Accurate energy functionals for evaluating electron correlation energies

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‧

2. Outline (提綱) • History and context. • Theory. • Example 1. Homogeneous Electron Gas. • Example 2. Metal slabs. • Conclusions and perspectives. +

3. Earlier achievements

4. 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

5. 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)

6. 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

7. Development of quantum mechanics 1913 Quantum theory of the atom. Nobel Prize in Physics, 1922 Niels Bohr (1885-1962)

8. Paul Dirac (1902-1984) 1938 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)

9. Development of quantum mechanics 1926 1932 1933 W. Heisenberg (1901-1976) Erwin Schrodinger (1887-1961)

10. Applications in solids 1928 1952 Forbidden region Felix Bloch 1905-1983

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. 1928 - 1932

12. 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)

13. More milestones (According to P. Coleman) • BCS theory of superconductivity. • Renormalization group. • Quantum hall effect, integer and fractionary. • Heavy fermions. • High temperature superconductivity.

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.

15. Theory First principles electronics structure calculation

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 solvethe 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)

17. Currently available methods • Configuration Interaction. Quantum Monte Carlo. (Wave function) • Many-body perturbation theory. (Green’s function) • Kohn-Sham Density Functional Theory (Density).

18. Configuration Interaction(Wave function method) +

19. Currently available methods • Configuration Interaction. Quantum Monte Carlo. (Wave function) • Many-body perturbation theory. (Green’s function) • Kohn-Sham Density Functional Theory (Density).

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 • 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. +

21. Currently available methods • Configuration Interaction. Quantum Monte Carlo. (Wave function) • Many-body perturbation theory. (Green’s function) • Kohn-Sham Density Functional Theory (Density).

22. KS-DFT formalism • It provides an independent particle scheme that describes the exact ground state density and energy.

23. KS-DFT formalism • Given the KS orbitals of the system we have.

24. KS-DFT formalism • The effective potential associated to the fictitious system is

25. The effective potential associated to the fictitious system is KS-DFT formalism • The effective potential associated to the fictitious system is

29. Example 1

30. Homogeneous Electron Gas Independent electron approximation

31. Exchange energy

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) 此事古難全

33. Ground-state energy of HEG Phys. Rev. Lett. 45, 566 (1980)

34. Exchange-Correlation energy

35. Structure factor

36. Density-density response function. (or Polarization)

37. Density-density response function. (or Polarization) RPA response function

38. Density-density response function. (or Polarization) Exact response function

39. Density-density response function. (or Polarization) Hubbard response function Hubbard local field factor

40. Hubbard vertex correction Considers the Coulomb repulsion between electrons with antiparallel spins.

41. Many-body effects Local field factor ~ TDDFT fxc kernel Let’s remember that

42. Approximations for fxc • The simplest form is ALDA • But it gives too poor energy when used with the ACFD formula. Reminder

43. HEG Correlation energies Phys. Rev. B 61, 13431, (2000)

44. Energy optimized kernels • Dobson and Wang. • Optimized Hubbard. where

45. Performance of kernels Phys. Rev. B 70, 205107 (2004)

46. Example 2

47. Jellium metal slabs

48. One Jellium Slab Thickness L = 6.4rs

49. Two slabs • Binding energies. (mHa/elec) • Surface energies. (erg/cm2)

50. Interaction energies Thickness L = 3rs and rs = 1.25