TimeDependent Density Functional Theory TDDFT part2 Takashi NAKATSUKASA

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. 8. 30 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 t 0 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 Ψ 0.

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

3 D 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 3 D space is discretized in lattice Each Kohn-Sham orbital: N : Number of particles Mr : Number of mesh points ｙ Mt : Number of time slices K. Yabana, G. F. Bertsch, Phys. Rev. B 54, 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 1. Weak instantaneous external perturbation 2. Calculate time evolution of 3. Fourier transform to energy domain ω [ Me. V ]

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

Comparison with measurement (linear optical absorption) TDDFT accurately describe optical absorption Dynamical screening effect is significant PZ+LB 94 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 C 3 H 6 isomer molecules Nakatsukasa & Yabana, Chem. Phys. Lett. 374 (2003) 613. • TDLDA cal with LB 94 in 3 D 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 [ e. V ]

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 3 D space is discretized in lattice Single-particle orbital: 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. C 71 (2005) 024301 X [ fm ]

50 σ [ mb ] 16 O Leistenschneider et al, PRL 86 (2001) 5442 22 O Berman & Fultz, RMP 47 (1975) 713 σ [ mb ] 0 50 28 O 20 40 SGII parameter set σ [ mb ] 0 50 0 E 1 resonances 16, 22, 28 O in 0 0 20 E [ Me. V ] Г=0. 5 Me. V 40 Note: Continnum is NOT taken into account !

18 O 16 O Prolate 10 20 30 Ex [ Me. V ] 40

10 20 24 Mg 26 Mg Prolate Triaxial 30 Ex [ Me. V ] 40 10 20 30 Ex [ Me. V ] 40

28 Si 30 Si Oblate 10 20 30 Ex [ Me. V ] 40 40

44 Ca Prolate 48 Ca 40 Ca 10 20 30 Ex [ Me. V ] 10 10 20 30 Ex [ Me. V ] 40

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

Choice of external fields

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

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

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

14 Be Peak at E〜 6 Me. V

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 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 68 Se 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

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