Lecture 17. Density Functional Theory (DFT)

1. Lecture 17. Density Functional Theory (DFT) • References • A bird’s-eye view of density functional theory (Klaus Capelle) • http://arxiv.org/PS_cache/cond-mat/pdf/0211/0211443.pdf • Nobel lecture: Electronic structure of matter — wave functions and density functionals (Walter Kohn; 1998) • http://prola.aps.org/pdf/RMP/v71/i5/p1253_1

2. Postulate #1 of quantum mechanics • The state of a quantum mechanical system is completely specified by the wavefunction or state function that depends on the coordinates of the particle(s) and on time. • The probability density to find the particle in the volume element located at r at time t is given by . (Born interpretation) • The wavefunction must be single-valued, continuous, finite, and normalized (the probability of find it somewhere is 1). • = <|> Probability density

3. Wave Function vs. Electron Density • Probability density of finding any electron within a volume element dr1 N electrons are indistinguishable Probability density of finding electron 1 with arbitrary spin within the volume element dr1 while the N-1 electrons have arbitrary positions and spin • Wavefunction • Function of 3N variables (r1, r2, …, rN) • Not observable • Function of three spatial variables r • Observable (measured by diffraction) • Possible to extend to spin-dependent electron density

4. Electron Density as the Basic Variable • Wavefunction as the center quantity • Cannot be probed experimentally • Depends on 4N (3N spatial, N spin) variables for N-electron system • Can we replace the wavefunction by a simpler quantity? • Electron density (r) as the center quantity • Depends on 3 spatial variables independent of the system size

5. Density suffices. • Unique definition of the molecular system (through Schrödinger equation) • (N, {RA}, {ZA})  Hamiltonian operator wavefunction properties • N = number of electrons • {RA} = nuclear positions • {ZA} = nuclear charges • Unique definition of the molecular system (through density, too) • (N, {RA}, {ZA})  electron density properties • (r) has maxima (cusps) at {RA}

6. Electron Density as the Basic Variable1st Attempt: Thomas-Fermi model (1927) • Kinetic energy based on the uniform electron gas (Coarse approximation) • Classical expression for nuclear-electron and electron-electron interaction • (Exchange-correlation completely neglected) • The energy is given completely in terms of the electron density(r). • The first example of density functional for energy. • No recourse to the wavefunction.

7. Slater’s Approximation of HF Exchange: X method (1951) • Approximation to the non-local exchange contribution of the HF scheme • Interaction between the charge density and the Fermi hole (same spin) • Simple approximation to the Fermi hole (spherically symmetric) • Exchange energy expressed as a density functional • Semi-empirical parameter  (2/3~1) introduced to improve the quality (from uniform electron gas) X or Hartree-Fock-Slater (HFS) method

8. Thomas-Fermi-Dirac Model • Combinations of the above two: • Thomas-Fermi model for kinetic & classical Coulomb contributions • Modified X model for exchange contribution • Pure density functionals • NOT very successful in chemical application

9. The Hohenberg-Kohn Theorems (1964) • Reference • P. Hohenberg and W. Kohn, Phys. Rev. (1964) 136, B864 • http://prola.aps.org/pdf/PR/v136/i3B/pB864_1

10. Hohenberg-Kohn Theorem #1 (1964) Proof of Existence

11. There cannot be two different Vext (thus two different wavefuntion ) that yield the same ground state electron density (r). Hohenberg-Kohn Theorem #1 (1964) Proof of Existence The ground state electron density (r) in fact uniquely determines the external potential Vext and thus the Hamilton operator H and thus all the properties of the system.

12. Proof

13. Hohenberg-Kohn Functional • Since the complete ground state energy is a functional of the ground state electron density, so must be its individual components. system-independent, i.e. independent of (N,{RA},{ZA}) Hohenberg-Kohn functional

14. Hohenberg-Kohn Functional: Holy Grail of DFT

15. Finding Unknown Functional: Major Challenge in DFT • The explicit form of the functionals lies completely in the dark. • Finding explicit forms for the unknown functionals represent the major challenge in DFT. Kinetic energy Non-classical contribution Self-interaction, exchange, correlation Classical coulomb interaction

16. Hohenberg-Kohn Theorem #2 (1964) Variational Principle • FHK[] delivers the lowest energyif and only if the input density • is the trueground state density 0. • * Limited only to the ground state energy. No excited state information! Proof from the variational principle of wavefunction theory

17. Variational Principle in DFT Levy’s Constrained Search (1979) • Use the variation principle in wavefunction theory (Chapter 1) • Do it in two separate steps: • Search over the subset of all the antisymmetric wavefunctions X that yield a particular density X upon quadrature  Identify Xmin which delivers the lowest energy EXfor the given densityX • Search over all densities (=A,B,…,X,…)  Identify the density  for which the wavefunction min from (Step 1) delivers the lowest energy of all. Search over all allowed, antisymmetric N-electron wavefunction

18. Variational Principle in DFT • Determined simply by the density • Independent of the wavefunction • The same for all the wavefunctions • integrating to a particular density Universal functional

19. HK Theorem in Real Life? Pragmatic Point of View • The variational principle applies to the exact functional only. • The true functional is not available. • We use an approximation for F[]. • The variational principle in DFT does not hold any more in real life. • The energies obtained from an “approximate” density functional theory can be lower than the exact ones! • Offers no solution to practical considerations. Only of theoretical value.

20. The Kohn-Sham Approach (1965) • Reference • W. Kohn and L.J. Sham, Phys. Rev. (1965) 140, A1133 • http://prola.aps.org/abstract/PR/v140/i4A/pA1133_1

21. Implement Hohenberg-Kohn Theorems: Thomas-Fermi? • Hohenberg-Kohn theorems • Hohenberg-Kohn universal functional • Thomas-Fermi(-Dirac) model for kinetic energy: fails miserably • “No molecular system is stable with respect to its fragments!” Classical coulomb known Explicit forms remain a mystery. (from uniform electron gas)

22. Hartree-Fock, a Single-Particle Approach: Better than TF Section IV.C, Nobel lecture: Electronic structure of matter — wave functions and density functionals (Walter Kohn; 1998)

23. Better Model for the Kinetic Energy: Orbitals & Non-Interacting Reference System (HF for DFT?) Section IV.C, Nobel lecture: Electronic structure of matter — wave functions and density functionals (Walter Kohn; 1998) • A single Slater determinant constructed from N spin orbitals (HF scheme) • Approximation to the true N-electron wavefunction • Exact wavefunction of a fictitious system of N non-interacting electrons (fermions) under an effective potential VHF • The kinetic energy is exactly expressed as • Use this expression in order to compute the major fraction of the kinetic energy of the interacting system at hand

24. Non-Interacting Reference System: Kohn-Sham Orbital • Hamiltonian with an effective localpotential Vs (no e-e interaction) • The ground state wavefunction (Slater determinant) • One-electron Kohn-Sham orbitals determined by • with the one-electron Kohn-Sham operator • satisfy Vs chosen to satisfy the density condition Ground state density of the real target system of interacting electrons

25. ( ) The Kohn-Sham One-Electron Equations • If we are not able to accurately determine the kinetic energy through an explicit functional, we should be a bit less ambitious and • concentrate on computing as much as we can of the true kinetic energy exactly; and then • deal with the remainder in an approximate manner. • Non-interacting reference system with the same density as the real one • Exchange-Correlation energy (Junkyard of all the unknowns) Kohn-Sham orbitals

26. The Kohn-Sham Equations. Vs and SCF • Energy expression • Variational principle (minimize E under the constraint ) Density-based Wavefunction -based only term unknown iterative solution SCF where

27. The Kohn-Sham Approach: Wave Function is Back! A bird’s-eye view of density functional theory (Klaus Capelle), Section 4 http://arxiv.org/PS_cache/cond-mat/pdf/0211/0211443.pdf

28. The Kohn-Sham Equation is “in principle” exact! • Hartree-Fock: • By using a single Slater determinant which can’t be the true wavefunction, the approximation is introduced right from the start • Kohn-Sham: • If the exact forms of EXC and VXC were known (which is not the case), it would lead to the exact energy. • Approximation only enters when we decide on the explicit form of the unknown functional, EXC and VXC. • The central goal is to find better approximations to those exchange-correlation functionals. Section IV.C, Nobel lecture: Electronic structure of matter — wave functions and density functionals (Walter Kohn; 1998)

29. The Kohn-Sham Procedure I where

30. The Kohn-Sham Procedure II

31. The Kohn-Sham Procedure III

32. The Exchange-Correlation Energy: Hartree-Fock vs. Kohn-Sham • Hatree-Fock • Kohn-Sham

33. The Quest for Approximate Exchange-Correlation (XC) Functionals • The Kohn-Sham approach allows an exact treatment of most of the contributions to the electronic energy. • All remaining unknown parts are collective folded into the “junkyard” exchange-correlation functional (EXC). • The Kohn-Sham approach makes sense only if EXC is known exactly, which is unfortunately not the case. • The quest for finding better and better XC functionals (EXC) is at the very heart of the density functional theory.

34. Is There a Systematic Strategy? • Conventional wavefunction theory • The results solely depends on the choice of the approximate wavefunction. • The true wavefunction can be constructed by • Full configuration interaction (infinite number of Slater determinants) • Complete (infinite) basis set expansion • Never realized just because it’s too complicated to be ever solved, • but we know how it can be improved step by step in a systematic manner • Density functional theory • The explicit form of the exact functional is a total mystery. • We don’t know how to approach toward the exact functional. • There is no systematic way to improve approximate functionals. • However, there are a few physical constraints for a reasonable functional.

35. (constant everywhere) The Local Density Approximation (LDA) • Model • Hypothetical homogeneous, uniform electron gas • Model of an idealized simple metal with a perfect crystal • (the positive cores are smeared out to a uniform background charge) • Far from realistic situation (atom,molecule) with rapidly varying density • The only system for which we know EXC exactly • (Slater or Dirac exchange functional in Thomas-Fermi-Dirac model)

36. The Local Density Approximation (LDA) • Exchange • Correlation • From numerical simulations, (VWN)

38. Gradient Expansion Approximation (GEA) • LDA: not sufficient for chemical accuracy, solid-state application only • includes only (r), the first term of Taylor expansion • In order to account for the non-homogeneity, • let’s supplement with the second term, (r) (gradient) • Works when the density is not uniform but very slowly varying • Does not perform well (Even worse than LDA) •  Violates basic requirements of true holes (sum rules, non-positiveness) • (LDA meets those requirements.)

39. Generalized Gradient Approximation (GGA) • Contains the gradients of the charge density and the hole constraints • Enforce the restrictions valid for the true holes • When it’s not negative, just set it to zero. • Truncate the holes to satisfy the correct sum rules. Reduced density gradient of spin  Local inhomogeneity parameter

40. Exact Exchange Approach? • Exchange contribution is bigger than correlation contribution. • Exchange energy of a Slater determinant can be calculated exactly (HF). • Exact HF exchange + approximate functionals only for correlation • (parts missing in HF) • Good for atoms, Bad for molecules (32 kcal/mol G2 error) *HF 78 kcal/mol • Why? • The resulting total hole has the wrong characteristics (not localized). • “Local” Slater exchange from uniform electron gas seems a better model. This division is artificial anyway. “delocalized” (due to a single Slater determinant) “local” model functional (should be delocalized to compensate Ex)

41. Slater’s Approximation of HF Exchange: X method (1951) • Approximation to the non-local exchange contribution of the HF scheme • Interaction between the charge density and the Fermi hole (same spin) • Simple approximation to the Fermi hole (spherically symmetric) • Exchange energy expressed as a density functional • Semi-empirical parameter  (2/3~1) introduced to improve the quality (from uniform electron gas) X or Hartree-Fock-Slater (HFS) method

42. 0 linear Becke’s Hybrid Functionals: Adiabatic Connection where Non-interacting, Exchange only (of a Slater determinant), Exact 1 Fully-interacting, Unknown, Approximated with XC functionals Empirical Fit (Becke93) (2-3 kcal/mol G2 error) Half-and-half (Becke93) (6.5 kcal/mol G2 error) “B3LYP”

44. Summary: XC Functionals • LDA: Good structural properties, Fails in energies with overbinding • GGA (BP86, BLYP, BPW91, PBE): Good energetics (< 5 kcal/mol wrt G2) • Hybrid (B3LYP): The most satisfactory results

45. Self-Interaction Problem • Consider a one-electron system (e.g. H) • There’s absolutely no electron-electron interaction. • The general KS equation should still hold. • Classical electron repulsion • To remove this wrong self-interaction error, we must demand • None of the current approximate XC functions (which are set up independent of J[]) is self-interaction free.  0 even for one-electron system (naturally taken care of in HF)

46. Self-Interaction in HF • Coulomb term J when i = j (Coulomb interaction with oneself) • Beautifully cancelled by exchange term K in HF scheme • HF scheme is free of self-interaction errors.  0 = 0

47. Self-Interaction Error J[]+EXC[]

48. HK Theorem in Real Life? • The variational principle applies to the exact functional only. • The true functional is not available. • We use an approximation for F[]. • The variational principle in DFT does not hold any more in real life. • The energies obtained from an “approximate” density functional theory can be lower than the exact ones!