Post on 17-Jan-2016
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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