Electronic excitations
KS band structure
Absorption
Extended systems
Linear-response excitations
from finite to extended systems
Silvana Botti
1 LSI,
CNRS-CEA-École Polytechnique, Palaiseau, France
2 LPMCN,
CNRS-Université Lyon 1, France
3 European
Theoretical Spectroscopy Facility
September 3, 2008 – Benasque, TDDFT school
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Outline
1
Electronic excitations within linear response
2
From KS equations to band structures
3
Optical absorption: different computational schemes
4
More on extended systems
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Outline
1
Electronic excitations within linear response
2
From KS equations to band structures
3
Optical absorption: different computational schemes
4
More on extended systems
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Spectroscopy
Electronic excitations:
Optical absorption
Photoemission
Electron energy loss
Inverse photoemission
Inelastic X-ray scattering
...
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Linear response
For a sufficiently small perturbation, the response of the system can be
expanded into a Taylor series with respect to the perturbation.
The linear coefficient linking the response to the perturbation is called
response function. It is independent of the perturbation and depends only
on the system.
We will consider only the first order (linear) response.
We will not consider strong field interaction (e.g. intense lasers).
We will consider non-magnetic materials.
Example
R
R
Density-density response function: δρ(r, t) = dt 0 dr0 χρρ (r, t, r0 , t 0 )vext (r0 , t 0 )
R 0R 0
Dielectric tensor: D(r, t) = dt dr (r, t, r0 , t 0 )E(r0 , t 0 )
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Microscopic dielectric Function
Question
How can we calculate the microscopic dielectric functions?
Answer
They are determined by the elementary excitations of the medium:
interband and intraband transitions, as well as collective excitations.
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Outline
1
Electronic excitations within linear response
2
From KS equations to band structures
3
Optical absorption: different computational schemes
4
More on extended systems
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Kohn-Sham equations
∇2
−
+ vKS (r) ϕi (r) = εi ϕi (r)
2
n (r) =
PN
i
|ϕi (r) |2 = electron density
εi = KS eigenvalues, ϕi (r) = KS single-particle orbitals
Hartree potential
.
vKS (r) = v (r) + vH (r) + vxc (r)
%
unknown exchange-correlation (xc) potential
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Can we calculate spectra within static DFT?
DFT allows calculations of electronic GROUND STATE properties in
an accurate (usually within 1-2%) and efficient way:
total energy, phase stability,
atomic structure, lattice parameters (X-ray diffraction),
electronic density (Scanning tunneling microscopy),
elastic constants,
phonon frequencies (IR, Raman, Neutron scatterning).
Ground-state DFT is not designed to access EXCITED STATES:
quasiparticle electronic structures (photoemission spectra),
band gap (photoemission gap and optical gap),
electronic excitations spectra (absorption, EELS, IXSS, etc.).
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Can we calculate spectra within static DFT?
DFT allows calculations of electronic GROUND STATE properties in
an accurate (usually within 1-2%) and efficient way:
total energy, phase stability,
atomic structure, lattice parameters (X-ray diffraction),
electronic density (Scanning tunneling microscopy),
elastic constants,
phonon frequencies (IR, Raman, Neutron scatterning).
Ground-state DFT is not designed to access EXCITED STATES:
quasiparticle electronic structures (photoemission spectra),
band gap (photoemission gap and optical gap),
electronic excitations spectra (absorption, EELS, IXSS, etc.).
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Can we calculate spectra within static DFT?
DFT allows calculations of electronic GROUND STATE properties in
an accurate (usually within 1-2%) and efficient way:
total energy, phase stability,
atomic structure, lattice parameters (X-ray diffraction),
electronic density (Scanning tunneling microscopy),
elastic constants,
phonon frequencies (IR, Raman, Neutron scatterning).
Ground-state DFT is not designed to access EXCITED STATES:
quasiparticle electronic structures (photoemission spectra),
band gap (photoemission gap and optical gap),
electronic excitations spectra (absorption, EELS, IXSS, etc.).
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Can we calculate spectra within static DFT?
DFT allows calculations of electronic GROUND STATE properties in
an accurate (usually within 1-2%) and efficient way:
total energy, phase stability,
atomic structure, lattice parameters (X-ray diffraction),
electronic density (Scanning tunneling microscopy),
elastic constants,
phonon frequencies (IR, Raman, Neutron scatterning).
Ground-state DFT is not designed to access EXCITED STATES:
quasiparticle electronic structures (photoemission spectra),
band gap (photoemission gap and optical gap),
electronic excitations spectra (absorption, EELS, IXSS, etc.).
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Kohn-Sham one-electron band structure
The one-electron band structure is the dispersion of the energy levels as a
function of k in the Brillouin zone.
The Kohn-Sham eigenvalues and eigenstates are not one-electron
energy states for the electron in the solid.
However, it is common to interpret the solutions of Kohn-Sham
equations as one-electron states: the result is often a good
representation, especially concerning band dispersion.
Gap problem: the KS band structure underestimates systematically
the band gap (often by more than 50%).
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
GaAs band structure
8
6
4
L3,c
Γ15,c
X3,c
2
X1,c
L1,c
Γ1,c
Γ15,v
Energy (eV)
0
-2
L3,v
Experimental gap: 1.53 eV
DFT-LDA gap: 0.57 eV
X5,v
Applying a scissor operator
(0.8 eV) we can correct the
band structure.
See the lectures of Rex Godby!
-4
-6
L2,v
X3,v
-8
GaAs
X1,v
-10
L1,v
-12
Γ1,v
L
Γ
Linear-response excitations
X
K
Γ
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Outline
1
Electronic excitations within linear response
2
From KS equations to band structures
3
Optical absorption: different computational schemes
4
More on extended systems
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
An intuitive Picture: Absorption
c
unoccupied states
occupied states
v
Independent particle KS picture using εKS
and ϕKS
i
i
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
The simplest way: independent-particle transitions
c
unoccupied states
occupied states
v
Fermi’s golden rule: χKS ∼
Linear-response excitations
P
v ,c
|hc|D|v i|2 δ(c − v − ω)
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
The simplest way
c
unoccupied states
occupied states
v
2 (ω) = 2
1 X
4π 2
lim
|mv ,c,k |2 δ(ck − v k − ω)
ΩNk ω 2 q→0 q2
v ,c,k
mv ,c,k = hc |q · v| v i velocity matrix elements
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Joint density of states
In the independent-particle approximation the dielectric function is
determined by two contributions: optical matrix elements and energy
levels.
2 (ω) = 2
1 X
4π 2
lim
|mv ,c,k |2 δ(ck − v k − ω)
ΩNk ω 2 q→0 q2
v ,c,k
If mv ,c,k can be considered constant then the spectrum is essentially given
by the joint density of states:
2 ∝ JDOS/ω 2 =
1 X
δ(ck − v k − ω)
Nk ω 2
v ,c,k
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
An intuitive Picture: Absorption
c
unoccupied states
occupied states
v
Photoemission process
hν − (Ekin + φ) = EN−1,v − EN,0 = −εv
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
An intuitive Picture: Absorption
c
unoccupied states
occupied states
v
Electron-hole interaction: Excitons!
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Beyond independent-particle picture – one way
c
unoccupied states
occupied states
v
Link to other lectures!
GW correction to energy levels (Rex Godby)
Excitons from Bethe-Salpeter equation (Matteo Gatti)
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Beyond independent-particle picture – another way
TDDFT and linear response (Hardy Gross)
Vext
macroscopic
Vind macroscopic and microscopic
↑
δn = χvext
χ = χKS + χKS (v + fxc ) χ
v = Coulomb potential, related to local field effects
fxc = quantum exchange-correlation effects
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Dealing with excitations: absorption
Here we consider TDDFT within linear response.
We want to calculate the absorption cross-section (or equivalently for a
solid the dielectric function),
σ(ω) =
4πω
Im {α(ω)}
c
(1)
where α(ω) is the dynamical polarizability. We have seen that all we need
is the density-density response function:
Z
Z
3
α(ω) = − d r d3 r 0 z χ(r, r0 , ω) z 0
χ measures how the density changes when we perturb the potential
Z
δn(r, ω) = d3 r 0 χ(r, r0 , ω)δvext (r0 , ω)
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Changing the perturbation
Different perturbations ⇒ different terms but similar equations!
Electric
V (r) = ri
(e.g., polarizabilities, absorption,
luminescence ...)
Magnetic
V (r) = Li
(e.g., susceptibilities, NMR ...)
Atomic Displacements
∂v (r)
V (r) =
∂Ri
(e.g., phonons ...)
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
I) Time propagation of the KS equations
Apply a perturbation of the form δvext σ (r, t) = −κ0 zδ(t) to the
ground state of a finite system.
At t = 0+ the Kohn-Sham orbitals are
ϕj (r, t = 0+ ) = eiκ0 z ϕ̃j (r) .
Propagate these occupied KS wave-functions for a (in)finite time.
The dynamical polarizability can be obtained from
Z
1
α(ω) = −
d3 r z δn(r, ω) .
κ0
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Time propagation of the KS equations
Main advantages:
Accurate photo-absorption spectrum of several finite systems.
Can be easily extended to study non-linear response.
Decreases storage requirements.
It allows the entire frequency-dependent dielectric function to be
calculated at once.
The scaling with the number of atoms is quite favorable.
Disadvantages:
The prefactor is fairly large as such calculations typically require ≈
10000 time-steps with a time-step of ≈ 10−3 fs.
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
II) The response function: TDDFT equations
The response function yields δn upon a infinitesimal change in the
potential:
Z
δn(r, ω) = d3 r 0 χ(r, r0 , ω)δvext (r0 , ω)
By construction of the time-dependent KS system, δn can also be
calculated from
Z
δn(r, ω) = d3 r 0 χKS (r, r0 , ω)δvKS (r0 , ω)
χKS is the density-density response function of the non-interacting KS
electrons. In terms of the stationary KS orbitals it reads
χKS (r, r0 , ω) = lim+
η→0
Linear-response excitations
∞
X
jk
(fk − fj )
ϕj (r)ϕ∗j (r0 )ϕk (r0 )ϕ∗k (r)
ω − (j − k ) + iη
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
The Dyson equation
Using the definition of the KS potential, we have
Z
Z
0
3 0 δn(r , ω)
δvKS (r, ω) = δvext (r, ω) + d r
+ d3 r 0 fxc (r, r0 , ω)δn(r0 , ω) ,
|r − r0 |
where the xc kernel fxc is the functional derivative
δvxc [n](r, t) fxc [n](r, r , t − t ) =
δn(r0 , t 0 ) n=n0
0
0
Combining the previous results, we arrive at the Dyson equation
χ(r, r0 , ω) = χKS (r, r0 , ω)
Z
Z
+ d3 x d3 x 0 χ(r, x, ω)
Linear-response excitations
1
0
+
f
(x,
x
,
ω)
χKS (x0 , r0 , ω)
xc
|x − x0 |
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
xc kernels: approximations
The key ingredient of linear-response within TDDFT is the xc kernel.
This is a very complicated functional of the density that encompasses all
non-trivial many-body effects of the system.
Example of common approximations:
Adiabatic LDA
fxcALDA (r, r, ω) = δ(r − r0 )
d HEG
v
(n(r))
dn xc
Adiabatic GGAs
. . . (many more kernels in the TDDFT book)
Adiabatic kernels are local in time.
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Common implementation for solids
The TDDFT equations in the linear-response regime can be cast in
numerous different forms. For solids, in most implementations:
The Dyson equation is solved routinely by projecting all quantities
onto a suitable set of basis functions (planewaves within
pseudotentials, or localized basis for all-electron calculations).
We get the matrix equation:
χ(ω) = χKS (ω) + χKS (ω) (v + fxc (ω)) χ(ω)
with χ(ω) = χG,G0 (ω)
To obtain an absorption spectrum or the loss function at a given
momentum transfer only one component of the matrix is required,
obtained by solving a linear system of equations, thus avoiding the
matrix inversion.
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
The poles of the response function
Lehmann representation of the density response function:
X h0| n̂(r) |mi hm| n̂(r0 ) |0i h0| n̂(r0 ) |mi hm| n̂(r) |0i 0
χ(r, r , ω) = lim+
−
ω − (Em − E0 ) + iη
ω + (Em − E0 ) + iη
η→0
m
χ (for a finite system) has poles at the excitation energies, Ω = Em − E0 ,
while χKS has poles at the KS eigenvalue differences, ωjk = j − k .
After some pages of algebra:
X
δjj 0 δkk 0 ωjk + fk 0 − fj 0 Kjk,j 0 k 0 (Ω) βj 0 k 0 = Ωβjk ,
j 0k 0
where we have defined the matrix element
Z
Z
1
3
3 0 ∗
0
Kjk,j 0 k 0 (ω) = d r d r ϕj (r)ϕk (r)
+ fxc (r, r , ω) ϕj 0 (r0 )ϕ∗k 0 (r0 )
|r − r0 |
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Single pole approximation
If the excitation is well described by one single-particle transition we can
neglect the off-diagonal terms of Kjk,j 0 k 0 :
ΩSPA = ω12 + 2<K12,12 .
The SPA also describes the spin-multiplet structure of otherwise
spin-unpolarized ground states through the spin-dependence of fxc :
Z
1
(1)
0
3 0 ∗
+ fxc (r, r , ω) ϕ1 (r0 )ϕ∗2 (r0 )
= ω12 + 2< d r d r ϕ1 (r)ϕ2 (r)
|r − r0 |
Z
Z
3
(2)
= ω12 + 2< d r d3 r 0 ϕ∗1 (r)ϕ2 (r)fxc
(r, r0 , ω)ϕ1 (r0 )ϕ∗2 (r0 ) ,
Z
ωsing
ωtrip
3
where
(1)
fxc
= (fxc↑↑ + fxc↑↓ ) /2
Linear-response excitations
(2)
fxc
= (fxc↑↑ − fxc↑↓ ) /2
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Casida’s formulation
Parametrization of the linear change of the density:
X
δn(r, ω) =
[ξia (ω)ϕ∗a (r)ϕi (r) + ξai (ω)ϕa (r)ϕ∗i (r)] ,
ia
where i denotes an occupied and a a virtual state. After some algebra, one
can prove that the excitation energies can be determined as solutions of
the pseudo-eigenvalue problem
X
√
√
δaa0 δii 0 (j − k )2 + 2 a − i Kai,a0 i 0 (Ω) a0 − i 0 βa0 i 0 = Ω2 βai .
a0 i 0
The eigenvalues of this equation are the square of the excitation energies,
while the eigenvectors can be used to calculate the oscillator strengths.
This is the formula implemented in quantum-chemistry codes.
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
III) The Sternheimer equation
Hartree-Fock: Coupled Hartree-Fock method
DFPT: Density functional perturbation theory
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
The Sternheimer equation
n
o
(1)
(0)
H (0) − m ± ω + iη ψm (r, ±ω) = −Pc H (1) (±ω)ψm (r)
with
H
(1)
Z
(ω) = V (r) +
n(1) (r0 , ω)
d r
+
|r − r0 |
3 0
Z
d3 r 0 fxc (r, r0 ) n(1) (r0 , ω)
and
n(1) (r, ω) =
occ. n h
X
i∗
h
i∗
o
(0)
(1)
(1)
(0)
ψm (r) ψm (r, ω) + ψm (r, −ω) ψm (r)
m
X. Andrade et al., J. Chem. Phys. 126, 184106 (2007).
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
The Sternheimer equation
Main advantages:
(Non-)Linear system of equations solvable by standard methods.
Only the occupied states enter the equation.
Scaling is N 2 , where N is the number of atoms.
Easy to use to extend to hyperpolarizabilities.
Disadvantages:
It is hard to converge close to a resonance.
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
IV) Superoperators and Lanczos methods
The dynamical polarizability is represented by a matrix continued
fraction whose coefficients can be obtained from a Lanczos method:
hP1 (ω − L)−1 Q1 i =
1
ω − a1 + b2 ω−a12 +... c2
This method uses only occupied states.
Calculations scale favorably with the system size.
Up to now only test applications.
B. Walker et al., Phys. Rev. Lett. 96, 113001 (2006).
D. Rocca et al., J. Chem. Phys. 128, 154105 (2008).
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Summary for finite systems
A large majority of the calculations is done using the ALDA
approximation.
In many cases (low Z atoms) ALDA gives essentially the same results
as AGGA.
The results are quite good, often of the same quality of more
demanding approaches, like CI using only single-excitations, or BSE.
In some systems the ALDA underestimates the onset of absorption
mainly due to the incorrect asymptotics of the LDA potential.
The ALDA also fails to reproduce the excitations of stretched H2 ,
Na2 , etc.
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Outline
1
Electronic excitations within linear response
2
From KS equations to band structures
3
Optical absorption: different computational schemes
4
More on extended systems
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Absorption spectra
Question
Which level of approximation should I use if I am interested in comparing
to experiments?
Answer
There is not a unique answer.
It depends on the system and on the kind of spectroscopy.
In some cases, the independent-particle approximation already gives
results good enough.
It is often necessary to go beyond the independent particle
approximation.
See the lectures of Matteo Gatti!
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
EELS of graphite
For a q in the plane the
independent-particle approximation
agrees with experiment.
What happens when q 6= 90◦ ?
Remind: M (q) = −11(q)
00
A. G. Marinopoulos et al., Phys. Rev. Lett. 89, 076402 (2002).
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Absorption spectrum of silicon
50
E2
Si
E1
EXP
RPA
SO
ε2(ω)
40
E1 and E2 peaks
are red-shifted.
excitonic effects on
E1 are missing.
30
Scissor operator
does not help:
blue-shifted peaks.
20
10
0
3
4
5
6
ω [eV]
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Absorption spectrum of silicon
50
EXP
BSE
TDLDA
TDDFT - LRC
Si
Adiabatic LDA for
xc kernel does not
improve the
spectrum.
ε2(ω)
40
30
BSE gives the
correct result.
20
BSE-derived kernels
give the correct
results.
10
0
3
4
5
6
ω [eV]
S. Botti et al., Rep. Prog. Phys. 70, 357 (2007).
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
Independent-particle absorption spectrum of silicon
Si
60
50
E2
EXP
RPA
SO
E1
ε2(ω)
40
Comparison with
Hartree-Fock
calculation
30
20
10
0
3
4
5
6
7
8
9
10
11 12
13 14
15
ω [eV]
F. Bruneval et al., J. Chem. Phys. 124, 144113 (2006)
Linear-response excitations
Silvana Botti
Electronic excitations
KS band structure
Absorption
Extended systems
For extended systems
The ALDA approximation does not improve the independent-particle
absorption spectra.
In many cases ALDA gives essentially the same results as RPA.
BSE and BSE-derived xc kernels + GW energies correct the spectra.
The main problem of ALDA is the wrong asymptotic limit.
Linear-response excitations
Silvana Botti