accurate energy functionals for evaluating electron correlation energies 鄭載佾...
Post on 21-Dec-2015
227 views
TRANSCRIPT
![Page 1: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/1.jpg)
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‧
![Page 2: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/2.jpg)
2
Outline (提綱 )
• History and context.
• Theory.
• Example 1. Homogeneous Electron Gas.
• Example 2. Metal slabs.
• Conclusions and perspectives.
+
![Page 3: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/3.jpg)
Earlier achievements
![Page 4: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/4.jpg)
4
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
![Page 5: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/5.jpg)
5
Advent of new physics
M. Planck(1858-1947)
Quantization of energyNobel Prize in Physics, 1918
Photoelectric effectNobel Prize in Physics, 1921
![Page 6: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/6.jpg)
6
Robert Millikan (1868-1953)
Measurement of electron charge and photoelectric effect.Nobel Prize in Physics, 1923
Disintegration of radiactive elementsNobel Prize in Chemistry, 1908
![Page 7: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/7.jpg)
7
Development of quantum mechanics
Niels Bohr(1885-1962)
Quantum theory of the atom.Nobel Prize in Physics, 1922
![Page 8: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/8.jpg)
8
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
![Page 9: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/9.jpg)
9
Development of quantum mechanics
Erwin Schrodinger(1887-1961)
19331932
W. Heisenberg (1901-1976)
![Page 10: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/10.jpg)
10
Felix Bloch 1905-1983
Forbidden region
Applications in solids
1952
![Page 11: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/11.jpg)
11
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.
![Page 12: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/12.jpg)
12
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)
![Page 13: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/13.jpg)
13
More milestones
• BCS theory of superconductivity.
• Renormalization group.
• Quantum hall effect, integer and fractionary.
• Heavy fermions.
• High temperature superconductivity.
(According to P. Coleman)
![Page 14: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/14.jpg)
14
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.
![Page 15: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/15.jpg)
Theory
First principles electronics structure calculation
![Page 16: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/16.jpg)
16
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.”
![Page 17: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/17.jpg)
17
Currently available methods
• Configuration Interaction. Quantum Monte Carlo. (Wave function)
• Many-body perturbation theory.
(Green’s function)
• Kohn-Sham Density Functional Theory (Density).
![Page 18: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/18.jpg)
18
Configuration Interaction(Wave function method)
+
![Page 19: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/19.jpg)
19
Currently available methods
• Configuration Interaction. Quantum Monte Carlo. (Wave function)
• Many-body perturbation theory.
(Green’s function)
• Kohn-Sham Density Functional Theory (Density).
![Page 20: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/20.jpg)
20
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.
+
![Page 21: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/21.jpg)
21
Currently available methods
• Configuration Interaction. Quantum Monte Carlo. (Wave function)
• Many-body perturbation theory.
(Green’s function)
• Kohn-Sham Density Functional Theory (Density).
![Page 22: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/22.jpg)
22
KS-DFT formalism
• It provides an independent particle scheme that describes the exact ground state density and energy.
![Page 23: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/23.jpg)
23
KS-DFT formalism
• Given the KS orbitals of the system we have.
![Page 24: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/24.jpg)
24
KS-DFT formalism
• The effective potential associated to the fictitious system is
![Page 25: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/25.jpg)
25
KS-DFT formalism
• The effective potential associated to the fictitious system is
• The effective potential associated to the fictitious system is
![Page 26: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/26.jpg)
26
![Page 27: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/27.jpg)
27
![Page 28: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/28.jpg)
28
![Page 29: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/29.jpg)
Example 1
![Page 30: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/30.jpg)
30
Homogeneous Electron Gas
3123 nkF
22
222FF
F
k
m
k
Independent electron approximation
FSt 5
3
3
3
41Srn
![Page 31: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/31.jpg)
31
Exchange energy
F
qqpp
XX
ke
qp
e
NN
E
FF
2
,
2
4
321
![Page 32: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/32.jpg)
32
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
![Page 33: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/33.jpg)
33
Ground-state energy of HEG
Phys. Rev. Lett. 45, 566 (1980)
![Page 34: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/34.jpg)
34
Exchange-Correlation energy
;,2
1
0
int
1
0
rrgdrr
rnrd
dn HXC
21221
;,21
);,( rnrnrrnrnrn
rrg
![Page 35: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/35.jpg)
35
Structure factor
![Page 36: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/36.jpg)
36
Density-density response function. (or Polarization)
0G
0G
![Page 37: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/37.jpg)
37
Density-density response function. (or Polarization)
RPA response function
![Page 38: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/38.jpg)
38
Density-density response function. (or Polarization)
Exact response function
![Page 39: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/39.jpg)
39
Density-density response function. (or Polarization)
Hubbard response function
Hubbard local field factor
![Page 40: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/40.jpg)
40
Hubbard vertex correction
Considers the Coulomb repulsion between electrons with antiparallel spins.
![Page 41: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/41.jpg)
41
Many-body effects
Local field factor ~ TDDFT fxc kernel
• Let’s remember that
![Page 42: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/42.jpg)
42
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
![Page 43: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/43.jpg)
43
HEG Correlation energies
Phys. Rev. B 61, 13431, (2000)
![Page 44: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/44.jpg)
44
Energy optimized kernels
• Dobson and Wang.
• Optimized Hubbard. where
![Page 45: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/45.jpg)
45
Performance of kernels
Phys. Rev. B 70, 205107 (2004)
![Page 46: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/46.jpg)
Example 2
![Page 47: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/47.jpg)
47
Jellium metal slabs
![Page 48: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/48.jpg)
48
One Jellium SlabThickness L = 6.4rs
![Page 49: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/49.jpg)
49
Two slabs
• Surface energies. (erg/cm2)
• Binding energies. (mHa/elec)
![Page 50: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/50.jpg)
50
Interaction energies
Thickness L = 3rs and rs = 1.25
![Page 51: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/51.jpg)
51
Cancellation of errors
![Page 52: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/52.jpg)
Conclusion and perspectives
![Page 53: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/53.jpg)
53
Conclusions
![Page 54: 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‧](https://reader034.vdocuments.pub/reader034/viewer/2022042608/56649d5a5503460f94a3a467/html5/thumbnails/54.jpg)
54
Perspectives
• TDDFT for excited states
• Development of fxc kernels
• Transport and spectroscopic propertiescond-mat/0604317