In this lecture, I will describe how to calculate optical response functions using real-time simulations. In particular, I will discuss td-hartree, td-dft and similar approximations.
8. Probing symmetries
Probing Symmetry Properties of Few-Layer MoS2
and h-BN by Optical Second-Harmonic
Generation
Nano Lett. 13, 3329 (2013)
SHG can probe
magnetic transition
15. Real time spectroscopy in practice 1 3- /
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:
Δ 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:
16. Real time spectroscopy in practice 2 3- /
For a small perturbation we consider only the first term,
the linear response regime
Δ P(r ,t)=∫χ(t−t ' ,r ,r ')E(t ' r ')dt ' dr '+O(E2
)
Δ P(ω)=χ(ω)E(ω)=(ϵ(ω)−1)E(ω)
And finally: ϵ(ω)=1+
Δ P(ω)
E(ω)
ϵ(ω)=
D(ω)
E(ω)
17. Real time spectroscopy in practice 3 3- /
1) Choose an external perturbation E(t)
2) Evolve the Schroedinger equation
3) We calculate the P(t) from (t)
4) Fourier transform P(t) and E(t) and get
i
d Ψ(t )
dt
=[H + E(t )]Ψ(t )
ϵ(ω)=1+
Δ P(ω)
E(ω)
18. Motivations
Better scaling for large system
Polarization and
Hamiltonian depend only
from valence bands. No need
of conduction bands!
19. Motivations
Better scaling for large system
Theory and implementation
are much easier
Polarization and
Hamiltonian depend only on
valence bands. No need of
conduction bands!
20. 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
21. 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(E4
)
22. One code to rule all“ ”
correlation effects
Equation of motions
are always the same
In order to include
correlation effects just
change the Hamiltonian
Notice that the present
approach is limited to
single-particle Hamiltonians.
H=H1+H2+H3+...
24. 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
25. The Hamiltonian II
timedependent 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)
Or equivalent:
HTDH =H KS(ρ0)+Vh (ρ−ρ0)
The density is written as:
ρ(r ,t)=∑i=1
N v
|Ψ(r ,t)|
2
26. The Hamiltonian III
TDDFT
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 RungeGross theorem guarantees that this is an exact
theory for isolated systems
27. Dephasing
Gauge-independent decoherence models for
solids in external fields
M. S. Wismer and V. S. Yakovlev
Phys. Rev. B 97, 144302 (2018)
The previous Hamiltonian are Hermitian
without any time-dependence
(expect the external field)
This means they do not introduce any dephasing!
Dephasing as non-local
operator in the
Hamiltoanian
Dephasing in post-(pre)
processing
~P(t)=P(t)e−λ t
See Octopus code or
Y.Takimoto, Phd thesis (2008)
31. 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)
32. Non-linear response in extended systems: a real-time approach
Claudio Attaccalite
https://arxiv.org/abs/1609.09639
34. 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)ϕj ,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
?
35. 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
36. Optical Absorption: IP
Non Interacting System
δρNI=χ
0
δVtot χ
0
=∑
ij
ϕi(r)ϕj
*
(r)ϕi
*
(r')ϕj(r ')
ω−(ϵi−ϵj)+ i η
Hartree, Hartree-Fock, dft.
=ℑχ0=∑
ij
∣〈 j∣D∣i〉∣2
δ(ω−(ϵj −ϵi))
ϵ''
(ω)=
8 π
2
ω2 ∑
i, j
∣〈ϕi∣e⋅̂v∣ϕj 〉∣2
δ(ϵi−ϵj−ℏ ω)
Absorption by independent
Kohn-Sham particles
Particles are interacting!
37. V ext=
0
V extV HV xc
q ,=
0
q ,
0
q,vf xc q ,q ,
TDDFT is an exact
theory for neutral
excitations!
Time Dependent DFT
V eff (r ,t)=V H (r ,t)+ V xc (r ,t)+ V ext (r ,t)
Interacting System
Non Interacting System
Petersilka et al. Int. J. Quantum Chem. 80, 584 (1996)
I= NI=
I
Vext
0=
NI
V eff
... by using ...
=
0
1
V H
V ext
V xc
V ext
v
f xc
i
∂ ̂ρk (t)
∂t
=[ HKS , ̂ρk ]=[ Hk
0
+V eff
, ̂ρk ]