Quantification in Materials Modeling 2015
2 Parameter free calculations. Ab initio theory Often excellent agreement with experiment Lukanov, Garrity, PRB (2012)
3 Parameter free calculations. Ab initio theory Often excellent agreement with experiment Experimental STM Sr on Ge (001) surface Lukanov, Garrity, PRB (2012)
4 Parameter free calculations. Ab initio theory Often excellent agreement with experiment Experimental STM Sr on Ge (001) surface DFT Low Energy Structure Lukanov, Garrity, PRB (2012)
5 Parameter free calculations. Ab initio theory Often excellent agreement with experiment Experimental STM Sr on Ge (001) surface Simulated STM DFT Low Energy Structure Lukanov, Garrity, PRB (2012)
6 Parameter free calculations. Ab initio theory Often excellent agreement with experiment Experimental STM Sr on Ge (001) surface Simulated STM DFT Low Energy Structure Lukanov, Garrity, PRB (2012)
7 Ab initio theory - questions How accurate are our equations / physical approximations? Validation with experiment Lots of work on this popular topic to study NOT the topic of this work.
8 Ab initio theory - questions How accurate are our equations / physical approximations? Validation with experiment Lots of work on this popular topic to study NOT the topic of this work. How precisely are we solving equations? Numerical verification Typical numerical errors? Do our codes agree? Who is correct? Often ignored.
9 What is Si lattice constant in GGA? It depends on time, apparently: And Si is the easiest element to treat!! Historical published Si lattice constants with PBE. Graphic Kurt Lejaeghere et al
10 Outline Density Functional Theory Background Solving Kohn-Sham eqns. All-electron methods Pseudopotentials Testing different basis sets GBRV pseudopotential library/tests Designing pseudopotentials Tradeoffs Practical example for lab
11 Electronic Structure Calculations We want to solve Schrodinger s equation: (For electrons)
12 Electronic Structure Calculations We want to solve Schrodinger s equation: (For electrons) Energy Electron coordinates Many-body Hamiltonian Many-body Wavefunction
13 Electronic Structure Calculations We want to solve Schrodinger s equation: (For electrons)
14 Electronic Structure Calculations We want to solve Schrodinger s equation: (For electrons) Kinetic Electron-ion Electron-electron
15 Electronic Structure Calculations We want to solve Schrodinger s equation: (For electrons) Contains essentially all of chemistry
16 Electronic Structure Calculations
17 Electronic Structure Calculations
18 Electronic Structure Calculations
19 Density Functional Theory
20 Kohn-Sham Equations
21 Kohn-Sham Equations Single particle eigenvalues, wavefunctions External Non-interacting kinetic Classical electrostatic (Hartree) Everything else (exchange-correlation)
22 Kohn-Sham Equations Given Atomic Positions Solve for n(r), Energy Forces
23 Exchange-correlation Functional The approximation in DFT Otherwise formally exact Many approximations exist, varying accuracy
24 Exchange-correlation Functional The approximation in DFT Otherwise formally exact Many approximations exist Local Density Approximation (LDA) Treat E xc like you have a locally uniform electron gas
25 Exchange-correlation Functional
26 Accuracy (LDA / GGA) Structural Properties: Lattice constants ~ 1% Bulk modulus ~10% Silicon LDA Energy vs. Vol Yin Cohen PRB (1982)
27 Accuracy (LDA / GGA) Structural Properties: Lattice constants ~ 1% Bulk modulus ~10% Atomization energy ~30 kcal/mol = 1.2 ev/atom LDA ~8 kcal/mol = 0.35 ev/atom GGA Solid formation energy ~0.25 ev/atom GGA Less when materials similar
28 Accuracy (LDA / GGA) Structural Properties: Lattice constants ~ 1% Bulk modulus ~10% Atomization energy ~30 kcal/mol = 1.2 ev/atom LDA ~8 kcal/mol = 0.35 ev/atom GGA Solid formation energy ~0.25 ev/atom GGA Electronic structure Metals good Insulators Qualitatively good, gaps off ~50% Silicon LDA Band Structure Yin Cohen PRB (1982) Gap is 50% too small
29 DFT Use DFT is the workhorse electronic structure method Used extensively in physics, materials science, chemistry, biology Useful combination of accuracy / speed (scales like N 3 ) Used as atomistic level of multi-scale calculations More accurate methods start from DFT GW, DMFT, QMC
30 How to solve Kohn-Sham? N el coupled differential eqns. Should just be a numerical challenge Choose a basis, Transform to linear algebra problem
31 How to solve Kohn-Sham? N el coupled differential eqns. Should just be a numerical challenge Choose a basis, Transform to linear algebra problem Can we just use plane-waves? Complete, periodic, converges systematically w/ G cut
32 Problem with naive plane-waves Strongly different energy/length scales Core vs valence energies 4 orders of magnitude
33 Problem with naive plane-waves
34 Problem with naive plane-waves
35 All-electron calculations Solve this issue by separating space Core regions vs. interstitial Use different basis sets in both regions Atomic-like Plane-waves Issue need to match boundaries, which requires energies
36 LAPW solution Linearized Augmented Plane waves (WIEN2k, etc) Advantages: Accurate if done carefully Disadvantages: Have to set E to expand around Have to set R MT Expensive, hard for user
37 Pseudopotentials
38 Pseudopotentials Zero outside r c
39 Pseudopotentials
40 Steps for generating PSP s
41 Steps for generating PSP s
42 Steps for generating PSP s
43 Steps for generating PSP s Black magic well, hopefully not anymore
44 A clarification Don t think of psp as statistical fit of a model Think of as improvable basis set.
45 A clarification Don t think of psp as statistical fit of a model Think of as improvable basis set. PSP s should: Be insensitive to minor parameter variations Reproduce many properties at once Work in many chemical environments (transferable) Be computationally inexpensive
46 Some nice properties If we require the following: PS and AE WF's match outside r c PS and AE eigenvalues match PS WF is normalized Hamann, et. al. Phys. Rev. Lett. 43, (1979)
47 Some nice properties If we require the following: Then: PS and AE WF's match outside r c PS and AE eigenvalues match PS WF is normalized Get good scattering properties: Matching even outside r c to second order in ε-ε nl Reproduces atomic system by construction Hopefully transferable to solid state Hamann, et. al. Phys. Rev. Lett. 43, (1979)
48 Scattering properties Modern PSP have multiple projectors: Force matching log derivatives at additional energies
49 PSP pros and cons Advantages: Computationally inexpensive Easy to use Disadvantages: Frozen-core approximation Hard to design/test
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