The calculation of the universal density functional by Lieb maximization

The calculation of the universal density functional by Lieb maximization Trygve Helgaker, Andy Teale, and Sonia Coriani Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry, University of Oslo, Norway Dipartimento di Scienze Chimiche, Università degli Studi di Trieste, Trieste, Italy 2010 Girona Workshop on Theoretical Chemistry (GWTC) Hotel Carlemany, Girona, Spain, October 18 20, 2010 T. Helgaker (CTCC, University of Oslo) Calculation of the universal density functional GWTC 2010 1 / 22

The universal density functional Consider the ground-state energy with interaction strength λ: E 0 [v] = inf Ψ T + i v(r i ) + λv ee Ψ the Rayleigh Ritz variation principle Ψ We may summarize density-functional theory in two variation principles: ( F λ [ρ] = sup Eλ [v] (v ρ) ) the Lieb variation principle v ( E λ [v] = inf Fλ [ρ] + (v ρ) ) the Hohenberg Kohn variation principle ρ these are alternative attempts at sharpening the same inequality into an equality F λ [ρ] E λ [v] (v ρ) E λ [v] F λ [ρ] + (v ρ) Fenchel s inequality The Hohenberg Kohn variation principle underlies all applications of DFT in chemistry its success hinges on the accurate modelling of F λ [ρ] The Lieb variation principle provides a tool for studying F λ [ρ] numerically we have implemented the Lieb variation principle at various levels ab initio theory our motivation is to benchmark approximate F λ [ρ] and to develop new ones Previous work: studies by Colonna and Savin (1999) for few-electron atoms work by Wu and Yang (2003) on Lieb maximizations Helgaker et al. (CTCC, University of Oslo) Lieb s universal density functional GWTC 2010 2 / 22

Conjugate functionals E[v] F [ρ] The ground-state energy E[v] is concave in v by the Rayleigh Ritz variation principle it can therefore be exactly represented by its convex conjugate F [ρ]: E[v] F [ρ] F Ρ sup v E v v Ρ F Ρ F Ρ Ρ min v max E v E v E v inf Ρ F Ρ v Ρ v Ρ v Ρ Each variation principle represents a Legendre Fenchel transformation the essential point is the concavity of E[v] rather than the details of the Schrödinger equation we may therefore construct F [v] also for approximate E[v] if these are concave Helgaker et al. (CTCC, University of Oslo) Lieb s universal density functional GWTC 2010 3 / 22

The concave envelope E[v] F [ρ] co E[v] However, approximate E[v] may not be concave (not variationally minimized) it still generates a convex F [ρ], conjugate to the concave envelope co E[v] E[v] F Ρ sup v E v v Ρ F Ρ F Ρ Ρ min v max co E v E v E v E v inf Ρ F Ρ v Ρ v Ρ v Ρ The concave envelope co E[v] is the least concave upper bound to E[v] with this caveat, we may introduce all ab-initio levels of theory for E[v] into DFT as E[v] converges to the exact ground-state energy, so does co E[v] Helgaker et al. (CTCC, University of Oslo) Lieb s universal density functional GWTC 2010 4 / 22

Kohn Sham theory and the adiabatic connection The potential is parametrized as suggested by Wu and Yang (2003): v(r) = v ext(r) + (1 λ)v ref (r) + t ct gt(r) Gaussian expansion For given E λ [v] and ρ, we calculate the universal functional and its λ derivative: F λ [ρ] = E λ [v max] (v max ρ) Lieb maximization F λ [ρ] = E λ [vmax] first-order property We introduce Kohn Sham theory by expanding about λ = 0: F λ [ρ] = F 0 [ρ] + λf 0 [ρ] + E c,λ[ρ] = T s[ρ] + λ(j[ρ] K[ρ]) + E c,λ [ρ] The correlation energy is the only term that depends on λ in a nontrivial manner: E c,λ [ρ] = F λ [ρ] F 0 [ρ] = Ψ λ V ee Ψ λ Ψ 0 V ee Ψ 0 AC integrand The correlation energy is negative and concave in λ, here illustrated for the neon atom: 0 5 0.10 area Ec Ρ 0.15 0.20 0.25 0.30 Ec, Λ Ρ ' E c,λ Ρ area Tc Ρ 0.35 Helgaker et al. (CTCC, University of Oslo) Lieb s universal density functional GWTC 2010 5 / 22

Basis-set convergence of the AC integrand Helium atom at the FCI/aug-cc-pVXZ levels of theory 1.02 E XC aug cc pv6z 1.06631 a.u. E XC aug cc pv6z FCI 1.06631 a.u. 1.04 WΛ 1.06 1.08 1.10 0.2 0.4 0.6 0.8 0 Helgaker et al. (CTCC, University of Oslo) Dynamical correlation GWTC 2010 6 / 22

Hartree Fock DFT The HF energy has a tiny correlation contribution ( 2.1 mh for water) at λ = 1, the total energy is minimized at λ = 0, the kinetic energy is minimized for the same density The associated orbital-relaxation energy is proportional to λ 2 the AC correlation curve is therefore very nearly a straight line 8.956 E c Ρ WXC,Λ a.u. 8.957 8.958 T c Ρ 8.959 0.2 0.4 0.6 0.8 1.0 Λ Helgaker et al. (CTCC, University of Oslo) Dynamical correlation GWTC 2010 7 / 22

The HOMO LUMO gap The HOMO LUMO gap increases with increasing λ LUMO 2 0.5 Eigenvalue a.u. 0.5 1.0 3p 433 3s 0.1100 2p 0.8063 LUMO 1 LUMO HOMO 3s 0.2094 3p 0.2063 2p 0.8507 1.5 2s 1.6739 HOMO 1 2s 1.9310 2.0 0.2 0.4 0.6 0.8 1.0 Virtual orbitals increase in energy as nonlocal exchange replaces local exchange Occupied orbitals decrease in energy because of relaxation Helgaker et al. (CTCC, University of Oslo) Dynamical correlation GWTC 2010 8 / 22

Dynamical correlation energy Dynamical correlation is quadratic in λ (by perturbation theory) the AC curves should therefore be linear in λ (by differentiation) The MP2 (red) and CCSD (blue) AC curves of water bend upwards why? 9.0 9.1 E c Ρ WXC,Λ a.u. 9.2 9.3 T c Ρ 9.4 9.5 0.2 0.4 0.6 0.8 1.0 Λ Helgaker et al. (CTCC, University of Oslo) Dynamical correlation GWTC 2010 9 / 22

The increasing importance of density constraint Second-order perturbation theory suggests the following model: E MP2 (λ) = ijab λ 2 ij ab 2 ε a(λ)+ε b (λ) ε i (λ) ε j (λ) λ2 g 2 h+λg = λg λg h+λg a quadratic dependence damped by an increasing HOMO LUMO gap Differentation yields a two-parameter AC model adjusted to initial slope and end point 0.1 0.2 CCSD water WΛ a.u. 0.3 0.4 CCSD neon 0.5 0.6 0.2 0.4 0.6 0.8 1.0 Λ The wave function loses its ability to adjust with increasing λ indeed, for λ 0, the AC curve becomes horizontal (strictly correlated electrons) Helgaker et al. (CTCC, University of Oslo) Dynamical correlation GWTC 2010 10 / 22

Higher-order dynamical correlation Higher-order correlation corrections are proportional to λ 3 the triples correction to the AC curve depends quadratically on λ 9.0 9.1 E c Ρ WXC,Λ a.u. 9.2 9.3 9.4 T c Ρ CCSD CCSD T CCSD 9.5 CCSDT 0.2 0.4 0.6 0.8 1.0 Λ Helgaker et al. (CTCC, University of Oslo) Dynamical correlation GWTC 2010 11 / 22

From dynamical to static correlation: H 2 dissociation As H 2 dissociates, correlation changes from dynamical to static increasing curvature for small λ with decreasing HOMO LUMO gap Static correlation energy arises from (near) degeneracy of electronic configurations it is of first order in λ (first-order degenerate perturbation theory) the corresponding AC curve is therefore horizontal 5 1.4 bohr 0.10 0.15 0.20 5 bohr 0.25 10 bohr Helgaker et al. (CTCC, University of Oslo) Static correlation GWTC 2010 12 / 22 0

A two-parameter model for the H 2 molecule From a two-level CI model, we obtain the ground-state energy E CI (λ) = 1 2 E 1 2 E 2 + 4g 2 λ 2, E = h + gλ Differentiation gives a two-parameter model for the AC integrand good least-square fits (full lines) and fits to initial gradient and end point (dashed lines) 0 Wc,Λ a.u. 5 0.10 0.15 R 0.7 a.u. R 1.4 a.u. R 3.0 a.u. 0.20 R 5.0 a.u. R 7.0 a.u. 0.25 R 1 a.u. 0.2 0.4 0.6 0.8 1.0 Helgaker et al. (CTCC, University of Oslo) Static correlation GWTC 2010 13 / 22

Strongly interacting electrons (λ > 1) Which two input values (parameters) can be used to construct accurate AC curves? Initial slope is twice the second-order Görling Levy perturbation theory E (0) = 2E0 GL2 [ρ] Görling Levy theory is an order-by-order expansion in λ for fixed density The point-charge-plus-continuum (PC) model yields the end point E ( ) = W PC[ρ] developed by Seidl, Perdew and Kurth (2000) for strictly correlated electrons λ = used in the interaction-strength-interpolation (ISI) model of the AC integrand Helgaker et al. (CTCC, University of Oslo) Static correlation GWTC 2010 14 / 22

Benchmarking explicit exchange correlation functionals Dissociation of H 2 at the RHF, BLYP and FCI levels of theory the BLYP functional improves considerably on HF theory for dissociation what is the reason for this improvement? 0.5 0.6 0.7 0.8 HF 0.9 1.0 BLYP FCI 1.1 1.2 Helgaker et al. (CTCC, University of Oslo) Static correlation GWTC 2010 15 / 22

BLYP and FCI correlation curves for H 2 The BLYP functional treats correlation as dynamical at all bond distances it was designed for spin-unrestricted theory but is used here in a spin-restricted manner it hence ignores static correlation BLYP FCI BLYP FCI R 1.4 bohr R 3.0 bohr BLYP BLYP R 5.0 bohr FCI R 1 bohr FCI Helgaker et al. (CTCC, University of Oslo) Static correlation GWTC 2010 16 / 22

BLYP and FCI exchange correlation curves for H 2 The improved BLYP performance arises from an overestimation of exchange error cancellation between exchange and correlation reduces total error to about one third R 1.4 bohr R 3.0 bohr FCI exchange BLYP exchange BLYP correlation FCI correlation R 5.0 bohr R 1 bohr Helgaker et al. (CTCC, University of Oslo) Static correlation GWTC 2010 17 / 22

Generalized, range-dependent adiabatic connection Interactions may also be turned on in a range-dependent manner ( ) λ erf w g λ (r 1 λ r ij ij ) = 2 ( λ r ij π 1 λ ) ( exp 1 ( λ 3 1 λ ) 2 ) r 2 ij Λ 0.5 Λ 0.75 6 6 5 5 4 Λ 0.9 4 Λ 0.99 3 3 2 2 1 1 0 0 As λ increases, we build up the full interaction from the outer to inner region this provides an alternative view of correlation Savin et al. 1996, Yang 1998, Toulouse et al. 2004 Helgaker et al. (CTCC, University of Oslo) Range separation GWTC 2010 18 / 22

AC correlation curves for the He isoelectronic series Curves reveal increasing compactness with increasing Z 2 4 6 8 0.10 0.12 0.14 Z 1 Z 2 0 0.1 0.2 0.3 0.4 0.5 Z 4 Z 10 Note: long-range interactions to the left; short-range interactions to the right Helgaker et al. (CTCC, University of Oslo) Range separation GWTC 2010 19 / 22

Range separation: dissociation of H 2 We consider first the total AC curve, including Coulomb, exchange and correlation it moves towards small λ values with increasing separation at full separation, all total interactions are interatomic 2.0 1.5 1.4 bohr 3.0 bohr 1.0 0.5 2.0 5.0 bohr 2.0 1 bohr 1.5 1.5 1.0 1.0 0.5 0.5 Note: long-range interactions to the left; short-range interactions to the right Helgaker et al. (CTCC, University of Oslo) Range separation GWTC 2010 20 / 22

Range separation: dissociation of H 2 We consider next the correlation-only AC curve at short bond distance, the interactions are predominantly short-ranged at long distances, short- and long-ranged interactions partially cancel 0.4 0.2 1.4 bohr 0.4 0.2 3.0 bohr 0.2 0.4 0.6 0.4 0.2 5.0 bohr 0.4 0.2 1 bohr 0.2 0.4 0.6 Note: long-range interactions to the left; short-range interactions to the right Helgaker et al. (CTCC, University of Oslo) Range separation GWTC 2010 21 / 22

Summary and acknowledgments We have calculated the universal density functional by Lieb maximization what information can be gotten from the Lieb variation principle? can accurate calculations of AC help in the design of density functionals? An interesting project is to study the current dependence of F [ρ, j] we have code for gauge-origin-invariant calculations of E[v, A] in finite magnetic fields a generalized Legendre Fenchel transform will provide F [ρ, j] with charge and current densities this will help in benchmarking and developing approximate E xc[ρ, j] We would like to thank Andreas Savin and Paola Gori-Giorgi for discussions This work was supported by the Norwegian Research Council through Grant No. 171185 (A.M.T) Grant No. 179568/V30 Centre for Theoretical and Computational Chemistry (CTCC) Helgaker et al. (CTCC, University of Oslo) Summary and acknowledgments GWTC 2010 22 / 22