1. 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
CNS-EFES Summer RIKEN Nishina Hall

2. Time-dependent HK theorem
First theorem
Runge & Gross (1984)
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
The initial state is arbitrary.
This condition allows an impulse potential, but forbids adiabatic switch-on.

3. 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 Ψ(t)、Ψ’(t)
Continuity eq.

4. Problem: Two external potentials are different, when their expansion
has different coefficients at a certain order
Using eq. (1), show

5. 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) .
The universal functional is determined.
v-representative density is assumed.

6. 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 ρ(r,t） and the initial state Ψ0 .

7. Time-dependent quantities → Information on excited states
Energy projection
Finite time period 　　　　　　→ Finite energy resolution

8. Time Domain Energy Domain Basic equations Time-dep. Schroedinger eq.
Time-dep. Kohn-Sham eq.
dx/dt = Ax
Energy resolution
ΔE〜ћ/T
All energies
Boundary Condition
Approximate boundary condition
Easy for complex systems
Basic equations
Time-indep. Schroedinger eq.
Static Kohn-Sham eq.
Ax=ax (Eigenvalue problem)
Ax=b (Linear equation)
Energy resolution
ΔE〜0
A single energy point
Boundary condition
Exact scattering boundary condition is possible
Difficult for complex systems

9. Photoabsorption cross section of rare-gas atoms
Zangwill & Soven, PRA 21 (1980) 1561

10. TDHF(TDDFT) calculation in 3D real space
H. Flocard, S.E. Koonin, M.S. Weiss, Phys. Rev. 17(1978)1682.

11. 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

12. 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

13. 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

14. Real-time calculation of response functions
Weak instantaneous external perturbation
Calculate time evolution of
Fourier transform to energy domain
ω [ MeV ]

15. 1. External perturbation t=0
Real-time dynamics of electrons
in photoabsorption of molecules
1. External perturbation　 t=0
2. Time evolution of dipole moment E
at t=0
Ethylene molecule

16. TDDFT Exp Comparison with measurement (linear optical absorption)
TDDFT accurately describe optical absorption
Dynamical screening effect is significant
PZ+LB94
with
Dynamical screening
without
TDDFT
Exp + －
Without dynamical screening
(frozen Hamiltonian)
T. Nakatsukasa, K. Yabana, J. Chem. Phys. 114(2001)2550.

17. 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)
Isomer effects can be understood in terms of symmetry and anti-screening effects on bound-to-continuum excitations.
Cross section [ Mb ]
Photon energy [ eV ]

18. 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.

19. 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)
X [ fm ]

20. E1 resonances in 16,22,28O 16O 22O 28O 50 σ [ mb ]
Leistenschneider et al, PRL86 (2001) 5442 50
22O
Berman & Fultz, RMP47 (1975) 713
σ [ mb ] 50 20 40
28O
SGII parameter set
Г=0.5 MeV
Note: Continnum is NOT taken into account !
E1 resonances in 16,22,28O
σ [ mb ] 20 40
E [ MeV ]

21. 18O
16O
Prolate 10 30 20 40
Ex [ MeV ]

22. 26Mg 24Mg Ex [ MeV ] Ex [ MeV ] Prolate Triaxial 10 20 30 40 10 20 30

23. 28Si
30Si
Oblate
Oblate 10 20 30 40
Ex [ MeV ] 10 20 30 40
Ex [ MeV ]

24. 44Ca 48Ca 40Ca Ex [ MeV ] Ex [ MeV ] Ex [ MeV ] Prolate 10 20 30 10 20

25. Giant dipole resonance in stable and unstable nuclei
Classical image of GDR p n

26. Choice of external fields

27. Neutrons
16O
Time-dep. transition density
δρ> 0
δρ< 0
Protons

28. Skyrme HF for 8,14Be 8Be 14Be ∆r=12 fm R=8 fm Adaptive coordinate x z
Neutron Proton
S.Takami, K.Yabana, and K.Ikeda, Prog. Theor. Phys. 94 (1995) 1011.

29. 8Be
Solid: K=1
Dashed: K=0
14Be

30. Peak at E〜6 MeV
14Be

31. Picture of pygmy dipole resonance
Halo neutrons
Neutrons
Protons n
Core n p
Ground state
Low-energy resonance

32. Nuclear Data by TDDFT Simulation
T.Inakura, T.N., K.Yabana
Ground-state properties
Create all possible nuclei on computer
Investigate properties of nuclei which are impossible to synthesize experimentally.
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

33. 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

34. 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

35. Postdoctoral opportunity at RIKEN
Click on “Carrier Opportunity”
FPR (Foreign Postdoctoral Researcher)