Excitation energies from density-functional theory some failures and successes. Trygve Helgaker

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1 1 Excitation energies from density-functional theory some failures and successes Trygve Helgaker Centre for Theoretical and Computational Chemistry, University of Oslo, Norway Ola Lutnæs, Simen Reine, Erik Tellgren, CTCC, University of Oslo, Norway Michael Peach and David Tozer, University of Durham, UK Stinne Høst, Branislav Jansik, Poul Jørgensen, Jeppe Olsen, University of Aarhus, Denmark Sonia Coriani, CTCC and University of Trieste, Italy Pawe l Sa lek, Royal Institute of Technology, Sweden Filip Paw lowski, Kazimierz Wielki University, Poland New Frontiers In Theory-Based Chemical Research May 5, 2008 University of Oslo Norway

2 2 Overview This talk consists of two separate parts In the first part, we consider some very general aspects of DFT the Hohenberg Kohn theorem the Legendre Fenchel transform In the second (main) part, we consider excitation energies some general comments comparison with coupled-cluster theory local, Rydberg and charge-transfer excitations the role of exact exchange the CAM-B3LYP functional assessment of DFT for excitation energies a diagnostic for excitation energies excitation energies in large systems

3 3 The Hohenberg Kohn theorem Different external potentials give rise to different ground-state densities v 1 (r) v 2 (r) + c ρ 1 (r) ρ 2 (r) Hohenberg Kohn theorem the potential is therefore determined by the ground-state density v[ρ] + c We may therefore express the energy in terms of the density Z F HK [ρ] = E[v[ρ]] v[ρ](r)ρ(r)dr Hohenberg Kohn functional We obtain the energy by a minimization over all ground-state densities» Z E[v] = min F HK [ρ] + v(r)ρ(r) dr Hohenberg Kohn variation principle ρ The problem with this procedure is the restriction of ρ(r) to a ground-state density we have no way of identifying ground-state densities from other densities There exists a more general formulation of density-functional theory it removes the restriction to ground-state densities

4 4 The convexity of E[v] A function is said to be convex if it satisfies the relation f(cx 1 + (1 c)x 2 ) cf(x 1 ) + (1 c)f(x 2 ), 0 c 1 fx 1 c fx 1 1c fx 2 fc x 1 1cx 2 fx 2 x 1 c x 1 1cx 2 x 2 For a given external potential v(r), the electronic energy E[v] is given by E[v] = min Ψ Ψ Ĥ[v] Ψ variation principle by the variation principle E[v] is a convex functional in v(r) note: this holds only for the ground state Convexity of E[v] is the only requirement for DFT

5 5 The conjugate energy functionals E[v] and F[ρ] For any convex E[v], there exists a conjugate convex functional F[ρ] such that R F[ρ] = max `E[v] v(r)ρ(r)dr energy as a functional of density v R E[v] = min `F[ρ] + v(r)ρ(r)dr energy as a functional of potential ρ E[v] and F[ρ] are conjugate functionals (mutual Legendre Fenchel transforms) The potential v(r) and the density ρ(r) are conjugate variables they belong to dual linear spaces such that R v(r)ρ(r)dr is finite they satisfy the reciprocal relations δf[ρ] δρ(r) = v(r), δe[v] δv(r) = ρ(r) Such conjugate functionals are ubiquitous in physics Lagrangian L(v) and Hamiltonian H(p) in classical mechanics energies U(V, S), H(P, S), A(V, T) and G(P, T) in thermodynamics Legendre transforms were introduced into DFT by Lieb IJQC 24, 243 (1983) Eschrig The Fundamentals of Density Functional Theory (Teubner, Stuttgart 1996)

6 6 The density-functional of the electronic energy The energy may be represented as a functional either of v(r) or of ρ(r): R F[ρ] = max `E[v] v(r)ρ(r)dr v R E[v] = min `F[ρ] + v(r)ρ(r)dr ρ note: there are no problems with domains for ρ and v these are complete dual linear spaces The Schrödinger equation does not enter the relationship between E[v] and F[ρ] to set up F[ρ], you only need to know E[v] for all possible potentials E[v] and F[ρ] contain exactly the same information We are free to choose the functional that suits our task best in molecular mechanics, we choose to parameterize E[v] in density-functional theory, we parameterize F[ρ] The Hohenberg Kohn theorem also relies on the variation principle ground-state ρ(r) determines v(r) uniquely however, it is not needed to establish F[ρ]

7 7 The functional form of F[ρ] is unknown Kohn Sham theory the most difficult part is the kinetic energy In Kohn Sham theory, we partition the energy as F[ρ] = T s [ρ] + J[ρ] + E XC [ρ] where the contributions are T s [ρ] = max `E0 [v] R v(r)ρ(r)dr noninteracting kinetic energy v ZZ ρ(r1 )ρ(r 2 ) J[ρ] = dr 1 dr 2 Coulomb energy r 12 E XC [ρ] = F[ρ] T s [ρ] J[ρ] exchange correlation energy The exchange correlation energy is usually taken to have the forms Z EXC LDA [ρ] = f(ρ(r))dr local-density approximation (GGA) Z EXC GGA [ρ] = f(ρ(r, ρ(r))dr generalized gradient approximation (GGA) a large number of functional forms for E XC [ρ] have been suggested they are essentially all local (LDA) or semi-local (GGA) functionals

8 8 DFT in chemistry Kohn Sham theory made DFT a semi-quantitative tool Local-density approximation exchange correlation modeled after the uniform electron gas widely applied in condensed-matter physics not sufficiently accurate in chemistry The introduction of Becke s gradient correction to exchange (1988) changed the situation the accuracy became sufficient to compete in chemistry indeed, surprisingly high accuracy for energetics Beyond Kohn Sham theory, orbital dependence is introduced into exchange some proportion of exact exchange (Becke, 1993) hybrid DFT Progress has to a large extent been semi-empirical nonempirical functionals empirical functionals However, the way forward is unknown Jackob s ladder (cmp. the ladder of coupled-cluster theory)

9 9 Reaction Enthalpies (kj/mol) B3LYP CCSD(T) exp. CH 2 + H 2 CH (2) C 2 H 2 + H 2 C 2 H (2) C 2 H 2 + 3H 2 2CH (2) CO + H 2 CH 2 O (1) N 2 + 3H 2 2NH (1) F 2 + H 2 2HF (1) O 3 + 3H 2 3H 2 O (2) CH 2 O + 2H 2 CH 4 + H 2 O (1) H 2 O 2 + H 2 2H 2 O (2) CO + 3H 2 CH 4 + H 2 O (1) HCN + 3H 2 CH 4 + NH (3) HNO + 2H 2 H 2 O + NH (1) CO 2 + 4H 2 CH 4 + 2H 2 O (1) 2CH 2 C 2 H (3)

10 10 Excitation energies Over the last decade, DFT has become the workhorse of chemistry its popularity stems from its ability to provide good structures and energetics DFT is widely used also for molecular properties response theory geometric, electric, magnetic perturbations static and dynamic perturbations linear and nonlinear response For most properties, the application of DFT has been a success for example, it created a revolution in the calculation of spin spin coupling constants Excitation energies are an interesting case important chemical property of molecules accessible from linear-response theory some excitations are well reproduced, others poorly the failures highlight deficiencies in the exchange correlation potential

11 11 Small molecules: comparison with coupled-cluster theory The best way to benchmark DFT is usually to compare with wave-function results For small molecules, very accurate coupled-cluster results are known coupled-cluster hierarchy converges systematically Hartree Fock, CCSD, CC3 theories We have compared singlet excitation energies and polarizabilities HF, CO and H 2 O d-aug-cc-pvtz (CO and H 2 O) and t-aug-cc-pvtz (HF) Salek et al., Mol. Phys. 103, 439 (2005) As always, the coupled-cluster convergence is uniform excitation energies decrease as higher excitations are included The CC3 results should be within 0.1 ev of the exact results Hartree Fock results differ from CC3 results by 8.6 ev CCSD results differ from CC3 results by 0.2 ev

12 12 Comparison with coupled-cluster theory for HF HF (grey), CCSD (red), CC3 (black) LDA (yellow), BLYP (green), B3LYP (blue)

13 13 Comparison with coupled-cluster theory for CO HF (grey), CCSD (red), CC3 (black) LDA (yellow), BLYP (green), B3LYP (blue)

14 Comparison with coupled-cluster theory for H 2 O HF (grey), CCSD (red), CC3 (black) LDA (yellow), BLYP (green), B3LYP (blue) 14

15 15 Comparison with coupled-cluster theory for CO, HF, and H 2 O Statistics for errors calculated excitation energies relative to the CC3 results (%) HF CCS CC2 CCSD LDA BLYP B3LYP std max Quality of the DFT results is only modest mean absolute errors: LDA 18%, BLYP 20%, B3LYP 12% all excitation energies are underestimated Note the role of exact exchange Hartree Fock theory overestimates excitation energies B3LYP (20% exact exchange) performs much better than LDA and BLYP the introduction of exchange reduces error considerably We have not here analyzed the results in terms of valence and Rydberg excitations

16 16 Deficiencies of DFT excitation energies Standard pure DFT functionals suffer from some known deficiencies The correct description of Rydberg excitation requires a correct asymptotic form the potential falls off too fast the asymptotic behaviour should be this gives too low Rydberg excitations lim r v XC(r) = 1 r A similar dependence is seen in charge-transfer (CT) excitations an electron is transferred from one part of a molecule to another there is a resulting exchange attraction 1/R between hole and electron local functionals are incapable of capturing this behaviour see Dreuw et al., JCP 119, 2943 (2003)

17 17 The role of exact exchange The deficiencies of DFT is not shared by Hartree Fock theory exact exchange provides the correct behaviour The correct behaviour is ensured by introduced full exact exchange in DFT unfortunately, this destroys more important things such as energetics delocalized exchange hole, difficult static correlation problem local exchange does not introduce static correlation The B3LYP functional uses a fixed amount of exchange, for all interelectronic separations let us introduce additional flexibility 1 = α + β erf(µr 12) + 1 [α + β erf(µr 12)], 0 α + β 1, 0 α, β 1 r 12 r 12 r 12 {z } {z } exact exchange functional exchange α exact exchange at zero separation increased to α + β at infinite separation Tawada et al. JCP 120, 8425 (2004), Yanai et al. CPL 393, 51 (2004) assessment: Peach et al. PCCP 8, 558 (2006)

18 18 Proportion of exact exchange in different functionals HF LC CAMB3LYP B3LYP

19 19 The application of CAM-B3LYP to excitation energies The CAM-B3LYP functional uses the following split (Yanai et al. CPL 393, 51 (2004)) 1 = α + β erf(µr 12) + 1 [α + β erf(µr 12)], α = 0.19, β = 0.46, µ = 0.33 r 12 r 12 r 12 {z } {z } exact exchange functional exchange The CAM-B3LYP functional improves significantly excitation energies errors in Rydberg excitation energies for CO and N 2 reduced by a factor of two errors CT excitation energies in a dipeptide reduced by the same amount errors in polarizabilities are likewise reduced NMR properties are unaffected However, atomization energies and vibrational frequencies become less accurate mean absolute errors in atomization energies are doubled note: α and β determined through a fit to atomization energies (G2-1 set) The CAM-B3LYP parameters are not optimal for excitation energies correct asymptotic form with β = 0.81 halves excitation errors again energetics now completely unacceptable

20 20 Distance-dependence of CT excitations charge-transfer excitation energy of C 2 H 4 C 2 F 4

21 21 Assessment of calculated excitation energies We have made an extensive assessment of DFT excitation energies Peach et al. JCP 128, (2008) 59 singlet excitations of 18 theoretically challenging main-group molecules O H O O O H O N N N N N N N H O H H H O H dipeptide β-dipeptide tripeptide n=1 5 N NC N acenes (n=1 5) N-phenylpyrrole DMABN H H H H n=2 5 PA oligomers (n=2 5) N 2 H 2 CO CO HCl DMABS = 4-(N,N-dimethylamino) benzonitrile

22 22 Tripeptide excitations O H O N N N H O H excitation type PBE B3LYP CAM exp. n 2 π2 local n 1 π1 local n 3 π3 local π 1 π2 CT π 2 π3 CT n 1 π2 CT n 2 π3 CT π 1 π3 CT n 1 π3 CT

23 23 Statistics for 59 excitation energies PBE B3LYP CAM local std max Rydberg std max CT std max

24 24 A diagnostic for excitation energies At present, DFT functionals for excitation energies are unsatisfactory valence excitations are fine (off by a few tenths of an ev) Rydberg, CT and core excitations are not poor When do we need to exercise caution? When can we trust our results? problems arise when excitations occur between orbitals with no spatial overlap Excitations represented by rotations κ ai between all occupied and virtual orbitals to measure spatial overlap between s and i, we introduce Z O ia = φ i (r) φ a (r) dr without moduli all overlaps would be trivially zero We now measure spatial overall using the quantity P ai Λ = κ2 ai O ai P, 0 Λ 1 ai κ2 ai a small value of Λ signifies a long-range excitation a large value of Λ signifies a short-range excitation Peach et al. JCP 128, (2008)

25 25 Excitation energies against Λ for PBE With this GGA functional, performance for Λ < 0.6 is erratic local excitations (Λ > 0.4) are well reproduced, within a few tenths of an ev Rydberg excitations (Λ < 0.3) are systematically underestimated by a few evs CT excitations cover a surprisingly large Λ range (DMABN)

26 26 Excitation energies against Λ for B3LYP With the introduction of exact exchange, performance improves local excitations are lifted and cover a slightly larger Λ range errors in Rydberg excitations are reduced by a factor of two likewise, an improvement in CT excitations is observed

27 27 Excitation energies against Λ for CAM-B3LYP Coulomb attenuation gives a uniform description of local and nonlocal excitations! however, errors as large as 1 ev are still observed

28 28 Some conclusions Extensive assessment of PBE, B3LYP and CAM-B3LYP for excitation energies Errors in excitation energies correlated to spatial orbital overlap Best overall behaviour provided by the attenuated CAM-B3LYP functional no correlation observed between errors and spatial overall a good quality, balanced description this functional is recommended for excitation-energy calculations The PBE functional exhibits a clear correlation between errors and spatial overlap to a lesser extent, the B3LYP functional exhibits the same behaviour We propose the following diagnostic test: a PBE excitation with Λ < 0.4 is likely to be in significant error a B3LYP excitation with Λ < 0.3 is likely to be in significant error The term charge transfer is ambiguous such excitations span a very large range of overlaps small for peptide systems, large for DMABN

29 29 Excitation energies in large systems We are working towards large systems (thousands of atoms) this requires a rewrite of many algorithms in quantum chemistry linear scaling from sparsity of matrix elements We represent the system entirely in terms of the density matrix D(X) = exp(x)d exp( X), X T = X symmetrically orthonormalized set of atomic orbitals Direct minimization of energy with respect to X E = min X E(D(X)) a more controlled optimization than RH diagonalizations with DIIS no molecular orbitals are generated Response theory may be formulated in terms of D(X) unperturbed system represented by X = 0 perturbations represented by X 0 the calculation of excitation energies and polarizabilities possible for large systems

30 30 Response theory We consider a system described by the one-electron density matrix D(X) = exp( X)D exp(x) (X = 0 for unperturbed system) and define the Hessian and metric operators in terms of their transformations E [2] (X) = (F vv F oo )X + X(F vv F oo ) + G vo ([D,X]) G ov ([D,X]) S [2] (X) = X ov X vo When perturbed by V ω of frequency ω, the system responds to first order as E [2] (X ω ) ωs [2] (X ω ) = [D,V ω ] linear response matrix equation perturbed density matrix: D ω = [D,X ω ] perturbed expectation values: Â; ˆV ω ω = TrA[D,X ω ] In the absence of a perturbation V ω = 0, we obtain an eigenvalue equation E [2] (X n ) = ω n S [2] (X n ) RPA matrix eigenvalue equation transition density matrix: D 0n = [D,X n ] 0 Â n transition moments: = TrA[D,Xn ] Coriani et al., J. Chem. Phys. 126, (2007) Ochsenfeld, Head-Gordon, Weber, Niklasson, and Challacombe (static properties)

31 31 Excitation energies of linear hydrocarbons and of D 6h graphite sheets The lowest 20 singlet and 20 triplet LDA/4-31G excitation energies as functions of n nalkanes Cn H 2 n2 0.6 polyenes Cn H n polyynes Cn H D 6 h graphite sheets In the linear chains, the lowest excitation energy decreases roughly as a + bn 1. BLYP/6-31G % % polyethylenes polyenes polyynes In the graphite sheets, the lowest excitation energy is proportional to n 1/2.

32 32 BLYP vs. LDA In general, BLYP performs in much the same manner as LDA for excitation energies: the BLYP excitation energies are typically within 1 or 2 me h of the LDA energies the differences between LDA and BLYP are smaller for longer chains BLYP increases the triplet singlet separation slightly Plots of the LDA/6-31G and BLYP/6-31G excitation energies in the polyynes: 0.09 singlet 0.09 triplet In general, LDA and GGA functionals underestimate excitation energies for HF, CO, and H 2 O, the underestimation is 15% to 20% this problem is exacerbated in extended conjugated systems the introduction of nonlocality (beyond GGA) may be very important in providing quantitatively correct excitation energies, in particular in conjugated systems

33 33 Polarizabilities of linear alkanes and alkenes To illustrate, we have calculated longitudinal polarizabilities in linear polymeric chains HF and DFT α and α/n in 6-31G basis, plotted against the number of carbons N 6 K alkanes Α 15 4 K 10 alkanes ΑN 2 K K alkenes Α 300 alkenes ΑN LDA BLYP 80 K 40 K HF B3LYP CAM CAM The alkenes are about an order of magnitude more polarizable than the alkanes all models agree on alkanes (α/n-limit: HF 14.4; LDA 16.3) widely different results for alkenes (α/n-limit: HF 97; LDA 427)

34 34 The importance of exact exchange for longitudinal polarizabilities Without a good description of long-range exchange, the systems become too polarizable 600 Hartree 0% LDA 0% BLYP 0% B3LYP 20% 100 HF 100% CAMB3LYP 19%65% CAMB3LYP 19%100% the Hartree model neglects all exchange and overestimates by a factor of eight pure DFT has a poor long-range exchange and overestimates by a factor of four hybrid functionals improve the situation, introducing some exact exchange compromise solution: standard DFT at short range, full exchange at long range

35 35 CAMB3LYP/6-31G excitation energies of alanine residue peptides HOMOLUMO gap lowest excitation energy & lowest Hessian eigenvalue lowest excitation energy almost identical with lowest Hessian eigenvalue CAMB3LYP has 19% short-range and 65% long-range exact exchange

36 36 B3LYP/6-31G excitation energies of alanine residue peptides B3LYP (with 20% exact exchange) behaves differently, in an unphysical manner beyond 27 alanine residues, the lowest excitation energy becomes very small RH/DIIS converges to an excited state (first-order saddle point) Polyalanine, excitation energies DIIS ARH Energy / a.u Alanine residues for LDA and GGA functionals, the crossing occurs even earlier local DFT functionals unable to account for long-range exchange in large molecules

37 37 B3LYP/6-31G polyalanine HOMO LUMO gap and lowest Hessian eigenvalue The B3LYP energy minimum behaves in a strange manner the lowest Hessian eigenvalue goes through a minimum for 27 residues the RH/DIIS solution behaves smoothly with increasing chain length Energy / a.u DIIS - HOMO-LUMO gap ARH - HOMO-LUMO gap DIIS - Hessian ARH - Hessian Alanine residues

38 38 B3LYP/6-31G polyalanine polarizabilities Polyalanine, dipole polarizability DIIS ARH Energy / a.u Alanine residues

Strategies for large-scale molecular self-consistent field calculations. Trygve Helgaker. Teoretisk og beregningsorientert kjemi

1 Strategies for large-scale molecular self-consistent field calculations Trygve Helgaker Centre for Theoretical and Computational Chemistry, University of Oslo, Norway Filip Paw lowski, Simen Reine, Erik