Transcript Nuclear Density Functional Theory

Time-Dependent Density
Functional Theory (TDDFT)
Takashi NAKATSUKASA
Theoretical Nuclear Physics Laboratory
RIKEN Nishina Center
• Density-Functional Theory (DFT)
• Time-dependent DFT (TDDFT)
• Applications
2008.8.29 CNS-EFES Summer School @ RIKEN Nishina Hall
Quarks, Nucleons, Nuclei, Atoms, Molecules
atom
nucleon
nucleus
e
molecule
q
N
q
N
q
α
α
Strong Binding
“Strong” Binding
clustering
deformation
rotation
vibration
rare gas
cluster
matter
“Weak” binding
“Weak” binding
Density Functional Theory
• Quantum Mechanics
– Many-body wave functions;


(r1 ,, rN )
• Density Functional Theory
– Density clouds;

F[  (r )]
The many-particle system can be described by a functional of
density distribution in the three-dimensional space.
Hohenberg-Kohn Theorem (1)
Hohenberg & Kohn (1964)
The first theorem
Density ρ(r) determines v(r) ,
except for arbitrary choice of zero point.

A system with a one-body potential vr 

H v  H   vri 
Hv gsv  Egsv gsv
i
2

 
pi

  vri    w(ri , rj )
i 2m
i
i j


gs


v
r




(r
)
Existence of one-to-one mapping：
v
v
Strictly speaking, one-to-one or one-to-none
v-representative

v
① vr   gs
Here , we assume the non-degenerate g.s.
Reductio ad absurdum: Assuming


different vr  and v' r  produces the same ground state gsv

v
v
v
( H  V )gs  E gs gs
V   vri 
i
) ( H  V ' )   E 
v
gs
v'
gs
v
gs

V '   v' ri 
i
(V  V ' )gsv  ( E gsv  E gsv ' )gsv
V and V’ are identical except for constant. → Contradiction
② gsv  v
( H  V )gsv  E gsv gsv
( H  V ' )gsv '  E gsv ' gsv '
Again, reductio ad absurdum


assuming different states gsv , gsv' with vr , v' r  produces the same
density  v
Egsv  gsv H  V gsv
 gsv ' H  V gsv '
 


v
v'
Egs  Egs   dr vr   v' r   v r 
H  V  H  V '(V  V ' )
 

v
v
gs V gs   dr vr  v r 
Replacing V ↔ V’
 


E gsv '  E gsv   dr v' r   vr   v r 
 Egsv  Egsv'  Egsv  Egsv' Contradiction !
Here, we assume that the density
v
is v-representative.
For degenerate case, we can prove one-to-one


vr   v (r )
Hohenberg-Kohn Theorem (2)
The second theorem
There is an energy density functional and the variational principle
determines energy and density of the ground state.
Any physical quantity must be a functional of density.

vr   gsv  v

Many-body wave function  r  is a functional of densityρ(r).
From theorem (1)
Energy functional for
external potential v (r)
Ev v   Evgs  Ev  
  
Ev    FHK      (r )v( r )dr
Ev      H  V  
Variational principle holds for vrepresentative density
FHK   : v-independent universal functional
The following variation leads to all the ground-state properties.

  
 F      (r )v(r )dr  


 
  ( r ) dr  N  0
In principle, any physical quantity of the ground state should be a
functional of density.
Variation with respect to many-body wave functions
↓
Variation with respect to one-body density
↓
Physical quantity


(r1 ,, rN )

 (r )

A[  (r )]  [  ] Aˆ [  ]
v-representative→ N-representative
Levy (1979, 1982)
The “N-representative density” means that it has a corresponding
many-body wave function.
Ritz’ Variational Principle
Min  r1 ,..,rN  H  r1 ,..,rN   gs r1 ,..,rN 
Hgs r1 ,..,rN   E gs gs r1 ,..,rN 
Decomposed into two steps
 Min  H  
Min  H   Min


 r  
  r 

F  r   Min  H 
  r 
Positive smoothρ(r) is N-representative.
Gilbert (1975), Lieb (1982)
Harriman’s construction (1980)
For 1-dimensional case (x1 ≤ x ≤ x2), we can construct a Slater
determinant from the following orbitals.
k ( x)   ( x) / N 1/ 2 exp2ik q( x), k  0,  1,  2,
1
q( x) 
N

x
x1
 ( x' )dx'
Problem 1: Prove that a Slater determinant with the N different orbitals
1
( x1 ,, xN ) 
deti ( x j )i , j 1,, N
N!
gives the density  (x)
(i) Show the following properties:
(1)
k ( x )   ( x ) / N
2
dq  ( x)

0
dx
N
0  q( x1 )  q( x)  q( x2 )  1
(ii) Show the orthonormality of orbitals:
i  j   ij
(iii) Prove the Slater determinant (1) produces
 (x)
Density functional theory at finite temperature
Canonical Ensemble
N   pi Ni Ni
i
 


1
F  r   Min trN  H  ln N 
N    r 


 
Grand Canonical Ensemble
   p Ni Ni Ni
N
i
 


1
F  r   Min t rN  H  ln N 
  r 


 




   F  r     (r )v(r )   dr
How to construct DFT
Model of Thomas-Fermi-Dirac-Weizsacker
Missing shell effects
Local density approximation (LDA) for kinetic energy is a serious problem.
Kinetic energy functional without LDA
Kohn-Sham Theory (1965)
Essential idea
Calculate non-local part of kinetic energy utilizing a non-interacting
reference system (virtual Fermi system).
Introduction of reference system
Estimate the kinetic energy in a non-interacting system with a potential
2

pi
T  Vs  
  vs ri 
i 2m
i

TS  v r   gsv T gsv
(T  Vs )gsv  E gsv gsv
The ground state is a Slater determinant with the lowest N orbitals:

 p2
  


1
v

 vs r i r    ii r 
gs 
deti rj 
N!
 2m

2
N
N

 2

p
 v r    i r 
TS  v r    i
i
2m
i 1
i 1
v → N-representative

TS  r   Min  T   Min D T D
   r 
D    r 
Minimize Ts[ρ] with a constraint on ρ(r)
Levy & Perdew (1985)
2
N

 2
 
 
p
* 
    i
i    (r ) i (r )   (r )dr    ij  i (r ) j (r )dr
2m
i 1
ij
 i


0
* 
i (r )
 2 2
  

 
   (r ) i (r )    ij j (r )
j
 2m

Orbitals that minimize Ts[ρ] are eigenstates of a singleparticle Hamiltonian with a local potential.
If these are the lowest N orbitals v → v-representative
Other N orbitals → Not v-representative
Kohn-Sham equation





F  r   TS  r   F  r   TS  r 
Veff  r 
2

p
  i
i  Veff  r  includes effects of interaction as well as
2m
a part of kinetic energy not present in Ts
i
 2
 
  r    i r  ,
i


i  j   ij 

Perform variation with respect to density in terms of orbitals Фi

 * 

 







E

r


d
r

r

(
r
)



ij 
i
j
ij   0
* 
i 
i



V
2 2

 i  eff i    ij j
2m

j
KS canonical equation
Veff
2 2

 i 
i   ii
2m

Problem 2: Prove that the following self-consistent procedure gives the
minimum of the energy:
  
E[  ]  Ts [  ]  Veff [  ]   vext (r ) (r )dr
(1)

 

 1 2
  veff (r )  vext (r )i (r )   ii (r )

 2m

(2)
N

 2
 Veff [  ]
~
 (r )   i (r )  veff (r ) 

 (r )  ( r ) ~ ( r )
i 1
(3) Repeat the procedure (1) and (2) until the convergence.
* Show
E[  ]
0

 (r )  ( r ) ~ ( r )
assuming the convergence.
KS-DFT for electrons
Veff    F    TS  


   (r )  (r ' )
e
 J    E xc     dr dr '    E xc  
2
r r'
2
Exc    T    Ts    Vee    J  
Exchange-correlation energy
It is customary to use the LDA for the exchange-correlation energy.
Its functional form is determined by results of a uniform electron gas:
High-density limit (perturbation)
Low-density limit (Monte-Carlo calculation)
In addition, gradient correction, self-energy correction can be added.
Spin polarization
→
Local spin-densty approx. (LSDA)
Exc  ,  
Example for Exchange-correlation energy
Perdew-Zunger (1981):
Based on high-density limit given by Gell-Mann & Brueckner
low-density limit calculated by Ceperley (Monte Carlo)
Exc    T    Ts    Vee    J  
 Ex  Ec
4
E x  0.4582
3rs
Local (Slater) approximation
 A ln rs  B  Crs ln rs  Drs
Ec  
  1  1 rs   2 rs


(rs  1)
(rs  1)
In Atomic unit
Application to atom
& molecules
E(R)
De
re
R
ωe
Optical constants of di-atomic molecules calculated with LSD
LSD=Local Spin Density
LDA=Local Density Approx.
Atomization energy
Errors in atomization energies (eV)
Exp
Li2
C2H2
1.04 eV
17.6 eV
20 simple
molecules
HF
-0.94
-4.9
3.1
LDA
-0.05
2.4
1.4
Gradient terms
GGA
-0.2
0.4
0.35
Kinetic terms
τ
-0.05
-0.2
0.13
Nuclear Density Functional
Hohenberg‐Kohn’s theorem




E  n r ,  p r  
min


   n  r ,  p  r 
H
Kohn-Sham equation (q = n, p)
2


p
iq  vq  n r ,  p r  iq   iqiq
2m

 2
 q r    iq r 


i  1, N ( Z ) for q  n( p)
i



vq  n r ,  p r  


 q








E  n r ,  p r   TS  n r ,  p r  
Skyrme density functional
Vautherin & Brink, PRC 5 (1972) 626
Historically, we derive a density functional with the HartreeFock procedure from an effective Hamiltonian.


 
 
 
1
V12  t0 (1  x0 P ) (r1  r2 )  t1  (r1  r2 )kˆ 2  kˆ'2  (r1  r2 )
2
   
  
  
 t2 k ' (r1  r2 )k  iW0  1   2   k ' (r1  r2 )k
V123
 
 
 t3 (r1  r2 ) (r2  r3 )
Uniform nuclear
matter with N=Z
or
 


r
 1 r 
D
V12  t3 1  P  (r1  r2 )  

 2 
E 3  2 k F2 3
1
3

 t0   t3  2  3t1  5t2 k F2
A 5 2m 8
16
80
( E / A) 6  2 k F2 9
15 2 3
2


K k


t


t


3
t

5
t

k
0
3
1
2
F
k F2
5 2m 4
8
4
2
F
Necessary to determine all the parameters.


 q r     i r ,  , q 
2
i ,
 
E   dr H r 
 
2

 q r     i r ,  , q 
i ,

 
 

* 
J q r    i    i r ,  , q   i r ,  ' , q      '
i , , '

N=Z nuclei (without Time-odd terms)
 
 2 1
 

2  3 2 1
1
3
3
H r  
 r   t0   3t1  5t2   9t1  5t2    t3   W0   J
2m
8
16
64
16
4
Nuclei with N≠Z (without Time-odd)

 2  1  1  2 
1
H r  
 r   t0 1  x0     x0    n2   p2
2m
2  2 
2


  14 t
1  t 2   



1
t2  t1  n n   p p 
8
2 2 1
1
1
1
 t 2  3t1   3t1  t 2  n  n   p  p   t 2  t1  J n  J p  t3  n  p 
16
32
16
4
 
 
 
1
 W0   J   n  J n   p   J p
2


DFT Nuclear Mass
Error for known nuclei (MeV)
BetheWeizsäcker
FRDM
(1995)
Skyrme
HF+BCS
HFB
3.55
0.68
2.22
0.67
Moller-Nix
Parameters: about 60
Tajima et al (1996)
Param.: about 10
Goriely et al (2002)
Param.: about 15
Recent developments
Lunney, Pearson, Thibault, RMP 75 (2003) 1021
Bender, Bertsch, Heenen, PRL 94 (2005) 102503
Bertsch, Sabbey, Unsnacki, PRC 71 (2005) 054311
k ( x)   ( x) / N 1/ 2 exp2ik q( x), k  0,  1,  2,
Answer 1:
1
q( x) 
N
We have

x
x1
 ( x' )dx'
k ( x )   ( x ) / N
2
dq  ( x)

0
dx
N
0  q( x1 )  q( x)  q( x2 )  1
These are orthonormal.

x2
x1
 ( x)l ( x)dx  
*
k
x2
 ( x)
exp2i q ( x)(l  k )dx
N
x2 dq( x )

exp2i q ( x)(l  k )dx
x1
dx
x1
  exp2i (l  k )q dq   kl
1
0
Using these properties, it is easy to prove that the Slater determinant
constructed with N orbitals of these produces ρ(x).
Answer 2:


 Ts  (r ' ) 
 Veff
E[  ]
   * 
 dr '  c.c.
  vext (r )
 (r ) i i (r ' )  (r )
 (r )
* 


 1 2   i (r ' ) 
   
' i (r ' )
 dr '  c.c. veff (r )  vext (r )
 2m
  (r )
i
* 


 i (r ' ) 


    i  veff (r ' )  vext (r ' )i (r ' )
 dr '  c.c. veff (r )  vext (r )
 (r )
i
 2
 i (r ' )   (r ' )

 


   i
 dr '  
 veff (r ' )  vext (r ' )dr '  veff (r )  vext (r )
 (r )
 (r )
i
0
*
i