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‧

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Outline (提綱 )

• 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

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

M. Planck(1858-1947)

Quantization of energyNobel Prize in Physics, 1918

Photoelectric effectNobel Prize in Physics, 1921

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Robert Millikan (1868-1953)

Measurement of electron charge and photoelectric effect.Nobel Prize in Physics, 1923

Disintegration of radiactive elementsNobel Prize in Chemistry, 1908

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Development of quantum mechanics

Niels Bohr(1885-1962)

Quantum theory of the atom.Nobel Prize in Physics, 1922

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Development of quantum mechanics

Louis De Broglie (1892-1987)

1929

W. Pauli(1900-1958)1945

E. Fermi (1901-1954)1938

Paul Dirac(1902-1984)1938

Statistical mechanics of electrons

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Development of quantum mechanics

Erwin Schrodinger(1887-1961)

19331932

W. Heisenberg (1901-1976)

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Felix Bloch 1905-1983

Forbidden region

Applications in solids

1952

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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

• Bohr & Mottelson. Collective model of nucleus. (1953)

• Bohm & Pines. Random Phase Approximation. (1953)

• Gell-Mann & Brueckner. Many body perturbation theory. (1957)

(According to D. Pines)

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More milestones

• BCS theory of superconductivity.

• Renormalization group.

• Quantum hall effect, integer and fractionary.

• Heavy fermions.

• High temperature superconductivity.

(According to P. Coleman)

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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.

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.

Annals of Physics 8, 272 (1960)

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.”

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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 single-

particle-like approximate eigenstates of the system: the quasiparticles

E E

Im G

non-interacting

interacting

1 2

• Can formulate an iterative expansion of the self-energy 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

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Homogeneous Electron Gas

3123 nkF

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222FF

F

k

m

k

Independent electron approximation

FSt 5

3

3

3

41Srn

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Exchange energy

F

qqpp

XX

ke

qp

e

NN

E

FF

2

,

2

4

321

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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)

XSTOTC t

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

;,2

1

0

int

1

0

rrgdrr

rnrd

dn HXC

21221

;,21

);,( rnrnrrnrnrn

rrg

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Structure factor

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Density-density response function. (or Polarization)

0G

0G

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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

rrnfnwrrf HEGXCXC ][][,

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

0

0

1

0

ˆˆˆTr2

1

wdudC

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)

Example 2

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Jellium metal slabs

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One Jellium SlabThickness L = 6.4rs

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Two slabs

• Surface energies. (erg/cm2)

• Binding energies. (mHa/elec)

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Interaction energies

Thickness L = 3rs and rs = 1.25

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Cancellation of errors

Conclusion and perspectives

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Conclusions

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Perspectives

• TDDFT for excited states

• Development of fxc kernels

• Transport and spectroscopic propertiescond-mat/0604317