In this talk I will present real-time spectroscopy and different code to perform this kind of calculations. This presentation can be download here: http://www.attaccalite.com/wp-content/uploads/2022/03/RealTime_Lausanne_2022.odp

1. Real-time Spectroscopy
for solids
Claudio Attaccalite
CNRS/CINaM, Aix-Marseille Universite (FR)
Theoretical Spectroscopy Lectures
2022, Lausanne
Repetita iuvant
(repeating does good)

2. The first experiment beyond
linear response
P(r ,t)=P0+χ
(1)
E+χ
(2)
E
2
+O(E
3
)
First experiments on linear-optics
by P. Franken 1961
Ref: Nonlinear Optics and
Spectroscopy
The Nobel Prize in Physics 1981
Nicolaas Bloembergen

3. Ref: supermaket
Why non-linear optics?
..applications..

4. Ref: supermaket
Why non-linear optics?
..applications..

5. Why non-linear optics?
..applications..
New green laser appeared in 2012
without non-linear crystals

6. Probe “invisible” excitations
The Optical Resonances in
Carbon
Nanotubes Arise
from Excitons
Feng Wang, et al.
Science 308, 838 (2005);
..research..

7. Real-time spectroscopy is
much similar to experiments

8. What is real-time
spectroscopy?
Choose a
perturbation
E(t)=δ(t−t0) E0
E(t)=sin(ωt)E0

9. What is real-time
spectroscopy?
Choose a
perturbation
E(t)=δ(t−t0) E0
E(t)=sin(ωt)E0
Time-evolution
of an effective
Schrodinger
equation
Ψ(t+Δt)=Ψ(t)−i Δ t H Ψ (t )
Ψ(t=0)=ΨGS

10. What is real-time
spectroscopy?
Choose a
perturbation
E(t)=δ(t−t0) E0
E(t)=sin(ωt)E0
Time-evolution
of an effective
Schrodinger
equation
Ψ(t+Δt)=Ψ(t)−i Δ t H Ψ (t )
Ψ(t=0)=ΨGS
Analyze the
results

11. Real-time spectroscopy in practice
D(r ,t)=E(r ,t)+P(r ,t)
Materials equations:
Electric
Displacement
Electric Field
Polarization
∇⋅E(r ,t)=4 πρtot (r ,t)
∇⋅D(r ,t)=4 πρext (r ,t)
From Gauss's law:
P(r ,t)=∫χ(t−t ' ,r ,r')E(t ' r')dt ' dr '+∫dt
1
dt
2
χ
2
(...)E(t
1
)E(t
2
)+O(E
3
)
In general:

12. Linear response from
real-time simulations
For a small perturbation we consider only the first term,
the linear response regime
P(r ,t)=∫χ(t−t ' ,r ,r')E(t ' r')dt ' dr '+O(E
2
)
P(ω)=χ(ω)E(ω)=(ϵ(ω)−1)E(ω)
And finally: ϵ(ω)=1+
P(ω)
E(ω)
ϵ(ω)=
D(ω)
E(ω)

13. For a small perturbation we consider only the first term,
the linear response regime
P(r ,t)=∫χ(t−t ' ,r ,r')E(t ' r')dt ' dr '+O(E
2
)
P(ω)=χ(ω)E(ω)=(ϵ(ω)−1)E(ω)
And finally: ϵ(ω)=1+
P(ω)
E(ω)
ϵ(ω)=
D(ω)
E(ω)
Frequency
space
spectroscopy
ϵ(ω)=1+χ(ω)
Linear response from
real-time simulations

14. Better scaling for large system
Polarization and
Hamiltonian depend only on
valence bands.
No need of conduction
bands!
Advantages

15. Advantages
Better scaling for large system
Theory and implementation
are much easier
Polarization and
Hamiltonian depend only on
valence bands.
No need of conduction
bands!

16. One code to rule all
spectroscopy responses
χ(2)
(ω;ω1, ω2)
P(ω)=P0+χ
(1)
(ω)E1(ω)+χ
(2)
E1(ω1) E2(ω2)+χ
(3)
E1 E2 E3+O(E
4
)
SFG
DFG
SHG

17. One code to rule all
spectroscopy responses
χ
(3)
(ω; ω1, ω2, ω3)
THG
P(ω)=P0+χ
(1)
(ω)E1(ω)+χ
(2)
E1(ω1) E2(ω2)+χ
(3)
E1 E2 E3+O(E
4
)

18. And even more..
Non-perturbative
phenomena (HHG)
Coupling with ionic motion,
other response functions..
Pump and probe

19. What to propagate?
(Current) Density Density Matrix
Green's function Wave function

20. Time-dependent DFT
In principle TD-DFT is an exact formulation of time-dependent quantum
mechanics for isolated systems (not for periodic systems!!!)
Adiabatic approximation (i.e. without memory effects):

21. The Hamiltonian I
independent particles
H KS(ρ0)=T+V ion+V h(ρ0)+V xc (ρ0) We start from
the Kohn-Sham Hamiltonian
If we keep fixed the
density in the Hamiltoanian
to the ground-state one
we get the independent
particle approximation
In the Kohn-Sham basis
this reads:
H KS(ρ0)=ϵi
KS
δi, j

22. The Hamiltonian II
time-dependent Hartree
(RPA)
HTDH =T+Vion+V h(ρ)+V xc (ρ0)
If we keep fixed the
density in Vxc but not in
Vh. We get the
time-dependent Hartree or
RPA (with local fields)
The density is written as:
ρ(r ,t)=∑i=1
N v
|Ψ(r ,t)|
2

23. The Hamiltonian III
TD-DFT
HTDH =T+Vion+V h(ρ)+V xc (ρ)
We let density fluctuate in
both the Hartree and the Vxc
tems
We get the TD-DFT for
solids
The Runge-Gross theorem guarantees that this is an exact
theory for isolated systems

24. TD-DFT in practice

25. Real-time Green’s functions
Kadanoff-Baym
equations
From Green’s functions to density matrix
Generalized Kadanoff-Baym Ansatz (GKBA)
Correspond to the equilibrium
self-energy

26. Electronic relaxation
3 – Measurement
process
1 – Photo-excitation
process
2 – relaxation towards
quasi-equilibrium
Sangalli, D., & Marini,
A. EPL (Europhysics Letters), 110(4), 47004.(2015)

27. The Gauge problem
H =
p2
2 m
+r E+V (r) Length gauge:
H =
1
2 m
( p−e A)2
+V (r) Velocity gauge:

28. The Gauge problem
H =
p2
2 m
+r E+V (r) Length gauge:
H =
1
2 m
( p−e A)2
+V (r) Velocity gauge:
Quantum Mechanics is gauge invariant,
both gauges must give the same results

29. The Gauge problem
H =
p2
2 m
+r E+V (r) Length gauge:
H =
1
2 m
( p−e A)2
+V (r) Velocity gauge:
Quantum Mechanics is gauge invariant,
both gauges must give the same results
… but in real calculations the each gauge choice
has its advantages and disadvantages

30. The Gauge problem
The guage transformation connect the different guages
A2=A1+∇ f
ϕ2=ϕ1−
1
c
∂ f
∂ t
ψ2=ψ1 e
−ie f (r, t)/ℏ c
vector potential
scalar potential
wave-function phase
f (r ,t) Arbitrary scalar
function

31. The Gauge problem
The guage transformation connect the different guages
A2=A1+∇ f
ϕ2=ϕ1−
1
c
∂ f
∂ t
Non-local operators do not commute
with gauge transformation
vector potential
scalar potential
wave-function phase
f (r ,t) Arbitrary scalar
function
Vnl ,Σxc ,ΔGW
,etc .…
ψ2=ψ1 e
−ie f (r, t)/ℏ c

32. The Gauge problem
H =
p2
2m
+r E+V (r)+V nl (r ,r ' )
H =
1
2m
( p−e A)
2
+V (r)+V nl (r ,r ' )
In presence of a non-local
operator
these Hamiltonians
Are not equivalent anymore
≠
W. E. Lamb, et al. Phys. Rev. A 36, 2763 (1987)
M. D. Tokman, Phys. Rev. A 79, 053415 (2009)

33. The Gauge problem
H =
p2
2m
+r E+V (r)+V nl (r ,r ' )
H =
1
2m
( p−e A)
2
+V (r)+V nl (r ,r ' )
≠
moral of the story:
non-local potential should be introduce
in length gauge and then transformed as
W. E. Lamb, et al. Phys. Rev. A 36, 2763 (1987)
M. D. Tokman, Phys. Rev. A 79, 053415 (2009)
H =
1
2m
( p−e A)
2
+V (r)+e
i Ar
V nl (r ,r ' )e
−i Ar '
In presence of a non-local
operator
these Hamiltonians
Are not equivalent anymore

34. The Gauge problem
H =
p2
2 m
+r E+V (r)
Length gauge:
H =
1
2 m
( p−e A)2
+V (r)
Velocity gauge:
● Non-local operators can be easily introduced
● The dipole operator <r> is ill-defined in solids
you need a formulation in term of Berry-phase
● Non-local operators acquires a dependence
from the external field
● The momentum operator <p> is well defined
also in solids
In recent years different wrong papers using velocity gauge
have been published (that I will not cite here)

35. Real-time Spectroscopy
codes
QDD
QDD
NESSi

36. The code families
TD-DFT
Density Matrix
and/or
Green’s
Functions
QDD
QDD
NESSi

37. Special features of TD-DFT codes
Absorbing
boundary conditions
QDD
QDD

38. Special features of TD-DFT codes
Absorbing
boundary conditions
QDD
QDD
Quantum
dissipation
QDD
QDD
former

39. Special features of TD-DFT codes
Absorbing
boundary conditions
QDD
QDD
Ehrenfest dynamics
Quantum
dissipation
QDD
QDD
former

40. Special features of TD-DFT codes
Absorbing
boundary conditions
QDD
QDD
Ehrenfest dynamics Couple TD-DFT and Maxwell
equations
Quantum
dissipation
QDD
QDD
former

41. Effective Schrodinger equation
PRB 88, 235113(2013)
H=
independent particles
+quasi-particle corrections
+time-dependent Hartree (RPA)
+TD-DFT in length gauge
+screend Hartree-Fock (excitonic effects)

42. Green’s function codes
Single-time
Green’s functions
(Kadanoff-Baym Ansatz)
[Density Matrix]

43. Green’s function codes
Two-time Green’s functions:
NESSi
DMFT
GW and
second Bohr
Single-time
Green’s functions
(Kadanoff-Baym Ansatz)
[Density Matrix]

44. Yambo code
http://yambo-code.org
Octopus code
https://octopus-code.org/
Salmon code
https://salmon-tddft.jp/
Exciting
http://exciting.wikidot.com/
Thank you
for your attention
Books
SIESTA
https://departments.icmab.es/leem/siesta/
NESSi
http://www.nessi.tuxfamily.org/
CHEERS
https://arxiv.org/abs/1810.03322
NWChem
https://www.nwchem-sw.org/
QDD
https://git.irsamc.ups-tlse.fr/reinhard/QDDold
BerkelyGW
https://berkeleygw.org/
Code References

45. Non-linear optics in molecules
Non-linear optics can calculated in the same way of TD-DFT as it
is done in OCTOPUS or RT-TDDFT/SIESTA codes.
Quasi-monocromatich-field
p-nitroaniline
Y.Takimoto, Phd thesis (2008)

46. How to calculated the dielectric constant
i
∂ ̂
ρk (t)
∂t
=[Hk +V
eff
, ̂
ρk ] ̂
ρk (t)=∑i
f (ϵk ,i)∣ψi,k 〉〈 ψi,k∣
The Von Neumann equation
(see Wiki http://en.wikipedia.org/wiki/Density_matrix)

r t ,
r'
t'
=

ind
r ,t
ext r' ,t '
=−i〈[ r ,t r' t ']〉
We want to calculate:
We expand X in an independent particle basis set
χ(⃗
r t ,⃗
r'
t'
)= ∑
i, j,l,m k
χi, j,l,m, k ϕi, k (r)ϕ j ,k
∗
(r)ϕl,k (r')ϕm ,k
∗
(r')
χi, j,l,m, k=
∂ ̂
ρi, j, k
∂Vl,m ,k
Quantum Theory of the
Dielectric Constant in Real Solids
Adler Phys. Rev. 126, 413–420 (1962)
What is Veff
?

47. Independent Particle
Independent Particle Veff
= Vext
∂
∂Vl ,m,k
eff
i
∂ρi, j ,k
∂t
= ∂
∂Vl ,m, k
eff
[Hk+V eff
, ̂
ρk ]i, j, k
Using:
{
Hi, j ,k = δi, j ϵi(k)
̂
ρi, j, k = δi, j f (ϵi,k)+
∂ ̂
ρk
∂V
eff
⋅V eff
+....
And Fourier transform respect to t-t', we get:
χi, j,l,m, k (ω)=
f (ϵi,k)−f (ϵj ,k)
ℏ ω−ϵj ,k+ϵi ,k+i η
δj ,l δi,m
i
∂ ̂
ρk (t)
∂t
=[Hk +V eff
, ̂
ρk ]
χi, j,l,m, k=
∂ ̂
ρi, j, k
∂Vl,m ,k