Ab-initio studies of the adiabatic connection in density-functional theory 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 Analysis, Manipulation and Simulation: New Developments in Theory 3rd EuChemMS Chemistry Congress Congress Center Nürnberg, Nürnberg, Germany, September 2, 2010 T. Helgaker (CTCC, University of Oslo) Calculation of the universal density functional EuChemMS 1 / 24
Outline 1 Lieb s formulation of density functional theory 2 Lieb maximization 3 The adiabatic connection 4 Dynamical correlation 5 Static correlation 6 Summary Helgaker et al. (CTCC, University of Oslo) Overview EuChemMS 2 / 24
The universal density functional In 1979, Levy formulated DFT in terms of a constrained-search optimization ( E λ [v] = inf Fλ [ρ] + (v ρ) ) the Hohenberg Kohn variation principle ρ F λ [ρ] = min Ψ T + λw Ψ the Levy variation principle Ψ ρ we have here introduced the interaction-strength parameter λ In 1983, Lieb gave a more symmetric but perhaps less intuitive formulation of DFT: ( E λ [v] = inf Fλ [ρ] + (v ρ) ) the Hohenberg Kohn variation principle ρ ( F λ [ρ] = sup Eλ [v] (v ρ) ) the Lieb variation principle v an unconstrained rather than constrained optimization of F λ [ρ] directly from E λ [v] equivalent to Levy s functional generalized to ensembles In Lieb s theory, E λ [v] and F λ [ρ] are related by Fenchel s inequality: E λ [v] F λ [ρ] + (v ρ) F λ [ρ] E λ [v] (v ρ) both variation principles attempt to sharpen this inequality into an equality these attempts are only successful for ground-state densities and potentials We shall apply the Lieb variation principle to standard ab-initio models of E λ [v] first application of Lieb s method to many-electron and molecular systems previous applications by Colonna and Savin (1999) to few-electron atoms Helgaker et al. (CTCC, University of Oslo) Lieb s universal density functional EuChemMS 3 / 24
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 EuChemMS 4 / 24
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 EuChemMS 5 / 24
Lieb s theory for approximate energies Lieb s theory may be applied to any exact or approximate energy that is concave: Fλ mod [ρ] LF Eλ mod [v] examples: the lowest state of any spin symmetry examples: all variationally determined energies such as E HF FCI λ [v] and Eλ [v] However, F mod λ [ρ] encapsulates information only about the concave part of Eλ mod[v] Eλ mod [v] LF Fλ mod [ρ] LF co E mod λ [v] E mod [v] λ Nothing more is needed for approximate ground-state energies such as MP2 and CCSD the approximate energy co E MP2 MP2 λ [v] is just as good as Eλ [v] We cannot hope to represent excited-state energies by a universal density functional the nth excited state E n λ [v] is in general not concave however, the sum of electronic energies up to the nth excited state is concave n E n λ [v] = E i λ [v] i=0 concave Helgaker et al. (CTCC, University of Oslo) Approximate energies EuChemMS 6 / 24
Lieb maximizations For a given density ρ(r) and chosen level of theory E λ [v], we wish to calculate F λ [ρ] = max v ( Eλ [v] (v ρ) ) by maximizing the right-hand side with respect to variations in the potential v(r) The potential is parameterized as suggested by Wu and Yang 2003: where the three terms are v c(r) = v ext(r) + (1 λ)v ref (r) + ct gt(r) t the physical, external potential v ext(r) the Fermi Amaldi (or other) reference potential to ensure correct asymptotic behaviour v ref (r) = (1 N 1) ρ(r ) r r dr an expansion in Gaussians g t(r) with coefficients c t we use large orbital basis sets, typically augmented with diffuse functions The optimization is carried out using Newton and quasi-newton methods there are no global convergence problems: a concave functional has at most one maximum 5 10 iterations with the exact Hessian to gradient norm 10 6 10 15 iterations with an approximate noninteracting Hessian All code is implemented in DALTON for HF, MP2, CCD, CCSD, and CCSD(T) Teale, Coriani and Helgaker, J. Chem. Phys. 130, 104111 (2009); ibid. 132, 164115 (2010) Helgaker et al. (CTCC, University of Oslo) Lieb maximization EuChemMS 7 / 24
The adiabatic connection We are going to consider the dependence of F λ [ρ] on λ: F λ [ρ] = min Ψ T + λw Ψ = Ψ ρ Ψ ρ λ T + λw Ψ ρ λ The functional F λ [ρ] increases monotonically with λ in a concave manner: F Λ Ρ W Λ Ρ F 0 Ρ F Λ Ρ F 0 Ρ 0.5 1. It can be represented by a monotonically decreasing right-continuous integrand W λ [ρ]: λ F λ [ρ] = F 0 [ρ] + W µ[ρ] dµ 0 the integrand W λ [ρ] is discontinuous where F λ [ρ] is nondifferentiable Assuming adiabaticity, the integrand is taken to be continuous: W λ [ρ] = F λ [ρ] = Ψ ρ λ W λ Ψρ AC integrand at each λ, we shall calculate both F λ [ρ] and its derivative F λ [ρ] Helgaker et al. (CTCC, University of Oslo) The adiabatic connection EuChemMS 8 / 24
Computational overview 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 next perform the Kohn Sham decomposition by introducing F 0 [ρ] and F 0 [ρ]: F 0 [ρ] = T s[ρ] noninteracting kinetic energy F 0 [ρ] = J[ρ] + Ex[ρ] Coulomb and exchange energies F λ [ρ] = T s[ρ] + λj[ρ] + λe x[ρ] + E c,λ [ρ] correlation energy The correlation energy is the only term that depends on λ in a nontrivial manner: F λ [ρ] = J[ρ] + Ex[ρ] + E c,λ [ρ] AC integrand The correlation energy is negative and concave in λ, here illustrated for the neon atom: 0.00 0.0 0.05 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) The adiabatic connection EuChemMS 9 / 24
Hartree Fock DFT The HF energy has a tiny correlation contribution ( 2.1 mh for water) the orbitals change with increasing λ but always provide the same density at λ = 0, the kinetic energy is minimized; at λ = 1, the total energy is minimized the associated orbital-relaxation energy is proportional to λ 2 8.956 E c Ρ WXC,Λ a.u. 8.957 8.958 T c Ρ 8.959 0.0 0.2 0.4 0.6 0.8 1.0 Λ The AC correlation curve is therefore very nearly a straight line in HF-OEP theory, the energy is 2.2 mh higher Görling and Ernzerhof (1995) surprisingly small energy differences between KS and HF wave functions of the same density Helgaker et al. (CTCC, University of Oslo) Dynamical correlation EuChemMS 10 / 24
The HOMO LUMO gap In HF theory, the wave function is one-determinantal at all 0 λ 1 the KS operator changes smoothly into the Fock operator, reproducing the HF density at all λ: ˆf λ = ˆf KS + λ [ˆkx v ] x(r) v c,λ (r) The HOMO LUMO gap increases with increasing λ LUMO 2 0.5 Eigenvalue a.u. 0.0 0.5 1.0 3p 0.0433 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 0.0 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 EuChemMS 11 / 24
The MP2 and CCSD correlation energies The MP2 and CCSD curves of water are not straight but bend slightly upwards the MP2 curve lies below the CCSD curve 9.0 9.1 E c Ρ WXC,Λ a.u. 9.2 9.3 T c Ρ 9.4 9.5 0.0 0.2 0.4 0.6 0.8 1.0 Λ A linear curve would reflect a λ 2 dependence of the correlation energy why does the curve bend? Helgaker et al. (CTCC, University of Oslo) Dynamical correlation EuChemMS 12 / 24
Dynamical correlation: doubles contribution 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 w 2 h+λg = λw λw h+λg quadratic dependence damped by an increasing HOMO LUMO gap We obtain a two-parameter AC model by differentiation and setting w = g we adjust h and g to reproduce exactly initial slope and end point 0.0 0.1 0.2 CCSD water WΛ a.u. 0.3 0.4 CCSD neon 0.5 0.6 0.0 0.2 0.4 0.6 0.8 1.0 Λ For λ 0, the AC curve becomes horizontal the wave function can no longer adjust: the electrons are as far apart as allowed by the density Helgaker et al. (CTCC, University of Oslo) Dynamical correlation EuChemMS 13 / 24
Dynamical correlation: triples contribution The CCSD(T) curve for water lies slightly below the CCSD curve the difference increases with increasing λ 9.0 0.000 9.1 E c Ρ 0.005 WXC,Λ a.u. 9.2 9.3 0.010 0.015 W CCSD T Ρ W CCSD Ρ 9.4 T c Ρ CCSD 0.020 9.5 CCSD T 0.025 0.0 0.2 0.4 0.6 0.8 1.0 Λ 0.0 0.2 0.4 0.6 0.8 1.0 An inspection of the triples energy correction suggests the following form E T (λ) = λ3 w 3 (h + λg) 2 E T (λ) = cλ2 + O(λ 3 ) 0.0256λ 2 an almost perfect fit with a one-parameter quadratic curve for λ 0, the triples AC integrand becomes horizontal Helgaker et al. (CTCC, University of Oslo) Dynamical correlation EuChemMS 14 / 24
From dynamical to static correlation: dissociation of H 2 Static correlation energy arises from (near) degeneracy of electronic configurations it is proportional to the interaction constant λ the corresponding AC curve is therefore horizontal As H 2 dissociates, correlation changes from dynamical to static purely dynamical correlation at equilibrium curvature increases as HOMO LUMO gap decreases with increasing bond length purely static correlation at dissociation 0.05 1.4 bohr 0.10 0.15 0.20 5 bohr 0.25 10 bohr We here (and in the following) assume spin-restricted theory Helgaker et al. (CTCC, University of Oslo) Static correlation EuChemMS 15 / 24
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 + 4w 2 λ 2, E = h + gλ Differentiation gives a simple two-parameter model for the AC integrand with w = g E CI (λ) = 1 2 g g(h + gλ) + 4w 2 λ 2 (h + gλ) 2 + 4w 2 λ 2 good least-square fits (full lines) and fits to initial gradient and end point (dashed lines) works for both dynamical and static correlation 0.00 Wc,Λ a.u. 0.05 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 10.0 a.u. 0.0 0.2 0.4 0.6 0.8 1.0 Helgaker et al. (CTCC, University of Oslo) Static correlation EuChemMS 16 / 24
The design of DFT functionals from AC models There has been much work on modeling AC integrands: Ernzerhof (1996), Burke et al. (1997), Seidl et al. (2000), Mori-Sánchez et al. (2007) In the absence of accurate data, there is little to differentiate the proposed AC forms such data is provided by the Lieb variation principle none of the suggested forms reproduce our data Part of the motivation for studying the AC is to develop new functionals simple forms can be turned directly into functionals by integration (Becke HH) All such AC forms are only as good as their input data we must also provide a recipe for producing input data We shall now consider one such simple model, based on strongly interacting electrons Helgaker et al. (CTCC, University of Oslo) Static correlation EuChemMS 17 / 24
Strongly interacting electrons (λ > 1) The AC curve of the CI model contains two adjustable parameters: E (λ) = 1 2 g g(h+gλ)+4g2 λ 2 (h+gλ), E(1) = 1 2 +4g 2 λ 2 0 E (µ) dµ 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 similar to bare-nucleus perturbation theory for fixed potential 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 Fixing h and g to reproduce 2E0 GL2 [ρ] and W PC [ρ], we obtain the dashed AC curves below Helgaker et al. (CTCC, University of Oslo) Static correlation EuChemMS 18 / 24
Attractive electrons (λ < 0) Attractive electrons have recently been considered by Seidl and Gori-Giorgi (2010) we have calculated AC curves for the physical H 2 density in the range 1 λ 1 0.4 R 10.0 a.u. 0.3 0.2 WΛ W0 a.u. 0.1 0.0 0.1 0.2 1.0 0.5 0.0 0.5 1.0 Λ Repulsive electrons (λ > 0): all AC curves become horizontal as the electrons become strictly correlated (for fixed density) covalent H 2 density Attractive electrons (λ < 0): all AC curves become equally sloped as the electrons move together (for fixed density) ionic H 2 density Helgaker et al. (CTCC, University of Oslo) Static correlation EuChemMS 19 / 24
Benchmarking explicit exchange correlation functionals Accurate studies of F λ [ρ] gives information about the exact functional this information may be used to benchmark approximate functionals example: comparison of the BLYP functional with the exact one 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 EuChemMS 20 / 24
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 10.0 bohr FCI Helgaker et al. (CTCC, University of Oslo) Static correlation EuChemMS 21 / 24
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 10.0 bohr Helgaker et al. (CTCC, University of Oslo) Static correlation EuChemMS 22 / 24
Summary and acknowledgments We have calculated the universal density functional by Lieb maximization Newton and quasi-newton methods standard levels of theory: HF, MP2, CCSD, CCSD(T), BLYP many-electron atoms and molecules We have presented accurate adiabatic-connection curves and discussed their shape dynamical and static correlation accurate modeling of AC curves repulsive and attractive electrons benchmarking of approximate 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 EuChemMS 23 / 24