dft and tddft - · pdf filedensity functional theory (dft) time dependent density functional...

Density Functional Theory (DFT)Time Dependent Density Functional Theory

The working equationsPerformance

Properties

DFT and TDDFT



Properties

Contents

1 Density Functional Theory (DFT)

2 Time Dependent Density Functional Theory

3 The working equations

4 Performance

4 Properties



Properties

Basic literature:Jensen: Introduction to Computational Chemistry, Wiley andSons.Koch, Holthausen: A Chemist’s Guide to Density FunctionalTheorey, Wiley.Parr, Yang: Density Functional Theory of Atoms andMolecules, Oxford.



Properties

The wavefunction is the central concept of quantum mechanics,since it describes the quantum mechanical state.

it determines all other properties through the calculation ofexpectation values.However, it is not a measurable quantity.It is a function of 3 N coordinates. Is such a detailedinformation required or is this an ’information overkill’ ?.It becomes an increasingly complex task to construct betterwavefunctions:orbital φ(r)→ product Ansatz → Slater determinant → CI!



Properties

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 isNOT required to calculate the desired expectation values.



Properties

Density and wavefunction

ρ(r1) = N∫|Ψ(r1...rN)|2dV2...dVN

However, this is not the way to go, since the determination of thetrue 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]?



Properties

==> 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 featureof a certain system? I.e., are the densities coming from differentexternal potentials (= core potentials in QC) different? Only then,the energy of a system can be uniquely determined by the density!



Properties

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 theexpectation values, this theorem assures that any quantummechanical observable is completely determined by the density.



Properties

Hohenberg and Kohn (HK) Theorems

HK2: There exists a functional E [ρ] with (ρ0: ground statedensity):

E [ρ] ≥ E0,

E [ρ0] = E0

Therefore, the derivative:

δE [ρ]

δρ= 0

results in an equation, from which the ground state density can bedetermined.



Properties

Total energy functional

E [ρ] = Ts [ρ] + Ene [ρ] + J[ρ] + Exc [ρ],

J[ρ] Hartree energy.Exc = Ex + Ec : exchange-correlation (XC) energy functional.Ts : kinetic energy of non-interaction particles==> will be evaluated from Slater determinant as in HFDFT is a single determinal method: fails in multi-referencecases



Properties

Kohn-Sham (KS) Theorem: non-interacting electrons

Let ρ0 be the ground state density of the interacting electrons.Then there exists a local potential veff [ρ0] for thenon-interacting electrons, leading to the same density ρ0 viasolution of the KS equations:[

−12∇2 + veff [ρ]

]φi = εiφi , ρ0(r) =

∑i

|φi |2

KS effective potential: veff [ρ] =δEpotδρ ,

veff [ρ] =∑α

ZαRα − r

dr +12

∫ρ(r ′)r − r ′

dr ′ + vxc [ρ]

XC-potential: vxc [ρ] = δExcδρ



Properties

XC functionals

LDA (Local Density Approximation: from electron gas):

Ex = C∫ρ4/3(r)d3r

GGA (Generalized Gradient Approximation):

Ex = C∫ρ4/3(r)F (s)d3r , s =

∇ρρ4/3

Various approximations for X and C:BP, BLYP and PBE being the most popular.Hybride Functionals:

Ehx = (1− c)EGGA

x + cEHFx

Usually, 20-30% HF-X work well (B3LYP: c=0.2)



Properties

Performance

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 vxc’near-sightedness’ of Exc



Properties

Asymptotics of vxc : Fast decay of LDA (GGA...) exchange potential

Figure: Elliot, Burke, Furche: arXiv:cond-mat/0703590v1 2007

Eigenvalue spectrum quantitatively incorrect:Ionization threshold too low.Rydberg states unbound (underestimated).



Properties

LDA and GGA are local functionals,

however, should be non-local as e.g. HF exchange:

Locality: consider two weakly interacting fragments:

ρ = ρ1 + ρ2

Then, if the densities do not overlap, the local functionalsvanish:

Exc = 0

This is in particular a problem for VdW interactions, whichDFT-GGA is not able to handle.(hybrides change only Ex , but Ec is the problem here!)(see e.g. JCP114 (2001) 5149)



Properties

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



Properties

Overestimation of polarizabiliy (Champagne et al. JCP 109, 10489)due to ’short-sightedness’ of Ex .

Figure: Gritsenko, Champagne,Gisbergen, Baerends



Properties

This has severe implications for many properties, e.g. protonaffinities:(J. Computer-Aided Mol. Design, 20 (2006) 511)

And will be of particular relevance for charge transfer excitations inTDDFT.



Properties

M. E. Casida, in: Recent Advances in Density FunctionalMethods, Vol. 1, ed. D. P. Chong (World Scientific,Singapore, 1995)M.E. Casida, in Recent Developments and Applications ofModern Density Functional Theory, edited by J.M. Seminario(Elsevier, Amsterdam, 1996), p. 391. Time-DependentDensity Functional Response Theory of Molecular Systems:Theory, Computational Methods and FunctionalsTime-Dependent Density Functional Theory, Edited by M.A.L.Marques, C.A. Ullrich, F. Nogueira, A. Rubio, K. Burke, andE.K.U. Gross, Lecture Notes in Physics Vol. 706 (Springer:Berlin, 2006).Elliot, Burke and Furche, arXiv:condmat/0703590v (broadreview, theory, applications)Dreuw and Head-Gordon, Chem. Rev. (2005) 105, 4009.(Comparison to CI etc., Biology, CT)



Properties

Early implementations:Bauernschmitt and Ahlrichs (1996) CPL 256, 454.(Turbomole)Jamorski et al., (1996)JCP 104, 5134. (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, 8218. (Gaussian)Hirata and Head Gordon (1999) CPL 313, 291. (TDA)



Properties

Introduction

DFT is a ground state theory, i.e. there is no general way, to applythe variational principles to excited states. They can be calculated

using the ∆ SCF procedurefor excited states with different symmetry (like triplets)

The applicability and accuracy, however, is limited.



Properties

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, whichdescribe the response of a system to an external perturbation.The polarisability α describes the reaction of a system to anexternal field ~F , inducing a dipole moment:

~δµ = α~F



Properties

A time dependent external field

~F (t) = ~F0sinωt

will induce a time dependent dipole moment.

~δµ(t) = δ~µ0sinωt.

Resonance → absorption of a photon

In linear response:

α =∂µ

∂FFourier transform of α(t): mean polarizability

α(ω) =∑I

fIωI − ω

has poles at the excitation energies ωI



Properties

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 (upto a time dependent constant, phase factor)



Properties

HK 2 equivalent

Energy not conserved → variational principle not available!DFT analog to the time-dependent Schrödinger equations isderived via action principle (A[ρ], action: energy x time).

Equation of motion:

i∂

∂tΨi (r , t) =

[T + veff [ρ](r , t)

]Ψi (r , t)

with

veff [ρ](r , t) = vext [ρ](r , t) +

∫ρ(r ′, t)

|r − r ′|dr ′ + vxc [ρ](r , t)



Properties

Approximation for XC

vxc [ρ] =δAxc

δρ

vxc [ρ], as a derivative of the Action functional is non-local in time,i.e. depends on all times t’<t ( memory effect!). The correctfunctional form, as in ground states DFT, is not known. Therefore,usually the adiabatic local density approximation (ALDA) is applied:

δAxc

δρ→ δExc

δρ

and the same functionals as for the ground state are applied. Thisapproximation is valid in the limit of slowly varying externalpotential (instantaneous response of density to externalperturbation).



Properties

(Linear) density response

v eff (r , t) = v eff ,0(r) + vapp(r , t)

Density depends on potential:

ρ[v eff ](r , t) = ρ[v eff ,0 + vapp](r , t)

Expansion:

ρ(r , t) = ρ0(r) + ρ1(r , t) + ...

= ρ0(r) +

∫ ∫χ(r , r ′, t, t ′)vapp(r ′, t ′)dr ′dt ′ + ...

Derivative of density wrt to external potential vapp:

χ(r , r ′, t, t ′) =δρ(r , t)

δvapp(r ′, t ′)



Properties


rewrite as:

δρ(r , t) =

∫ ∫χ(r , r ′, t, t ′)vapp(r ′, t ′)dr ′dt ′

Ψ0 is the ground state wave function (exact!),a+i and ai the creation and annihilation operators.Excited state wf (for j to i excitation):

|Φij >= a+

i aj |Φ0 >



Properties


Density matrix representation: Pij =< Φ0|a+i aj |Φ0 >,

Write the potential as: vapp =∑

kl vappkl a+

k al

δPij(t) =∑kl

∫χij ,kl (t − t ′)vappkl (t ′)dt ′.

Fouriertransform:

δPij(ω) =∑kl

χij ,kl (ω)vappkl (ω).



Properties


Turning on the extra potential vapp, the KS potential changes:

δv eff = vapp + δv scf

v scf being the Hartree and XC potential. Therefore, in DFT theresponse is:

δPij(ω) =∑kl

χij ,kl (ω)δv effkl (ω),

This means, the applied potential induces a polarization, which inturn changes the KS potential, which as well has to be included inthe density response (due to ALDA, this is instantaneous!).

δPij(ω) =∑kl

χij ,kl (ω)(vappkl (ω) + δv scfkl (ω)

),



Properties


Calculate δv scf (ω) :

δv scf =

∫δv scf

δρδρ,

orδv scfij (ω) =

∑ij ,kl

Kij ,kl (ω)δPkl (ω)

The coupling matrix Kij ,kl is the Fourier transform of the derivativeof the SCF potential wrt the density matrix, with ALDA we get:

Kij ,kl =

∫ ∫ψ∗i (r)ψj(r)

(1

|r − r ′|+δvxcδρ

)ψ∗l (r ′)ψk(r ′)drdr ′



Properties


Therefore:

δPij(ω) =∑kl

χij ,kl (ω)

(vappkl (ω) +

∑mn

Kkl ,mn(ω)δPmn(ω)

),

This is a self-consistent equation for δPij(ω), but first we have toworry about the response function:



Properties

Response function for the non-interacting KS problem

Solution of the KS equations: hKS |ψi >= εi |ψi > the responsefunction reduces to (|ΦI > become Slater determinants):

χij ,kl (ω) = δikδjlnj − ni

ω − (εi − εj)

ni : occupation numbers(nj − ni ) 6= 0 only for hole-particle (hp) or for particle-hole (ph)excitations!



Properties

Linear response and TDDFT

δPij(ω) =∑kl

χij ,kl (ω)

vappkl (ω) +∑ij ,kl

Kij ,kl (ω)δPkl (ω)

=

∑kl

δikδjlnj − ni

ω − (εi − εj)

(vappkl (ω) +

∑mn

Kkl ,mn(ω)δPmn(ω)

)

=nj − ni

ω − (εi − εj)

(vappij (ω) +

∑mn

Kij ,mn(ω)δPmn(ω)

)

ω − (εi − εj)nj − ni

δPij(ω)−∑mn

Kij ,mn(ω)δPmn(ω) = vappij (ω)



Properties

The working equations

∑mn

(ω − (εi − εj)

nj − niδinδjm − Kij ,mn(ω)

)δPmn(ω) = vappij (ω)

This equations can be solved to get the density response, withwhich e.g. the polarizability can be calculated.

To simplify, order the eqns. into ph and hp parts. Since δPmn(ω) ishermitian, the hp and ph parts can be written as complexconjugate. This leads to matrix equations of the following form:



Properties

[(A BB A

)− ω

(1 00 −1

)](δPδP∗

)=

(δvappδv∗app

)

Aijσ,klτ = (εkτ − εlτ )δikδjlδσ,τ − Kijσ,klτ

Bijσ,klτ = −Kijσ,klτ

Note: Eqns. have been extended by the spin indices.



Properties

The working equations

Neglecting the external potential vapp for a moment, the left handside can be transformed into a pseudo Eigenvalue equation:

ΩFI = ω2I FI (1)

with ωI being the exact excitation energies and

Ω = ω2ijσδikδjlδσ,τ − 2

√ωijσKijσ,klτ

√ωklτ

Kijσ,klτ =

∫ ∫ψ∗σi (r)ψσj (r)

(1

|r − r ′|+δvσxcδρτ

)ψ∗τl (r ′)ψτk (r ′)drdr ′



Properties

Polarizabilities

The polarizabilities can be calculated as:

αxz = 2~xS−1/2 (Ω− ω21)−1 S−1/2~z ,

therefore, solution of eq. 1 determines the poles, hence the trueexcitation energies.



Properties

Excitation energies from linear response

Equation to solve:∑ijσ

[ω2ijσδikδjlδστ + 2

√ωijKijσ,klτ

√ωkl]

F Iklτ = ω2

I F Iijσ

ωI : ’true’ excitation energies, exact for exact DFT functionalsωij = εi − εj evaluated as KS eigenvalues.Couplingmatrix K leads to correction of single-particleenergies.

Kijσ,klτ =

∫ ∫ ′ψi (r)ψj(r)

(1

|r − r′|+

δ2Exc

δρσ(r)δρτ (r′)

)ψk(r′)ψl (r′)drdr′

K is the derivative of the KS potential with respect to thedensity, it is positive for singlets, negative for triplets.



Properties

Excitation energies from linear response

Equation to solve:∑ijσ

[ω2ijσδikδjlδστ + 2

√ωijKijσ,klτ

√ωkl]

F Iklτ = ω2

I F Iijσ

The ωij = εi − εj are ’corrected’ by:Kijσ,ijσ: the changed Coulomb and XC interactions due toexcitationKijσ,klτ ’coupling’ to other excitations ’kl’: mixing ofexcitations.



Properties

Accuracy and Problems

Typical accuracy: 0.3-0.5 eVFor 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



Properties

Problem of double excitations:

Doubly excited states of e.g. polyenes seem to be well described,however, simple counting argument (Casida eq.) shows, that theyare not covered. Need for frequency dependent kernel.

Cave et al.,(2004) CPL 389, 39. Maitrea et al., JCP 120, 5932.Casida (2005) JCP 122, 054111

However, not available to ordinary users.



Properties

Higher excited states:

Wrong LDA (GGA...) asymptotics → inaccurate KS εi ,Kijσ,klτ small → excitations dominated by ∆ε

Elliot, Burke, Furche: arXiv:cond-mat/0703590v1 2007

Asymptotically corrected functionals:Casida and Salahub (2000) JCP 122 8918

Gruening et al, (2001) JCP 114 652.

Tozer and Handy (1998) JCP 109 10180.



Properties

CT excitations

Casida et al., (2000) JCP 113, 7062.Dreuw et al., (2003) JCP 119, 2943.Grimme and Parac (2003) ChemPhysChem 3, 292.Wanko et al, (2004) JCP 120, 1674.Bernasconi et al., (2004) CPL 394, 141.



Properties

CT excitations

for long range charge transfer (CT) excitation:

ωCT ≈ ID − AA − 1/R

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



Properties

CT excitations

ωCT ≈ ID − AA − 1/R

ω = εa − εi + 2[ΨgΨ∗u|ΨgΨ∗u] + [ΨgΨ∗u|fxc |ΨgΨ∗u]

Problem 1: ID = εi , but only for asymptotic correct functionalsProblem 2: but AA 6= εa in DFT ( JCP 121, 655)Problem 3: [ΨgΨ∗u|ΨgΨ∗u] is exchange integral, decaysexponentially, and ...



Properties

CT excitations

From TD-HF: [ΨgΨ∗u|fxc |ΨgΨ∗u]→ [ΨgΨ∗g |ΨuΨ∗u],i.e., the derivative of the exchange functional becomes coulomb-like.

Problem of local functionals:Decay with overlap of the two functions Ψg (r) and Ψ∗u(r),over which is integrated: fxc = Cδ(r − r ′)→: the fxc part decays quickly due to approximate kernel.the kernel is short ranged and does not show the desired 1/Rbehavior!except: exact HF exchange is mixed in (hybrides): this leadsto the 1/R dependence:Problem: B3lYP: only 20 % HF ex.



Properties

CT excitations

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



Properties

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.



Properties

Suggested cure: range seperated functionals.

Figure: Jaquemin et al., (2008) JCTC 4, 123.

However, not really there yet!



Properties

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.



Properties

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.



Properties

CT excitations

Geometry dependence: twist (Wanko et al. JPCB 109 (2005) 3606)

wrong dependence on dihedral angle.



Properties

CT excitations

Response to external electric fields (Wanko et al. JPCB 109 (2005)3606)

does not change much with environment: not to use forQM/MM



Properties

CT excitations

Bacteriorhodopsin (2.18eV) and its retinal chromophor (1.9 eV).



Properties

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



Properties

CT excitations

Absorption and emission spectra of terphenyl

as expected, TD-DFT systematically off



Properties

CT excitations

Absorption and emission spectra of terphenyl: sampling DFTBexcited states with DFTB and ZINDO calculated

TD-DFTB spectrum too narrow: wrong dependence on BLA



Properties

Summary:accuracy for ’easy’ excitations: 0.5 eVRydberg states: wrong asymptotic behavior → asymptoticcorrectiondouble excitations → special formalismsCT and ionic exciations → functionals to be developed

rough guide:

use for low lying single excitations, where the density does notchange’ too’ muchalways test or look for tests done



Properties

Properties: Gradients and NCAS

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



Properties

Forces

First implementation: Van Caillie, Amos,(2000) CPL 317, 159.Variational formalism: Furche (JCP 114, 5982) showed, thatthe equations of motion for the reduced density matrix can bederived from a Lagrangian.Excitation energies in TDDFT can be obtained as stationarypoints of the Functional F (similar to GS DFT!)Properties can be calculated as derivatives of this functionalwith respect to a parameter (Ri → forces)Gradients: Furche and Ahlrichs (2002) JCP 117, 7433.



Properties

NACS

Chernyak and Mukamel (2000) JCP 112 3572. (NACS)Baer, (2002) CPL 364, 75. (NACS)Levine et al., (2006) Mol. Phys. 104, 1039. (CI, NACS)Cordova et al., (2007) JCP 127 164111. (Surfaces, CI)

NACS: limited to S0-SI !



Properties

TDDFT-SH studies

Craig et al, (2005) PRL 95, 163001, Habenicht et al., (2006)PRL 96, 187401Comment: Maitrea, (2006) JCP 125, 014110Tapavicza, (2007) PRL 98, 023001

Doing dynamics, something always happens, however, does it do itdue to the right reason?Since TDDFT has so many problems, a careful comparison with abinitio methods is required.



Properties

Photochemistry studies

Compute surfaces and compare with MR methods.Wanko et al. (2004) JCP 120, 1674.Levine et al., (2006) Molecular Physics 104, 1039.Cordova et al., (2007) JCP 127, 164111.

Result: desaster.



Properties

Polyenes

underestimation of ionic states can flip the order of states→wrong surfaces = wrong photochemistryProbem of single determinantal wavefunction at CI: wrongdescription of ground stateno double excitations: geometry of CI for these states wrong→no pyramidalization, CI purely twisted structureCT states, e.g. in retinal: wrong bond alternation in S1,completely wrong surface!



Properties

Polyenes

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



Properties

Polyenes

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



Properties

Polyenes

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



Properties

Single excitations

E.g. GFP, retinal, PYP,Interestingly, TD-DFT predicts small gap at CI, i.e. descriptionseems to be right at first sight!variation of state energy at CI much larger compared to abinitio methodsintersection space has wrong dimension: N-1 instead of N-2!→ no CI between ground and excited states in TDDFT!



Properties

Oxirane

no CT excitationsRydberg statesbiradical structure (ground state probem)triplett instability: use TDA (neglect B matrix in workingequations) to decouple from ground state.

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