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 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

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.

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

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

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

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

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

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

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.

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

Density functional theory at finite temperature Canonical Ensemble Grand Canonical Ensemble

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).

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

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

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

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.

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)

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

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.

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

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

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.

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

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) 102503 Bertsch, Sabbey, Unsnacki, PRC 71 (2005) 054311

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).

Answer 2: