Introduction to TDDFT Linear-Response TDDFT in Frequency-Reciprocal space on a Plane-Waves basis: the DP (Dielectric Properties) code

Valerio Olevano CNRS, Institut Néel, Grenoble, France 2 Résumé

● Motivation

● TDDFT

● Linear-Response TDDFT

● Frequency-Reciprocal space TDDFT

● TDDFT on a Plane Waves basis: the DP code

● Approximations: RPA, ALDA, new kernels

● Results

● Conclusions

[email protected] Condensed Matter: 3 a Formidable Many-Body Problem

● Well known interaction: electromagnetic (like in QED)

● All our ignorance in Cond-Mat has roots into the many-body problem.

[email protected] 4 the Many-Body Problem

● Already a problem in Classical Physics: 3-body problem (Euler, Lagrange, …) and on

● In Quantum Mechanics even 2 bodies is a problem

[email protected] The Many-Body problem 5

kinetic e-ions many-body interaction e-e interaction

where N can be up to the Avogadro number 1023!!!

many-body wavefunction

[email protected] 6 The Many-Body problem

Factorizable Hamiltonian

Solvable single-particle Schrödinger equation

Many-body ground-state total energy

Total many-body ground state wavefunction Condensed Matter 7 Theory

Models Ab Initio (first principles) simplify and search (if possible) numerical solution but analytical solutions microscopic Hamiltonian ● Quantum Chemistry: ● Anderson CI ● Hubbard CC DMFT Subject Headings: … ● Kondo ● QMC ● Ising ● Density-Functional: ● Heisenberg DFT TDDFT ● Many-body Green function: adjustable GW parameters BSE approximations (U, J, ...) (LDA, ...) 8 Why do we need ab initio theories?

1) To understand and explain observed phenomena; 2) To offer experimentalists reference data; 3) To predict properties before the synthesis, the experiment.

Purposes of the:

[email protected] 9 Which ab initio theory for which properties/spectroscopy?

[email protected] 10 Properties of Matter

Electronic Ground State Electronic Excited States properties: properties:

● Total energy ● Electronic structure ● Atomic structure ● Band plot, gap

● Phase stability ● Metal/Insulator char.

● ● Electronic density Optical properties ● Dielectric properties ● Magnetic order ● Transport ● Elastic constant ● e-ph and Superconductivity ● Phonons 11 What DFT can predict (normal error 1~2% in the 99% of cond-mat systems)

● Atomic Structure, Lattice Parameters (XRD)

● Total Energy, Phase Stability ● Electronic Density (SEM, XRS, STM) ● Elastic Constants ● Phonon Frequencies (IR, Neutron scattering) that is, all Ground State Properties! and this also in Strongly Correlated systems!:

Vanadium Oxide, VO DFT-LDA nlcc DFT-LDA semic EXP [Longo et al.] 2 a 5.659 Å 5.549 Å 5.7517 ± 0.0030 Å lattice b 4.641 Å 4.522 Å 4.5378 ± 0.0025 Å parameters c 5.420 Å 5.303 Å 5.3825 ± 0.0025 Å M. Gatti et al., PRL 2007  121.46 121.73˚ 122.646˚ ± 0.096 12 What DFT cannot predict

● Electronic Structure, Bandplot

● Bandgap, Metal/Insulator/Semiconductor

● Optical and Dielectric properties that is, all Excited State Properties!

You can use DFT to predict these properties, but don't blame DFT if it fails!

[email protected] 13 Which ab initio theory for which excited state properties/spectroscopy

[email protected] 14 Excitations: Charged vs Neutral

Charged Excitations Neutral Excitations N → N+1 (or N-1) N → N (Photoemission Spectroscopy) (Optical and Dielectric e h Spectroscopy)

c h c

v v

Inverse Photoemission Optical Absorption

[email protected] 15 Excitations: Charged vs Neutral

Charged Excitations Neutral Excitations N → N+1 (or N-1) N → N

c c electron-electron electron-electron interaction interaction + electron-hole interaction v v

Inverse Photoemission Optical Absorption

[email protected] 16 Possible ab-initio Theories for the Excited States

● TDDFT (Time-Dependent Density-Functional Theory) in the Approximations: ● RPA Neutral ● TDLDA Excitations ● beyond ● MBPT (Many-Body Theory) in the Approximations: ● GW Charged Excitations ● BSE (Bethe-Salpeter Equation) Neutral Excitations

[email protected] 17 Neutral Excitations Spectroscopies (for TDDFT)

[email protected] 18 Optical Properties incident reflected h photon photon c

cv v sample

transmitted photon h

d e t e c t o r

[email protected] 19 Optical Spectroscopies

Aspnes and Studna Lautenschlager et al. PRB 27, 985 (1983). PRB 36, 4821 (1987).

Optical Absorption Refraction Index

Aspnes and Studna PRB 27, 985 (1983).

[email protected] Reflectance 20 Dielectric Properties (EELS)

- e c e- electrons cv e-

v

- sample e c e-

e-

d e v p t e c plasmon t o r

[email protected] 21 Synchrotron Radiation (IXSS) X X rays c

cv v sample

c

d e t e c t v p o  r ESRF, Grenoble plasmon

[email protected] 22 Energy-Loss Spectroscopies

Stiebling Z. Phys. B 31, 355 (1978).

Energy Loss (EELS)

Sturm et al. Schuelke and Kaprolat PRL 48, 1425 (1982), PRL 67, 879 (1991) (ESRF, Grenoble) (ESRF, Grenoble).

Inelastic X-ray Scattering Coherent Inelastic Spectroscopy (IXSS) Scattering Spectroscopy (CIXS)

[email protected] 23 TDDFT

[email protected] 24 What is TDDFT? ● TDDFT is an extension of DFT; it is a DFT with time-dependent external potential:

Static Ionic Potential Hohenberg-Kohn Static Ionic Potential theorem + External Perturbation Runge-Gross (E.M. field) theorem

● Fundamental degree of freedom: Time-Dependent electronic Density (r,t)

(instead of the total many-body wavefunction (r1,...,rN,t))

[email protected] 25 TDDFT milestones

● Runge and Gross (1984): rigorous basis of TDDFT.

● Gross and Kohn (1985): TDDFT in Linear Response.

[email protected] 26 The Runge Gross theorem    PRL 52, 997 (1984) i∂t  1 t=T V 1 t W  1 t    i∂t  2 t =T V 2 tW  2 t 

v1 t≠v 2 t ct  ⇒ 1t ≠2 t 

(t) Any observable is functional v of the (time-dependent) density: 1   o t=〈  t∣o∣ t〉=o[]t  v 2  

t

[email protected] 27 Runge Gross theorem caveats 1) Any observable is functional of the o t=o [, 0]t  density and of the initial state 0=(t0) 2) The Runge-Gross theorem is proven under the requirement of v(t) expanded v(t) What about this case? around t0. And if you run into problems already in describing the (non- equilibrium) initial state, you can't go on with TDDFT. Previous demonstrations required periodic v(t) or a small td t t perturbation (linear response), or later 0 (Laplace transformable switch-on v(t) This also is not potentials+initial ground-state). But there Runge-Gross described is no general proof of the Runge-Gross theorem.

t t 0

[email protected] 28 TDDFT vs DFT Zoological Comparative Anatomy

[email protected] DFT vs TDDFT 29 Hohenberg-Kohn: Runge-Gross:

vr⇔ r  vr ,t ⇔ r , t

The Total Energy: The Action:

t1  ∣  ∣ 〈∣H∣〉=E [] ∫ dt 〈t  i∂t−H t  t〉= A[]

t0 are unique functionals of the density. The extrema of the: The stationary points of the: Total Energy Action  E []  A[] =0 =0  r  r ,t  give the exact density of the system: r  r ,t  DFT vs TDDFT 30 Kohn-Sham: Runge-Gross:

N N KS 2 KS 2 r =∑∣i r ∣ r ,t =∑∣i r ,t∣ i=1 i=1

KS r '  Exc [] r ' ,t   A [] v r=v r ∫ dr '  v KS r , t=vr , t∫ dr '  xc ∣r−r '∣  r ∣r−r '∣  r , t

KS KS KS KS KS KS KS H r i r=i i r i∂t i r ,t =H r , ti r , t Kohn-Sham equations

(t) W (t) W=0 v vKS   real system Kohn-Sham system t t TDDFT further caveats 31

1) The functional Axc[] is defined only for v- representable densities. It is undefined for  which do not correspond to some  A[]  A [] potential v → problems when variations =0 xc  r ,t   r ,t  A[] with respect to arbitrary densities

are required in order to search for δ AKS [ρ] If exists =v KS (r , t) stationary points. δρ(r ,t) Then we can define the 2) A functional A[] with (t) on the real time, Legendre transform: is ill defined, since we would run in a A [v ]=− A[]∫ dr dt r ,tvr , t causality-symmetry paradox → we can  A [v] =r , t solve the problem by working with t on the  vr , t Keldysh contour. Response functions are symmetric on the contour and become  A  r ,t  causal on physical time. =  vr , t vr 't '  vr 't ' t 0 t symmetric causal Van Leeuwen, PRL 80, 1280 (1998). (=0 for t'>t)

[email protected] 32 TDDFT in Linear Response

[email protected] TDDFT in Linear Response 33 Gross and Kohn (1985)

● If:

• with:

Strong (Laser) perturbations excluded!

TDDFT = DFT + Linear Response (to the time-dependent perturbation)

[email protected] 34 Runge-Gross Theorem for Linear Response TDDFT

DFT:

TDDFT:

LR-TDDFT:

[email protected] 35 Calculation Scheme

DFT calculation (ABINIT code)

LR-TDDFT calculation (DP code)

[email protected] 36 Polarizability  External Perturbation (variation in the external potential)

Induced Density (variation in the density induced by v) implicit definition of the polarizability 

Polarizability  Fundamental quantity associated to the cond-mat system - v is also indicated as v = v purpose of the TDDFT calculation    ext

[email protected] LR-TDDFT Kohn-Sham scheme: 37 the Independent Particle Polarizability (0)

Let's introduce a ficticious, Kohn-Sham non-interacting system, such that:

definition of the polarizability of the Kohn-Sham system:

Kohn-Sham polarizability (or independent-particle polarizability)

It is also indicated as KS = = (0) = 0  s  

[email protected] Variation in the Kohn-Sham 38 effective Total Potential

We write the Kohn-Sham effective potential variation KS (= ) v vs as the external + screening, composed of an Hartree and an XC potential:

Kohn-Sham or Effective Perturbation

The variation in the density induces a variation in the Hartree and in the exchange-correlation potentials which screen the external perturbation:

Exchange-Correlation Kernel definition

[email protected] the Independent Particle Polarizability 39 KS=(0) By variation KS (= ) of the Kohn-Sham equations, one obtains the v vs Linear-Response variation of the density  and then an expression for KS in terms of the Kohn­Sham energies and wavefunctions:

Independent-Particle Polarizability (Adler-Wiser)

In Frequency-Reciprocal space:

[email protected] 40 as a function of KS From:

The polarizability in terms of the independent-particle polarizability is:

Polarizability 

Coulombian (Local-Fields) Exchange-Correlation Kernel

Also explicitly:

[email protected] 41 Dielectric Function

 =  vtot vext vH classical potential → test-particle  vtot

 vtot= vext vH  vxc quantistic potential → test-electron

definition of the Dielectric Function 

Test-Particle Dielectric Function

−1 Test-Electron Dielectric Function te =1vc f xc 

[email protected] Valerio Olevano, Introduction to TDDFT From the Microscopic to the 42 Macroscopic Dielectric Function

M r ,r '=r ,r '

1  q ,= Macroscopic Dielectric Function  M −1  00 q , 

−1 G G' q , In a periodic system

Macroscopic Dielectric Observables Function  

LF NLF M ≠00= M

[email protected] 43 Local-Fields Effects (LF)

tot −1 ext Effect of the non diagonal elements  v G =∑ G G'  vG ' G ' (density inhomogeneities)

+  −1  ~  vtot  vext

NLF Macroscopic Dielectric Function  M q ,=00 q , without local­fields effects (NLF)

[email protected] 44 TDDFT: fundamental equations

Dielectric Observables Function 

Polarizability 

Coulombian (Local-Fields) Exchange-Correlation Kernel

Independent-Particle Polarizability (Adler-Wiser)

Exchange-Correlation Kernel

approximations required

[email protected] LR-TDDFT Calculation Scheme 45 Résumé

DFT DFT DFT  KS KS

 KS f xc





M

[email protected] 46 DP code

[email protected] 47 dp code (dielectric properties)

• The thing: Linear-Response TDDFT code in Frequency-Reciprocal Space on PW basis. • Purpose: Dielectric and Optical Properties (Absorption, Reflectivity, Refraction, EELS, IXSS, CIXS,.. • Systems: bulk, surfaces, clusters, molecules, atoms (through supercells) made of insulator, semiconductor and metal elements. • Approximations: RPA, ALDA, GW-RPA, LRC, non-local kernels, …, with and without LF (Local Fields). • Machines: Linux and other UNIX flavours, even Windows porting. • Libraries: BLAS, Lapack, CXML, ESSL, IMSL, ASL, Goedecker, FFTW. • Interfaces: ABINIT

http://www.dp-code.org

[email protected] Frequency-Reciprocal space: 48 why and where could be convenient

● Reciprocal Space → Infinite Periodic Systems (Bulk, but also Surfaces, Wires, Tubes with the use of Supercells);

● Frequency Space → Spectra.

In the case of isolated systems (atoms, molecules) it is more convenient a real space-time approach (e.g. Octopus code)

[email protected] DP Flow Diagram 49 KS KS i DFT-code interface i G FFT-1 KS i r

ij r = *i r j r FFT

ij G

 G  * G' 0 q , =  f − f  ij ij G G' ∑ij i j i− j − INV f xc G G' q , 

−1 G G' q , http://www.dp-code.org

LF NLF M q , M q , 50 DP tricks

−1  If we only need  00 = 1 − v 0  00 that is only 00 then instead of solving (inverting a full matrix):

0 0  −1 0  G G'=1− vc− f xcGG' '  G'' G'

we solve the linear system for only the first column of G' 0

 0 0 0  1− v c− f xc G G'  G' 0=G 0

O(N2) instead of O(N3)

[email protected] http://www.dp-code.org DP Parameters 51 KS KS i i G N = npwwfn G FFT-1 KSr N = nbands i b

 ij r = *i r j r FFT

 ij G

 G  * G' 0 q , =  f − f  ij ij G G' ∑ij i j i− j − INV f xc G G' q ,  N = npwmat G −1 G G' q , N = nomega  LF NLF M q , M q , DP performances: 52 CPU scaling and Memory usage

● CPU scaling for 0 2  : N b N k N  N GN r log N r  ● CPU scaling for −1 2  : N  N G

● 2 Memory occupation N N G Nr N k N b

http://www.dp-code.org

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http://www.dp-code.org 55 XC Approximations

[email protected] 56 Exchange-Correlation Kernel f : xc the RPA approximation

● Random Phase Approximation = neglect of the exchange-correlation effects (in the response)

● RPA

[email protected] 57 RPA Approximation RPA (without Local Fields) = sum over independent transitions (application of Fermi’s Golden Rule to an independent particle system)

confined system solid c f

h cv h  fi

v i

Fermi’s Golden Rule

Optical Absorption KS wavefunctions KS energies Valerio Olevano, CNRS 58 Adiabatic Local Density Approximation (TDLDA)

● The Adiabatic Local Density Approximation

● ALDA

local in r-space static (no memory effects)

[email protected] 59 TDDFT: Results

RPA ALDA

Energy Loss ok ok Isolated Optical Prop ok but.. ok but..

Energy Loss ok ok Solids Optical Prop no no

Energy Loss -> -Im ­1 Optical Properties -> Im 

[email protected] 60 TDDFT Results: Energy Loss

[email protected] 61 TDDFT and EELS in Solids

• TDLDA (but also RPA) in good agreement with experiment; • Importance of Local-Field (LF) effects.

[email protected] 62 Local Field Effects in EELS

• RPA is enough. But when inhomogeneities are present, Local-Field effects should be absolutely taken into account. • Quantitative Agreement

A. Marinopoulos et al. PRL 89, 76402 (2002) [email protected] IXSS and CIXS 63 and other synchrotron-radiation spectroscopies

Weissker et al.

V. Olevano, thesis • In Solids all Dielectric Properties related to the Energy- Loss function are well described by TDDFT in RPA with an improvement in ALDA. [email protected] 64 IXSS synchrotron-radiation spectroscopy

Weissker et al.

• In Solids all Dielectric Properties related to the Energy- Loss function are well described by TDDFT in RPA with an improvement in ALDA.

[email protected] 65 TDDFT Results: Optical Properties

[email protected] TDDFT RPA 66 Optical Properties in Nanotubes

● RPA is qualitatively able to interpret observed structures in optical spectra

[email protected] LF Effects in 67 in C and BN Nanotubes

● LF explain depolarization effects both for C and BN Nanotubes in perpendicular polarization optical spectra

[email protected] LF Effects in 68 in Si Nanowires

PRB 72, 153310 (2005).

● LF explain depolarization effects for Nanowires in perpendicular polarization optical spectra

[email protected] 69 Optical Absorption in Clusters

RPA NLF EXP TDLDA • Fair agreement (improvable) of the TDLDA result with the Experiment.

RPA LF

G. Onida et al, Rev. Mod. Phys. 74, 601 (2002).

[email protected] 70 TDDFT Excitation Energies

eigenvalues = poles of  λ 2 λ Casida equation ∑ Ωtt ' (ω=E λ) At'=E λ At (same equation as t ' RPA of nuclear physics) eigenvectors = oscillator strengths

2 Hxc tt'=t tt '2 t t ' f tt'

t =c−v Kohn-Sham excitation energies

Hxc f tt ' =∫ dr 1 dr 2  *c r 1v r1 [vc r 1 ,r 2  f xc r1 ,r 2 , ] *v' r2 c' r2  4-points Hartree+xc kernel

[email protected] 71 TDDFT Excitation Energies

eigenvalues = excitations = poles of 

Casida equation (same equation as RPA of nuclear physics)

eigenvectors (oscillator strengths)

Kohn-Sham energies

Coulomb interaction

TDDFT kernel

xc-kernel (must be approximated: [email protected] Adiabatic LDA) 72 Optical Absorption on Helium atom

● Exact-DFT+Exact-TDDFT must of course reproduce the Exact result.

● Approximated TDLDA on top of Exact-DFT introduces the right singlet-triplet exchange split (but thank to the v term) and performs reasonably well.

● TDLDA performance: 0.2 eV error. Petersilka, Gross and Burke (2000)

[email protected] 73 Optical Properties in Solids: Si

• The TDLDA cannot reproduce Optical Properties in Solids. • We miss both:

1 Self-Energy (electron-electron) effects (red shift of the entire spectrum)

2 Excitonic (electron-hole) effects (underestimation of the low-energy part)

[email protected] 74 GW: optical properties in Solids

● GW corrects the red-shift but still misses the Excitonic Effects

[email protected] 75 MBPT: GW, BSE and Excitonic Effects c c

c exciton

h cv h W h cv cv

v v v RPA GW BSE

e- P = + + + O(2) h+ Polarizability

[email protected] Bethe-Salpeter Equation: 76 optical properties in Solids

● Almost quantitative agreement BSE - experiment

[email protected] 77 Solid Argon: Hydrogen series n=1

n=2 n=3

2 • Exciton ~ Hydrogen atom  En ~1/n Balmer-like series • BSE can reproduce even bound Excitons What can we do to solve 78 the TDDFT kernel problem?

● If a new Approximation in TDDFT could be established, combining TDDFT’s simplicity with MBPT’s reliability… ● Hints for this new Approximation: compare critically MBPT with TDDFT fundamental equations.

[email protected] 79 New Approximations: LRC (Long Range Contribution only)

Nanoquanta kernel (or mapping BSE on TDDFT)

JGM-based NLDA (Jellium-with-Gap model based bi-local density approximation)

[email protected] 80 LRC Approximation

long-range coulombian

ALDA: local kernel

Long Range Contribution only

−1 =−4.6 ∞ Inversely proportional to the screening

[email protected] 81 TDDFT, LRC approximation

S. Botti et al. ● Thanks to the LRC approximation, TDDFT works also on Optical Properties in Solids

[email protected] 82 Nanoquanta kernel

G W -1 0

L. Reining, V. Olevano, A. Rubio and G. Onida, (2001) G. Adragna and R. Del Sole, (2001) F. Sottile, V. Olevano and L. Reining, (2004) A. Marini, R. Del Sole, A. Rubio, (2004) U. Von Barth, N. E. Dahlen, R. Van Leeuwen and G. Stefanucci, (2006) R. Stubner, I. Tokatly and O. Pankratov, (2006) M. Gatti,, V. Olevano, L. Reining and I. Tokatly, (2007)

[email protected] 83 Solid Argon: Bound Excitons

n=1

n=2 n=3

F. Sottile et al. 2005

With the Nanoquanta kernel, TDDFT describes even Bound Excitons 84 TDDFT JGM NLDA approximation

P. E. Trevisanutto et al., PRB (2013).

[email protected] 85 Conclusions

● TDDFT is a valid theory of Condensed Matter Theoretical Physics to calculate from first principles excited-state properties; ● The agreement with the experiment is good, but the choice of the right xc-approximation with respect to the given spectroscopy is crucial: ● RPA with LF is able to reproduce EELS spectra at q=0;

● TDLDA improves upon RPA on EELS (and also IXSS, CIXS) spectra at high q;

● TDLDA seems also to improve upon RPA on optical spectra in finite systems;

● More refined kernels (LRC, Nanoquanta) are required to reproduce optical spectra in solids, especially in presence of strong excitonic effects and bound excitons. ● Perspectives: ● Improve the algorithms, simplify the orbital expressions of the kernel.

[email protected] Thank you for your attention! 87 Non-Linear Optics Second Order Independent-Particle Polarizability 2  KS

 2q  −q  −q  f − f   f − f  2 2q ,q ,q , , = ij jl li i l − l j KS ∑ijl 2− j −i [ −l−i − j −l ] Where:

−iqr ij q=〈i∣e ∣ j 〉

i=n ,k j=n', k2q l=n'',kq

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