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

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

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

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

② assuming different states with produces the same density Replacing V ↔ V’ Contradiction ! Again, reductio ad absurdum 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 Ritz’ Variational Principle Decomposed into two steps The “N-representative density” means that it has a corresponding many-body wave function. Levy (1979, 1982)

Positive smoothρ(r) is N-representative. Gilbert (1975), Lieb (1982) Harriman’s construction (1980) For 1-dimensional case (x 1 ≤ x ≤ x 2 ), we can construct a Slater determinant from the following orbitals.

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

Density functional theory at finite temperature Canonical Ensemble Grand Canonical Ensemble

How to construct DFT Model of Thomas-Fermi-Dirac-Weizsacker Missing shell effects Local density approximation (LDA) for kinetic energy is a serious problem. Kohn-Sham Theory (1965) Kinetic energy functional without LDA 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 T s [ρ] with a constraint on ρ(r) Orbitals that minimize T s [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 Levy & Perdew (1985)

Kohn-Sham equation includes effects of interaction as well as a part of kinetic energy not present in T s Perform variation with respect to density in terms of orbitals Ф i KS canonical equation

Problem 2: Prove that the following self-consistent procedure gives the minimum of the energy: (3) Repeat the procedure (1) and (2) until the convergence. (1) (2) * 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) In Atomic unit Local (Slater) approximation

Application to atom & molecules rere DeDe ωeωe R E(R) LSD=Local Spin Density LDA=Local Density Approx. Optical constants of di-atomic molecules calculated with LSD

Atomization energy Li 2 C2H2C2H2 20 simple molecules Exp 1.04 eV17.6 eV HF LDA GGA τ Errors in atomization energies (eV) 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. Uniform nuclear matter with N=Z or 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 HFB Error for known nuclei (MeV) Moller-Nix Parameters: about 60 Tajima et al (1996) Param.: about 10 Recent developments Lunney, Pearson, Thibault, RMP 75 (2003) 1021 Bender, Bertsch, Heenen, PRL 94 (2005) Bertsch, Sabbey, Unsnacki, PRC 71 (2005) Goriely et al (2002) Param.: about 15

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 1:

Answer 2: