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

One-to-one mapping between time-dependent density ρ(r,t) and time-dependent potential v(r,t)

Runge & Gross (1984)

except for a constant shift of the potential

Condition for the external potential:

Possibility of the Taylor expansion around finite time t0

0

0)(!

1),(

k

kk ttv

ktv rr

The initial state is arbitrary.

This condition allows an impulse potential, but forbids adiabatic switch-on.

First theorem

Different potentials, v(r,t) , v’(r,t), make time evolution from the same initial state into Ψ(t) 、 Ψ’(t)

)()(),(ˆ)(),( ttHttt

i

rjrj

)()()( ttHtt

i

Schrödinger equation:

Current density follows the equation

0)(')(),('),()(

)(),(),('),(

0

00

1

rrrrr

rrrjrj

kktt

k

k

ktt

k

vvtvtvt

w

wtttt

),('),( tt rjrj ),('),( tt rr Continuity eq.

kforvv kk 0)()( ' rr

(1)

Problem: Two external potentials are different, when their expansion

has different coefficients at a certain order

Using eq. (1), show

0)(')(),('),()(

)(),(),('),(

0

00

1

rrrrr

rrrjrj

kktt

k

k

ktt

k

vvtvtvt

w

wtttt

kforvv kk 0)()( ' rr

0

0)(!

1),(

k

kk ttv

ktv rr

The universal density functional exists, and the variational principle determines the time evolution.

Second theorem

1

0

)]([)()]([][t

tttH

titdtS

From the first theorem, we have ρ(r,t) ↔Ψ(t). Thus, the variation of the following function determines ρ(r,t) .

1

0

),(),(][~

][t

ttvtddtSS rrr

The universal functional is determined.][~ S

v-representative density is assumed.

Time-dependent KS theoryAssuming non-interacting v-representability

N

ii trtr

1

2,,

1

0

)]([)]([][][

),(

][),]([

),(),]([2

),( 22

t

t DD

s

isi

tTt

itSS

t

Stv

ttvm

tt

i

rr

rrr

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 Kohn-Sham (TDKS) equation

Time-dependent quantities→ Information on excited states

nn

tiEn

nnn

nectc )()0(

n

nnniEt EEcdtet )()(

2

1

nn

n

ntiEt

EE

cdteet

22

2

)2()(

2)(

2

1

Energy projection

Finite time period 　　　　　　→ Finite energy resolution

1~T

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 conditi

on• 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 co

ndition is possible• Difficult for complex systems

Time Domain Energy Domain

Photoabsorption cross section of rare-gas atoms

Zangwill & Soven, PRA 21 (1980) 1561

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

TDHF(TDDFT) calculation in 3D real space

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

X

ｙ

3D space is discretized in lattice

Each Kohn-Sham orbital:

N : Number of particles

Mr : Number of mesh points

Mt : Number of time slices

Nitt MtnMrknkii ,,1,)},({),( ,1

,1

rr

),(),()],([)],([)(2

1),( extxcHion

2 ttVttVVm

tt

i ii rrrrrr

Time-Dependent Kohn-Sham equation

K. Yabana, G.F. Bertsch, Phys. Rev. B54, 4484 (1996).

ri~

T. Nakatsukasa, K. Yabana, J. Chem. Phys. 114, 2550 (2001).

Calculation of time evolution

Time evolution is calculated by the finite-order Taylor expansion

)(

!

)2(

)(')]'(exp)(

tn

tthti

tdtthitt

in

n

i

tt

ti

Violation of the unitarity is negligible if the time step is small enough:

1max t

max 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

dtetFtd

FdB ti

)(ˆ)(Im1)ˆ;(

)(ˆ)( tFt

)(ˆ)(ext tFtV

)(ˆ)( tFt

ω [ MeV ]

d

FdB )ˆ;(

Real-time dynamics of electronsin photoabsorption of molecules

zyxitrtV iext ,,),(, r

1. External perturbation 　 t=0

2. Time evolution of dipole moment

Ethylene molecule

),( trtd ii rE

at t=0

T. Nakatsukasa, K. Yabana, J. Chem. Phys. 114(2001)2550.

Comparison with measurement (linear optical absorption)

ExpTDDFT

Without dynamical screening(frozen Hamiltonian)

TDDFT accurately describe optical absorptionDynamical screening effect is significant

),()],([),( trtrnhtrt

i ii

),()]([),( 0 trrnhtrt

i ii

withwithout

Dynamical screening

tEext

++

+

++

+

+

－－

－－

－

－

－

tEind

PZ+LB94

Photoabsorption cross section in C3H6 isomer molecules

Photon energy [ eV ]

Cro

ss s

ectio

n [

Mb

]

• 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.

Nakatsukasa & Yabana, Chem. Phys. Lett. 374 (2003) 613.

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

X [ fm ]

y [

fm ]

3D space is discretized in lattice

Single-particle orbital:

N: Number of particles

Mr: Number of mesh points

Mt: Number of time slices

Nitt MtnMrknkii ,,1,)},({),( ,1

,1

rr

),()()](,,,,[),( exHF ttVthtt

i iti rJsjr �

Time-dependent Hartree-Fock equation

Spatial mesh size is about 1 fm.

Time step is about 0.2 fm/c

Nakatsukasa, Yabana, Phys. Rev. C71 (2005) 024301

ri~

0 20 400

50

0

050

50

E [ MeV ]

σ [

mb

]σ

[ m

b ]

σ [

mb

]16O

22O

28OSGII parameter set

Г=0.5 MeV

Note: Continnum is NOT taken into account !

Leistenschneider et al, PRL86 (2001) 5442

Berman & Fultz, RMP47 (1975) 713

0 20 40

E1 resonances in 16,22,28O

Ex [ MeV ]

10 4020 30

18O16OProlate

Ex [ MeV ]10 4020 30

Ex [ MeV ]10 4020 30

24Mg 26MgProlate Triaxial

Ex [ MeV ]10 4020 30

Ex [ MeV ]10 4020 30

28Si 30SiOblateOblate

40Ca

44Ca

48Ca

Ex [ MeV ]10 4020 30 Ex [ MeV ]

10 4020 30

Ex [ MeV ]10 20 30

Prolate

Giant dipole resonance instable and unstable nuclei

npClassical image of GDR

Choice of external fields

Neutrons

Protons

δρ> 0

δρ< 0

16O

ppp tt 0)()(

nnn tt 0)()( Time-dep. transition density

Skyrme HF for 8,14Be

S.Takami, K.Yabana, and K.Ikeda, Prog. Theor. Phys. 94 (1995) 1011.

R=8 fm

∆r=12 fm

Adaptive coordinate

8Be

14Be

Neutron Proton

x

z

x

y

x

z

Solid: K=1Dashed: K=0

8Be

14Be

nnn tt 0)()( ppp tt 0)()(

Peak at E 〜 6 MeV14Be

Picture of pygmy dipole resonance

n

p n

Ground state

Protons Neutrons

Low-energy resonance

Halo neutrons

Core

Nuclear Data by TDDFT Simulation

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

n

Real-time response of neutron-rich nuclei

Photoabsorption cross sections

Ground-state properties

)()()()( extKS ttVthtt

i ii

TDDFT Kohn-Sham equation

T.Inakura, T.N., K.Yabana

Non-linear regime (Large-amplitude dynamics)

N.Hinohara, T.N., M.Matsuo, K.MatsuyanagiQuantum tunneling dynamics in nuclear shape-coexistence phenomena in 68Se

Cal Exp

SummarySummary

(Time-dependent) Density functional theory assures (Time-dependent) Density functional theory assures the existence of functional to reproduce exact many-the existence of functional to reproduce exact many-body dynamics.body dynamics.Any physical observable is a functional of density.Any physical observable is a functional of density.Current functionals rely on the Kohn-Sham schemeCurrent functionals rely on the Kohn-Sham schemeApplications are wide in variety; Nuclei, Atoms, moleApplications are wide in variety; Nuclei, Atoms, molecules, solids, …cules, solids, …We show TDDFT calculations of photonuclear cross We show TDDFT calculations of photonuclear cross sections using a Skyrme functional. sections using a Skyrme functional. Toward theoretical nuclear data tableToward theoretical nuclear data table

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