Time-Dependent Density Functional Theory (TDDFT) part-2

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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 _at_ RIKEN Nishina
Hall
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Time-dependent HK theorem
Runge Gross (1984)
First theorem
One-to-one mapping between time-dependent density
?(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
The initial state is arbitrary. This condition
allows an impulse potential, but forbids
adiabatic switch-on.
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Schrödinger equation
Current density follows the equation
(1)
Different potentials, v(r,t) , v(r,t), make time
evolution from the same initial state into
?(t)??(t)
Continuity eq.
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Problem Two external potentials are different,
when their expansion has different coefficients
at a certain order Using eq. (1), show
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Second theorem
The universal density functional exists, and the
variational principle determines the time
evolution.
From the first theorem, we have ?(r,t) ??(t).
Thus, the variation of the following function
determines ?(r,t) .
The universal functional is
determined.
v-representative density is assumed.
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Time-dependent KS theory
Assuming non-interacting v-representability
Time-dependent Kohn-Sham (TDKS) equation
Solving the TDKS equation, in principle, we can
obtain the exact time evolution of many-body
systems. The functional depends on ?(r,t) and the
initial state ?0 .
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Time-dependent quantities? Information on
excited states
Energy projection
Finite time period ? Finite energy
resolution
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Energy Domain
Time Domain

• Basic equations
• Time-dep. Schroedinger eq.
• Time-dep. Kohn-Sham eq.
• dx/dt Ax
• Energy resolution
• ?E??/T
• All energies
• Boundary Condition
• Approximate boundary condition
• Easy for complex systems

• Basic equations
• Time-indep. Schroedinger eq.
• Static Kohn-Sham eq.
• Axax (Eigenvalue problem)
• Axb (Linear equation)
• Energy resolution
• ?E?0
• A single energy point
• Boundary condition
• Exact scattering boundary condition is possible
• Difficult for complex systems

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Photoabsorption cross section of rare-gas atoms
Zangwill Soven, PRA 21 (1980) 1561
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TDHF(TDDFT) calculation in 3D real space
H. Flocard, S.E. Koonin, M.S. Weiss, Phys. Rev.
17(1978)1682.
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3D lattice space calculationApplication 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
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Real-space TDDFT calculations
Time-Dependent Kohn-Sham equation
3D space is discretized in lattice Each Kohn-Sham
orbital
N Number of particles Mr Number of mesh
points Mt Number of time slices
y
K. Yabana, G.F. Bertsch, Phys. Rev. B54, 4484
(1996).
T. Nakatsukasa, K. Yabana, J. Chem. Phys. 114,
2550 (2001).
X
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Calculation of time evolution
Time evolution is calculated by the finite-order
Taylor expansion
Violation of the unitarity is negligible if the
time step is small enough
The maximum (single-particle) eigenenergy in the
model space
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Real-time calculation of response functions

1. Weak instantaneous external perturbation
2. Calculate time evolution of
3. Fourier transform to energy domain

? MeV
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Real-time dynamics of electrons in
photoabsorption of molecules
1. External perturbation t0
2. Time evolution of dipole moment
E at t0
Ethylene molecule
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Comparison with measurement (linear optical
absorption)
TDDFT accurately describe optical
absorption Dynamical screening effect is
significant
PZLB94
with
Dynamical screening
without
TDDFT
Exp
Without dynamical screening (frozen Hamiltonian)
T. Nakatsukasa, K. Yabana, J. Chem. Phys.
114(2001)2550.
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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 lt 6 Å)
• Isomer effects can be understood in terms of
  symmetry and anti-screening effects on
  bound-to-continuum excitations.

Cross section Mb
Photon energy eV
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Nuclear response functionDynamics of low-lying
modes and giant resonances
Skyrme functional is local in coordinate space
? Real-space calculation Derivati
ves are estimated by the finite difference method.
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Skyrme TDHF in real space
Time-dependent Hartree-Fock equation
3D space is discretized in lattice Single-particle
orbital
N Number of particles Mr Number of mesh
points Mt Number of time slices
y fm
Spatial mesh size is about 1 fm. Time step is
about 0.2 fm/c
Nakatsukasa, Yabana, Phys. Rev. C71 (2005) 024301
X fm
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E1 resonances in 16,22,28O
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16O
Leistenschneider et al, PRL86 (2001) 5442
s mb
0
50
22O
Berman Fultz, RMP47 (1975) 713
s mb
0
50
20
40
0
28O
SGII parameter set ?0.5 MeV Note Continnum is
NOT taken into account !
s mb
0
0
20
40
E MeV
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18O
16O
Prolate
10
30
40
20
Ex MeV
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26Mg
24Mg
Triaxial
Prolate
10
40
20
30
10
40
20
30
Ex MeV
Ex MeV
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28Si
30Si
Oblate
Oblate
10
40
20
30
Ex MeV
10
40
20
30
Ex MeV
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44Ca
Prolate
48Ca
40Ca
10
20
30
Ex MeV
10
40
20
30
Ex MeV
10
40
20
30
Ex MeV
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Giant dipole resonance instable and unstable
nuclei
Classical image of GDR
p
n
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Choice of external fields
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Neutrons
16O
Time-dep. transition density
d?gt 0
d?lt 0
Protons
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Skyrme HF for 8,14Be
?r12 fm
R8 fm
8Be
Adaptive coordinate
14Be
Neutron Proton
S.Takami, K.Yabana, and K.Ikeda, Prog. Theor.
Phys. 94 (1995) 1011.
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8Be
Solid K1 Dashed K0
14Be
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Peak at E?6 MeV
14Be
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Picture of pygmy dipole resonance
Halo neutrons
Neutrons
Protons
n
Core
n
p
Ground state
Low-energy resonance
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Nuclear Data by TDDFT Simulation
T.Inakura, T.N., K.Yabana
Ground-state properties

1. Create all possible nuclei on computer
2. Investigate properties of nuclei which are
   impossible to synthesize experimentally.
3. Application to nuclear astrophysics, basic data
   for nuclear reactor simulation, etc.

n
Photoabsorption cross sections
TDDFT Kohn-Sham equation
n
Real-time response of neutron-rich nuclei
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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
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Summary

• (Time-dependent) Density functional theory
  assures the existence of functional to reproduce
  exact many-body 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

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Opportunity FPR (Foreign Postdoctoral Researcher)