1. Time-Dependent Density Functional Theory (TDDFT)
Takashi NAKATSUKASA
Theoretical Nuclear Physics Laboratory
RIKEN Nishina Center
Density-Functional Theory (DFT)
Time-dependent DFT (TDDFT)
Applications
CNS-EFES Summer RIKEN Nishina Hall

2. Quarks, Nucleons, Nuclei, Atoms, Molecules
nucleus e
molecule q N N q q α α
Strong Binding
“Strong” Binding
clustering
rare gas
deformation rotation vibration
cluster matter
“Weak” binding
“Weak” binding

3. Density Functional Theory
Quantum Mechanics
Many-body wave functions;
Density Functional Theory
Density clouds;
The many-particle system can be described by a functional of density distribution in the three-dimensional space.

4. Hohenberg-Kohn Theorem (1)
The first theorem
Hohenberg & Kohn (1964)
Density ρ(r) determines v(r) ,
except for arbitrary choice of zero point.
A system with a one-body potential
Existence of one-to-one mapping：　　　　　　 　　　　
Strictly speaking, one-to-one or one-to-none
　　　v-representative

5. ① Here , we assume the non-degenerate g.s.
Reductio ad absurdum: Assuming
different　　　　and　　　　produces the same ground state　　　
V and V’ are identical except for constant. → Contradiction

6. ② assuming different states with produces the same density
Again, reductio ad absurdum
assuming different states with 　　　　　　 produces the same density
Replacing V ↔ V’
Contradiction !
Here, we assume that the density　　　　is v-representative.
For degenerate case, we can prove one-to-one

7. 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.
From theorem (1)
Many-body wave function is a functional of densityρ(r).　
Energy functional for external potential v (r)
Variational principle holds for v-representative density
: v-independent universal functional

8. The following variation leads to all the ground-state properties.
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

9. v-representative→ N-representative
Levy (1979, 1982)
The “N-representative density” means that it has a corresponding many-body wave function.
Ritz’ Variational Principle
Decomposed into two steps

10. 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.

11. Problem 1: Prove that a Slater determinant with the N different orbitals
gives the density
(1)
(i) Show the following properties:
(ii) Show the orthonormality of orbitals:
(iii) Prove the Slater determinant (1) produces

12. Density functional theory at finite temperature
Canonical Ensemble
Grand Canonical Ensemble

13. How to construct DFT Kohn-Sham Theory (1965)
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).

14. Introduction of reference system
Estimate the kinetic energy in a non-interacting system with a potential
The ground state is a Slater determinant with the lowest N orbitals:
v → N-representative

15. Minimize Ts[ρ] with a constraint on ρ(r)
Levy & Perdew (1985)
Orbitals that minimize Ts[ρ] are eigenstates of a single-particle Hamiltonian with a local potential.
If these are the lowest N orbitals v →　v-representative
Other N orbitals　 →　Not v-representative

16. Kohn-Sham equation
includes effects of interaction as well as a part of kinetic energy not present in Ts
Perform variation with respect to density in terms of orbitals Фi
KS canonical equation

17. Problem 2: Prove that the following self-consistent procedure gives the minimum of the energy:
(1)
(2)
(3) Repeat the procedure (1) and (2) until the convergence.
* Show assuming the convergence.

18. KS-DFT for electrons 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)

19. 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)
Local (Slater) approximation
In Atomic unit

20. Application to atom & molecules
E(R) re R De ωe
Optical constants of di-atomic molecules calculated with LSD
LSD=Local Spin Density
LDA=Local Density Approx.

21. Atomization energy Li2 C2H2 HF -0.94 -4.9 3.1 LDA -0.05 2.4 1.4 GGA
Errors in atomization energies (eV)
Li2
C2H2
20 simple molecules
Exp
1.04 eV
17.6 eV HF
-0.94
-4.9
3.1
LDA
-0.05
2.4
1.4
GGA
-0.2
0.4
0.35 τ
0.13
Gradient terms
Kinetic terms

22. Nuclear Density Functional
Hohenberg‐Kohn’s theorem
Kohn-Sham equation (q = n, p)

23. Skyrme density functional
Vautherin & Brink, PRC 5 (1972) 626
Historically, we derive a density functional with the Hartree-Fock procedure from an effective Hamiltonian. or
Uniform nuclear matter with N=Z
Necessary to determine all the parameters.

24. N=Z nuclei (without Time-odd terms)
Nuclei with N≠Z (without Time-odd)

25. DFT Nuclear Mass Bethe-Weizsäcker 3.55 FRDM (1995) 0.68 Skyrme HF+BCS
Error for known nuclei (MeV)
Bethe-Weizsäcker
3.55
FRDM
(1995)
0.68
Skyrme
HF+BCS
HFB
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)
Bertsch, Sabbey, Unsnacki, PRC 71 (2005)

26. Answer 1:
We have
These are orthonormal.
Using these properties, it is easy to prove that the Slater determinant constructed with N orbitals of these produces ρ(x).

27. Answer 2: