TimeDependent Density Functional Theory TDDFT

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Time-Dependent Density Functional Theory (TDDFT)
Takashi NAKATSUKASA Theoretical Nuclear Physics
Laboratory RIKEN Nishina Center

• Density-Functional Theory (DFT)
• Time-dependent DFT (TDDFT)
• Applications

2008.8.30 CNS-EFES Summer School _at_ RIKEN Nishina
Hall
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Quarks, Nucleons, Nuclei, Atoms, Molecules
atom
nucleon
e
nucleus
molecule
q
N
N
q
q
a
a
Strong Binding
Strong Binding
rare gas
clustering
deformation rotation vibration
cluster matter
Weak binding
Weak binding
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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.
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Hohenberg-Kohn Theorem (1)
The first theorem
Hohenberg Kohn (1964)
Density ?(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
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?
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
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?
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
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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 density?(r).
Energy functional for external potential v (r)
Variational principle holds for v-representative
density
v-independent universal functional
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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
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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
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Positive smooth?(r) is N-representative.
Gilbert (1975), Lieb (1982)
Harrimans construction (1980) For 1-dimensional
case (x1 x x2), we can construct a Slater
determinant from the following orbitals.
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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
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Density functional theory at finite temperature
Canonical Ensemble
Grand Canonical Ensemble
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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).
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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
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Minimize Ts? with a constraint on ?(r)
Levy Perdew (1985)
Orbitals that minimize Ts? 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
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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 ?i
KS canonical equation
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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.
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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)
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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
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Application to atom molecules
E(R)
re
R
De
?e
Optical constants of di-atomic molecules
calculated with LSD
LSDLocal Spin Density LDALocal Density Approx.
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Atomization energy
Errors in atomization energies (eV)
Gradient terms
Kinetic terms
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Nuclear Density Functional
Hohenberg-Kohns theorem
Kohn-Sham equation (q n, p)
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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 NZ
Necessary to determine all the parameters.
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NZ nuclei (without Time-odd terms)
Nuclei with N?Z (without Time-odd)
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DFT Nuclear Mass
Error for known nuclei (MeV)
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
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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 ?(x).
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Answer 2