Transcript 基研研究会2005．11 - University of Tokyo

Time-Dependent Density
Functional Theory (TDDFT)
part-2
Takashi NAKATSUKASA
Theoretical Nuclear Physics Laboratory
RIKEN Nishina Center
• Density-Functional Theory (DFT)
• Time-dependent DFT (TDDFT)
• Applications
2008.9.1 CNS-EFES Summer School @ RIKEN Nishina Hall
Time-dependent HK theorem
Runge & Gross (1984)
First theorem
One-to-one mapping between time-dependent density ρ(r,t)
and time-dependent potential v(r,t)
except for a constant shift of the potential
Condition for the external potential:
Possibility of the Taylor expansion around finite time t0

1
k
v(r, t )   vk (r)t  t0 
k 0 k!
The initial state is arbitrary.
This condition allows an impulse potential, but forbids adiabatic switch-on.

i
 (t )  H (t )  (t )
t
Schrödinger equation:
Current density follows the equation



i j(r, t )   (t ) ˆj(r ), H (t )  (t )
t
(1)
Different potentials, v(r,t) , v’(r,t), make time evolution from the same
initial state into Ψ(t)、Ψ’(t)
vk (r)  vk' (r)  0 for k

 
 t 
k 1
 j(r, t )  j' (r, t )t t

wk (r )   
 t 
k
0
   (r, t0 )wk (r )
 v(r, t )  v' (r, t )t t
 j(r, t )  j' (r, t )
0
Continuity eq.
 vk (r )  v'k (r )  0
 (r, t )   ' (r, t )
Problem: Two external potentials are different, when their expansion

1
k
v(r, t )   vk (r)t  t0 
k 0 k!
has different coefficients at a certain order
vk (r)  vk' (r)  0
for k
Using eq. (1), show

 
 t 
k 1
 j(r, t )  j' (r, t )t t

wk (r )   
 t 
k
0
   (r, t0 )wk (r )
 v(r, t )  v' (r, t )t t
0
 vk (r )  v'k (r )  0
Second theorem
The universal density functional exists, and the variational
principle determines the time evolution.
From the first theorem, we have ρ(r,t) ↔Ψ(t). Thus, the variation of the
following function determines ρ(r,t) .

S[  ]   dt [  ]( t ) i  H (t ) [  ]( t )
t0
t
t1
~
S[  ]  S [  ]   dt dr (r, t )v(r, t )
t1
t0
The universal functional
~
S [ ]
is determined.
v-representative density is assumed.
Time-dependent KS theory
Assuming non-interacting v-representability
Time-dependent Kohn-Sham (TDKS) equation
N

 2
 r , t    i r , t 
i 1
 2 2


i i (r, t )   
  vs [  ](r, t ) i (r, t )
t
 2m

 S [ ]
vs [  ](r, t ) 
 (r, t )
S [  ]  S[  ]  
t1
t0

 D [  ](t ) i  T  D [  ](t )
t
Solving the TDKS equation, in principle, we can obtain the exact time
evolution of many-body systems.
The functional depends on ρ(r,t） and the initial state Ψ0 .
Time-dependent quantities
→ Information on excited states
(0)   cn  n

(t )   cneiEnt  n
n
n
Energy projection
1
2



 (t ) eiEt dt   cn  n  ( E  En )
n
Finite time period T ~ 1  → Finite energy resolution
1
2



iEt  t 2
(t ) e e
2
dt  
n
2
2
n  ( E  En )  ( 2)
cn
Time Domain
Energy Domain
Basic equations
• Time-dep. Schroedinger eq.
• Time-dep. Kohn-Sham eq.
Basic equations
• Time-indep. Schroedinger eq.
• Static Kohn-Sham eq.
• dx/dt = Ax
• Ax=ax (Eigenvalue problem)
• Ax=b (Linear equation)
Energy resolution
ΔE〜ћ/T
All energies
Energy resolution
ΔE〜0
A single energy point
Boundary Condition
• Approximate boundary
condition
• Easy for complex systems
Boundary condition
• Exact scattering boundary
condition is possible
• Difficult for complex systems
Photoabsorption cross
section of rare-gas atoms
Zangwill & Soven, PRA 21 (1980) 1561
TDHF(TDDFT) calculation in 3D real space
H. Flocard, S.E. Koonin, M.S. Weiss, Phys. Rev. 17(1978)1682.
3D lattice space calculation
Application of the nuclear
Skyrme-TDHF technique to
molecular systems
Local density approximation (except for Hartree term)
→Appropriate for coordinate-space representation
Kinetic energy is estimated with the finite difference method
Real-space TDDFT calculations
Time-Dependent Kohn-Sham equation

 1

 i (r, t )   
 2  Vion (r )  VH [  (r, t )]   xc [  (r, t )]  Vext (r, t )  i (r, t )
t
 2m

3D space is discretized in lattice
Mt
Each Kohn-Sham orbital: i (r, t )  {i (rk , tn )}nk 11,,
Mr , i  1,, N
N : Number of particles
Mr : Number of mesh points
Mt : Number of time slices
ｙ
i
~ r 
 i
K. Yabana, G.F. Bertsch, Phys. Rev. B54, 4484 (1996).
T. Nakatsukasa, K. Yabana, J. Chem. Phys. 114, 2550
(2001).
X
Calculation of time evolution
Time evolution is calculated by the finite-order Taylor
expansion
t  t

 i (t  t )  exp  i  h(t ' )]dt'  i (t )
 t


 it h(t   t 2) n
n
n!
i
(t )
Violation of the unitarity is negligible if the time step is
small enough:
t  max  1
 max
The maximum (single-particle) eigenenergy in the model space
Real-time calculation of
response functions
1. Weak instantaneous external
perturbation
(t ) Fˆ (t )
Vext (t )  Fˆ (t )
2. Calculate time evolution of
(t ) Fˆ (t )
3. Fourier transform to energy domain
dB(; Fˆ )
1
it
ˆ
  Im  (t ) F (t ) e dt
d

dB(; Fˆ )
d
ω [ MeV ]
Real-time dynamics of electrons
in photoabsorption of molecules
1. External perturbation t=0
Vext r, t    ri (t ), i  x, y, z
2. Time evolution of dipole moment
d i t    ri  (r, t )
E
at t=0
Ethylene molecule
Comparison with measurement (linear optical absorption)
TDDFT accurately describe optical absorption
Dynamical screening effect is significant
PZ+LB94
i




 i (r , t )  h[n(r , t )] i (r , t )
t
with
Dynamical screening
without
i




 i (r , t )  h[n0 (r )] i (r , t )
t
TDDFT
Exp
Without dynamical screening
(frozen Hamiltonian)
T. Nakatsukasa, K. Yabana, J. Chem. Phys. 114(2001)2550.
－－
－
－
－
－
－
Eext t 
++
+
+
+
++
Eind t 
Photoabsorption cross section in C3H6 isomer molecules
Nakatsukasa & Yabana, Chem. Phys. Lett. 374 (2003) 613.
• TDLDA cal with LB94 in 3D real space
• 33401 lattice points (r < 6 Å)
Cross section [ Mb ]
• Isomer effects can be understood in terms of symmetry and antiscreening effects on bound-to-continuum excitations.
Photon energy [ eV ]
Nuclear response function
Dynamics of low-lying modes
and giant resonances
Skyrme functional is local in coordinate space
→ Real-space calculation
Derivatives are estimated by the finite difference method.
Skyrme TDHF in real space
Time-dependent Hartree-Fock equation

~ r 
 i



i  i (r , t )  hHF [  , , j, s, J ]( t )  Vext (t )  i (r , t )
t
3D space is discretized in lattice
Single-particle orbital:
Mt
i (r, t )  {i (rk , tn )}nk 11,,
Mr , i  1,, N
N: Number of particles
y [ fm ]
Mr: Number of mesh points
Mt: Number of time slices
Spatial mesh size is about 1 fm.
Time step is about 0.2 fm/c
Nakatsukasa, Yabana, Phys. Rev. C71 (2005) 024301
X [ fm ]
50
σ [ mb ]
16O
Leistenschneider et al,
PRL86 (2001) 5442
0
50
22O
σ [ mb ]
Berman & Fultz, RMP47 (1975) 713
0
50
0
28O
20
40
σ [ mb ]
SGII parameter set
E1 resonances
16,22,28O
in
0
0
20
E [ MeV ]
Г=0.5 MeV
40
Note: Continnum is NOT
taken into account !
18O
16O
Prolate
10
20
30
Ex [ MeV ]
40
10
20
24Mg
26Mg
Prolate
Triaxial
30
Ex [ MeV ]
40
10
20
30
Ex [ MeV ]
40
28Si
30Si
Oblate
Oblate
10
20
30
Ex [ MeV ]
10
20
30
Ex [ MeV ]
40
40
44Ca
Prolate
48Ca
40Ca
10
20
30
Ex [ MeV ]
10
10
20
30
Ex [ MeV ]
40
20
30
Ex [ MeV ]
40
Giant dipole resonance in
stable and unstable nuclei
Classical image of GDR
p
n
Choice of external fields
Neutrons
16O
 n (t )  n (t )  0 n
Time-dep. transition density
δρ> 0
δρ< 0
 p (t )   p (t )  0 p
Protons
Skyrme HF for
8,14Be
∆r=12 fm
R=8 fm
8Be
Adaptive coordinate
y
z
x
14Be
x
z
x
Neutron
Proton
S.Takami, K.Yabana, and K.Ikeda, Prog. Theor.
Phys. 94 (1995) 1011.
8Be
Solid: K=1
Dashed: K=0
14Be
14Be
Peak at E〜6 MeV
 n (t )  n (t )  0 n  p (t )   p (t )  0 p
Picture of pygmy dipole resonance
Halo neutrons
Protons
Neutrons
n
Core
p
Ground state
n
Low-energy resonance
Nuclear Data by TDDFT Simulation
T.Inakura, T.N., K.Yabana
1. Create all possible nuclei on
computer
2. Investigate properties of nuclei which
are impossible to synthesize
experimentally.
n
3. Application to nuclear astrophysics,
basic data for nuclear reactor
simulation, etc.
Ground-state properties
Photoabsorption cross sections
n
TDDFT Kohn-Sham equation
i

 i (t )  hKS  (t )  Vext (t )   i (t )
t
Real-time response
of neutron-rich nuclei
Non-linear regime
(Large-amplitude dynamics)
N.Hinohara, T.N., M.Matsuo, K.Matsuyanagi
Quantum tunneling dynamics in nuclear
shape-coexistence phenomena in 68Se
Cal
Exp
Summary
(Time-dependent) Density functional theory assures
the existence of functional to reproduce exact manybody dynamics.
Any physical observable is a functional of density.
Current functionals rely on the Kohn-Sham scheme
Applications are wide in variety; Nuclei, Atoms,
molecules, solids, …
We show TDDFT calculations of photonuclear cross
sections using a Skyrme functional.
Toward theoretical nuclear data table
Postdoctoral opportunity at RIKEN
http://www.riken.jp/
Click
on “Carrier Opportunity”
FPR (Foreign Postdoctoral Researcher)