Accurate%20energy%20functionals%20for%20evaluating%20electron%20correlation%20energies

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Accurate energy functionals for evaluating
electron correlation energies

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Outline (??)

• History and context.
• Theory.
• Example 1. Homogeneous Electron Gas.
• Example 2. Metal slabs.
• Conclusions and perspectives.

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Earlier achievements
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Discovery of the electron

• 1897

• 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 (1856-1940) discovers the
electron. (Cambridge, UK)
Nobel Prize in Physics, 1906
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Advent of new physics

• 1900

Quantization of energy Nobel Prize in Physics,
1918

• 1905

Photoelectric effect Nobel Prize in Physics, 1921
M. Planck (1858-1947)
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• 1909

Measurement of electron charge and photoelectric
effect. Nobel Prize in Physics, 1923
Robert Millikan (1868-1953)

• 1911

Disintegration of radiactive elements Nobel Prize
in Chemistry, 1908
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Development of quantum mechanics

• 1913

Quantum theory of the atom. Nobel Prize in
Physics, 1922
Niels Bohr (1885-1962)
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Development of quantum mechanics

• 1921

• 1925

1929
Statistical mechanics of electrons
W. Pauli (1900-1958) 1945
E. Fermi (1901-1954) 1938
Louis De Broglie (1892-1987)
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Development of quantum mechanics

• 1926

1933
1932
Erwin Schrodinger (1887-1961)
W. Heisenberg (1901-1976)
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Applications in solids

• 1928

1952
Forbidden region
Felix Bloch 1905-1983
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First attempts in electronic structre calculation

• 1928 - 1932

• 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, 177393, 1972.

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Theory

• First principles electronics structure calculation

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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.
• (Greens 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.
• (Greens 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 Greens function
• The spectral function, Im G, tells you about the
  single-particle-like approximate eigenstates of
  the system the quasiparticles

• 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.
• (Greens 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

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Example 1
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Homogeneous Electron Gas
Independent electron approximation
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Exchange energy
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Correlation energy

• RPA. Bohm and Pines. (1953)
• Gell-Mann and Brueckner. ( 1957)
• Sawada. (1957)
• Hubbard. (1957)
• Nozieres and Pines. (1958)
• Quinn and Ferrel. (1958)
• Ceperley and Alder. (1980)

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Ground-state energy of HEG
Phys. Rev. Lett. 45, 566 (1980)
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Exchange-Correlation energy
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Structure factor
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Density-density response function. (or
Polarization)
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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

• Lets remember that

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Approximations for fxc

• The simplest form is ALDA

• But it gives too poor energy when used with the
  ACFD formula.

Reminder
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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
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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
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Conclusions
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Perspectives

• TDDFT for excited states
• Development of fxc kernels
• Transport and spectroscopic properties

• cond-mat/0604317