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
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
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
)
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
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)
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
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