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

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 2008.9.1 CNS-EFES Summer School @ 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 potentialv(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 stateΨ0 .

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

8. 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 Energy Domain Time Domain

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 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

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

16. + － － + － + － + － + － + + － 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. Nakatsukasa & Yabana, Chem. Phys. Lett. 374 (2003) 613. Photoabsorption cross section in C3H6 isomer molecules • 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 functionDynamics 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) 024301 X [ fm ]

20. 50 16O E1 resonances in 16,22,28O Leistenschneider et al, PRL86 (2001) 5442 σ [ mb ] 0 50 22O Berman & Fultz, RMP47 (1975) 713 σ [ mb ] 0 50 20 40 0 28O SGII parameter set Г=0.5 MeV Note: Continnum is NOT taken into account ! σ [ mb ] 0 0 20 40 E [ MeV ]

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

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

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

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

25. Giant dipole resonance instable 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 z y z x x x ∆r=12 fm R=8 fm 8Be Adaptive coordinate 14Be 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 Low-energy resonance Ground state

32. T.Inakura, T.N., K.Yabana Ground-state properties Nuclear Data by TDDFT Simulation n 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. 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 http://www.riken.jp/ Click on “Carrier Opportunity” FPR (Foreign Postdoctoral Researcher)