Density Functional Theory (DFT) Time Dependent Density Functional Theory The working equations Performance Properties.

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1 DFT and TDDFT

2 Contents 1 Density Functional Theory (DFT)

3 Basic literature: Jensen: Introduction to Computational Chemistry, Wiley and Sons. Koch, Holthausen: A Chemist s Guide to Density Functional Theorey, Wiley. Parr, Yang: Density Functional Theory of Atoms and Molecules, Oxford.

4 The wavefunction is the central concept of quantum mechanics, since it describes the quantum mechanical state. it determines all other properties through the calculation of expectation values. However, it is not a measurable quantity. It is a function of 3 N coordinates. Is such a detailed information required or is this an information overkill?. It becomes an increasingly complex task to construct better wavefunctions: orbital φ(r) product Ansatz Slater determinant CI!

5 In contrast, the electron density ρ(r) is an observable, can be determined e. g. by X-ray. is a function of three coordinates (x,y,z). it can be shown, that the information of 3N coordinates is NOT required to calculate the desired expectation values.

6 Density and wavefunction ρ(r 1 ) = N Ψ(r 1...r N ) 2 dv 2...dV N However, this is not the way to go, since the determination of the true N-particle wf is the complicated task! Can we determine the density directly? Can we get an energy depending on the density only E[ρ]?

7 ==> this would be a Density-Functional: Function of a function ==> how to determine? Need energy functional and then minimize as in HF: Variational principle ==> but most important question: is the density an unique feature of a certain system? I.e., are the densities coming from different external potentials (= core potentials in QC) different? Only then, the energy of a system can be uniquely determined by the density!

8 Hohenberg and Kohn (HK) Theorems HK1: the map G: v(r) ρ(r) is invertible. I.e. there is a one-to-one correspondence of potential and density, therefore, it is uniquely defined through the external potential. Since the potential uniquely determines the wf and the wf the expectation values, this theorem assures that any quantum mechanical observable is completely determined by the density.

9 Hohenberg and Kohn (HK) Theorems HK2: There exists a functional E[ρ] with (ρ 0 : ground state density): E[ρ] E 0, Therefore, the derivative: E[ρ 0 ] = E 0 δe[ρ] δρ = 0 results in an equation, from which the ground state density can be determined.

10 Total energy functional E[ρ] = T s [ρ] + E ne [ρ] + J[ρ] + E xc [ρ], J[ρ] Hartree energy. E xc = E x + E c : exchange-correlation (XC) energy functional. T s : kinetic energy of non-interaction particles ==> will be evaluated from Slater determinant as in HF DFT is a single determinal method: fails in multi-reference cases

11 Kohn-Sham (KS) Theorem: non-interacting electrons Let ρ 0 be the ground state density of the interacting electrons. Then there exists a local potential v eff [ρ 0 ] for the non-interacting electrons, leading to the same density ρ 0 via solution of the KS equations: [ 1 ] v eff [ρ] φ i = ɛ i φ i, ρ 0 (r) = i φ i 2 KS effective potential: v eff [ρ] = δepot v eff [ρ] = α Z α R α r dr XC-potential: v xc [ρ] = δexc δρ δρ, ρ(r ) r r dr + v xc [ρ]

12 XC functionals LDA (Local Density Approximation: from electron gas): E x = C ρ 4/3 (r)d 3 r GGA (Generalized Gradient Approximation): E x = C ρ 4/3 (r)f (s)d 3 r, s = ρ ρ 4/3 Various approximations for X and C: BP, BLYP and PBE being the most popular. Hybride Functionals: E h x = (1 c)e GGA x + ce HF x Usually, 20-30% HF-X work well (B3LYP: c=0.2)

13 LDA, GGA...: Accuracy for Geometries, vib. frequencies quite good. Energies: LDA, some GGA s show severe overbinding hybrid functionals. See Koch/Holthausen for more details. Probems due to the approximate nature of the functionals: Self-interaction error (SIC) asymptotics of v xc near-sightedness of E xc

14 Asymptotics of v xc : Fast decay of LDA (GGA...) exchange potential Figure: Elliot, Burke, Furche: arxiv:cond-mat/ v Eigenvalue spectrum quantitatively incorrect: Ionization threshold too low. Rydberg states unbound (underestimated).

15 LDA and GGA are local functionals, however, should be non-local as e.g. HF exchange: Locality: consider two weakly interacting fragments: ρ = ρ 1 + ρ 2 Then, if the densities do not overlap, the local functionals vanish: E xc = 0 This is in particular a problem for VdW interactions, which DFT-GGA is not able to handle. (hybrides change only E x, but E c is the problem here!) (see e.g. JCP114 (2001) 5149)

16 Overestimation of polarization in extended conjugated chains (Champagne et al. JCP 109, 10489) due to short-sightedness (locality) of E x.

17 Overestimation of polarizabiliy (Champagne et al. JCP 109, 10489) due to short-sightedness of E x. Figure: Gritsenko, Champagne,Gisbergen, Baerends

18 This has severe implications for many properties, e.g. proton affinities: (J. Computer-Aided Mol. Design, 20 (2006) 511) And will be of particular relevance for charge transfer excitations in TDDFT.

19 M. E. Casida, in: Recent Advances in Density Functional Methods, Vol. 1, ed. D. P. Chong (World Scientific, Singapore, 1995) M.E. Casida, in Recent Developments and Applications of Modern Density Functional Theory, edited by J.M. Seminario (Elsevier, Amsterdam, 1996), p Time-Dependent Density Functional Response Theory of Molecular Systems: Theory, Computational Methods and Functionals Time-Dependent Density Functional Theory, Edited by M.A.L. Marques, C.A. Ullrich, F. Nogueira, A. Rubio, K. Burke, and E.K.U. Gross, Lecture Notes in Physics Vol. 706 (Springer: Berlin, 2006). Elliot, Burke and Furche, arxiv:condmat/ v (broad review, theory, applications) Dreuw and Head-Gordon, Chem. Rev. (2005) 105, (Comparison to CI etc., Biology, CT)

20 Early implementations: Bauernschmitt and Ahlrichs (1996) CPL 256, 454. (Turbomole) Jamorski et al., (1996)JCP 104, (DeMon) M. Petersilka, U. J. Grossmann und E. K. U. Gross, Phys. Rev. Lett. 76, 1212 (1996). (alternative formulation) Stratmann et al., (1998) JCP 109, (Gaussian) Hirata and Head Gordon (1999) CPL 313, 291. (TDA)

21 Introduction DFT is a ground state theory, i.e. there is no general way, to apply the variational principles to excited states. They can be calculated using the SCF procedure for excited states with different symmetry (like triplets) The applicability and accuracy, however, is limited.

22 Introduction Alternative: time dependent density functional theory (TD-DFT) and linear response theory (LR): TD-DFRT. Basic idea: Calculate response properties: E.g the polarizabilities, which describe the response of a system to an external perturbation. The polarisability α describes the reaction of a system to an external field F, inducing a dipole moment: δµ = α F

23 A time dependent external field F (t) = F 0 sinωt will induce a time dependent dipole moment. δµ(t) = δ µ 0 sinωt. Resonance absorption of a photon In linear response: α = µ F Fourier transform of α(t): mean polarizability α(ω) = I f I ω I ω has poles at the excitation energies ω I

24 HK 1 equivalent Generalize DFT for time-dependent phenomena HK 1+2 for time dependent external fields: Runge & Gross (1984, PRL 52, 997): v(r, t) ρ(r, t) The time dependent potential uniquely determines the density (up to a time dependent constant, phase factor)

25 HK 2 equivalent Energy not conserved variational principle not available! DFT analog to the time-dependent Schrödinger equations is derived via action principle (A[ρ], action: energy x time). Equation of motion: with i t Ψ i(r, t) = [ ˆT + v eff [ρ](r, t)] Ψ i (r, t) v eff [ρ](r, t) = v ext [ρ](r, t) + ρ(r, t) r r dr + v xc [ρ](r, t)

26 Approximation for XC v xc [ρ] = δa xc δρ v xc [ρ], as a derivative of the Action functional is non-local in time, i.e. depends on all times t <t ( memory effect!). The correct functional form, as in ground states DFT, is not known. Therefore, usually the adiabatic local density approximation (ALDA) is applied: δa xc δρ δe xc δρ and the same functionals as for the ground state are applied. This approximation is valid in the limit of slowly varying external potential (instantaneous response of density to external perturbation).

27 (Linear) density response Density depends on potential: Expansion: v eff (r, t) = v eff,0 (r) + v app (r, t) ρ[v eff ](r, t) = ρ[v eff,0 + v app ](r, t) ρ(r, t) = ρ 0 (r) + ρ 1 (r, t) +... = ρ 0 (r) + χ(r, r, t, t )v app (r, t )dr dt +... Derivative of density wrt to external potential v app : χ(r, r, t, t ) = δρ(r, t) δv app (r, t )

28 (Linear) density response rewrite as: δρ(r, t) = χ(r, r, t, t )v app (r, t )dr dt Ψ 0 is the ground state wave function (exact!), â + i and â i the creation and annihilation operators. Excited state wf (for j to i excitation): Φ i j >= â + i â j Φ 0 >

29 (Linear) density response Density matrix representation: P ij =< Φ 0 â + i â j Φ 0 >, Write the potential as: ˆv app = kl v app kl â + k âl δp ij (t) = χ ij,kl (t t )v app kl (t )dt. kl Fouriertransform: δp ij (ω) = kl χ ij,kl (ω)v app kl (ω).

30 (Linear) density response Turning on the extra potential v app, the KS potential changes: δv eff = v app + δv scf v scf being the Hartree and XC potential. Therefore, in DFT the response is: δp ij (ω) = χ ij,kl (ω)δvkl eff (ω), kl This means, the applied potential induces a polarization, which in turn changes the KS potential, which as well has to be included in the density response (due to ALDA, this is instantaneous!). δp ij (ω) = kl ( χ ij,kl (ω) v app kl (ω) + δvkl scf (ω) ),

31 (Linear) density response Calculate δv scf (ω) : or δv δv scf scf = δρ δρ, δv scf ij (ω) = ij,kl K ij,kl (ω)δp kl (ω) The coupling matrix K ij,kl is the Fourier transform of the derivative of the SCF potential wrt the density matrix, with ALDA we get: ( 1 K ij,kl = ψi (r)ψ j (r) r r + δv ) xc ψl δρ (r )ψ k (r )drdr

32 (Linear) density response Therefore: δp ij (ω) = kl χ ij,kl (ω) ( v app kl (ω) + K kl,mn (ω)δp mn (ω) mn ), This is a self-consistent equation for δp ij (ω), but first we have to worry about the response function:

33 Response function for the N-electron problem Ψ 0 is the ground state wave function (exact!), â + i and â i the creation and annihilation operators. Excited state wf (for j to i excitation): Φ i j >= â + i â j Φ 0 > Time dependent perturbation theory ( Φ I >, exact MB eigenfunctions): H 0 Φ I >= E I Φ I > χ ij,kl (ω) = I < Φ 0 â + j â i Φ I >< Φ I â + k âl Φ 0 > ω (E I E 0 ) < Φ 0 â + k âl Φ I >< Φ I â + i â j Φ 0 > ω + (E I E 0 )

34 Response function for the non-interacting KS problem Solution of the KS equations: h KS ψ i >= ɛ i ψ i > the response function reduces to ( Φ I > become Slater determinants): n j n i χ ij,kl (ω) = δ ik δ jl ω (ɛ i ɛ j ) n i : occupation numbers (n j n i ) 0 only for hole-particle (hp) or for particle-hole (ph) excitations!

35 Linear response and TDDFT δp ij (ω) = kl = kl = χ ij,kl (ω) v app n j n i δ ik δ jl ω (ɛ i ɛ j ) ( n j n i ω (ɛ i ɛ j ) kl (ω) + K ij,kl (ω)δp kl (ω) ij,kl ( v app kl (ω) + K kl,mn (ω)δp mn (ω) mn ) v app ij (ω) + K ij,mn (ω)δp mn (ω) mn ) ω (ɛ i ɛ j ) δp ij (ω) n j n i mn K ij,mn (ω)δp mn (ω) = v app ij (ω)

36 ( ω (ɛi ɛ j ) mn n j n i ) δ in δ jm K ij,mn (ω) δp mn (ω) = v app ij (ω) This equations can be solved to get the density response, with which e.g. the polarizability can be calculated. To simplify, order the eqns. into ph and hp parts. Since δp mn (ω) is hermitian, the hp and ph parts can be written as complex conjugate. This leads to matrix equations of the following form:

37 [( A B B A ) ( 1 0 ω 0 1 )] ( δp δp ) = ( δvapp δv app ) A ijσ,klτ = (ɛ kτ ɛ lτ )δ ik δ jl δ σ,τ K ijσ,klτ B ijσ,klτ = K ijσ,klτ Note: Eqns. have been extended by the spin indices.

38 Neglecting the external potential v app for a moment, the left hand side can be transformed into a pseudo Eigenvalue equation: ΩF I = ω 2 I F I (1) with ω I being the exact excitation energies and Ω = ω 2 ijσδ ik δ jl δ σ,τ 2 ω ijσ K ijσ,klτ ωklτ K ijσ,klτ = ( 1 ψi σ (r)ψj σ (r) r r + δv xc σ ) δρ τ ψl τ (r )ψk τ (r )drdr

39 Polarizabilities The polarizabilities can be calculated as: α xz = 2 xs 1/2 ( Ω ω 2 1 ) 1 S 1/2 z, therefore, solution of eq. 1 determines the poles, hence the true excitation energies.

40 Excitation energies from linear response Equation to solve: [ ω 2 ijσ δ ik δ jl δ στ + 2 ] ω ij K ijσ,klτ ωkl F I klτ = ωi 2 Fijσ I ijσ ω I : true excitation energies, exact for exact DFT functionals ω ij = ɛ i ɛ j evaluated as KS eigenvalues. Couplingmatrix K leads to correction of single-particle energies. ( 1 K ijσ,klτ = ψ i (r)ψ j (r) r r + δ 2 ) E xc δρ σ (r)δρ τ (r ψ k (r )ψ l (r )drdr ) K is the derivative of the KS potential with respect to the density, it is positive for singlets, negative for triplets.

41 Excitation energies from linear response Equation to solve: [ ω 2 ijσ δ ik δ jl δ στ + 2 ] ω ij K ijσ,klτ ωkl F I klτ = ωi 2 Fijσ I ijσ The ω ij = ɛ i ɛ j are corrected by: K ijσ,ijσ : the changed Coulomb and XC interactions due to excitation K ijσ,klτ coupling to other excitations kl : mixing of excitations.

42 Accuracy and Problems Typical accuracy: ev For low lying excitations with small change in density! Problems: Double excitations: not covered! Higher excitations: Rydberg states. Charge transfer (CT) states. Single determinate ground state (e.g. CI with ground state) Triplet instability

43 Problem of double excitations: Doubly excited states of e.g. polyenes seem to be well described, however, simple counting argument (Casida eq.) shows, that they are not covered. Need for frequency dependent kernel. Cave et al.,(2004) CPL 389, 39. Maitrea et al., JCP 120, Casida (2005) JCP 122, However, not available to ordinary users.

44 Higher excited states: Wrong LDA (GGA...) asymptotics inaccurate KS ɛ i, K ijσ,klτ small excitations dominated by ɛ Elliot, Burke, Furche: arxiv:cond-mat/ v Asymptotically corrected functionals: Casida and Salahub (2000) JCP Gruening et al, (2001) JCP Tozer and Handy (1998) JCP

45 CT excitations Casida et al., (2000) JCP 113, Dreuw et al., (2003) JCP 119, Grimme and Parac (2003) ChemPhysChem 3, 292. Wanko et al, (2004) JCP 120, Bernasconi et al., (2004) CPL 394, 141.

46 Example: H 2 Lets consider H 2 (Casida review): ω T ɛ u ɛ g + 2 Ψ g (r)ψ 1 u(r) r r Ψ g (r )Ψ u(r )d r d 3 r ( v up up ) + Ψ g (r)ψ xc vxc u(r) Ψ g (r )Ψ ρ up ρ u(r )d r d 3 r dn = ɛ + 2[Ψ g Ψ u Ψ g Ψ u] + [Ψ g Ψ u f xc Ψ g Ψ u] ɛ: Kohn-Sham Orbital energy difference [Ψ g Ψ u Ψ g Ψ u] exchange integral, exponentially decaying [Ψ g Ψ u f xc Ψ g Ψ u] =?

47 CT excitations for long range charge transfer (CT) excitation: ω CT I D A A 1/R

48 CT excitations ω CT I D A A 1/R ω = ɛ a ɛ i + 2[Ψ g Ψ u Ψ g Ψ u] + [Ψ g Ψ u f xc Ψ g Ψ u] Problem 1: I D = ɛ i, but only for asymptotic correct functionals Problem 2: but A A ɛ a in DFT ( JCP 121, 655) Problem 3: [Ψ g Ψ u Ψ g Ψ u] is exchange integral, decays exponentially, and...

49 CT excitations From TD-HF: [Ψ g Ψ u f xc Ψ g Ψ u] [Ψ g Ψ g Ψ u Ψ u], i.e., the derivative of the exchange functional becomes coulomb-like. Problem of local functionals: Decay with overlap of the two functions Ψ g (r) and Ψ u(r), over which is integrated: f xc = Cδ(r r ) : the f xc part decays quickly due to approximate kernel. the kernel is short ranged and does not show the desired 1/R behavior! except: exact HF exchange is mixed in (hybrides): this leads to the 1/R dependence: Problem: B3lYP: only 20 % HF ex.

50 CT excitations Figure: Dreuw, Head-Gorden, Chem. Rev. 105, 4009

51 CT excitations Same problem for ionic states in large conjugated systems. Figure: Grimme and Parac, ChemPhysChem 3, 292 Not solvable with hybrides, other states deteriorate.

52 Suggested cure: range seperated functionals. Figure: Jaquemin et al., (2008) JCTC 4, 123. However, not really there yet!

53 CT excitations Bacteriorhodopsin (2.18eV) and its retinal chromophor (1.9 ev). after photon absorption, excited states reisomerization. color tuning: what induces the blue-shift.

54 Ground states and excitation energies (Wanko et al. JPCB 109 (2005) 3606) TD-DFT too high (0.4 ev). wrong dependence on bond length alternation.

55 CT excitations Geometry dependence: twist (Wanko et al. JPCB 109 (2005) 3606) wrong dependence on dihedral angle.

56 CT excitations Response to external electric fields (Wanko et al. JPCB 109 (2005) 3606) does not change much with environment: not to use for QM/MM

57 CT excitations Bacteriorhodopsin (2.18eV) and its retinal chromophor (1.9 ev).

58 CT excitations TD-DFT is color blind. TD-DFT too high (0.4 ev). virtually no response to environment. wrong response to geometrical distorsion. all rhodopsins same color

59 CT excitations Absorption and emission spectra of terphenyl as expected, TD-DFT systematically off

60 CT excitations Absorption and emission spectra of terphenyl: sampling DFTB excited states with DFTB and ZINDO calculated TD-DFTB spectrum too narrow: wrong dependence on BLA

61 Summary: accuracy for easy excitations: 0.5 ev Rydberg states: wrong asymptotic behavior asymptotic correction double excitations special formalisms CT and ionic exciations functionals to be developed rough guide: use for low lying single excitations, where the density does not change too much always test or look for tests done

62 : Gradients and NCAS The excitation energies in TD-DFT are calculated post-scf, i.e. no wavefunction or density for the excited state is calculated (available). Therefore, the calculation of forces is not straight forward as e.g. in ab initio method. The NACS can not be computed as in ab initio theory using the ground and excited states wavefunctions.

63 Forces First implementation: Van Caillie, Amos,(2000) CPL 317, 159. Variational formalism: Furche (JCP 114, 5982) showed, that the equations of motion for the reduced density matrix can be derived from a Lagrangian. Excitation energies in TDDFT can be obtained as stationary points of the Functional F (similar to GS DFT!) can be calculated as derivatives of this functional with respect to a parameter (R i forces) Gradients: Furche and Ahlrichs (2002) JCP 117, 7433.

64 NACS Chernyak and Mukamel (2000) JCP (NACS) Baer, (2002) CPL 364, 75. (NACS) Levine et al., (2006) Mol. Phys. 104, (CI, NACS) Cordova et al., (2007) JCP (Surfaces, CI) NACS: limited to S 0 -S I!

65 TDDFT-SH studies Craig et al, (2005) PRL 95, , Habenicht et al., (2006) PRL 96, Comment: Maitrea, (2006) JCP 125, Tapavicza, (2007) PRL 98, Doing dynamics, something always happens, however, does it do it due to the right reason? Since TDDFT has so many problems, a careful comparison with ab initio methods is required.

66 Photochemistry studies Compute surfaces and compare with MR methods. Wanko et al. (2004) JCP 120, Levine et al., (2006) Molecular Physics 104, Cordova et al., (2007) JCP 127, Result: desaster.

67 Polyenes underestimation of ionic states can flip the order of states wrong surfaces = wrong photochemistry Probem of single determinantal wavefunction at CI: wrong description of ground state no double excitations: geometry of CI for these states wrong no pyramidalization, CI purely twisted structure CT states, e.g. in retinal: wrong bond alternation in S1, completely wrong surface!

68 Polyenes (Wanko et al. (2004) JCP 120, 1674)

69 Polyenes Octatrien (Wanko et al. (2004) JCP 120, 1674)

70 Polyenes Butadien (Wanko et al. (2004) JCP 120, 1674)

71 Single excitations E.g. GFP, retinal, PYP, Interestingly, TD-DFT predicts small gap at CI, i.e. description seems to be right at first sight! variation of state energy at CI much larger compared to ab initio methods intersection space has wrong dimension: N-1 instead of N-2! no CI between ground and excited states in TDDFT!

72 Oxirane no CT excitations Rydberg states biradical structure (ground state probem) triplett instability: use TDA (neglect B matrix in working equations) to decouple from ground state.

Time-Dependent Density-Functional Theory

Summer School on First Principles Calculations for Condensed Matter and Nanoscience August 21 September 3, 2005 Santa Barbara, California Time-Dependent Density-Functional Theory X. Gonze, Université Catholique