TDDFT: applications and special topics
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Transcript of TDDFT: applications and special topics
TDDFT: applications and special topics
Carsten A. UllrichUniversity of Missouri
APS March Meeting, DenverMarch 2, 2014
2 Outline
PART I:
How to do a time-dependent Kohn-Sham calculation, and what to get out of it
PART II:
TDDFT in the linear-response regime:Double and charge-transfer excitations,spectroscopic properties in materials
PART III:
The future of TDDFT
3 The Runge-Gross theorem of TDDFT
N
kj kj
N
jj
j tVtHrr
r1
2
1,
2)(ˆ
1
2Consider an N-electronsystem with Hamiltonian
tV ,r )(t tn ,r Hti ˆ
0fixed
)(ˆ)( tnt
t,rjuniquelydetermined
uniquelydetermined
RG step 1 RG step 2
Therefore: tV ,r tn ,r1:1
4
ttVtVtVtt
i jxcHextj ,,,,2
,2
rrrrr
Consider an N-electron system, starting from a stationary state.
Solve a set of static KS equations to get a set of N ground-state orbitals:
Njtjj ,...,1,, 0)0( rr
rrrrr )0()0(0
2
2 jjjxcHext VV,tV
The N static KS orbitals are taken as initial orbitals and will be propagated in time:
N
jj ttn
1
2,, rr Time-dependent density:
Time-dependent Kohn-Sham scheme
5
0t T time
start withselfconsistent
KS ground state
propagateuntil here
I. Propagate TtttVi jold
KSj ,, 02
2
II. With the density 2j
j ttn calculate the new KS potential
tnVtnVtVtV xcHextnew
KS
III. Selfconsistency is reached if TtttVtV newKS
oldKS ,, 0
for all Ttt ,0
Time-dependent self-consistency (I)
6
tett jtHi
j ,,ˆ
rr Propagate a time step :t
Crank-Nicolson algorithm:
2ˆ1
2ˆ1ˆ
tHi
tHie tHi
tHtttHt ji
ji ,ˆ1,ˆ1 22 rr
Problem: H must be evaluated at the mid point 2tt But we know the density only for times t
Numerical time propagation
7
Predictor Step:
tj ttHttj )1()1( ˆ
nth Corrector Step:
tj ttHtt nnj )1()1( ˆ
ttHtH
ttHn
)(
21 ˆˆ
2ˆ
Selfconsistency is reached if remains unchanged for)(tn Ttt ,0upon addition of another corrector step in the time propagation.
Time-dependent self-consistency (II)
8
1
2
3
Prepare the initial state, usually the ground state, by a static DFT calculation. This gives the initial orbitals: )0,()0( rj
Solve TDKS equations selfconsistently, using an approximatetime-dependent xc potential which matches the static one usedin step 1. This gives the TDKS orbitals: ),(),( tntj r r
Calculate the relevant observable(s) as a functional of ),( tn r
Summary of TDKS scheme: 3 steps
9 Observables: the time-dependent density
The simplest observable is the time-dependent density itself: ),( tn r
Electron density map of the myoglobinmolecules, obtained using time-resolvedX-ray scattering. A short-lived CO groupappears during the photolysis process[Schotte et al., Science 300, 1944 (2003)]
10 Observables: the particle number
Unitary time propagation: spaceall Ntnrd ),(3 r
During an ionizationprocess, chargemoves away fromthe system.
Numerically, we candescribe this on a finite grid with anabsorbing boundary.
The number of bound/escaped particles at time t is approximately given by
boundescvolumeanalyzingbound NNNtnrdN ,),(3 r
11 Observables: moments of the density
dipole moment: zyxtnrrdtd ,,,),()( 3 r
sometimes one wants higher moments, e.g. quadrupole moment:
),()3()( 23 tnrrrrdtq r
One can calculate the Fourier transform of the dipole moment:
f
i
t
t
ti
fi
dtetdtt
d )(
1)(
Dipole power spectrum:
3
1
2)()(
tdD
12 Example: Na9+ cluster in a strong laser pulse
off resonance on resonance
not muchionization
a lot ofionization
Intensity: I=1011 W/cm2
13 Example: dipole power spectrum of Na9+ cluster
14 Implicit density functionals
We have learned that in TDDFT all quantum mechanical observables become density functionals:
)]([ˆ)]([)]([ tntntn
Some observables (e.g., the dipole moment), can easily beexpressed as density functionals. But there are also difficult cases!
► Probability to find the system in a k-fold ionized state
)()()( 00 tttPl
kl
klk
Projector on eigenstateswith k electrons in thecontinuum
15 Implicit density functionals
►Photoelectron kinetic-energy spectrum
dEtdEEPN
k
kE
t
2
1
)(lim)(
►State-to-state transition amplitude (S-matrix)
)(lim, tS ft
fi
All of the above observables are easy to express in terms of thewave function, but very difficult to write down as explicit density functionals.
Not knowing any better, people often calculate them approximately usingthe KS Slater determinant instead of the exact wave function. This isan uncontrolled approximation, and should only be done with great care.
16 Ionization of a Na9+ cluster in a strong laser pulse
25 fs laser pulses0.87 eV photon energyI=4x1013 W/cm2
For implicit observablessuch as ion probabilitiesone needs to make twoapproximations:
(1) for the xc potential in the TDKS calculation
(2) for calculating the observable from the TDKS orbitals.
17 Outline
PART I:
How to do a time-dependent Kohn-Sham calculation, and what to get out of it
PART II:
TDDFT in the linear response regime:double and charge transfer excitations,spectroscopic properties in materials
PART III:
The future of TDDFT
18 The Casida formalism for excitation energies
Y
X
10
01
Y
X
AK
KA**
Excitation energies follow from eigenvalue problem(Casida 1995):
rrrrrr
rr
aixcaiaiia
aiiaiaaaiiaiia
frdrdK
KA
,,1
,*33
,
,,
For real orbitals we can rewrite this as
ai
aiaiaiiaiaaiaiaaii ZZK 2,
2 2
121212122 2121221 HxcHxc ff 2-level approx.:
19 The Casida formalism for excitation energies
The Casida formalism gives, in principle, the exact excitation energiesand oscillator strengths. In practice, three approximations are required:
► KS ground state with approximate xc potential
► The inifinite-dimensional matrix needs to be truncated
► Approximate xc kernel (usually adiabatic):
r
rrr
n
Vf
statxcadia
xc
,
advantage: can use any xc functional from static DFT (“plug and play”)disadvantage: no frequency dependence, no memory
→ missing physics
20 Excited states with TDDFT: general trends
● Even with the simplest functionals, TDDFT gives significant improvement over KS excitation energies
● Try to do your ground-state calculation with a good xc functional. Then, the KS excitations will already be close to the exact excitations, and the TDDFT corrections can be small.
● The degree of accuracy of TDDFT for excited-state energies, forces and geometries is basically the same as that which can be achieved with the same functionals for the ground state.
● However, there are challenging cases, which we will discuss next.
21 Single versus double excitations
22 Single versus double excitations
j
jnjn C
The nth excited many-body state can be expanded in a series ofSlater determinants:
► “Single Excitation”: the expansion of is dominated by some singly excited Slater determinant.
► “Double Excitation”: the expansion of is dominated by some doubly excited Slater determinant.
► There are many-body states that have both singly and doubly excited character, so that a clear assignment cannot be made.
n
n
23
),(),,(||
1,,),( 1
331 rrr
rrrrr
nfrdrdn xcs
Single versus double excitations
Has poles at KS single excitations. The exact response function hasmore poles (single, doubleand multiple excitations).
Gives dynamical corrections tothe KS excitation spectrum. Shifts the single KS poles to thecorrect positions, and createsnew poles at double andmultiple excitations.
► Adiabatic approximation (fxc does not depend on ω): only single excitations!► ω-dependence of fxc will generate additional solutions of the Casida equations, which corresponds to double/multiple excitations.
24 Double excitations in linear polyenes
butadiene
● Lowest excitation, is optically dark and has significant double-excitation character● The necessary configuration mixing requires ω-dependent xc kernel● One can construct such an xc kernel explicitly for cases where there are close lying single and double excitations that are well isolated from the rest of the spectrum (“Dressed” TDDFT), see Maitra et al., JCP 120, 5932 (2004)
gg AA 11 21
25 Charge-transfer excitations
Zincbacteriochlorin-Bacteriochlorin complex(light-harvesting in plantsand purple bacteria)
TDDFT error: 1.4 eV
TDDFT predicts CT states energetically well below local fluorescing states. Predicts CT quenching of the fluorescence. Not observed!Dreuw and Head-Gordon, JACS 126, 4007 (2004)
26 Charge-transfer excitations: large separation
RAI ad
exactct
1
(ionization potential of donor minuselectron affinity of acceptor plus Coulomb energy of the charged fragments)
What do we get in TDDFT? Let’s look at the two-level approximation:
)()(),,()()(2 32 rrrrrr
dH
aLHxc
dH
aL
dH
aL
SPAct
frdrd
The highest occupied orbital of the donor and the lowest unoccupiedorbital of the acceptor have exponentially vanishing overlap!
dH
aL
TDDFTct
For all (semi)local xc approximations,TDDFT significantly underestimatescharge-transfer energies!
27 Charge-transfer excitations: exchange
R
rdrd
HFdH
HFaL
dH
dH
aL
aLHFd
HHFa
LTDHFct
1
||
)()()()(
,,
32,,
rr
rrrr
and use Koopmans theorem!
TDHF reproduces -1/R behavior of charge-transfer energies correctly. Therefore, hybrid functionals (such as B3LYP) will give some improvement over LDA and GGA.
Even better are the so-called range-separated hybrids: Let
||
|)|(1
||
|)|(
||
1
rr
rr
rr
rr
rr
ff )erfc()(,)( xxfexf x
GGAc
HFLRx
GGASRxxc EEEE
28 Charge-transfer excitations: long-range correction
Long-range corrected (LC) functionals (i.e., range-sparated hybrids)perform very well for charge-transfer excitations.
Tawada et al., JCP 120,8425 (2004)
29
What are the properties of the unknown exact xc kernel that must be well-modelled to get long-range CT energies correct ?
Exponential dependence on the fragment separation R,
fxc ~ exp(aR)
For transfer between open-shell species, need strong frequency-dependence.
Gritsenko & Baerends (PRA, 2004), Maitra (JCP, 2005), Tozer (JCP, 2003) Tawada et al. (JCP, 2004)
Step in Vxc re-aligns the 2 atomic HOMOs near-degeneracy of molecular HOMO & LUMO static correlation, crucial double excitations frequency-dependence!
(It’s a rather ugly kernel…)
“LiH”
step
Long-range charge-transfer excitations
30 Metals: particle-hole continuum and plasmons
In ideal metals, all single-particle states inside the Fermi sphereare filled. A particle-hole excitation connects an occupied single-particle state inside the sphere with an empty state outside.
From linear response theory, one can show that the plasmon dispersiongoes as
...)( 2 qq pl m
nepl
24
pl
31 Plasmon excitations in bulk metals
Quong and Eguiluz, PRL 70, 3955 (1993)
● In general, excitations in (simple) metals very well described by ALDA. ●Time-dependent Hartree (=RPA) already gives the dominant contribution
● fxc typically gives some (minor) corrections (damping!)
●This is also the case for 2DEGs in doped semiconductor heterostructures
Al
Gurtubay et al., PRB 72, 125114 (2005)
Sc
32 Plasmon excitations in metal clusters
Yabana and Bertsch, PRB 54,4484 (1996)
Surface plasmons (“Mie plasmon”) in metal clusters are very well reproducedwithin ALDA. Plasmonics: mainly using classical electrodynamics, not quantum response
Calvayrac et al., Phys. Rep.337, 493 (2000)
33 Insulators: three different gaps
Band gap: xcKSgg EE ,
Optical gap:exciton
gopticalg EEE 0
The Kohn-Sham gap approximates the opticalgap (neutral excitation),not the band gap!
34 Elementary view of Excitons
Mott-Wannier exciton:weakly bound, delocalizedover many lattice constants
An exciton is a collective interband excitation:single-particle excitations arecoupled by Coulomb interaction
Real space: Reciprocal space:
35 Excitonic features in the absorption spectrum
● Sharp peaks below the onset of the single-particle optical gap
● Redistribution of oscillator strength: enhanced absorption close to the onset of the continuum
36
G. Onida, L. Reining, A. Rubio, RMP 74, 601 (2002)S. Botti, A. Schindlmayr, R. Del Sole, L. Reining, Rep. Prog. Phys. 70, 357 (2007)
RPA and ALDA both bad!
►absorption edge red shifted (electron self-interaction)
►first excitonic peak missing (electron-hole interaction)
Silicon
Why does ALDA fail?
Optical absorption of insulators
37 Linear response in periodic systems
),(),()(
),(),(),(
21211
21
kkk
kkk
GGGGGGG
GGGGGGGG
2
1
xc
ss
fV
Optical properties are determined by the macroscopic dielectric function:
2~)( nmac (Complex index of refraction)
For cubic symmetry, one can prove that
1
00
1
0),(lim)(
GGGG k
kmac
),()(),(1 kkk GGGGGGG VTherefore, one needs the
inverse dielectric matrix:
38 The xc kernel for periodic systems
rGq
qGG
GG,
rGq qrr )(,
)( ),(),,(
i
FBZHxc
iHxc efef
TDDFT requires the following matrix elements as input:
00
00
,,
)(,
)(, ),(
GqkkGqkk
rGqGG
q GG
rGqGG kkqkk
iaia
ii
aHxcFBZ
ai
iaiia ieafaeiK
Most important: long-range limit of “head” :)0( GG)0( q
qkk0q
qr ai
i aei20,
1),(
qf exact
xc q00 q
but .const),(0, q00 q ALDA
xcf Therefore, no excitonsin ALDA!
39
● LRC (long-range corrected) kernel (with fitting parameter α):
GGGG Gqq
2, ||
LRCxcf
Long-range xc kernels for solids
● “bootstrap” kernel (S. Sharma et al., PRL 107, 186401 (2011)
)0,(
)0,(),(
00
1
, q
qq GG
GGs
bootxcf
● Functionals from many-body theory: (requires matrix inversion)
exact exchangeexcitonic xc kernel from Bethe-Salpeter equation
40 Excitons with TDDFT: “bootstrap” xc kernel
S. Sharma et al., PRL 107, 186401 (2011)
41
► TDDFT works well for metallic and quasi-metallic systems already at the level of the ALDA. Successful applications for plasmon modes in bulk metals and low-dimensional semiconductor heterostructures.
► TDDFT for insulators is a much more complicated story:
● ALDA works well for EELS (electron energy loss spectra), but not for optical absorption spectra
● Excitonic binding due to attractive electron-hole interactions, which require long-range contribution to fxc
● At present, the full (but expensive) Bethe-Salpeter equation gives most accurate optical spectra in inorganic and organic materials (extended or nanoscale), but TDDFT is catching up.
● Several long-range XC kernels have become available (bootstrap, meta-GGA), with promising results. Stay tuned!
Extended systems - summary
42 Outline
PART I:
How to do a time-dependent Kohn-Sham calculation, and what to get out of it
PART II:
TDDFT in the linear response regime:Double and charge transfer excitations,spectroscopic properties in materials
PART III:
The future of TDDFT
43 The future of TDDFT: biological applications
C.A. Rozzi, S.M. Falke, N. Spallanzani, A. Rubio, E. Molinari, D. Brida, M. Maiuri, G.Cerullo, H. Schramm, J. Christoffers, and C. Lienau, Nature Commun. 4, 1602 (2013)
(TD)DFT can handle big systems (103—106 atoms).
Many applications to large organic systems (DNA, light-harvestingcomplexes, organic solar cells) will become possible.
Charge-transfer excitations and van der Waals interactions canbe treated from first principles.
Carotene-porphyrin-fullerenelight-harvesting triad:time-dependent simulationof charge-transfer processincluding ionic motion(Ehrenfest dynamics)
44 The future of TDDFT: materials science
K. Yabana, S. Sugiyama, Y. Shinohara, T. Otobe, and G.F. Bertsch, PRB 85, 045134 (2012)
● Combined solution of TDKS and Maxwell’s equations● Strong fields acting on crystalline solids: dielectric breakdown, coherent phonons, hot carrier generation● Coupling of electron and nuclear dynamics allows description of relaxation and dissipation (TDDFT + Molecular Dynamics)
Si
Vacuum Si
45 The future of TDDFT: open formal problems
► Development of nonadiabatic xc functionals (needed for double excitations, dissipation, etc.)
► TDDFT for open systems: nanoscale transport in dissipative environments. Some theory exists, but applications so far restricted to simple model systems
► Strongly correlated systems. Mott-Hubbard insulators, Kondo effect, Coulomb blockade. Requires subtle xc effects (discontinuity upon change of particle number)
► Formal extensions: finite temperature, relativistic effects…
TDDFT will remain an exciting field of research for many years to come!
46 TDDFT computer codes
Commercial: ADF, Gaussian, Jaguar, Q-Chem, Turbomole, VASP, WIEN2k,…
Open-source: ABINIT, DP, EXC, Exciting, GAMESS, NWChem, Octopus, Parsec, Quantum Espresso, SIESTA, CPMD, Smeagol, YAMBO, ELK,…
47 Literature
Time-dependent Density-FunctionalTheory: Concepts and Applications(Oxford University Press 2012)
“A brief compendium of TDDFT” Carsten A. Ullrich and Zeng-hui YangarXiv:1305.1388Brazilian Journal of Physics 44, 154 (2014)
C.A. Ullrich homepage:http://web.missouri.edu/~ullrichc