Comparison of ehange-correlation functionals: from LDA to GGA and beyond Martin Fuchs Fritz-Haber-Institut der MPG, Berlin, Germany Density-Functional Theory Calculations for Modeling Materials and Bio-Molecular Properties and Functions - A Hands-On Computer Course, 3 October - 5 November 25, IPAM, UCLA, Los Angeles, USA
Density-functional theory DFT is an exact theory of the ground state of an interacting many-particle system: E = min n N E v[n] E = min Ψ N Ψ Ĥv Ψ by Hohenberg-Kohn theorem: v(r) v[n(r)] Total energy density functional (electrons: w(r, r ) = 1/ r r ) E v [n] = min Ψ n Ψ ˆT + Ŵ Ψ + n(r)v(r)dτ = F [n] + n(r)v(r) dτ Non-interacting case: w(r, r ) F [n] min Ψ n Ψ ˆT Ψ = Φ[n] ˆT Φ[n] = T s [n] Φ[n] is a Slater determinant Interacting case rewritten: E v [n] = T s [n] + E [n] + 1 2 n(r)w(r, r )n(r )dτdτ + n(r)v(r) dτ XC energy functional: kinetic correlation energy & ehange and Coulomb correlation energy E [n] =: Ψ[n] ˆT Ψ[n] T s [n] + Ψ[n] Ŵ Ψ[n] 1 2 R n(r)w(r, r )n(r )dτdτ
Kohn-Sham scheme Minimization of E v [n] with n(r) N determines groundstate: δe v [n] δn(r) = µ δf [n] δn(r) + v(r) = δt s[n] δn(r) + vks ([n]; r) = µ... effective non-interacting system Kohn-Sham independent-particle equations: h i 2 2 + vks (r) φ i (r) = ɛ i φ i (r) with v KS (r) = w(r, r )n(r )dτ + δe [n] δn(r) + v(r), density n(r) = P i f i φ i (r) 2, occupancies ɛ i < µ : f i = 1, ɛ i = µ : f i 1, ɛ i > µ : f i = (aufbau principle). non-interacting kinetic energy treated exactly T s [n] = R P i f i φ i (r) 2 dτ classical electrostatics (e-n & e-e) treated exactly only need to approximate E [n] and v ([n]; r) = δe [n] δn(r) applied is spin density functional theory: n, treated as separate variables determines accuracy in practice! beware: v KS (r) is a local operator. Direct approximation of whole F [n] possible too: DFT with beware: generalized Kohn-Sham schemes 1 and nonlocal effective potentials, e.g. Hartree-Fock eqs. 1 Seidl, Görling, Vogl, Majewski, Levy, Phys Rev B 53, 3764 (1996).
E LDA [n] = n(r)e hom (n(r))dτ Local-density approximation local dependence on density or Wigner-Seitz radius r s = ( 4π 3 n) 1/3. XC energy per electron is that of the homogeneous electron gas, e [n] = e hom (n) n=n(r) ehange part e hom x (n) known analytically (n) known analytically for r s and r s numerically exact for 2 < r s < 1 from QMC data 1... as parametrization interpolating over all r s PW91, Perdew-unger, VWN 2 correlation part e hom c e (Ry) 1. 2. 3..1 1. 1. 1. r s (bohr) workhorse in DFT applications to solids real systems are far from jellium-like homogeneity... why does LDA work at all?... where does it fail?... how to improve beyond it? 1 Ceperley, Alder, Phys Rev Lett 45, 566 (198). 2 Perdew, Wang, Phys Rev B 45, 13244 (1992); Perdew, unger (198); Vosko, Wilk, Nussair (198).
Performance of the LDA structural, elastic, and vibrational properties often good enough crystal bulk lattice constants accurate to within 3%, usually underestimated bulk moduli somewhat too large, > 1% error not uncommon for d-metals phonons somewhat too stiff binding energies are too negative (overbinding), up to several ev cohesive energies of solids, but formation enthalpies often o.k. molecular atomization energies, mean error (148 molecules G2 set) 3 ev activation energies in chemical reactions unreliable too small/absent, e.g. for H 2 on various surfaces (Al, Cu, Si,...) relative stability of crystal bulk phases can be uncertain SiO 2 high pressure phase more stable than zero pressure phase underestimated transition pressure e.g. for diamond β-tin phase transitions in Si & Ge magnetic phases electronic structure can be usefully interpreted (density of states, band structures), eept for band gaps (a more fundamental issue than LDA!)
View on XC through the XC hole Definition of the XC energy by a coupling constant integration of the e-e interaction 1 : E([n]; λ) = Ψ λ [n] ˆT + ˆV λ + λŵ Ψ λ[n] = min Ψ λ n... by Hohenberg-Kohn theorem: n(r) v λ ([n]; r) for any λ 1 λ = non-interacting case/ Kohn-Sham potential, v λ= ([n]; r) = v KS ([n]; r) λ = 1 external potential, v λ=1 ([n]; r) = v(r) groundstate energy: E([n]; 1) = E([n]; ) + 1 d E([n]; λ) dλ dλ E [n] = R 1 Ψ λ[n] λŵ Ψ λ[n] dλ 1 2 R n(r)w(r, r )n(r )dτdτ X, XX = 1 2 w(r, r )n(r)n(r ) Ψ λ[n] P p,q ˆΨ p (r) ˆΨ q (r ) ˆΨ q (r ) ˆΨ p (r) Ψ λ [n] n(r)n(r ) dτdτ PSfrag XC in terms of the pair correlation function g λ ([n]; r, r replacements ) XC in terms of the adiabatic connection integrand (scaled e-e potential energy) may distinguish ehange and correlation, e.g. E = E x + E c g(r, r ) r r 1 cf. DFT books by Dreizler and Gross & Yang and Parr
... coupling constant averaged XC hole λ-integration implies coupling constant averaged pair correlation function g([n]; r, r ) = 1 g λ ([n]; r, r )dλ Identify the XC energy as E [n] = 1 2 dτ n(r) dτ n(r ){ g([n]; r, r ) 1} w(r, r ) interpretation: the electron density n(r) interacts with the electron density of the XC hole n ([n]; r, r ) = n(r ){ g([n]; r, r ) 1} Pauli elusion principle & Coulomb repulsion local density approximation corresponds to n ([n]; r, r ) = n(r){ g hom (n(r); r r ) 1} PSfrag replacements always centered at reference electron & spherical XC hole (n). + correlation ehange.5 homogeneous electron gas 1 2 3 4 5 6 k F r r (NB: correlation part varies with k F )
From the XC hole to the XC energy Focus on E [n] = R n(r)e ([n]; r)dτ (... component of the total energy) w = 1 r r = 1 u for e only the angle averaged XC hole matters (and LDA hole is always spherical) n (r, r + u) = X lm n lm (r, u)y lm(ω u ) E [n] = 1 2 n (r, r + u) n(r) dτ u dτ 1 u 2 n(r) n (r, u) u u 2 du dτ for E only the system & angle averaged XC hole matters XC energy in terms of averaged XC hole 1 E [n] = N 2 n (u) = 1 N n (u) u n(r)n (r, u)dτ u 2 du = N average XC energy per electron LDA & GGA approximate average holes rather closely work mostly o.k. 1 Perdew et al., J Chem Phys 18, 1552 (1998).
Sum rule and other constraints on the XC hole sum rule (constrains global behavior) n ([n]; r, r )dr = 1... average hole too LDA: R n(r)n hom (n(r), r, r )dr = 1 on-top hole n () (fixes value at u = ) LDA: accurate (exact in some limits) for correlation, exact for ehange cusp condition (constrains behavior around u = ) LDA: correct n (u) = n () + n() u u= average XC hole approximate sum rule on top cusp exact 1 2 3 4 5 separation u LDA works well because the LDA (average) XC hole is that of a physical system, jellium Beware: pointwise behavior of n LDA (r, r ) may be incorrect (e.g. outside metal surface), system averaging unweights tail and near-nucleus regions XC potential v ([n]; r) for LDA can be locally poor, but again less so for the system average
Generalized Gradient Approximation for E ➊ Gradient expansion of XC energy and (later) hole: n ([n]; r, u) n (r, u = ) + u n u= +..., and generalized to imposing constraints to meet e.g. n x (r, u) <, R n x (r, u)dr = 1, by real space cutoffs numerical GGA scaling relations and bounds on E by analytic approximation to numerical GGA parameter-free GGA by Perdew-Wang PW91 simplified in PBE GGA contains LDA & retains all its good features ➋ Earlier: analytic model or ansatz + empirical parameter(s) Langreth-Mehl (C), Becke 86(X) + Lee-Yang-Parr(C) BLYP,... ➌ Alternative: analytic ansatz + (many) fitted parameters (e.g. fit to thermochemical data) see accuracy limit of GGA functionals, can be better & worse than PBE Generic GGA XC functional E GGA [n] = n e LDA x (n) F GGA (n, s) n=n(r) dτ, s = n 2k F n n=n(r) Enhancement factor F (n, s) over LDA ehange: function of density and scaled gradient understanding how GGA s work Calculations with GGA s are not more involved than with LDA, eept that v [n; r] = v (n, i n, i j n) n(r)
Differences in present GGA s ments enhancement factor 1.8 1.6 1.4 1.2 1. r s =1 5 1 5 1 PBE r s = PBE XC 1 2 3 scaled gradient PSfrag replacements enhancement factor scaled gradient enhancement factor hang et al. revised revpbe 1.8 1.6 1.4 1.2 r s =1 5 1 5 1 hang 1. 1 2 3 scaled gradient PSfrag replacements enhancement factor scaled gradient enhancement factor 1.8 1.6 1.4 1.2 1. BLYP r s =1 5 1 5 1 BLYP 1 2 3 scaled gradient... WC-PBE Wu & Cohen 5 xpbe Xu & Goddard 4 HCTH Handy et al. 1 RPBE Hammer et al. 99 revpbe hang et al. 98 PBE GGA Perdew et al. 96 PW91 Perdew & Wang 91 PW91 Perdew & Wang 91 BP Becke & Perdew 88 BLYP Becke & Lee et al. 88... revpbe & BLYP more nonlocal than PBE GGA molecules: more accurate atomization energies solids: bondlengths too large, cohesive energies too small LYP correlation incorrect for jellium more local GGAs will make lattice constants smaller (e.g. Tinte et al. PRB 58, 11959 (1998)), but make binding energies more negative
... how GGA s change total energies Analysis in terms of selfconsistent LDA density shows how GGAs work: E GGA tot [n GGA ] = E LDA tot [n LDA ] + E GGA [n LDA ] E LDA [n LDA ] + O(n GGA n LDA ) 2 Spectral decomposition in terms of s: E GGA E LDA XC = s= h e GGA i (r) e LDA (r) δ(s s(r))dτds energy per atom (a.u.).5..5 X+C PBE correlation rev PBE ehange Al fcc 1 2 3 4 scaled gradient s PSfrag replacements energy per atom (ev)..2.5 fcc Al PBE revpbe differential integrated 1 2 3 4 scaled gradient s -.3 (.5) ev -.5 (.7) ev only s 4 contribute, similar analysis can be made for n(r) for more see e.g. upan et al., PRB 58, 11266 (1998)
Cohesive properties in GGA error (%) Bulk lattice constants GGA increase due to more repulsive core-valence XC. 5 5 Na Al NaCl C Si PW91 GGA inconsistent GGA LDA Cu Ge SiC AlAs GaAs expt. error (ev) Cohesive energies GGA reduction mostly valence effect. 1.5 1..5..5 Na Al C NaCl Si PW91 GGA Cu Ge SiC AlAs GaAs LDA inconsistent GGA expt. for comparison of LDA, GGA, and Meta-GGA see Staroverov, Scuseria, Perdew, PRB 69, 7512 (24)
... energy barriers: H 2 on Cu(111) barrier to dissociative adsorption - PW91-GGA.7 ev - LDA <.1 ev total energy of free H 2 - PW91-GGA 31.8 ev ( expt.) - LDA 3.9 ev energy TS E des a E ads a?? Cu + H 2 : LDA-GGA 1 ev reaction coordinate Hammer et al. PRL 73, 14 (1994)
Phase transition β-tin - diamond in Si Spectral decomposition in terms of n 2. diamond-structure Relative volume / r s 1.5 1. β-tin c a a β-tin structure.5. diamond 1 2 3 4 Density parameter r s [bohr] diamond: a = 1.26 bohr, semiconductor β-tin: c/a =.552, metallic
Phase transition β-tin - diamond in Si Gibbs construction: E β t + p tv β t = E dia t + p t V dia t, p exp t = 1.3... 12.5 GPa 25 energy (ev/atom).6.4.2. β tin LDA diamond GGA.6.7.8.9 1. 1.1 relative volume V/V exp transition pressure (GPa) 2 15 1 5 expt LDA PBE PW91 revpbe BP.2.3.4.5.6 energy change (ev/atom) BLYP GGA increases transition pressure, inhomogeneity effect use of LDA-pseudopotentials insufficient Moll et al, PRB 52, 255 (1995); DalCorso et al, PRB 53, 118 (1996); McMahon et al, PRB 47, 8337 (1993).
Performance of PBE GGA vs. LDA atomic & molecular total energies are improved GGA corrects the LDA overbinding: average error for G2-1 set of molecules: +6 ev HF 1.5 ev LSDA.5 ev PBE-GGA.2 ev PBE hybrid ±.5 ev goal, better cohesive energies of solids improved activation energy barriers in chemical reactions (but still too low) improved description of relative stability of bulk phases more realistic for magnetic solids useful for electrostatic hydrogen bonds GGA softens the bonds increasing lattice constants decreasing bulk moduli no consistent improvement LDA yields good relative bond energies for highly coordinated atoms, e.g. surface energies, diffusion barriers on surfaces GGA favors lower coordination (larger gradient!), not always enough where LDA has a problem, e.g. CO adsorption sites on transition metal surfaces significance of GGA? GGA workfunctions for several metals turn out somewhat smaller than in LDA one-particle energies/bands close to LDA Van der Waals (dispersion) forces not included! Comparing LDA and (different) GGAs gives an idea about possible errors!
Beyond GGA: orbital dependent XC functionals Kohn-Sham non-interacting system make φ i [n] density-functionals: n v KS [n; r] φ i LDA and GGA are explicit density-functionals E [n] Implicit density-functionals formulated in terms of φ i [n]? more flexible for further improvements self-interaction free: E [n] N=1 =, v ( r ) = 1 r, nonspherical n (r, r )...? Start with exact ehange : Ehange-energy E x [{φ i }] = 1 2 P φ i,j i (r) φ j (r )φ i (r )φ j (r) dτdτ r r same as in Hartree-Fock, but 2 2 vks (r) φ i (r) = ɛ i φ i (r) KS and HF orbitals different! Groundstate... in KS-DFT: optimized effective potential OEP method, local KS potential in Hartree-Fock: variation with respect to orbitals yield HF eqs., nonlocal effective potential E HF E KS EXX
Hybrid functional: mixing exact ehange with LDA/GGA XC = EXX + LDA correlation is less accurate (underbinds) than XC = LDA for molecules challenge: correlation functional that is compatible with exact ehange? hybrid functionals to interpolate adiabatic connection, U λ [n] E [n] = 1 U [n](λ) dλ U [n](λ) = Ψ λ [n] Ŵ Ψ λ[n] U Hartree [n] molecular dissociation: U = molecules - atoms... looks like U (λ) E x 1 λ coupling strength LDA,GGA too negative T c E c LDA, GGA good Ψ λ= = Φ KS KS system Ψ λ=1 physical system
How they are defined:... hybrid functionals How hydbrids and GGAs work: use exact ehange for λ = and a local functional for λ = 1 Hybrid functional = interpolation n o E hyb = E GGA + a E x E GGA x mixing parameter a =.16....3 from fitting thermochemical data a = 1/4 by 4th order perturbation theory PBE = PBE1PBE B3LYP 3-parameter combination of Becke X-GGA, LYP C-GGA, and LDA molecular dissociation energies on average within 3 kcal/mol.1 ev (but 6 times larger errors can happen) Adiabatic connection N 2 2N U (λ) (ev) 2-2 -4-6 "exact" PBE GGA ehange only.5 1 coupling strength λ XC contributions to binding: PBE hybrid (ev) E b E x E c PBE -1.58-1.82-2.49 PBE -9.62-1.1-2.49 exact -9.75 1.39-5.7... X and C errors tend to cancel! Becke, J Chem Phys 98, 5648 (1993); Perdew et al, J Chem Phys 15, 9982 (1996). comprehensive comparison: Staroverov, Scuseria, Tao, Perdew, J Chem Phys 119, 12129 (23)
XC revisited: role of error cancellation between X and C... or: when GGAs or hybrids are not enough see e.g. Baerends and Gritsenko, JCP 123, 6222 (25)
Meta-GGA s... orbital dependent functionals XC energy functional E MGGA [n] = n(r)e MGGA (n, n, t s ) n(r) dτ kinetic energy density of non-interacting electrons t s (r) = 1 2 P occ i f i φ i (r) 2 OEP or HF style treatment of δe MGGA [φ i ] δφ i (r) =... 1 2 e t s φ i TPSS: Tao, Perdew, Stavroverov, Scuseria, Phys Rev Lett 91, 14641 (23): + v x finite PKB: Perdew, Kurth, upan, Blaha, Phys Rev Lett 82, 2544 (1999): XC non-empirial, LDA limit Van Voorhis, Scuseria, J Chem Phys 19, 4 (1998): XC highly fitted, no LDA limit Colle, Salvetti, Theoret Chim Acta 53, 55 (1979): C, no LDA limit BLYP GGA Becke, J Chem Phys 19, 292 (1998): XC + exact ehange fitted TPSS accomplishes a consistent improvement over (PBE) GGA PKB improved molecular binding energy, but worsened bond lengths in molecules & solids hybrid functionals on average still more accurate for molecular binding energies TPSS provides sound, nonempirical basis for new hybrids TPSSh next step: MGGA correlation compatible with exact ehange?
ACFDT XC: including unoccupied Kohn-Sham states Adiabatic connection: KS system λ = physical system λ = 1 E [n] = R 1 Ψ λ Ŵee Ψ λ dλ U[n] Fluctuation-dissipation theorem: W (λ) =: Ψ λ Ŵee Ψ λ = 1 2 e 2 r r» π... using dynamical density response χ λ (iu, r, r ) du n( r)δ( r r ) d rd r From noninteracting Kohn-Sham to interacting response by TD-DFT χ (iu, r, r ) = 2R P i,j ϕ i ( r)ϕ j ( r)ϕ j ( r )ϕ i ( r ) i u (ɛ j ɛ i ) ɛ i [n] ϕ i ([n], r)... KS eigenvalues... KS orbitals χ λ (iu) = χ (iu) + χ (iu) K h λ (iu) χ λ(iu) Dyson equation 6dim. RPA: K h λ forces... using Coulomb and XC kernel from TD-DFT In principle ACFDT formula gives exact XC functional In practice starting point for fully nonlocal approximations = λ r r 1 and zero XC kernel... yields exact ehange and London dispersion combine with XC kernels, hybrids with usual XC functionals, split Coulomb interaction...
Status of RPA type functionals Molecular dissociation energies RPA for all e-e separations error in dissociation energy (ev) 2 1.5 -.5-1 -2 H 2 N 2 RPA+ RPA PBE hyb PBE GGA O 2 F 2 Si 2 HF CO CO 2 C 2 H 2 H 2 O... no consistent improvement over GGA... not too bad, without X and C error cancellation... better TD-DFT XC kernels needed! Stacked benzene dimer (Van der Waals complex) short-range: GGA + long-range: RPA Interaction energy (kcal/mol) 5 4 3 2 1 1 2 3 4 3 3.5 4 4.5 Separation (A)... includes dispersion forces! CCSD(T) MP2 vdw DF GGA(revPBE) GGA(PW91) Furche, PRB 64, 19512 (22), Fuchs and Gonze, PRB 65, 23519 (22) Dion et al., PRL 92, 24641 (24)
Summary LDA & GGA are de facto controlled approximations to the average XC hole GGA remedies LDA shortcomings w.r.t. total energy differences but may also overcorrect (e.g. lattice parameters) still can & should check GGA induced corrections for plausibility by... simple arguments like homogeneity & coordination... results from quantum chemical methods (Quantum Monte Carlo, CI,... )... depends on actual GGA functional hybrid functionals mix in exact ehange (B3LYP, PBE,... functionals) orbital dependent, implicit density functionals: exact Kohn-Sham ehange, Meta-GGA & OEP method, functionals from the adiabatic-connection fluctuation-disspation formula Always tell what XC functional you used, e.g. PBE-GGA (not just GGA)... helps others to understand your results... helps to see where XC functionals do well or have a problem