Why density functional theory works and how to improve upon it

1. Why density functional theory works and how to improve upon it Kieron Burke & Donghyung Lee, Attila Cangi, Peter Elliott, John Snyder, Lucian Constantin UC Irvine Physics and Chemistry http://dft.uci.edu NAMET

2. Outline • Overview • Some details NAMET

3. Modest statements • The most important problem I’ve ever worked on • Possible payoffs • Understanding of asymptotic approximations • Complete transformation of society • Explains many things about many areas • Semiclassical expansions • DFT and approximations like Thomas-Fermi • Ties together • Math • Physics • Chemistry • Engineering NAMET

4. Insults • Physicists • Is it possible that your most precious elegant little theories (e.g., many-body theory with Feynman diagrams) are a stupid approach to electronic structure? • Chemists • Would you rather continue with LCSF (linear combinations of successful functionals) or actually derive stuff? • Applied mathematicians • Do you want to spend the rest of your life proving things only 6 people care about, or would you rather do something useful? NAMET

5. Electronic structure problem What atoms, molecules, and solids exist, and what are their properties? Pitt1

6. Properties from Electronic Ground State • Make Born-Oppenheimer approximation • Solids: • Lattice structures and constants, cohesive energies, phonon spectra, magnetic properties, … • Molecules: • Bond lengths, bond angles, rotational and vibrational spectra, bond energies, thermochemistry, transition states, reaction rates, (hyper)-polarizabilities, NMR, … Pitt1

7. Big picture Biochemistry WKB Gutwiller trace 1d or 2d Chemistry Empiricism Becke, Truhlar Exact conditions Perdew, Levy TF theory Lieb et al Atoms Astrophysics, protein folding, soil science,… Materials science Condensed matter physics Modern DFT Kohn-Sham EXC[n↑,n↓] NAMET

8. Thomas/Fermi Theory 1926 • Around since 1926, before QM • Exact energy: E0 = T + Vee + V • T = kinetic energy • Vee = electron-electron repulsion • V = All forces on electrons, such as nuclei and external fields • Thomas-Fermi Theory (TF): • T ≈ TTF = 0.3 (3p)2/3∫dr n5/3(r) • Vee≈ U = Hartree energy = ½ ∫dr ∫dr’ n(r) n(r’)/|r-r’| • V = ∫drn(r) v(r) • Minimize E0[n] for fixed N • Properties: • Exact for neutral atoms as Z gets large (Lieb+Simon, 73) • Typical error of order 10% • Teller’s unbinding theorem: Molecules don’t bind. Pitt1

9. Modern Kohn-Sham era • 40’s and 50’s: John Slater began doing calculations with orbitals for kinetic energy and an approximate density functional for Exc[n] (called Xα) • 1964: Hohenberg-Kohn theorem proves an exact E0[n] exists (later generalized by Levy) • 1965: Kohn-Sham produce formally exact procedure and suggest LDA for Exc[n] Pitt1

10. ground-state density of interacting system = Kohn-Sham equations (1965) Pitt1

11. He atom in exact Kohn-Sham DFT Everything has (at most) one KS potential Dashed-line: EXACT KS potential Pitt1

12. Recipe for exact Exc[n] • Given a trial density n(r) • Find the v(r) that yields n(r) for interacting electrons • Find the vs(r) that yields n(r) for non-interacting electrons • Find vH(r) (easy) • vxc(r)=vs(r)-v(r)-vH(r) • Can also extract Exc=E-Ts-V-U • Much harder than solving Schrödinger equation. • In fact, QMA hard (Schuch and Verstraete. Nature Physics, 5, 732 (2009).) Pitt1

13. Local (spin)density approximation • Write Exc[n]=∫d3r exc(n(r)), where exc(n) is XC energy density of uniform gas. • Workhorse of solid-state physics for next 25 years or so. • Uniform gas called reference system. • Most modern functionals begin from this, and good ones recover this in limit of uniformity. Pitt1

14. Subsequent development • Must approximate a small unknown piece of the functional, the exchange-correlation energy Exc[n]. • 70’s-90’s: Much work (Langreth, Perdew, Becke, Parr) going from gradient expansion (slowly-varying density) to produce more accurate functionals, called generalized gradient approximations (GGA’s). • Early 90’s: • Approximations became accurate enough to be useful in chemistry • 98 Nobel to Kohn and Pople Pitt1

15. Commonly-used functionals • Local density approximation (LDA) • Uses only n(r) at a point. • Generalized gradient approx (GGA) • Uses both n(r) and |n(r)| • Should be more accurate, corrects overbinding of LDA • Examples are PBE and BLYP • Hybrid: • Mixes some fraction of HF • Examples are B3LYP and PBE0 Pitt1

16. Too many functionals Pitt1

17. Functional approximations • Original approximation to EXC[n] : Local density approximation (LDA) • Nowadays, a zillion different approaches to constructing improved approximations • Culture wars between purists (non-empirical) and pragmatists. • This is NOT OK. NAMET

18. Modern DFT development It’s tail must decay like -1/r It must have sharp steps for stretched bonds It keeps H2 in singlet state as R→∞ Pitt1

19. Things users despise about DFT • No simple rule for reliability • No systematic route to improvement • If your property turns out to be inaccurate, must wait several decades for solution • Complete disconnect from other methods • Full of arcane insider jargon • Too many functionals to choose from • Can only be learned from another DFT guru NAMET

20. Things developers love about DFT • No simple rule for reliability • No systematic route to improvement • If a property turns out to be inaccurate, can take several decades for solution • Wonderful disconnect from other methods • Lots of lovely arcane insider jargon • So many functionals to choose from • Must be learned from another DFT guru NAMET

21. Difference between Ts and EXC • Pure DFT in principle gives E directly from n(r) • Original TF theory of this type • Need to approximate TS very accurately • Thomas-Fermi theory of this type • Modern orbital-free DFT quest (See Trickey and Wesolowsi talks) • Misses quantum oscillations such as atomic shell structure • KS theory uses orbitals, not pure DFT • Made things much more accurate • Much better density with shell structure in there. • Only need approximate EXC[n]. NAMET

22. Kieron’s trail of tears Include turning points Bohr atoms: Vee=0 Langer uniformization Arbitrary 3d potential WKB HF atoms Include exchange Semiclassics in Coulomb potential Include correlation TF theory Lieb et al Atoms 1d particles in wavy box V->0 at ∞ All electronic structure calculations Real atoms NAMET

23. The big picture • We show local approximations are leading terms in a semiclassical approximation • This is an asymptotic expansion, not a power series • Leading corrections are usually NOT those of the gradient expansion for slowly-varying gases • Ultimate aim: Eliminate empiricism and derive density functionals as expansion in ħ. NAMET

24. Basic picture • Turning points produce quantum oscillations • Shell structure of atoms • Friedel oscillations • There are also evanescent regions • Each feature produces a contribution to the energy, larger than that of gradient corrections • For a slowly-varying density with Fermi level above potential everywhere, there are no such corrections, so gradient expansion is the right asymptotic expansion. • For everything else, need GGA’s, hybrids, meta-GGA’s, hyper GGA’s, non-local vdW,… NAMET

25. Pandora • Many difficulties in answering this question: • Semiclassical methods • Asymptotic expansions • Boundary layer theory NAMET

26. What we’ve done so far Semiclassical density, Elliott, PRL 2008 Corrections to local approx, Cangi, PRB 2010 Bohr atoms, Snyder, in prep Slowly varying densities, with Perdew, PRL 2006 Ionization in large Z limit, Constantin, sub. JCP, 2010 Exact conditions on TS, D. Lee et al, PRA, 2009 Derivation of B88, Elliott, Can J Chem, 2009 PBEsol, Perdew et al, PRL 2008. NAMET

27. A major ultimate aim: EXC[n] • Explains why gradient expansion needed to be generalized (Relevance of the slowly-varying electron gas to atoms, molecules, and solids J. P. Perdew, L. A. Constantin, E. Sagvolden, and K. Burke, Phys. Rev. Lett. 97, 223002 (2006).) • Derivation of b parameter in B88 (Non-empirical 'derivation' of B88 exchange functional P. Elliott and K. Burke, Can. J. Chem. 87, 1485 (2009).). • PBEsolRestoring the density-gradient expansion for exchange in solids and surfaces J.P. Perdew, A. Ruzsinszky, G.I. Csonka, O.A. Vydrov, G.E. Scuseria, L.A. Constantin, X. Zhou, and K. Burke, Phys. Rev. Lett. 100, 136406 (2008)) • explains failure of PBE for lattice constants and fixes it at cost of good thermochemistry • Gets Au- clusters right NAMET

28. Structural and Elastic Properties Errors in LDA/GGA(PBE)-DFT computed lattice constants and bulk modulus with respect to experiment Improvements of PBEsol → Fully converged results (basis set, k-sampling, supercell size) → Error solely due to xc-functional → GGA does not outperform LDA → characteristic errors of <3% in lat. const. < 30% in elastic const. → LDA and GGA provide bounds to exp. data → provide “ab initio error bars” Blazej Grabowski, Dusseldorf • Inspection of several xc-functionals is critical to estimate predictive power and error bars! NAMET

29. Essential question • When do local approximations become relatively exact for a quantum system? • What is nature of expansion? • What are leading corrections? NAMET

30. Need help • Asymptotic analysis • Semiclassical theory, including periodic orbits • Boundary layer theory • Path integrals • Green’s functions for many-body problems • Random matrix theory • E.g., who has done spin-decomposed TF theory? NAMET

31. What we might get • We study both TS and EXC • For TS: • Would give orbital-free theory (but not using n) • Can study atoms to start with • Can slowly start (1d, box boundaries) and work outwards • For EXC: • Improved, derived functionals • Integration with other methods NAMET

32. Outline • Overview • Some details NAMET

33. One particle in 1d NAMET

34. N fermions NAMET

35. Rough sums NAMET

36. Inversion NAMET

37. Higher orders NAMET

38. Test system v(x)=-D sinp(mπx) NAMET

39. Semiclassical density for 1d box TF Classical momentum: Classical phase: Fermi energy: Classical transit time: Elliott, Cangi, Lee, KB, PRL 2008 NAMET

40. Density in bumpy box • Exact density: • TTF[n]=153.0 • Thomas-Fermi density: • TTF[nTF]=115 • Semiclassical density: • TTF[nsemi]=151.4 • DN < 0.2% NAMET

41. Usual continua • Scattering states: • For a finite system, E > 0 • Solid-state: Thermodynamic limit • For a periodic potential, have continuum bands NAMET

42. A new continuum • Consider some simple problem, e.g., harmonic oscillator. • Find ground-state for one particle in well. • Add a second particle in first excited state, but divide ħ by 2, and resulting density by 2. • Add another in next state, and divide ħ by 3, and density by 3 • … • →∞ NAMET

43. Continuum limit Leading corrections to local approximations Attila Cangi, Donghyung Lee, Peter Elliott, and Kieron Burke, Phys. Rev. B 81, 235128 (2010). Attila Cangi NAMET

44. Example of utility of formulas • Worst case (N=1) • Note accuracy outside of turning points • No evanescent contributions in formula NAMET

45. Getting to real systems • Include real turning points and evanescent regions, using Langer uniformization • Consider spherical systems with Coulombic potentials (Langer modification) • Develop methodology to numerically calculate corrections for arbitrary 3d arrangements NAMET

46. Classical limit for neutral atoms • For interacting systems in 3d, increasing Z in an atom, keeping it neutral, approaches the classical continuum, ie same as ħ→0 NAMET

47. Ionization as Z→∞ Lucian Constantin Using code of Eberhard Engel NAMET

48. Z→∞ limit of ionization potential • Shows even energy differences can be found • Looks like LDA exact for EX as Z-> ∞. • Looks like finite EC corrections • Looks like extended TF (treated as a potential functional) gives some sort of average. • Lucian Constantin, John Snyder, JP Perdew, and KB, arXiv. Could we get accurate results with QMC? See Richard Needs, PRE, 2005. NAMET

49. Ionization density for large Z NAMET

50. Ionization density as Z→∞ NAMET