Download KS-DFT formalism

Accurate energy functionals
for evaluating electron
correlation energies
鄭載佾
國家理論科學研究中心物理組，
新竹‧
Outline (提綱)
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History and context.
Theory.
Example 1. Homogeneous Electron Gas.
Example 2. Metal slabs.
Conclusions and perspectives.
2
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Earlier achievements
Discovery of the electron
Could anything at first sight seem
more impractical than a body
which is so small that its mass is
an insignificant fraction of the
mass of an atom of hydrogen?
J.J. Thompson (18561940) discovers the
electron.
(Cambridge, UK)
Nobel Prize in Physics, 1906
4
Advent of new physics
Quantization of energy
Nobel Prize in Physics, 1918
Photoelectric effect
Nobel Prize in Physics, 1921
M. Planck
(1858-1947)
5
Measurement of electron charge
and photoelectric effect.
Nobel Prize in Physics, 1923
Robert Millikan
(1868-1953)
Disintegration of radiactive elements
Nobel Prize in Chemistry, 1908
6
Development of quantum
mechanics
Niels Bohr
(1885-1962)
Quantum theory of
the atom.
Nobel Prize in
Physics, 1922
7
Development of quantum
mechanics
1929
Louis De Broglie
(1892-1987)
Statistical mechanics of electrons
W. Pauli
(19001958)
1945
E. Fermi
(19011954)
Paul Dirac
(1902-1984)
1938
1938
8
Development of quantum
mechanics
1932
W. Heisenberg
(1901-1976)
1933
Erwin Schrodinger
(1887-1961)
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Applications in solids
1952
Forbidden region
Felix Bloch
1905-1983
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First attempts in electronic
structre calculation
• Egil Hylleraas. Configuration interaction,
correlated basis functions.
• Douglas Hartree and Vladimir Fock. Mean
field calculations.
• Wigner and Seitz. Cellular method.
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More milestones
(According to D. Pines)
• Bohr & Mottelson. Collective model of
nucleus. (1953)
• Bohm & Pines. Random Phase
Approximation. (1953)
• Gell-Mann & Brueckner. Many body
perturbation theory. (1957)
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More milestones
(According to P. Coleman)
• BCS theory of superconductivity.
• Renormalization group.
• Quantum hall effect, integer and fractionary.
• Heavy fermions.
• High temperature superconductivity.
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More is different
“At each level of complexity, entirely new
properties appear, and the understanding
of these behaviors requires research
which I think is as fundamental in its
nature as any other”
P. W. Anderson. Science, 177:393, 1972.
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Theory
First principles electronics
structure calculation
Quotation from H. Lipkin
“We can begin by looking at the fundamental paradox of the many-body
problem; namely that people who do not know how to solve the three-body
problem are trying to solve the N-body problem.
Our choice of wave functions is very limited; we only know how to use
independent particle wave functions. The degree to which this limitation
has invaded our thinking is marked by our constant use of concepts which
have meaning only in terms of independent particle wave functions: shell
structure, the occupation number, the Fermi sea and the Fermi surface, the
representation of perturbation theory by Feynman diagrams.
All of these concepts are based upon the assumption that it is reasonable to
talk about a particular state being occupied or unoccupied by a particle
independently of what the other particles are doing. This assumption is
generally not valid, because there are correlations between particles.
However, independent particle wave functions are the only wave functions
that we know how to use. We must therefore find some method to treat
correlations using these very bad independent particle wave functions.”
Annals of Physics 8, 272 (1960)
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Currently available methods
• Configuration Interaction. Quantum Monte
Carlo. (Wave function)
• Many-body perturbation theory.
(Green’s function)
• Kohn-Sham Density Functional Theory
(Density).
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Configuration Interaction
(Wave function method)
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+
Currently available methods
• Configuration Interaction. Quantum Monte
Carlo. (Wave function)
• Many-body perturbation theory.
(Green’s function)
• Kohn-Sham Density Functional Theory
(Density).
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Many-body theory
• Electronic and optical experiments often measure some
aspect of the one-particle Green’s function
• The spectral function, Im G, tells you about the singleparticle-like approximate eigenstates of the system: the
quasiparticles
Im G
non-interacting
interacting
E1 E2


• Can formulate an iterative expansion of the self-energy S
in powers of W, the screened Coulomb interaction, the
leading term of which is the GW approximation
• Can now perform such calculations computationally for
real materials, without adjustable parameters.
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Currently available methods
• Configuration Interaction. Quantum Monte
Carlo. (Wave function)
• Many-body perturbation theory.
(Green’s function)
• Kohn-Sham Density Functional Theory
(Density).
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KS-DFT formalism
• It provides an independent particle
scheme that describes the exact ground
state density and energy.
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KS-DFT formalism
• Given the KS orbitals of the system we
have.
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KS-DFT formalism
• The effective potential associated to the
fictitious system is
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KS-DFT formalism
• The effective potential associated to the
fictitious system is
• The effective potential associated to the
fictitious system is
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Example 1
Homogeneous Electron Gas
Independent electron approximation
3
tS   F
5
k F  3 n  3
1
2
2
2
2
k F k F
F 

2m
2
1 4
3
   rS
n 3
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Exchange energy
EX
1
X 

N
N

p  p F , q  qF
2e 2
3 e2k F

pq
4 
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Correlation energy
•
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•
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RPA. Bohm and Pines. (1953)
Gell-Mann and Brueckner. ( 1957)
Sawada. (1957)
Hubbard. (1957)
C
Nozieres and Pines. (1958)
Quinn and Ferrel. (1958)
   TOT  tS   X 

• Ceperley and Alder. (1980)

此事古難全
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Ground-state energy of HEG
Phys. Rev. Lett. 45, 566 (1980)
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Exchange-Correlation energy
1
 XC n   
0
d

 int     H
n r
  dr 
d g r, r;  

2 r  r 0
1
1
2n2 r, r;    nr1 nr2 
g ( r, r;  ) 
nr1 nr2 
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Structure factor
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Density-density response
function. (or Polarization)
G0
G0
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Density-density response
function. (or Polarization)
RPA response function
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Density-density response
function. (or Polarization)
Exact response function
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Density-density response
function. (or Polarization)
Hubbard response
function
Hubbard local field
factor
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Hubbard vertex correction
Considers the Coulomb repulsion between
electrons with antiparallel spins.
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Many-body effects
Local field factor ~ TDDFT fxc kernel
• Let’s remember that
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Approximations for fxc
• The simplest form is
ALDA
HEG

f XC r  r , w [n ]  f XC [n ]   r  r
• But it gives too poor energy when used
with the ACFD formula.
Reminder
1

1
 C   d  du Tr wˆ  ˆ   ˆ 0 
20 0
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HEG Correlation energies
Phys. Rev. B 61, 13431, (2000)
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Energy optimized kernels
• Dobson and Wang.
• Optimized Hubbard.
where
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Performance of kernels
Phys. Rev. B 70, 205107 (2004)
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Example 2
Jellium metal slabs
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One Jellium Slab
Thickness L = 6.4rs
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Two slabs
• Binding energies. (mHa/elec)
• Surface energies. (erg/cm2)
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Interaction energies
Thickness L = 3rs and rs = 1.25
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Cancellation of errors
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Conclusion and perspectives
Conclusions
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Perspectives
• TDDFT for excited states
• Development of fxc kernels
• Transport and spectroscopic properties
cond-mat/0604317
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