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Introduction to DFT Part 2

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Part 2 of a tutorial given in the Brazilian Physical Society meeting, ENFMC. Abstract: Density-functional theory (DFT) was developed 50 years ago, connecting fundamental quantum methods from early days of quantum mechanics to our days of computer-powered science. Today DFT is the most widely used method in electronic structure calculations. It helps moving forward materials sciences from a single atom to nanoclusters and biomolecules, connecting solid-state, quantum chemistry, atomic and molecular physics, biophysics and beyond. In this tutorial, I will try to clarify this pathway under a historical view, presenting the DFT pillars and its building blocks, namely, the Hohenberg-Kohn theorem, the Kohn-Sham scheme, the local density approximation (LDA) and generalized gradient approximation (GGA). I would like to open the black box misconception of the method, and present a more pedagogical and solid perspective on DFT.

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Introduction to DFT Part 2

  1. 1. XXXVIII ENFMC Brazilian Physical Society Meeting Introduction to density functional theory Mariana M. Odashima ENFMC
  2. 2. Problem HK-KS xc LDA Construction Challenges Final Remarks This tutorial Introduction to density-functional theory Context and key concepts (1927-1930) (Born-Oppenheimer, Hartree, Hartree-Fock, Thomas-Fermi) Fundamentals (1964-1965) (Hohenberg-Kohn theorem, Kohn-Sham scheme) Approximations (≈ 1980-2010) (local density and generalized gradient approximations (LDA and GGA), construction of functionals) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 1/76 ENFMC
  3. 3. Problem HK-KS xc LDA Construction Challenges Final Remarks Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 2/76 ENFMC
  4. 4. Problem HK-KS xc LDA Construction Challenges Final Remarks Dirac (1929) “The general theory of quantum mechanics is now almost complete (...) The underlying physi- cal laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. (...) It therefore becomes desirable that approxi- mate practical methods of applying quantum me- chanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation.” Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 3/76 ENFMC
  5. 5. Problem HK-KS xc LDA Construction Challenges Final Remarks The electronic structure problem Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 4/76 ENFMC
  6. 6. Problem HK-KS xc LDA Construction Challenges Final Remarks The electronic structure problem Quantum many-body problem of N interacting electrons: Ψel(r1, r2, ..., rN ) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 4/76 ENFMC
  7. 7. Problem HK-KS xc LDA Construction Challenges Final Remarks The electronic structure problem Quantum many-body problem of N interacting electrons: Ψel(r1, r2, ..., rN ) Paradigms: atom / electron gas Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 4/76 ENFMC
  8. 8. Problem HK-KS xc LDA Construction Challenges Final Remarks The electronic structure problem Quantum many-body problem of N interacting electrons: Ψel(r1, r2, ..., rN ) Paradigms: atom / electron gas Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 4/76 ENFMC
  9. 9. Problem HK-KS xc LDA Construction Challenges Final Remarks The electronic structure problem Quantum many-body problem of N interacting electrons: Ψel(r1, r2, ..., rN ) Paradigms: atom / electron gas Methods based on the wavefunction (Hartree-Fock, CI, Coupled Cluster, MP2, QMC) Methods based on the Green’s function, reduced density matrix, density (density functional theory) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 4/76 ENFMC
  10. 10. Problem HK-KS xc LDA Construction Challenges Final Remarks Hartree’s method Single-particle Schr¨odinger equation − 2 2m 2 + vext(r) + vH (r) ϕi(r) = iϕi(r) , Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 5/76 ENFMC
  11. 11. Problem HK-KS xc LDA Construction Challenges Final Remarks Hartree’s method Single-particle Schr¨odinger equation − 2 2m 2 + vext(r) + vH (r) ϕi(r) = iϕi(r) , Mean field potential vH (r) = e2 d3 r n(r ) |r − r | Hartree energy UH [n] = ΨH | ˆU|ΨH = e2 2 d3 r d3 r n(r)n(r ) |r − r | Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 5/76 ENFMC
  12. 12. Problem HK-KS xc LDA Construction Challenges Final Remarks Hartree’s method Single-particle Schr¨odinger equation − 2 2m 2 + vext(r) + vH (r) ϕi(r) = iϕi(r) , Mean field potential vH (r) = e2 d3 r n(r ) |r − r | Hartree energy UH [n] = ΨH | ˆU|ΨH = e2 2 d3 r d3 r n(r)n(r ) |r − r | . Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 5/76 ENFMC
  13. 13. Problem HK-KS xc LDA Construction Challenges Final Remarks Hartree-Fock Antisymmetrization in a Slater determinant ΨHF (r) = 1 √ N! ϕ1(x1) ϕ1(x2) · · · ϕ1(xN ) ϕ2(x1) ϕ2(x2) · · · ϕ2(xN ) ... ... ... ... ϕN (x1) ϕN (x2) · · · ϕN (xN ) Fock exchange energy (indirect) Ex = ΨHF | ˆU|ΨHF = − e2 2 i,j,σ dr dr ϕ∗ iσ(r)ϕ∗ jσ(r )ϕiσ(r )ϕjσ(r) |r − r | Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 6/76 ENFMC
  14. 14. Problem HK-KS xc LDA Construction Challenges Final Remarks Thomas-Fermi model Use the infinite gas of non-interacting electrons with a uniform density n = n(r) to evaluate the kinetic energy of atoms, molecules Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 7/76 ENFMC
  15. 15. Problem HK-KS xc LDA Construction Challenges Final Remarks Thomas-Fermi model Use the infinite gas of non-interacting electrons with a uniform density n = n(r) to evaluate the kinetic energy of atoms, molecules TTF [n] = tgas(n(r))n(r)d3 r Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 7/76 ENFMC
  16. 16. Problem HK-KS xc LDA Construction Challenges Final Remarks Our tutorial Introduction to density-functional theory Context and key concepts (1927-1930) (Born-Oppenheimer, Hartree, Hartree-Fock, Thomas-Fermi) Fundamentals (1964-1965) (Hohenberg-Kohn theorem, Kohn-Sham scheme) Approximations (≈ 1980-2010) (local density and generalized gradient approximations (LDA and GGA), construction of functionals) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 7/76 ENFMC
  17. 17. Problem HK-KS xc LDA Construction Challenges Final Remarks Back to our question Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 7/76 ENFMC
  18. 18. Problem HK-KS xc LDA Construction Challenges Final Remarks Back to our question a program ? a method? some obscure theory? Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 6/76 ENFMC
  19. 19. Problem HK-KS xc LDA Construction Challenges Final Remarks Density functional theory (DFT) Quantum theory based on the density n(r) wave functions Ψ(r1, r2, ...rN ) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 7/76 ENFMC
  20. 20. Problem HK-KS xc LDA Construction Challenges Final Remarks Density functional theory (DFT) Quantum theory based on the density n(r) wave functions Ψ(r1, r2, ...rN ) Single-particle Kohn-Sham equations Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 7/76 ENFMC
  21. 21. Problem HK-KS xc LDA Construction Challenges Final Remarks Density functional theory (DFT) Quantum theory based on the density n(r) wave functions Ψ(r1, r2, ...rN ) Single-particle Kohn-Sham equations Electronic structure boom: Nobel Prize to W.Kohn/J.Pople Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 7/76 ENFMC
  22. 22. Problem HK-KS xc LDA Construction Challenges Final Remarks Density functional theory (DFT) Quantum theory based on the density n(r) wave functions Ψ(r1, r2, ...rN ) Single-particle Kohn-Sham equations Electronic structure boom: Nobel Prize to W.Kohn/J.Pople Hohenberg-Kohn theorem: Ψ(r1, r2, ..., rN ) ⇔ n(r) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 7/76 ENFMC
  23. 23. Problem HK-KS xc LDA Construction Challenges Final Remarks Density functional theory (DFT) Quantum theory based on the density n(r) wave functions Ψ(r1, r2, ...rN ) Single-particle Kohn-Sham equations Electronic structure boom: Nobel Prize to W.Kohn/J.Pople Hohenberg-Kohn theorem: Ψ(r1, r2, ..., rN ) ⇔ n(r) Which means, Ψ(r) = Ψ[n(r)] observables = observables[n(r)] Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 7/76 ENFMC
  24. 24. Problem HK-KS xc LDA Construction Challenges Final Remarks Hohenberg-Kohn (1964) Phys. Rev. 136 B864 (1964). Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 8/76 ENFMC
  25. 25. Problem HK-KS xc LDA Construction Challenges Final Remarks After THK From the ground-state density it is possible, in principle, to calculate the corresponding wave functions and all its observables. However: the Hohenberg-Kohn theorem does not provide any means to actually calculate them. We have DFT in theory, now, in practice?... Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 9/76 ENFMC
  26. 26. Problem HK-KS xc LDA Construction Challenges Final Remarks After THK From the ground-state density it is possible, in principle, to calculate the corresponding wave functions and all its observables. However: the Hohenberg-Kohn theorem does not provide any means to actually calculate them. We have DFT in theory, now, in practice?... Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 9/76 ENFMC
  27. 27. Problem HK-KS xc LDA Construction Challenges Final Remarks After THK arXiv:1403.5164 “By the late fall of 1964, Kohn was thinking about alternative ways to transform the theory he and Hohenberg had developed into a practical scheme for atomic, molecular, and solid state calculations. Happily, he was very well acquainted with an approximate approach to the many-electron problem that was notably superior to the Thomas-Fermi method, at least for the case of atoms. This was a theory proposed by Douglas Hartree in 1923 which exploited the then just-published Schr¨odinger equation in a heuristic way to calculate the orbital wave functions φk(r), the electron binding energies k, and the charge density n(r) of an N-electron atom. Hartree’s theory transcended Thomas-Fermi theory primarily by its use of the exact quantum-mechanical expression for the kinetic energy of independent electrons.” Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 10/76 ENFMC
  28. 28. Problem HK-KS xc LDA Construction Challenges Final Remarks After THK Kohn believed the Hartree equations could be an example of the HK variational principle. He knew the self-consistent scheme and that it could give an approximate density So he suggested to his new post-doc, Lu Sham, that he try to derive the Hartree equations from the Hohenberg-Kohn formalism. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 11/76 ENFMC
  29. 29. Problem HK-KS xc LDA Construction Challenges Final Remarks After THK Kohn believed the Hartree equations could be an example of the HK variational principle. He knew the self-consistent scheme and that it could give an approximate density So he suggested to his new post-doc, Lu Sham, that he try to derive the Hartree equations from the Hohenberg-Kohn formalism. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 11/76 ENFMC
  30. 30. Problem HK-KS xc LDA Construction Challenges Final Remarks After THK Kohn believed the Hartree equations could be an example of the HK variational principle. He knew the self-consistent scheme and that it could give an approximate density So he suggested to his new post-doc, Lu Sham, that he try to derive the Hartree equations from the Hohenberg-Kohn formalism. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 11/76 ENFMC
  31. 31. Problem HK-KS xc LDA Construction Challenges Final Remarks Kohn-Sham approach/scheme Auxiliary non-interacting system Single-particle equations − 2 2 2m + vKS (r) ϕk(r) = kϕk(r) Effective potential vKS (r) = vext(r) + vH (r) + vxc(r) Formally: constraint on the ground-state density Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 12/76 ENFMC
  32. 32. Problem HK-KS xc LDA Construction Challenges Final Remarks Kohn-Sham kindergarden Interacting (complicated) Ficticious non-interacting under effective field Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 13/76 ENFMC
  33. 33. Problem HK-KS xc LDA Construction Challenges Final Remarks Outline 1 Review of our problem 2 Review of HK-KS 3 Exchange-correlation 4 LDA and GGA 5 Construction 6 Challenges 7 Final Remarks Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 13/76 ENFMC
  34. 34. Problem HK-KS xc LDA Construction Challenges Final Remarks Exchange-correlation Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 13/76 ENFMC
  35. 35. Problem HK-KS xc LDA Construction Challenges Final Remarks Exchange-correlation arXiv:1403.5164 “As trained solid-state physicists, Hohenberg and Kohn knew that the entire history of research on the quantum mechanical many-electron problem could be interpreted as attempts to identify and quantify the physical effects described by this universal density functional.” For example, many years of approximate quantum mechanical calculations for atoms and molecules had established that the phenomenon of exchange - a consequence of the Pauli exclusion principle - contributes significantly to the potential energy part of U[n]. Exchange reduces the Coulomb potential energy of the system by tending to keep electrons with parallel spin spatially separated.”. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 14/76 ENFMC
  36. 36. Problem HK-KS xc LDA Construction Challenges Final Remarks Universal functional Energy functional: Kinetic + Coulomb + External E[n] = T[n] + U[n] + V [n] We can define a universal F[n] F[n] = T[n] + U[n] which is the same independent of your system. Our task is approximate U[n], the many-particle problem. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 15/76 ENFMC
  37. 37. Problem HK-KS xc LDA Construction Challenges Final Remarks Exchange-correlation arXiv:1403.5164 “As trained solid-state physicists, Hohenberg and Kohn knew that the entire history of research on the quantum mechanical many-electron problem could be interpreted as attempts to identify and quantify the physical effects described by this universal density functional. For example, many years of approximate quantum mechanical calculations for atoms and molecules had established that the phenomenon of exchange - a consequence of the Pauli exclusion principle - contributes significantly to the potential energy part of U[n].Exchange reduces the Coulomb potential energy of the system by tending to keep electrons with parallel spin spatially separated.” Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 16/76 ENFMC
  38. 38. Problem HK-KS xc LDA Construction Challenges Final Remarks Exchange-correlation arXiv:1403.5164 “As trained solid-state physicists, Hohenberg and Kohn knew that the entire history of research on the quantum mechanical many-electron problem could be interpreted as attempts to identify and quantify the physical effects described by this universal density functional. For example, many years of approximate quantum mechanical calculations for atoms and molecules had established that the phenomenon of exchange - a consequence of the Pauli exclusion principle - contributes significantly to the potential energy part of U[n]. Exchange reduces the Coulomb potential energy of the system by tending to keep electrons with parallel spin spatially separated.” Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 16/76 ENFMC
  39. 39. Problem HK-KS xc LDA Construction Challenges Final Remarks Coulomb energy Coulomb energy U[n] = UH [n] + Ex[n] + where UH [n] = e2 2 d3 r d3 r n(r)n(r ) |r − r | . is the electrostatic, mean field repulsion, and Ex[ϕ[n]] = − e2 2 i,j,σ d3 r d3 r ϕ∗ iσ(r)ϕ∗ jσ(r )ϕiσ(r )ϕjσ(r) |r − r | is the exchange energy due to the Pauli principle. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 17/76 ENFMC
  40. 40. Problem HK-KS xc LDA Construction Challenges Final Remarks Coulomb energy Coulomb energy U[n] = UH [n] + Ex[n] + where UH [n] = e2 2 d3 r d3 r n(r)n(r ) |r − r | . is the electrostatic, mean field repulsion, and Ex[ϕ[n]] = − e2 2 i,j,σ d3 r d3 r ϕ∗ iσ(r)ϕ∗ jσ(r )ϕiσ(r )ϕjσ(r) |r − r | is the exchange energy due to the Pauli principle. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 17/76 ENFMC
  41. 41. Problem HK-KS xc LDA Construction Challenges Final Remarks On correlation arXiv:1403.5164 Coulomb energy U[n] = UH [n] + Ex[n] + “The remaining potential energy part of U[n] takes account of short-range correlation effects. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 18/76 ENFMC
  42. 42. Problem HK-KS xc LDA Construction Challenges Final Remarks On correlation arXiv:1403.5164 Coulomb energy U[n] = UH [n] + Ex[n] + “The remaining potential energy part of U[n] takes account of short-range correlation effects. Correlation also reduces the Coulomb potential energy by tending to keep all pairs of electrons spatially separated.” Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 18/76 ENFMC
  43. 43. Problem HK-KS xc LDA Construction Challenges Final Remarks On correlation arXiv:1403.5164 Coulomb energy U[n] = UH [n] + Ex[n] + Ec[n] “The remaining potential energy part of U[n] takes account of short-range correlation effects. Correlation also reduces the Coulomb potential energy by tending to keep all pairs of electrons spatially separated.” Correlation energy: Ec < 0 Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 18/76 ENFMC
  44. 44. Problem HK-KS xc LDA Construction Challenges Final Remarks On correlation arXiv:1403.5164 Coulomb energy U[n] = UH [n] + Ex[n] + Ec[n] “Note for future reference that the venerable Hartree-Fock approximation takes account of the kinetic energy and the exchange energy exactly but (by definition) takes no account of the correlation energy”. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 19/76 ENFMC
  45. 45. Problem HK-KS xc LDA Construction Challenges Final Remarks On correlation arXiv:1403.5164 Coulomb energy U[n] = UH [n] + Ex[n] + Ec[n] “Note for future reference that the venerable Hartree-Fock approximation takes account of the kinetic energy and the exchange energy exactly but (by definition) takes no account of the correlation energy”. Hartree-Fock energy EHF [n] = Ts[ϕ[n]] + V [n] + UH [n] + Ex[ϕ[n]] Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 19/76 ENFMC
  46. 46. Problem HK-KS xc LDA Construction Challenges Final Remarks Exchange-correlation in DFT Kohn-Sham effective potential: vKS (r) = vext(r) + vH (r) + vxc(r) Our task is to find vxc, preferrably as a functional of the density. Orbital functionals bring non-locality (integrals over r and r ). So, in the Kohn-Sham DFT, we recast the many-particle problem in finding xc potentials. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 20/76 ENFMC
  47. 47. Problem HK-KS xc LDA Construction Challenges Final Remarks Exchange-correlation in DFT Total energy E[n] = T[n] + V [n] + U[n] = Ts[ϕi[n]] + V [n] + UH [n] + Exc[n] Some approximations: single-particle kinetic and Hartree. Leave the corrections (T − Ts and U − UH ) to the Exc. Ts[ϕi[n]] = − 2 2m N i d3 rϕ∗ i (r) 2 ϕi(r) UH [n] = e2 2 d3 r d3 r n(r)n(r ) | r − r | Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 21/76 ENFMC
  48. 48. Problem HK-KS xc LDA Construction Challenges Final Remarks Exchange-correlation energy The exchange-correlation energy Exc is the new clothing of the many-body problem Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 22/76 ENFMC
  49. 49. Problem HK-KS xc LDA Construction Challenges Final Remarks Exchange-correlation energy The exchange-correlation energy Exc is the new clothing of the many-body problem exchange: Pauli principle correlation: kinetic and Coulombic contributions beyond single-particle (one Slater determinant) xc = “nature’s glue” that binds matter together (Exc < 0) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 22/76 ENFMC
  50. 50. Problem HK-KS xc LDA Construction Challenges Final Remarks Exchange-correlation energy “Electrons moving through the density swerve to avoid one another, like shoppers in a mall.” “The resulting reduction of the potential energy of mutual Coulomb repulsion is the main contribution to the negative exchange-correlation energy. The swerving motion also makes a small positive kinetic energy contribution to the correlation energy” J.Perdew et al. in J. Chem. Theory Comput. 5, 902 (2009). Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 23/76 ENFMC
  51. 51. Problem HK-KS xc LDA Construction Challenges Final Remarks Exchange-correlation energy In Kohn-Sham DFT, the exchange-correlation energy Exc[n] holds the main difficulty of the many-body problem. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 24/76 ENFMC
  52. 52. Problem HK-KS xc LDA Construction Challenges Final Remarks Exchange-correlation energy In Kohn-Sham DFT, the exchange-correlation energy Exc[n] holds the main difficulty of the many-body problem. Now, how to construct an approximate Exc[n]? Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 24/76 ENFMC
  53. 53. Problem HK-KS xc LDA Construction Challenges Final Remarks Outline 1 Review of our problem 2 Review of HK-KS 3 Exchange-correlation 4 LDA and GGA 5 Construction 6 Challenges 7 Final Remarks Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 24/76 ENFMC
  54. 54. Problem HK-KS xc LDA Construction Challenges Final Remarks State-of-the-art Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 25/76 ENFMC
  55. 55. Problem HK-KS xc LDA Construction Challenges Final Remarks Back in 65 Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 26/76 ENFMC
  56. 56. Problem HK-KS xc LDA Construction Challenges Final Remarks Back in 65 Introduce KS equations Explore possible Exc Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 27/76 ENFMC
  57. 57. Problem HK-KS xc LDA Construction Challenges Final Remarks Density functional Starting point: electron gas Exc = d3 rexc[n]n(r) (exc: energy density per particle) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 28/76 ENFMC
  58. 58. Problem HK-KS xc LDA Construction Challenges Final Remarks Thomas-Fermi-Dirac spirit Using the paradigm of an uniform, homogeneous system to help with inhomogeneous problems E ≈ ETFD [n] = TLDA s [n] + UH [n] + ELDA x + V [n] . Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 29/76 ENFMC
  59. 59. Problem HK-KS xc LDA Construction Challenges Final Remarks Local density approximation (LDA) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 30/76 ENFMC
  60. 60. Problem HK-KS xc LDA Construction Challenges Final Remarks Local density approximation (LDA) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 30/76 ENFMC
  61. 61. Problem HK-KS xc LDA Construction Challenges Final Remarks Local density approximation (LDA) ELDA xc [n] = d3 r ehom xc (n(r)) ehom xc (n) = ehom x (n) + ehom c (n) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 30/76 ENFMC
  62. 62. Problem HK-KS xc LDA Construction Challenges Final Remarks Local density approximation (LDA) ELDA xc [n] = d3 r ehom xc (n(r)) ehom xc (n) = ehom x (n) + ehom c (n) For the homogeneous electron gas, we have the expression of the Dirac exchange energy ehom x (n) = − 3 4 3 π 1/3 n4/3 , Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 30/76 ENFMC
  63. 63. Problem HK-KS xc LDA Construction Challenges Final Remarks Local density approximation (LDA) ELDA xc [n] = d3 r ehom xc (n(r)) ehom xc (n) = ehom x (n) + ehom c (n) For the homogeneous electron gas, we have the expression of the Dirac exchange energy ehom x (n) = − 3 4 3 π 1/3 n4/3 , For ehom c ? Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 30/76 ENFMC
  64. 64. Problem HK-KS xc LDA Construction Challenges Final Remarks Local density approximation (LDA) ELDA xc [n] = d3 r ehom xc (n(r)) ehom xc (n) = ehom x (n) + ehom c (n) For the homogeneous electron gas, we have the expression of the Dirac exchange energy ehom x (n) = − 3 4 3 π 1/3 n4/3 , For ehom c ? Monte Carlo Quˆantico → parametrizations Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 30/76 ENFMC
  65. 65. Problem HK-KS xc LDA Construction Challenges Final Remarks Local density approximation (LDA) ELDA xc [n] = d3 r ehom xc (n(r)) ehom xc (n) = ehom x (n) + ehom c (n) For the homogeneous electron gas, we have the expression of the Dirac exchange energy ehom x (n) = − 3 4 3 π 1/3 n4/3 , For ehom c ? Monte Carlo Quˆantico → parametrizations ePW92 c = −2c0(1+α1rs)ln 1 + 1 2c1(β1r 1/2 s + β2rs + β3r 3/2 s + β4r2 s ) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 30/76 ENFMC
  66. 66. Problem HK-KS xc LDA Construction Challenges Final Remarks Parametrizations of the correlation energy E.g.: low-density limit of the electron gas ec(rs) = −e2 d0 rs + d1 r 3/2 s + d2 r4 s + ... rs → ∞ , Wigner’s parametrization (1934): eW c (rs) = − 0.44e2 7.8 + rs . W (Wigner-1934) BR (Brual Rothstein-1978) vBH (von Barth e Hedin-1972) GL (Gunnarson e Lundqvist-1976) VWN (Vosko, Wilk e Nusair-1980) PZ (Perdew e Zunger-1981) PW92 (Perdew e Wang-1992) EHTY (Endo et al-1999) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 31/76 ENFMC
  67. 67. Problem HK-KS xc LDA Construction Challenges Final Remarks Next step: Inhomogeneities, gradient of the density Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 32/76 ENFMC
  68. 68. Problem HK-KS xc LDA Construction Challenges Final Remarks Gradient expansion approximation (GEA) Systematic corrections to LDA for slowly varying densities Inhomogeneities captured by “reduced density gradients” Ex[n] = Ax d3 r n4/3 [1+µs2 +...] Ec[n] = d3 r n[ec(n)+β(n)t2 +...] where s = | n| 2kF n e t = | n| 2ksn Truncated expansion leads to violation of sum rules For atoms, exchange improves over LDA, but not correlation (gets even positive) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 33/76 ENFMC
  69. 69. Problem HK-KS xc LDA Construction Challenges Final Remarks Generalized gradient approximation (GGA) GEA successor; widened the applications of DFT in quantum chemistry EGGA xc [n] = d3 r f (n(r), n(r)) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 34/76 ENFMC
  70. 70. Problem HK-KS xc LDA Construction Challenges Final Remarks Generalized gradient approximation (GGA) GEA successor; widened the applications of DFT in quantum chemistry EGGA xc [n] = d3 r f (n(r), n(r)) Ma e Brueckner (1968): first GGA, empirical parameter corrects positive correlation energies Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 34/76 ENFMC
  71. 71. Problem HK-KS xc LDA Construction Challenges Final Remarks Generalized gradient approximation (GGA) GEA successor; widened the applications of DFT in quantum chemistry EGGA xc [n] = d3 r f (n(r), n(r)) Ma e Brueckner (1968): first GGA, empirical parameter corrects positive correlation energies Langreth e Mehl (1983): random-phase approximation helps corrections; correlation cutoff; semiempirical Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 34/76 ENFMC
  72. 72. Problem HK-KS xc LDA Construction Challenges Final Remarks Generalized gradient approximation (GGA) GEA successor; widened the applications of DFT in quantum chemistry EGGA xc [n] = d3 r f (n(r), n(r)) Ma e Brueckner (1968): first GGA, empirical parameter corrects positive correlation energies Langreth e Mehl (1983): random-phase approximation helps corrections; correlation cutoff; semiempirical Perdew e Wang (PW86): LM83 extended without empiricism, lower exchange errors of LDA to 1-10% Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 34/76 ENFMC
  73. 73. Problem HK-KS xc LDA Construction Challenges Final Remarks Generalized gradient approximation (GGA) GEA successor; widened the applications of DFT in quantum chemistry EGGA xc [n] = d3 r f (n(r), n(r)) Ma e Brueckner (1968): first GGA, empirical parameter corrects positive correlation energies Langreth e Mehl (1983): random-phase approximation helps corrections; correlation cutoff; semiempirical Perdew e Wang (PW86): LM83 extended without empiricism, lower exchange errors of LDA to 1-10% Becke (B88): correct assintotic behavior of exchange energy; fitted parameter from atomic energies Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 34/76 ENFMC
  74. 74. Problem HK-KS xc LDA Construction Challenges Final Remarks Generalized gradient approximation (GGA) GEA successor; widened the applications of DFT in quantum chemistry EGGA xc [n] = d3 r f (n(r), n(r)) Ma e Brueckner (1968): first GGA, empirical parameter corrects positive correlation energies Langreth e Mehl (1983): random-phase approximation helps corrections; correlation cutoff; semiempirical Perdew e Wang (PW86): LM83 extended without empiricism, lower exchange errors of LDA to 1-10% Becke (B88): correct assintotic behavior of exchange energy; fitted parameter from atomic energies PW91: same Becke’s Fxc idea, impose correlation cutoff, and a good parametrization of correlation (PW92). Attempts to obey as many universal constraints as possible. No empirical parameters. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 34/76 ENFMC
  75. 75. Problem HK-KS xc LDA Construction Challenges Final Remarks Generalized gradient approximation (GGA) GEA successor; widened the applications of DFT in quantum chemistry EGGA xc [n] = d3 r f (n(r), n(r)) Ma e Brueckner (1968): first GGA, empirical parameter corrects positive correlation energies Langreth e Mehl (1983): random-phase approximation helps corrections; correlation cutoff; semiempirical Perdew e Wang (PW86): LM83 extended without empiricism, lower exchange errors of LDA to 1-10% Becke (B88): correct assintotic behavior of exchange energy; fitted parameter from atomic energies PW91: same Becke’s Fxc idea, impose correlation cutoff, and a good parametrization of correlation (PW92). Attempts to obey as many universal constraints as possible. No empirical parameters. PBE GGA was announced as “GGA made simple”, PW91 substitute Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 34/76 ENFMC
  76. 76. Problem HK-KS xc LDA Construction Challenges Final Remarks State-of-the-art Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 35/76 ENFMC
  77. 77. Problem HK-KS xc LDA Construction Challenges Final Remarks Perdew-Burke-Ernzerhof GGA (1996) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 36/76 ENFMC
  78. 78. Problem HK-KS xc LDA Construction Challenges Final Remarks Visualizing GGAs non-locality Enhancement factor Fxc: EGGA xc [n] ≈ d3 r n Fxc(rs, ζ, s) ex(rs, ζ = 0) Captures the effects of correlation (through rs) spin polarization (ζ) density inhomogeneity (through the reduced density gradient s). Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 37/76 ENFMC
  79. 79. Problem HK-KS xc LDA Construction Challenges Final Remarks Example: PBE exchange FPBE x (s) = 1 + κ − κ 1 + µ κ s2 , µ = π2 βGE /3, so that there will be a cancellation of the exchange and correlation gradients, and the jellium result is recovered. βGE comes from the second-order gradient expansion in the limit of slowly-varying densities κ is fixed by the Lieb-Oxford bound s is the “reduced density gradient” s = | n| 2(3π2)1/3n4/3 = | n| 2kF n , which corresponds to a inhomogeneity parameter, measuring how fast the density changes in the scale of the Fermi wavelength 2π/kF . Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 38/76 ENFMC
  80. 80. Problem HK-KS xc LDA Construction Challenges Final Remarks Exchange enhancement factors Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 39/76 ENFMC
  81. 81. Problem HK-KS xc LDA Construction Challenges Final Remarks PBE: “GGA made simple” Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 40/76 ENFMC
  82. 82. Problem HK-KS xc LDA Construction Challenges Final Remarks Outline 1 Review of our problem 2 Review of HK-KS 3 Exchange-correlation 4 LDA and GGA 5 Construction 6 Challenges 7 Final Remarks Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 40/76 ENFMC
  83. 83. Problem HK-KS xc LDA Construction Challenges Final Remarks Two construction approaches Fitting empirical parameters E.g.: B3LYP (A. Becke on the right) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 41/76 ENFMC
  84. 84. Problem HK-KS xc LDA Construction Challenges Final Remarks Two construction approaches Fitting empirical parameters E.g.: B3LYP (A. Becke on the right) Inserting exact constraints (↔ J. Perdew) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 41/76 ENFMC
  85. 85. Problem HK-KS xc LDA Construction Challenges Final Remarks Two construction approaches Fitting empirical parameters E.g.: B3LYP (A. Becke on the right) Inserting exact constraints (↔ J. Perdew) n = uniform → LDA n ≈ uniform → LDA + O( ) = GEA Ex < 0, Ec 0 Uniform density scaling Spin scaling One-electron limit Derivative discontinuity Lower bounds Ex.: PW86, PW91, PBE Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 41/76 ENFMC
  86. 86. Problem HK-KS xc LDA Construction Challenges Final Remarks Constraint satisfaction Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 42/76 ENFMC
  87. 87. Problem HK-KS xc LDA Construction Challenges Final Remarks Constraint satisfaction Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 43/76 ENFMC
  88. 88. Problem HK-KS xc LDA Construction Challenges Final Remarks State-of-the-art Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 44/76 ENFMC
  89. 89. Problem HK-KS xc LDA Construction Challenges Final Remarks State-of-the-art Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 44/76 ENFMC
  90. 90. Problem HK-KS xc LDA Construction Challenges Final Remarks Beyond LDA and GGA Meta-GGA: + non-interacting kinetic energy density τ. Ex: TPSS, PKZB EMGGA xc [n] = d3 rf (n(r), n(r), τ[n]) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 45/76 ENFMC
  91. 91. Problem HK-KS xc LDA Construction Challenges Final Remarks Beyond LDA and GGA Meta-GGA: + non-interacting kinetic energy density τ. Ex: TPSS, PKZB EMGGA xc [n] = d3 rf (n(r), n(r), τ[n]) Hiper-GGA: + exact exchange energy density ex EHGGA xc [n] = d3 rf (n(r), n(r), τ[n], ex[n]) , Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 45/76 ENFMC
  92. 92. Problem HK-KS xc LDA Construction Challenges Final Remarks Beyond LDA and GGA Meta-GGA: + non-interacting kinetic energy density τ. Ex: TPSS, PKZB EMGGA xc [n] = d3 rf (n(r), n(r), τ[n]) Hiper-GGA: + exact exchange energy density ex EHGGA xc [n] = d3 rf (n(r), n(r), τ[n], ex[n]) , Hybrids: mix of exact exchange Ex with ELDA x and Eaprox c . Ex: B3LYP Ehib xc [n] = aEexact x + (1 − a)ELDA x [n] + Eaprox c Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 45/76 ENFMC
  93. 93. Problem HK-KS xc LDA Construction Challenges Final Remarks Beyond LDA and GGA functionals Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 46/76 ENFMC
  94. 94. Problem HK-KS xc LDA Construction Challenges Final Remarks Systematic improvement? Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 47/76 ENFMC
  95. 95. Problem HK-KS xc LDA Construction Challenges Final Remarks Systematic improvement? Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 47/76 ENFMC
  96. 96. Problem HK-KS xc LDA Construction Challenges Final Remarks Systematic trends? Consider Localized vs extended densities; covalent and ionic bonds Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 48/76 ENFMC
  97. 97. Problem HK-KS xc LDA Construction Challenges Final Remarks Systematic trends? Consider Localized vs extended densities; covalent and ionic bonds Systematic trends between LDA e PBE; between GGAs e hybrids Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 48/76 ENFMC
  98. 98. Problem HK-KS xc LDA Construction Challenges Final Remarks Systematic trends? Consider Localized vs extended densities; covalent and ionic bonds Systematic trends between LDA e PBE; between GGAs e hybrids Example: lattice constants Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 48/76 ENFMC
  99. 99. Problem HK-KS xc LDA Construction Challenges Final Remarks Systematic trends? Consider Localized vs extended densities; covalent and ionic bonds Systematic trends between LDA e PBE; between GGAs e hybrids Example: lattice constants Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 48/76 ENFMC
  100. 100. Problem HK-KS xc LDA Construction Challenges Final Remarks Systematic trends? Consider Localized vs extended densities; covalent and ionic bonds Systematic trends between LDA e PBE; between GGAs e hybrids Example: lattice constants Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 48/76 ENFMC
  101. 101. Problem HK-KS xc LDA Construction Challenges Final Remarks Outline 1 Review of our problem 2 Review of HK-KS 3 Exchange-correlation 4 LDA and GGA 5 Construction 6 Challenges 7 Final Remarks Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 48/76 ENFMC
  102. 102. Problem HK-KS xc LDA Construction Challenges Final Remarks DFT downsides Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 49/76 ENFMC
  103. 103. Problem HK-KS xc LDA Construction Challenges Final Remarks DFT downsides DFT is variational, not perturbative: no systematic improvement Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 49/76 ENFMC
  104. 104. Problem HK-KS xc LDA Construction Challenges Final Remarks DFT downsides DFT is variational, not perturbative: no systematic improvement Kohn-Sham quantities lack physical meaning Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 49/76 ENFMC
  105. 105. Problem HK-KS xc LDA Construction Challenges Final Remarks DFT downsides DFT is variational, not perturbative: no systematic improvement Kohn-Sham quantities lack physical meaning In principle, everything can be extracted from the density; however, there is no prescription for building the HK or xc density functional Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 49/76 ENFMC
  106. 106. Problem HK-KS xc LDA Construction Challenges Final Remarks DFA downsides (density-functional approximations) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 50/76 ENFMC
  107. 107. Problem HK-KS xc LDA Construction Challenges Final Remarks DFA downsides (density-functional approximations) No prescription for building the xc density functional Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 50/76 ENFMC
  108. 108. Problem HK-KS xc LDA Construction Challenges Final Remarks DFA downsides (density-functional approximations) No prescription for building the xc density functional Combining exact constraints: arbitrary forms Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 50/76 ENFMC
  109. 109. Problem HK-KS xc LDA Construction Challenges Final Remarks DFA downsides (density-functional approximations) No prescription for building the xc density functional Combining exact constraints: arbitrary forms Single-particle and electron gas paradigm may not be enough Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 50/76 ENFMC
  110. 110. Problem HK-KS xc LDA Construction Challenges Final Remarks DFA downsides (density-functional approximations) No prescription for building the xc density functional Combining exact constraints: arbitrary forms Single-particle and electron gas paradigm may not be enough Often we miss the condensed-matter richness: strong correlations, excitations, dispersion forces, relativistic effects Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 50/76 ENFMC
  111. 111. Problem HK-KS xc LDA Construction Challenges Final Remarks What typical functionals miss Strong correlations Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 51/76 ENFMC
  112. 112. Problem HK-KS xc LDA Construction Challenges Final Remarks What typical functionals miss Strong correlations Dispersion forces Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 51/76 ENFMC
  113. 113. Problem HK-KS xc LDA Construction Challenges Final Remarks What typical functionals miss Strong correlations Dispersion forces Band gaps Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 51/76 ENFMC
  114. 114. Problem HK-KS xc LDA Construction Challenges Final Remarks What typical functionals miss Strong correlations Dispersion forces Band gaps Charge-transfer Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 51/76 ENFMC
  115. 115. Problem HK-KS xc LDA Construction Challenges Final Remarks What is wrong in our approximations? Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 52/76 ENFMC
  116. 116. Problem HK-KS xc LDA Construction Challenges Final Remarks There are different problems that arise in common density functional approximations. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 53/76 ENFMC
  117. 117. Problem HK-KS xc LDA Construction Challenges Final Remarks There are different problems that arise in common density functional approximations. I will quickly comment two of them. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 53/76 ENFMC
  118. 118. Problem HK-KS xc LDA Construction Challenges Final Remarks There are different problems that arise in common density functional approximations. I will quickly comment two of them. Self-interaction error and delocalization error Derivative discontinuity Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 53/76 ENFMC
  119. 119. Problem HK-KS xc LDA Construction Challenges Final Remarks Self-interaction error Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 54/76 ENFMC
  120. 120. Problem HK-KS xc LDA Construction Challenges Final Remarks Self-interaction error Take your functional and evaluate it for a one-electron density. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 54/76 ENFMC
  121. 121. Problem HK-KS xc LDA Construction Challenges Final Remarks Self-interaction error Take your functional and evaluate it for a one-electron density. In principle, if you have one electron, there is no Coulomb interaction and you should have U[n(1) ] = 0 Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 54/76 ENFMC
  122. 122. Problem HK-KS xc LDA Construction Challenges Final Remarks Self-interaction error Take your functional and evaluate it for a one-electron density. In principle, if you have one electron, there is no Coulomb interaction and you should have U[n(1) ] = 0 this means that UH [n(1) ] + Ex[n(1) ] + Ec[n(1) ] = 0 Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 54/76 ENFMC
  123. 123. Problem HK-KS xc LDA Construction Challenges Final Remarks Self-interaction error Take your functional and evaluate it for a one-electron density. In principle, if you have one electron, there is no Coulomb interaction and you should have U[n(1) ] = 0 this means that UH [n(1) ] + Ex[n(1) ] + Ec[n(1) ] = 0 However, many common functionals have a spurious error, called self-interaction, leaving a small amount of extra charge. This is a problem that affects strongly correlated systems. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 54/76 ENFMC
  124. 124. Problem HK-KS xc LDA Construction Challenges Final Remarks Delocalization error Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 55/76 ENFMC
  125. 125. Problem HK-KS xc LDA Construction Challenges Final Remarks Delocalization error Consider a system of N electrons. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 55/76 ENFMC
  126. 126. Problem HK-KS xc LDA Construction Challenges Final Remarks Delocalization error Consider a system of N electrons. If I add or remove one electron, it was proved [Perdew et al 1982] that the total energy behaves linearly with N: Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 55/76 ENFMC
  127. 127. Problem HK-KS xc LDA Construction Challenges Final Remarks Delocalization error Consider a system of N electrons. If I add or remove one electron, it was proved [Perdew et al 1982] that the total energy behaves linearly with N: Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 55/76 ENFMC
  128. 128. Problem HK-KS xc LDA Construction Challenges Final Remarks Delocalization error Consider a system of N electrons. If I add or remove one electron, it was proved [Perdew et al 1982] that the total energy behaves linearly with N: However, common density functionals behave concavely, sometimes favoring fractional configurations. This affects problems of charge transfer in molecules or electronic transport. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 55/76 ENFMC
  129. 129. Problem HK-KS xc LDA Construction Challenges Final Remarks There are several illnesses that arise from the KS picture and common density functional approximations. I will quickly comment two of them. Self-interaction error and delocalization error Derivative discontinuity Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 56/76 ENFMC
  130. 130. Problem HK-KS xc LDA Construction Challenges Final Remarks Derivative discontinuity (I) As we observed, the derivative of energy changes discontinuosly when we change the particle number: Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 57/76 ENFMC
  131. 131. Problem HK-KS xc LDA Construction Challenges Final Remarks Derivative discontinuity (I) As we observed, the derivative of energy changes discontinuosly when we change the particle number: Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 57/76 ENFMC
  132. 132. Problem HK-KS xc LDA Construction Challenges Final Remarks Derivative discontinuity (I) As we observed, the derivative of energy changes discontinuosly when we change the particle number: Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 57/76 ENFMC
  133. 133. Problem HK-KS xc LDA Construction Challenges Final Remarks Derivative discontinuity and the fundamental gap Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 58/76 ENFMC
  134. 134. Problem HK-KS xc LDA Construction Challenges Final Remarks Derivative discontinuity and the fundamental gap The fundamental gap in solid-state physics (photoemission gap, 2x chemical hardness) is defined by Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 58/76 ENFMC
  135. 135. Problem HK-KS xc LDA Construction Challenges Final Remarks Derivative discontinuity and the fundamental gap The fundamental gap in solid-state physics (photoemission gap, 2x chemical hardness) is defined by Fundamental gap: Ionization potential - Electron affinity Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 58/76 ENFMC
  136. 136. Problem HK-KS xc LDA Construction Challenges Final Remarks Derivative discontinuity and the fundamental gap The fundamental gap in solid-state physics (photoemission gap, 2x chemical hardness) is defined by Fundamental gap: Ionization potential - Electron affinity Ionization potential: I = E(N−1)−E(N) = − ∂E ∂N N−δN Electron affinity: A = E(N)−E(N+1) = − ∂E ∂N N+δN Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 58/76 ENFMC
  137. 137. Problem HK-KS xc LDA Construction Challenges Final Remarks Derivative discontinuity in our energy functional In our density functional, the discontinuity will also appear E[n] = Ts[n] + UH [n] + V [n] + Exc[n] Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 59/76 ENFMC
  138. 138. Problem HK-KS xc LDA Construction Challenges Final Remarks Derivative discontinuity in our energy functional In our density functional, the discontinuity will also appear E[n] = Ts[n] + UH [n] + V [n] + Exc[n] The discontinuous kinetic part is called Kohn-Sham non-interacing gap, and the xc part is the derivative discontinuity, the many-body correction to the Kohn-Sham non-interacting gap. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 59/76 ENFMC
  139. 139. Problem HK-KS xc LDA Construction Challenges Final Remarks Derivative discontinuity in our energy functional In our density functional, the discontinuity will also appear E[n] = Ts[n] + UH [n] + V [n] + Exc[n] The discontinuous kinetic part is called Kohn-Sham non-interacing gap, and the xc part is the derivative discontinuity, the many-body correction to the Kohn-Sham non-interacting gap. ∆L = δExc[n] δn(r) N+δN − δExc[n] δn(r) N−δN Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 59/76 ENFMC
  140. 140. Problem HK-KS xc LDA Construction Challenges Final Remarks Derivative discontinuity in our energy functional In our density functional, the discontinuity will also appear E[n] = Ts[n] + UH [n] + V [n] + Exc[n] The discontinuous kinetic part is called Kohn-Sham non-interacing gap, and the xc part is the derivative discontinuity, the many-body correction to the Kohn-Sham non-interacting gap. ∆L = δExc[n] δn(r) N+δN − δExc[n] δn(r) N−δN The fundamental gap (I-A) is given by the sum ∆fund = ∆KS + ∆L Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 59/76 ENFMC
  141. 141. Problem HK-KS xc LDA Construction Challenges Final Remarks Kohn-Sham gap vs fundamental gap Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 60/76 ENFMC
  142. 142. Problem HK-KS xc LDA Construction Challenges Final Remarks Kohn-Sham gap vs fundamental gap Therefore the Kohn-Sham gap is not equal to the fundamental gap. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 60/76 ENFMC
  143. 143. Problem HK-KS xc LDA Construction Challenges Final Remarks Kohn-Sham gap vs fundamental gap Therefore the Kohn-Sham gap is not equal to the fundamental gap. Most functionals show no derivative discontinuity jump. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 60/76 ENFMC
  144. 144. Problem HK-KS xc LDA Construction Challenges Final Remarks Kohn-Sham gap vs fundamental gap Therefore the Kohn-Sham gap is not equal to the fundamental gap. Most functionals show no derivative discontinuity jump. Ex. LDA: adapted from PRL 107, 183002 (2011). Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 60/76 ENFMC
  145. 145. Problem HK-KS xc LDA Construction Challenges Final Remarks Kohn-Sham gap vs fundamental gap Therefore the Kohn-Sham gap is not equal to the fundamental gap. Most functionals show no derivative discontinuity jump. Ex. LDA: PRL 96, 226402 (2006). Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 61/76 ENFMC
  146. 146. Problem HK-KS xc LDA Construction Challenges Final Remarks Some observations on KS quantities Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 62/76 ENFMC
  147. 147. Problem HK-KS xc LDA Construction Challenges Final Remarks Some observations on KS quantities The price for the simplification of the problem is that Kohn-Sham is an auxiliary tool. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 62/76 ENFMC
  148. 148. Problem HK-KS xc LDA Construction Challenges Final Remarks Some observations on KS quantities The price for the simplification of the problem is that Kohn-Sham is an auxiliary tool. The KS mapping gives you the energy and ground-state density. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 62/76 ENFMC
  149. 149. Problem HK-KS xc LDA Construction Challenges Final Remarks Some observations on KS quantities The price for the simplification of the problem is that Kohn-Sham is an auxiliary tool. The KS mapping gives you the energy and ground-state density. There is no proof that the KS quantities have a physical meaning, with few exceptions. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 62/76 ENFMC
  150. 150. Problem HK-KS xc LDA Construction Challenges Final Remarks Some observations on KS quantities The price for the simplification of the problem is that Kohn-Sham is an auxiliary tool. The KS mapping gives you the energy and ground-state density. There is no proof that the KS quantities have a physical meaning, with few exceptions. The KS gap is not equal to the fundamental gap, and the eigenvalues are not quasiparticle spectra. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 62/76 ENFMC
  151. 151. Problem HK-KS xc LDA Construction Challenges Final Remarks Some observations on KS quantities Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 63/76 ENFMC
  152. 152. Problem HK-KS xc LDA Construction Challenges Final Remarks Some observations on KS quantities Nonetheless, the KS eigenvalues can be a very good approximation to the quasiparticle spectrum. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 63/76 ENFMC
  153. 153. Problem HK-KS xc LDA Construction Challenges Final Remarks Some observations on KS quantities Nonetheless, the KS eigenvalues can be a very good approximation to the quasiparticle spectrum. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 63/76 ENFMC
  154. 154. Problem HK-KS xc LDA Construction Challenges Final Remarks General recommendations Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 64/76 ENFMC
  155. 155. Problem HK-KS xc LDA Construction Challenges Final Remarks General recommendations Functionals families (LDA,GGA,MGGA,hybrids): Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 64/76 ENFMC
  156. 156. Problem HK-KS xc LDA Construction Challenges Final Remarks General recommendations Functionals families (LDA,GGA,MGGA,hybrids): Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 64/76 ENFMC
  157. 157. Problem HK-KS xc LDA Construction Challenges Final Remarks General recommendations Functionals families (LDA,GGA,MGGA,hybrids): Important to know the functional proposal and its improvements Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 64/76 ENFMC
  158. 158. Problem HK-KS xc LDA Construction Challenges Final Remarks General recommendations Functionals families (LDA,GGA,MGGA,hybrids): Important to know the functional proposal and its improvements Check previous literature on the atomic, bulk trends, their character and problems Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 64/76 ENFMC
  159. 159. Problem HK-KS xc LDA Construction Challenges Final Remarks General recommendations Functionals families (LDA,GGA,MGGA,hybrids): Important to know the functional proposal and its improvements Check previous literature on the atomic, bulk trends, their character and problems When possible, confrontation with experimental or highly accurate methods Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 64/76 ENFMC
  160. 160. Problem HK-KS xc LDA Construction Challenges Final Remarks Outline 1 Review of our problem 2 Review of HK-KS 3 Exchange-correlation 4 LDA and GGA 5 Construction 6 Challenges 7 Final Remarks Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 64/76 ENFMC
  161. 161. Problem HK-KS xc LDA Construction Challenges Final Remarks Timeline Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 65/76 ENFMC
  162. 162. Problem HK-KS xc LDA Construction Challenges Final Remarks Timeline Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 65/76 ENFMC
  163. 163. Problem HK-KS xc LDA Construction Challenges Final Remarks DFT Impact Citation Statistics from 110 Years of Physical Review (1893 - 2003) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 66/76 ENFMC
  164. 164. Problem HK-KS xc LDA Construction Challenges Final Remarks DFT Impact Citation Statistics from 110 Years of Physical Review (1893 - 2003) (Physics Today, p.49 Junho 2005) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 66/76 ENFMC
  165. 165. Problem HK-KS xc LDA Construction Challenges Final Remarks Back to the electronic structure spirit “Where solid-state physics has Fermi energy, chemical potential, band gap, density of states, and local density of states, quantum chemistry has ionization potential, electron affinity, hardness, softness, and local softness. Much more too. DFT is a single language that covers atoms, molecules, clusters, surfaces, and solids.” Robert Parr Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 67/76 ENFMC
  166. 166. Problem HK-KS xc LDA Construction Challenges Final Remarks Back to the electronic structure spirit “Where solid-state physics has Fermi energy, chemical potential, band gap, density of states, and local density of states, quantum chemistry has ionization potential, electron affinity, hardness, softness, and local softness. Much more too. DFT is a single language that covers atoms, molecules, clusters, surfaces, and solids.” Robert Parr Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 67/76 ENFMC
  167. 167. Problem HK-KS xc LDA Construction Challenges Final Remarks Back to the electronic structure spirit “Where solid-state physics has Fermi energy, chemical potential, band gap, density of states, and local density of states, quantum chemistry has ionization potential, electron affinity, hardness, softness, and local softness. Much more too. DFT is a single language that covers atoms, molecules, clusters, surfaces, and solids.” Robert Parr Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 67/76 ENFMC
  168. 168. Problem HK-KS xc LDA Construction Challenges Final Remarks Back to the electronic structure spirit “Where solid-state physics has Fermi energy, chemical potential, band gap, density of states, and local density of states, quantum chemistry has ionization potential, electron affinity, hardness, softness, and local softness. Much more too. DFT is a single language that covers atoms, molecules, clusters, surfaces, and solids.” Robert Parr Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 67/76 ENFMC
  169. 169. Problem HK-KS xc LDA Construction Challenges Final Remarks Back to the electronic structure spirit “Where solid-state physics has Fermi energy, chemical potential, band gap, density of states, and local density of states, quantum chemistry has ionization potential, electron affinity, hardness, softness, and local softness. Much more too. DFT is a single language that covers atoms, molecules, clusters, surfaces, and solids.” Robert Parr Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 67/76 ENFMC
  170. 170. Problem HK-KS xc LDA Construction Challenges Final Remarks 1964/65-2015 Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 68/76 ENFMC
  171. 171. Problem HK-KS xc LDA Construction Challenges Final Remarks 1964/65-2015 Hohenberg-Kohn ’64: Kohn-Sham ’65: Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 68/76 ENFMC
  172. 172. Problem HK-KS xc LDA Construction Challenges Final Remarks Walter Kohn Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 69/76 ENFMC
  173. 173. Problem HK-KS xc LDA Construction Challenges Final Remarks Walter Kohn Born in 1923, in a jew middle-class family Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 69/76 ENFMC
  174. 174. Problem HK-KS xc LDA Construction Challenges Final Remarks Walter Kohn Born in 1923, in a jew middle-class family World War II: fled to England with help of family friends -wishing to become a farmer Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 69/76 ENFMC
  175. 175. Problem HK-KS xc LDA Construction Challenges Final Remarks Walter Kohn Born in 1923, in a jew middle-class family World War II: fled to England with help of family friends -wishing to become a farmer First interned in British camps for “enemy aliens” Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 69/76 ENFMC
  176. 176. Problem HK-KS xc LDA Construction Challenges Final Remarks Walter Kohn Born in 1923, in a jew middle-class family World War II: fled to England with help of family friends -wishing to become a farmer First interned in British camps for “enemy aliens” In Canadian camps, supported by Red Cross, studies math Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 69/76 ENFMC
  177. 177. Problem HK-KS xc LDA Construction Challenges Final Remarks Walter Kohn Born in 1923, in a jew middle-class family World War II: fled to England with help of family friends -wishing to become a farmer First interned in British camps for “enemy aliens” In Canadian camps, supported by Red Cross, studies math Working as lumberjacks, earning 20 cents per day, buys Slater’s book “Chemical Physics” Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 69/76 ENFMC
  178. 178. Problem HK-KS xc LDA Construction Challenges Final Remarks Walter Kohn Born in 1923, in a jew middle-class family World War II: fled to England with help of family friends -wishing to become a farmer First interned in British camps for “enemy aliens” In Canadian camps, supported by Red Cross, studies math Working as lumberjacks, earning 20 cents per day, buys Slater’s book “Chemical Physics” Joins the Canadian army and gets a BS degree in Applied Mathematics Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 69/76 ENFMC
  179. 179. Problem HK-KS xc LDA Construction Challenges Final Remarks Walter Kohn Born in 1923, in a jew middle-class family World War II: fled to England with help of family friends -wishing to become a farmer First interned in British camps for “enemy aliens” In Canadian camps, supported by Red Cross, studies math Working as lumberjacks, earning 20 cents per day, buys Slater’s book “Chemical Physics” Joins the Canadian army and gets a BS degree in Applied Mathematics Finishes a crash master’s course and applies for PhDs Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 69/76 ENFMC
  180. 180. Problem HK-KS xc LDA Construction Challenges Final Remarks Walter Kohn Born in 1923, in a jew middle-class family World War II: fled to England with help of family friends -wishing to become a farmer First interned in British camps for “enemy aliens” In Canadian camps, supported by Red Cross, studies math Working as lumberjacks, earning 20 cents per day, buys Slater’s book “Chemical Physics” Joins the Canadian army and gets a BS degree in Applied Mathematics Finishes a crash master’s course and applies for PhDs Awarded a scholarship for Harvard; becomes PhD student of Julian Schwinger Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 69/76 ENFMC
  181. 181. Problem HK-KS xc LDA Construction Challenges Final Remarks Walter Kohn and Julian Schwinger Kohn met Schwinger only “a few times a year”. “It was during these meetings, sometimes more than 2 hours long, that I learned the most from him. (...) to dig for the essential; to pay attention to the experimental facts; to try to say something precise and operati- onally meaningful, even if one cannot calcu- late everything a priori; not to be satisfied un- til one has embedded his ideas in a coherent, logical, and aesthetically satisfying structure. (...) I cannot even imagine my subsequent sci- entific life without Julian’s example and tea- ching.” Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 70/76 ENFMC
  182. 182. Problem HK-KS xc LDA Construction Challenges Final Remarks Kohn’s scientific background Schwinger: Green’s functions, variational principles, scattering Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 71/76 ENFMC
  183. 183. Problem HK-KS xc LDA Construction Challenges Final Remarks Kohn’s scientific background Schwinger: Green’s functions, variational principles, scattering Van Vleck: entered solid-state physics Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 71/76 ENFMC
  184. 184. Problem HK-KS xc LDA Construction Challenges Final Remarks Kohn’s scientific background Schwinger: Green’s functions, variational principles, scattering Van Vleck: entered solid-state physics Rostocker: Green’s functions to solve the electron band structure problem (KKR) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 71/76 ENFMC
  185. 185. Problem HK-KS xc LDA Construction Challenges Final Remarks Kohn’s scientific background Schwinger: Green’s functions, variational principles, scattering Van Vleck: entered solid-state physics Rostocker: Green’s functions to solve the electron band structure problem (KKR) Bell Labs: semiconductor physics (transistor rush) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 71/76 ENFMC
  186. 186. Problem HK-KS xc LDA Construction Challenges Final Remarks Kohn’s scientific background Schwinger: Green’s functions, variational principles, scattering Van Vleck: entered solid-state physics Rostocker: Green’s functions to solve the electron band structure problem (KKR) Bell Labs: semiconductor physics (transistor rush) Luttinger; effective mass equation for the energy levels of impurity states in Silicon: “one-particle method” Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 71/76 ENFMC
  187. 187. Problem HK-KS xc LDA Construction Challenges Final Remarks Kohn’s scientific background Schwinger: Green’s functions, variational principles, scattering Van Vleck: entered solid-state physics Rostocker: Green’s functions to solve the electron band structure problem (KKR) Bell Labs: semiconductor physics (transistor rush) Luttinger; effective mass equation for the energy levels of impurity states in Silicon: “one-particle method” ... electronic transport; phonons; insulating state; Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 71/76 ENFMC
  188. 188. Problem HK-KS xc LDA Construction Challenges Final Remarks Kohn’s scientific background Schwinger: Green’s functions, variational principles, scattering Van Vleck: entered solid-state physics Rostocker: Green’s functions to solve the electron band structure problem (KKR) Bell Labs: semiconductor physics (transistor rush) Luttinger; effective mass equation for the energy levels of impurity states in Silicon: “one-particle method” ... electronic transport; phonons; insulating state; Mott: Thomas-Fermi for screening Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 71/76 ENFMC
  189. 189. Problem HK-KS xc LDA Construction Challenges Final Remarks Kohn’s scientific background Schwinger: Green’s functions, variational principles, scattering Van Vleck: entered solid-state physics Rostocker: Green’s functions to solve the electron band structure problem (KKR) Bell Labs: semiconductor physics (transistor rush) Luttinger; effective mass equation for the energy levels of impurity states in Silicon: “one-particle method” ... electronic transport; phonons; insulating state; Mott: Thomas-Fermi for screening de Gennes, Friedel: metals and alloys; Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 71/76 ENFMC
  190. 190. Problem HK-KS xc LDA Construction Challenges Final Remarks Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 72/76 ENFMC
  191. 191. Problem HK-KS xc LDA Construction Challenges Final Remarks “Kohn’s seminal papers (...) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 72/76 ENFMC
  192. 192. Problem HK-KS xc LDA Construction Challenges Final Remarks “Kohn’s seminal papers (...) are all most notable for their clarity and the simplicity of the mathematics one encounters. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 72/76 ENFMC
  193. 193. Problem HK-KS xc LDA Construction Challenges Final Remarks “Kohn’s seminal papers (...) are all most notable for their clarity and the simplicity of the mathematics one encounters. On many occasions, after reading through the material, I found myself saying something like “of course things go that way, I could have written this myself”. (...) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 72/76 ENFMC
  194. 194. Problem HK-KS xc LDA Construction Challenges Final Remarks “Kohn’s seminal papers (...) are all most notable for their clarity and the simplicity of the mathematics one encounters. On many occasions, after reading through the material, I found myself saying something like “of course things go that way, I could have written this myself”. (...) It is the case that the most important and fundamental new ideas and concepts in our field are very simple and obvious, once they have been set forth for the first time. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 72/76 ENFMC
  195. 195. Problem HK-KS xc LDA Construction Challenges Final Remarks “Kohn’s seminal papers (...) are all most notable for their clarity and the simplicity of the mathematics one encounters. On many occasions, after reading through the material, I found myself saying something like “of course things go that way, I could have written this myself”. (...) It is the case that the most important and fundamental new ideas and concepts in our field are very simple and obvious, once they have been set forth for the first time. I am reminded of remarks I have read recently in an essay by Steven Weinberg, who states that the very important and fundamental papers in physics are notable for their clarity. Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 72/76 ENFMC
  196. 196. Problem HK-KS xc LDA Construction Challenges Final Remarks “Kohn’s seminal papers (...) are all most notable for their clarity and the simplicity of the mathematics one encounters. On many occasions, after reading through the material, I found myself saying something like “of course things go that way, I could have written this myself”. (...) It is the case that the most important and fundamental new ideas and concepts in our field are very simple and obvious, once they have been set forth for the first time. I am reminded of remarks I have read recently in an essay by Steven Weinberg, who states that the very important and fundamental papers in physics are notable for their clarity. The new ideas are applied quickly because of this.” Douglas L. Mills Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 72/76 ENFMC
  197. 197. Problem HK-KS xc LDA Construction Challenges Final Remarks Acknowledgements (I) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 73/76 ENFMC
  198. 198. Problem HK-KS xc LDA Construction Challenges Final Remarks Acknowledgements (I) Klaus Capelle, UFABC, Brazil E.K.U. Gross, MPI-Halle,Germany Sam Trickey, QTP-Univ.Florida Caio Lewenkopf, UFF, Brazil Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 73/76 ENFMC
  199. 199. Problem HK-KS xc LDA Construction Challenges Final Remarks References Kohn’s Nobel lecture, Electronic structure of matter—wave functions and density functionals, (http://www.nobelprize.org/nobel_prizes/chemistry/ laureates/1998/kohn-lecture.html) A. Becke, Perspective: Fifty years of density-functional theory in chemical physics, (http://www.ncbi.nlm.nih.gov/pubmed/24832308) K. Capelle, A bird’s-eye view of density-functional theory, (http://www.scielo.br/scielo.php?script=sci_arttext&pid= S0103-97332006000700035) Perdew and Kurth, A Primer in Density Functional Theory, (http://www.physics.udel.edu/˜bnikolic/QTTG/NOTES/DFT/BOOK=primer_ dft.pdf) Perdew et al., Some Fundamental Issues in Ground-State Density Functional Theory: A Guide for the Perplexed http://pubs.acs.org/doi/full/10.1021/ct800531s Zangwill, The education of Walter Kohn and the creation of density functional theory, (http://arxiv.org/abs/1403.5164) M. M. Odashima, PHD Thesis (http://www.teses.usp.br/teses/disponiveis/76/76131/tde-14062010- 164125/pt-br.php) Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 74/76 ENFMC
  200. 200. Problem HK-KS xc LDA Construction Challenges Final Remarks References Electronic Structure Basic - Theory and Practical Methods. Richard M Martin, Cambridge (2008) Atomic and Electronic Structure of Solids. Efthimios Kaxiras, Cambridge (2003). Density Functional Theory - An Advanced Course. Eberhard Engel and Reiner M. Dreizler, Springer (2011). Many-Electron Approaches in Physics, Chemistry and Mathematics: A Multidisciplinary View. Eds. Volker Bach, Luigi Delle Site, Springer (2014). Many-Body Approach to Electronic Excitations - Concepts and Applications. Friedhelm Bechstedt, Springer (2015). Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 75/76 ENFMC
  201. 201. Problem HK-KS xc LDA Construction Challenges Final Remarks Acknowledgements To all ENFMC organizers and FAPERJ. Thank you for your attention! https://sites.google.com/site/mmodashima/ Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 76/76 ENFMC

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