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 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.

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.

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

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.

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 .

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

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

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 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

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

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

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

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.

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 ]

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 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 ]

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 ]

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

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

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

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

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

Choice of external fields

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

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.

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

Peak at E〜6 MeV 14Be

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

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

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 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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