Pseudopotentials: design, testing, typical errors

Transcription

1 Pseudopotentials: design, testing, typical errors Kevin F. Garrity 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 Problem: Numerically Intractable Many-body Ψ complex, high dimensional fn Cannot be solved directly for 10 e s

17 Electronic Structure Calculations Problem: Numerically Intractable Many-body Ψ complex, high dimensional fn Cannot be solved directly for 10 e s We are asking for too much info. Don t care about Ψ. Care about energies, forces.

18 Electronic Structure Calculations Problem: Numerically Intractable Many-body Ψ complex, high dimensional fn Cannot be solved directly for 10 e s We are asking for too much info. Don t care about Ψ. Care about energies, forces. Can we solve for something simpler? Yes. Charge density n(r)

19 Density Functional Theory Hohenberg-Kohn Theorem: Charge density n(r) contains same info as Ψ GS n(r) is a real function of just 3 spatial dimensions Energy is functional of n(r):

20 Kohn-Sham Equations Need a way to solve for E, n(r) Map onto non-interacting system: Where:

21 Kohn-Sham Equations Need a way to solve for E, n(r) Map onto non-interacting system: Single particle eigenvalues, wavefunctions External Non-interacting kinetic Classical electrostatic (Hartree) Everything else (exchange-correlation)

22 Kohn-Sham Equations Need a way to solve for E, n(r) Map onto non-interacting system: 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 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 This work: GGA Generalized gradient approximation Incorporates n(r), n r Not unique (use PBE)

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 r Ψ(r) r ( a. u)

34 rψ(r) Problem with naive plane-waves Pb length scales: Describe ~0.001 Ang G cut =2π/(10-13 m) PW Density 1 per m -3 BZ size m PW's Diagonalize x10 10 r ( a. u)

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 Only calculate valence states Replace core states + nuclear with effective potential Require Ψ to match AE outside r c Smoothly to zero inside

38 Pseudopotentials Only calculate valence states Replace core states + nuclear with effective potential Require Ψ to match AE outside r c Smoothly to zero inside Non-local: s,p,d feel different potentials V ps = V loc ps r + D l β lm >< β lm lm Zero outside r c

39 Pseudopotentials Only calculate valence states Replace core states + nuclear with effective potential Require Ψ to match AE outside r c Smoothly to zero inside

40 A clarification Don t think of psp as statistical fit of a model Think of as improvable basis set.

41 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

42 Steps for generating PSP s 1) Solve isolated atom - Separates into R nl (r) Y lm (θ, φ) - Use your chosen DFT approximations - Get all-electron Ψ,ε nl

43 Steps for generating PSP s 1) Solve isolated atom - Separates into R nl (r) Y lm (θ, φ) - Use your chosen DFT approximations - Get all-electron Ψ,ε nl 2) Choose r cut and then generate Ψ ps - Matches for r > r cut - Goes smoothly to zero inside r cut

44 Steps for generating PSP s 1) Solve isolated atom - Separates into R nl (r) Y lm (θ, φ) - Use your chosen DFT approximations - Get all-electron Ψ,ε nl 2) Choose r cut and then generate Ψ ps - Matches for r > r cut - Goes smoothly to zero inside r cut 3) Invert equations to get V ps - Should also be smooth

45 Steps for generating PSP s 1) Solve isolated atom - Separates into R nl (r) Y lm (θ, φ) - Use your chosen DFT approximations - Get all-electron Ψ,ε nl 2) Choose r cut and then generate Ψ ps - Matches for r > r cut - Goes smoothly to zero inside r cut 3) Invert equations to get V ps - Should also be smooth Black magic well, hopefully not anymore

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 V ps = V loc ps r + D l β lm >< β lm lm >1 projector per l

49 PSP pros and cons Advantages: Computationally inexpensive Easy to use Disadvantages: Frozen-core approximation Hard to design/test

50 Example-converging fcc Al Go to your uq4mm directory svn update cd Garrity/psp_tutorial use p espresso xas use p oncvpsp /do_al_pwcutoff.x 10 PW cutoff in Ryd

51 Example-converging fcc Al

52 Testing Basis Sets What level of precision is possible?

53 Is this still typical? Not anymore. Historical published Si lattice constants with PBE. Graphic Kurt Lejaeghere et al

54 Older PSP s vs a difficult test Older HGH and TM sets perform very poorly (>2% error) BR set improved (0-2% latt. const. error) Still not very good. Bennet Phys. Proc (2012)

55 Upcoming work on DFT numerical accuracy Major effort to systematically calculate and compare 15 codes, 40 basis sets many only recently available Focused on elemental crystals With best basis sets / convergence, agreement 1-2 mev/ atom, lattice constant 0.1%, bulk modulus 4% (This is inter-code comparison, not with expt.)

56 Delta test Summarize difference in equation of state: Calculate 7-9 energy vs volume points, fit equation of state Good agreement 1-2 mev/atom Not only possible test, but a reasonable one. K. Lejaeghere et al, Crit. Rev. in Sol. State and Mat. Sci 39, 1-24 (2014)

57 Delta comparison matrix, all-electron Highest convergence settings Good agreement with each other Detailed settings will be published

58 Delta comparison matrix all-codes Recent PSP sets are accurate Only slightly worse than AE This is only elements Need to test compounds Accuracy is not only factor: Speed Compatibility Availability See also

59

60 Typical psps: Motivation my psp set Created ad hoc, often unpublished Only designed / tested for one material Testing not public Quality of public potentials unknown Difficult to reproduce previous work Duplication of effort Doesn t work for high throughput computing

61 1-3 PRL , PRL , PRL 110, See references in Nature Materials 12, (2013), Comp. Mater. Sci 50, 2295 (2011) First principles high-throughput computing Explore many materials to optimize a property At Rutgers Piezoelectrics, ferroelectric, antiferroelectrics 1-3 aflowlib.org, materials project, etc Attempting to calculate ICSD, search for thermoelectrics, etc (~17,000 compounds so far) Searches for stable binary and ternary compounds, photovoltaics, catalysts, battery materials, hydrogen storage, etc 4-5 Most use VASP proprietary

62 Challenges of high-throughput Run 1000's of calculations Need library of atoms Need soft potentials single cutoff Need high transferability Unusual chemical environments Cannot test environments individually

63 My PSP set GBRV pseudopotential library Compatible with Quantum Espresso Abinit jdft, Castep, etc Designed for high-throughput calculations Tested in many structures Fast and accurate K.F. Garrity, J.W. Bennett, K.M. Rabe and D. Vanderbilt, Comput. Mater. Sci. 81, 446 (2014)

64 Specifications of GBRV psp set Optimized for high-throughput Includes H to Bi (except noble gases, f-block) 40 Ryd plane wave-cutoff Accurate stresses Highly transferable Public testing data Reduce duplication of effort Open-source Can be used/improved by community Methodology Ultrasoft pseudopotentials / PAW

65 Ultrasoft potentials / PAWs Relax norm-conserving constraint Add 'missing' charge back later Allows for softer potentials Norm-conserving Oxygen Ultrasoft Oxygen Vanderbilt PRB 41, 7892 (1990)

66 Solid state tests crucial Testing Procedure Test trade-offs of speed / accuracy Compare to all-electron (AE), same approximations Many chemical environments Covalent, ionic, metallic Cubic for convenience

67 Libraries to test GBRV - QE Ultrasoft library for quantum espresso Abinit PAW library for abinit Low 40 Ryd PW cutoff JTH - Abinit PAW library Low 40 Ryd PW cutoff PSLIB - qe-forge.org/gf/project/pslibrary/ Quantum Espresso PAW library High PW cutoff Ryd Low PW cutoff 40 Ryd VASP - PAW library for VASP code PW cutoff Ryd Proprietary

68 Test1: fcc/bcc lattice constants Isolates each atom Metallic bonding

69 Test2: rock salt lattice constants Ionic bonding Note higher errors for other sets

70 Test3: perovskite lattice constants Ionic bonding Test +3, +4, +5 oxidation states

71 Test4: Half-heusler lattice constant Complicated covalent/ionic bonding Shorter bond lengths

72 Test5: zinc blende delta test Covalent bonding Shorter bond lengths

73 Testing Summary My potentials are good combination: 1) Speed 2) Accuracy 3) Transferability

74 Testing Summary My potentials are good combination: 1) Speed 2) Accuracy 3) Transferability

75 Testing Summary My potentials are good combination: 1) Speed 2) Accuracy 3) Transferability

76 Typical errors summary Expected accuracy For well-designed and efficient potentials In unknown chemical environment half-heusler error distribution 0.15% rms error lattice constant 1-2 mev/atom delta test 5% bulk modulus 90% within +/- 0.2% lattice constant Poorly tested potentials can be very bad. Especially but not exclusively older potentials

77 Designing pseudopotentials

78 Constructing Potentials Things to adjust: -r c -semicore/valence -nlcc, etc Ultrasoft Oxygen Testing is key: -Average psp goes though iterations

79 Start I want a psp Initial psp (similar atom) Guided by: Periodic trends Trial and error Adjust r c s Look at log deriv. Increase r c Remove semicore Solid State Tests Reduce r c Add Semicore Adjust local potential Extra projectors Too Hard Good? Yes! Bad Testing The End

80 Examples from code you will use Optimized norm-conserving Vanderbilt pseudopotentials D. R. Hamann, PRB (2013) Nice new code, much easier than back in my day. Supports multiple projects, optimization.

81 Si input file Atom name, Z 3 Core 2 valence 1s2s2p 3s3p

82 Atomic states: 1s 2 2s 2 2p 6 3s 2 3p 2 Si input file

83 Si input file Maximum angular momentum L=2 means d

84 r c for s,p,d Si input file

85 Si input file q target for s,p,d q 2 / 2 = E cut

86 Local potential Cutoff r c s Si input file

87 Number of projectors Energy of extra projector For s,p,d Si input file

88 Si graphical output v_pseudo

89 Si graphical output n(r)

90 Si graphical output s AE/PS WF r c

91 Si graphical output excited

92 Si graphical output logd Good matching from -2 to +1 Ha important

93 Si graphical output convergence error Decay should be ~linear on this scale (1 Ha = 2 Ryd = 2* ev)

94 Si test silicon carbide zinc blende Calculated Vol = Ang 3 /atom, B0=211.1 GPa, B = 3.90 Reference Vol = Ang 3 /atom, B0=212.7 GPa, B = 3.69 Error (%) -0.36% 0.76% -5.6%

95 More difficult example Ti 3d state

96 More difficult example Ti 3d state r c Very localized, poor agreement below rc=2.4 a.u.

97 More difficult example Ti 3d state Error dominated by d state

98 More difficult example Ti 3d state r c If we decrease r c to 2.2, we get better agreement...

99 More difficult example Ti 3d state But the convergence is much worse. We can try to force it lower by adjusting the q target

100 More difficult example Ti 3d state Warning signs oscillatory behavior in d-channel Looks suspicious

101 More difficult example Ti 3d state We made problem worse! Cannot solve all problems through optimization This tradeoff will need to be tested in solid state calcs with application in mind.

102 Conclusions Kohn-Sham equations can be solved All-electron or pseudopotentials Pseudopotentials can be made fast and accurate 0.15% lattice constant error, 5% bulk modulus Careful testing is required Different structures, oxidation states Tradeoffs between accuracy and speed depends on application GBRV library provides good speed/accuracy combination In rest of lab, you will design/test pseudopotentials

103 Lab INSTRUCTIONS.txt in nanohub Or 1. Aluminum Run psp generator, test in 2 structures 2. Boron Adjust r c to be more accurate/harder, re-test 3. Indium Compare without/with semicore d state to potential, test 4. Gallium optional. Play with psp. 5. Phosphorous optional. Create input file from Al.

104 Tour of the periodic table

105 Main Group - Softest Soft s and p orbitals Well-separated from core Easist to pseudize

106 Main Group - Harder Localized 2p state No core p state Difficult to make soft Trade-off with accuracy Fine for solid-state Oxygen

107 Main Group semi-core d s and p states overlap with core d states Include d as semi-core

108 Alkaline metals Tend to ionize Require semi-core s, p Li, Be, Na, Mg Difficult to make soft K, Ca, Rb, Sr, Cs, Ba Sensitive to parameters

109 Early Transition Metals Open d-shell Very localized Magnetism, nlcc Semicore s, p Mn All-Electron

110 Late Transition Metals Some open d-shell Less overlap, fewer semi-core Only p (Rh,Pd,Ir,Pt) Only d (Cd,Au,Hg)

111

112

113 APW solution Augmented Plane waves Atomic-like in muffin tins PW's in interstitial Need E to match S to I But I want E... From S. Cottenier (2004)

114 Tools for PSPs Opium opium.sourceforge.net/ Norm-conserving optimized designed non-local potentials Abinit or QE Can be fully-relativistic Vanderbilt Ultrasoft Ultrasoft pseudopotentials QE or abinit PAWs Atomic code QE package quantum-espresso.org Norm-conserving, ultrasoft, PAW Can be fully-relativistic Cornell pseudopotential vault nninc.cnf.cornell.edu psp

115 Pseudopotentials for High Throughput Computation Kevin F. Garrity Joseph W. Bennett, Karin M. Rabe, David Vanderbilt Rutgers Group Meeting March 28, 2013

116 Advantages of GBRV Tied for best performance Lowest cutoffs (40 Ryd) PSLIB at 50 Ryd A few VASP potentials above 40 Ryd Open source, public testing Use with QE and abinit Understand / modify potentials Reproduce results

117 Vanderbilt PRB 41, 7892 (1990) Ultrasoft potentials Relax norm-conserving constraint (generalized eigenvalue) Add 'missing' charge back later Allows for softer potentials Multiple projectors Accuracy for semi-core states, localized states Very similar to PAWs Pros: Cons: Soft Accurate More difficult to code Not fully implemented

118 Density Functional Theory Hohenberg-Kohn Theorem: Charge density n(r) contains same info as ground state Ψ

119 Density Functional Theory Hohenberg-Kohn Theorem: Charge density n(r) contains same info as ground state Ψ H Ψ GS n(r) Normal Solving Order

120 Density Functional Theory Hohenberg-Kohn Theorem: Charge density n(r) contains same info as ground state Ψ H Ψ GS n(r) One-to-one correspondence Can get Ψ from n(r)

121 Density Functional Theory Hohenberg-Kohn Theorem: Charge density n(r) contains same info as ground state Ψ n(r) is a real function of just 3 spatial dimensions

122 Density Functional Theory Hohenberg-Kohn Theorem: Charge density n(r) contains same info as ground state Ψ n(r) is a real function of just 3 spatial dimensions Energy is functional of n(r):

123 Bad ways to get psps: Some grad student in my group made it 15 years ago I think he works in finance now

124 Bad ways to get psps: Some grad student in my group made it 15 years ago I think he works in finance now It was on the web without any testing Everything on the internet is true

125 Bad ways to get psps: Some grad student in my group made it 15 years ago I think he works in finance now It was on the web without any testing Everything on the internet is true I adjusted it to match experiment LDA/GGA shouldn t match experiment!