(If visualization is slow, please try downloading the file.) Part 1 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.

1. XXXVIII ENFMC Brazilian Physical Society Meeting
Introduction to
density functional theory
Mariana M. Odashima
ENFMC

2. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Today’s task:
DFT
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 1/64
ENFMC

3. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Today’s task:
DFT
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 1/64
ENFMC

4. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Today’s task:
DFT
is it ...
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 1/64
ENFMC

5. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Today’s task:
DFT
is it ... a program ?
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 1/64
ENFMC

6. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Today’s task:
DFT
is it ... a program ?
a method?
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 1/64
ENFMC

7. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Today’s task:
DFT
is it ... a program ?
a method?
some
obscure
theory?
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 1/64
ENFMC

8. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Some perspective
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 2/64
ENFMC

9. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Some perspective
(adapted from PhDComics)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 2/64
ENFMC

10. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Some perspective
(adapted from PhDComics)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 2/64
ENFMC

11. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Some perspective
(adapted from PhDComics)
Calm down!
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 2/64
ENFMC

12. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Some perspective
(adapted from Original “DFT song” written by V. Blum and K. Burke)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 3/64
ENFMC

13. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Some perspective
(adapted from Original “DFT song” written by V. Blum and K. Burke)
When I find my model’s unpredictive,
Walter Kohn just comes to me,
speaking words of wisdom, DFT.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 3/64
ENFMC

14. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Some perspective
(adapted from Original “DFT song” written by V. Blum and K. Burke)
When I find my model’s unpredictive,
Walter Kohn just comes to me,
speaking words of wisdom, DFT.
And in my hour of code-debugging,
he stands right in front of me,
saying ”you just gotta learn
your chemistree”.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 3/64
ENFMC

15. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Some perspective
(adapted from Original “DFT song” written by V. Blum and K. Burke)
When I find my model’s unpredictive,
Walter Kohn just comes to me,
speaking words of wisdom, DFT.
And in my hour of code-debugging,
he stands right in front of me,
saying ”you just gotta learn
your chemistree”.
(...)
LSD, PBE,
B3LYP, HSE.
I thought you were first principles, DFT...
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 3/64
ENFMC

16. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Some perspective
Adapted from Original “DFT song” written by V. Blum and K. Burke
When I find my model’s unpredictive,
Walter Kohn just comes to me,
speaking words of wisdom, DFT.
And in my hour of code-debugging,
he stands right in front of me,
saying ”you just gotta learn
your chemistree”.
(...)
LSD, PBE,
B3LYP, HSE.
I thought you were first principles, DFT...
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 3/64
ENFMC

17. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Some perspective
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 4/64
ENFMC

18. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Some perspective
Walter Kohn
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 4/64
ENFMC

19. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Some perspective
Walter Kohn Computer simulations
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 4/64
ENFMC

20. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Some perspective
Walter Kohn Computer simulations
Molecules, nanostructures
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 4/64
ENFMC

21. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Some perspective
Walter Kohn Computer simulations
Molecules, nanostructures Acronyms of functionals
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 4/64
ENFMC

22. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Today’s task:
Introduction to density-functional theory
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 5/64
ENFMC

23. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Today’s task:
Introduction to density-functional theory
Context and key concepts (1927-1930)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 5/64
ENFMC

24. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Today’s task:
Introduction to density-functional theory
Context and key concepts (1927-1930)
Fundamentals (1964-1965)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 5/64
ENFMC

25. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Today’s task:
Introduction to density-functional theory
Context and key concepts (1927-1930)
Fundamentals (1964-1965)
Approximations (≈ 1980-2010)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 5/64
ENFMC

26. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Today’s task:
Introduction to density-functional theory
Context and key concepts (1927-1930)
Fundamentals (1964-1965)
Approximations (≈ 1980-2010)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 5/64
ENFMC

27. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Disclaimer
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 6/64
ENFMC

28. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Disclaimer
“Density-functional theory is
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 6/64
ENFMC

29. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Disclaimer
“Density-functional theory is
a subtle,
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 6/64
ENFMC

30. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Disclaimer
“Density-functional theory is
a subtle,
seductive,
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 6/64
ENFMC

31. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Disclaimer
“Density-functional theory is
a subtle,
seductive,
provocative business.”
A. Becke
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 6/64
ENFMC

32. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Outline
1 Electronic structure
2 Hartree and Hartree-Fock methods
3 Thomas-Fermi model
4 Hohenberg-Kohn Theorem
5 Kohn-Sham Scheme
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 6/64
ENFMC

33. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Outline
1 Electronic structure
2 Hartree and Hartree-Fock methods
3 Thomas-Fermi model
4 Hohenberg-Kohn Theorem
5 Kohn-Sham Scheme
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 6/64
ENFMC

34. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
General context
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 7/64
ENFMC

35. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
General context
Quantum method to describe properties of materials
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 7/64
ENFMC

36. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
General context
Quantum method to describe properties of materials
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 7/64
ENFMC

37. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
General context
Quantum method to describe properties of materials
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 7/64
ENFMC

38. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
General context
Quantum method to describe properties of materials
Chemistry, Physics, Material science
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 7/64
ENFMC

39. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Modelling
Consider N electrons, P nuclei. Schr¨odinger Equation:
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 8/64
ENFMC

40. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Modelling
Consider N electrons, P nuclei. Schr¨odinger Equation:
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 8/64
ENFMC

41. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Modelling
Consider N electrons, P nuclei. Schr¨odinger Equation:
ˆHΨ = EΨ
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 8/64
ENFMC

42. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Modelling
Consider N electrons, P nuclei. Schr¨odinger Equation:
ˆHΨ = EΨ
onde
Ψ(r1, r2, ..., rN , R1, R2, ..., RP)
e
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 8/64
ENFMC

43. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Modelling
Consider N electrons, P nuclei. Schr¨odinger Equation:
ˆHΨ = EΨ
onde
Ψ(r1, r2, ..., rN , R1, R2, ..., RP)
e
ˆH =
N
i
−
2
2m
2
ri
+
P
i
−
2
2Mi
2
Ri
+
N
i<j
e2
|ri − rj|
−
N
i
P
j
Zje2
|ri − Rj|
+
P
i<j
ZiZje2
|Ri − Rj|
,
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 8/64
ENFMC

44. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Born-Oppenheimer Approximation
Simplification: separation of the nuclear and electronic scales
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 9/64
ENFMC

45. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Born-Oppenheimer Approximation
Simplification: separation of the nuclear and electronic scales
Born-Oppenheimer: nuclei fixed
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 9/64
ENFMC

46. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Born-Oppenheimer Approximation
Simplification: separation of the nuclear and electronic scales
Born-Oppenheimer: nuclei fixed
ˆH =
N
i
−
2
2m
2
ri
+
P
i
−
2
2Mi
2
Ri
+ +
N
i<j
e2
|ri − rj|
−
N
i
P
j
Zje2
|ri − Rj|
+
P
i<j
ZiZje2
|Ri − Rj|
,
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 9/64
ENFMC

47. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Born-Oppenheimer approximation
ˆH = ˆHel + ˆHnuc
where
ˆHel =
N
i
−
2
2m
2
ri
+
N
i<j
e2
|ri − rj|
−
N
i
P
j
Zje2
|ri − Rj|
= ˆT + ˆU + ˆV
ˆHnuc =
P
i<j
ZiZje2
|Ri − Rj|
= ˆVnuc .
¯Ψ(ri; Ri) = Ψel(ri; Ri) Φnuc(Ri) .
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 10/64
ENFMC

48. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Born-Oppenheimer approximation
ˆH = ˆHel + ˆHnuc
where
ˆHel =
N
i
−
2
2m
2
ri
+
N
i<j
e2
|ri − rj|
−
N
i
P
j
Zje2
|ri − Rj|
= ˆT + ˆU + ˆV
ˆHnuc =
P
i<j
ZiZje2
|Ri − Rj|
= ˆVnuc .
¯Ψ(ri; Ri) = Ψel(ri; Ri) Φnuc(Ri) .
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 10/64
ENFMC

49. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Born-Oppenheimer approximation
ˆH = ˆHel + ˆHnuc
where
ˆHel =
N
i
−
2
2m
2
ri
+
N
i<j
e2
|ri − rj|
−
N
i
P
j
Zje2
|ri − Rj|
= ˆT + ˆU + ˆV
ˆHnuc =
P
i<j
ZiZje2
|Ri − Rj|
= ˆVnuc .
¯Ψ(ri; Ri) = Ψel(ri; Ri) Φnuc(Ri) .
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 10/64
ENFMC

50. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Born-Oppenheimer approximation
ˆH = ˆHel + ˆHnuc
where
ˆHel =
N
i
−
2
2m
2
ri
+
N
i<j
e2
|ri − rj|
−
N
i
P
j
Zje2
|ri − Rj|
= ˆT + ˆU + ˆV
ˆHnuc =
P
i<j
ZiZje2
|Ri − Rj|
= ˆVnuc .
¯Ψ(ri; Ri) = Ψel(ri; Ri) Φnuc(Ri) .
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 10/64
ENFMC

51. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
The electronic structure problem
Many interacting electrons: Ψel(r1, r2, ..., rN )
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 11/64
ENFMC

52. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
The electronic structure problem
Many interacting electrons: Ψel(r1, r2, ..., rN )
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 11/64
ENFMC

53. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
The electronic structure problem
Many interacting electrons: Ψel(r1, r2, ..., rN )
Paradigms: Hydrogen atom (single particle problem);
uniform electron gas
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 11/64
ENFMC

54. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
The electronic structure problem
Many interacting electrons: Ψel(r1, r2, ..., rN )
Paradigms: Hydrogen atom (single particle problem);
uniform electron gas
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 11/64
ENFMC

55. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
The electronic structure problem
Many interacting electrons: Ψel(r1, r2, ..., rN )
Paradigms: Hydrogen atom (single particle problem);
uniform electron gas
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 11/64
ENFMC

56. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
The electronic structure problem
Many interacting electrons: Ψel(r1, r2, ..., rN )
Paradigms: Hydrogen atom (single particle problem);
uniform electron gas
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 11/64
ENFMC

57. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
The electronic structure problem
Many interacting electrons: Ψel(r1, r2, ..., rN )
Paradigms: Hydrogen atom (single particle problem);
uniform electron gas
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 11/64
ENFMC

58. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
The electronic structure problem
Many interacting electrons: Ψel(r1, r2, ..., rN )
Paradigms: Hydrogen atom (single particle problem);
uniform electron gas
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 11/64
ENFMC

59. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
The electronic structure problem
Many interacting electrons: Ψel(r1, r2, ..., rN )
Paradigms: Hydrogen atom (single particle problem);
uniform electron gas
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 11/64
ENFMC

60. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
The electronic structure problem
Many interacting electrons: Ψel(r1, r2, ..., rN )
Paradigms: Hydrogen atom (single particle problem);
uniform electron gas
(by K. Capelle)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 11/64
ENFMC

61. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
The electronic structure problem
Many interacting electrons: Ψel(r1, r2, ..., rN )
Paradigms: Hydrogen atom (single particle problem);
uniform electron gas
(by K. Capelle)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 11/64
ENFMC

62. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
The electronic structure problem
Many interacting electrons: Ψel(r1, r2, ..., rN )
Paradigms: Hydrogen atom (single particle problem);
uniform electron gas
(by K. Capelle)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 11/64
ENFMC

63. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
The electronic structure problem
Many interacting electrons: Ψel(r1, r2, ..., rN )
Paradigms: Hydrogen atom (single particle problem);
uniform electron gas
(by K. Capelle)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 11/64
ENFMC

64. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
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 12/64
ENFMC

65. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
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 12/64
ENFMC

66. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
The electronic structure problem
Quantum many-body problem
of N interacting electrons: Ψel(r1, r2, ..., rN )
under the static nuclei external potential.
Methods based on the wavefunction
(Hartree-Fock, CI, Coupled Cluster, MP2, QMC)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 12/64
ENFMC

67. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
The electronic structure problem
Quantum many-body problem
of N interacting electrons: Ψel(r1, r2, ..., rN )
under the static nuclei external potential.
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 12/64
ENFMC

68. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
The electronic structure problem
Quantum many-body problem
of N interacting electrons: Ψel(r1, r2, ..., rN )
under the static nuclei external potential.
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 12/64
ENFMC

69. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Parenthesis
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 13/64
ENFMC

70. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Parenthesis
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 13/64
ENFMC

71. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Timeline
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 14/64
ENFMC

72. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Timeline
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 14/64
ENFMC

73. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Timeline
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 14/64
ENFMC

74. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Dirac’s impressions (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 15/64
ENFMC

75. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Dirac’s impressions (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 15/64
ENFMC

76. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Dirac’s impressions (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 15/64
ENFMC

77. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Dirac’s impressions (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 15/64
ENFMC

78. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Dirac’s impressions (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 15/64
ENFMC

79. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Timeline
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 16/64
ENFMC

80. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
DFT predecessors
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 17/64
ENFMC

81. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
DFT predecessors
Hartree
Hartree-Fock
Thomas-Fermi-Dirac
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 17/64
ENFMC

82. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Outline
1 Electronic structure
2 Hartree and Hartree-Fock methods
3 Thomas-Fermi model
4 Hohenberg-Kohn Theorem
5 Kohn-Sham Scheme
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 17/64
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83. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree’s Method (1928)
In the late 1920’s, Douglas Hartree (phy-
sicist, mathematician, pioneer compu-
ter scientist) developed a self-consistent
approach to solve the single-electron
Schr¨odinger equation; each electron
would be under a mean field that accoun-
ted for the other electrons.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 18/64
ENFMC

84. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree’s Method (1928)
In the late 1920’s, Douglas Hartree (phy-
sicist, mathematician, pioneer compu-
ter scientist) developed a self-consistent
approach to solve the single-electron
Schr¨odinger equation; each electron
would be under a mean field that accoun-
ted for the other electrons.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 18/64
ENFMC

85. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree’s Method (1928)
In the late 1920’s, Douglas Hartree (phy-
sicist, mathematician, pioneer compu-
ter scientist) developed a self-consistent
approach to solve the single-electron
Schr¨odinger equation; each electron
would be under a mean field that accoun-
ted for the other electrons.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 18/64
ENFMC

86. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree’s Method (1928)
Schr¨odinger’s equation for independent electrons
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 19/64
ENFMC

87. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree’s Method (1928)
Schr¨odinger’s equation for independent electrons
−
2
2m
2
+ ˆV (r) ϕi(r) = iϕi(r) ,
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 19/64
ENFMC

88. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree’s Method (1928)
Schr¨odinger’s equation for independent electrons
−
2
2m
2
+ ˆV (r) ϕi(r) = iϕi(r) ,
Wave function: product of monoelectronic orbitals
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 19/64
ENFMC

89. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree’s Method (1928)
Schr¨odinger’s equation for independent electrons
−
2
2m
2
+ ˆV (r) ϕi(r) = iϕi(r) ,
Wave function: product of monoelectronic orbitals
ΨH (r) = ϕ1(r1)ϕ2(r2)ϕ3(r3)...ϕN (rN ) .
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 19/64
ENFMC

90. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree’s Method (1928)
Schr¨odinger’s equation for independent electrons
−
2
2m
2
+ ˆV (r) ϕi(r) = iϕi(r) ,
Wave function: product of monoelectronic orbitals
ΨH (r) = ϕ1(r1)ϕ2(r2)ϕ3(r3)...ϕN (rN ) .
Remaining electrons: effective field
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 19/64
ENFMC

91. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree’s Method (1928)
Schr¨odinger’s equation for independent electrons
−
2
2m
2
+ ˆV (r) ϕi(r) = iϕi(r) ,
Wave function: product of monoelectronic orbitals
ΨH (r) = ϕ1(r1)ϕ2(r2)ϕ3(r3)...ϕN (rN ) .
Remaining electrons: effective field
ˆV (r) = −
P
j
Zje2
|r − Rj|
+ e2
N
i
d3
r
|ϕi(r )|2
|r − r |
= ˆVion(r) + ˆVH (r) . n(r) =
N
i
|ϕi(r)|2
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 19/64
ENFMC

92. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree’s Method (1928)
Schr¨odinger’s equation for independent electrons
−
2
2m
2
+ ˆV (r) ϕi(r) = iϕi(r) ,
Wave function: product of monoelectronic orbitals
ΨH (r) = ϕ1(r1)ϕ2(r2)ϕ3(r3)...ϕN (rN ) .
Remaining electrons: effective field
ˆV (r) = −
P
j
Zje2
|r − Rj|
+ e2
N
i
d3
r
|ϕi(r )|2
|r − r |
= ˆVion(r) + ˆVH (r) . n(r) =
N
i
|ϕi(r)|2
Self-consistent solution
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 19/64
ENFMC

93. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree’s Method (1928)
Wave function
ΨH (r) = ϕ1(r1)ϕ2(r2)ϕ3(r3)...ϕN (rN ) .
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 20/64
ENFMC

94. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree’s Method (1928)
Wave function
ΨH (r) = ϕ1(r1)ϕ2(r2)ϕ3(r3)...ϕN (rN ) .
Total energy:
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 20/64
ENFMC

95. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree’s Method (1928)
Wave function
ΨH (r) = ϕ1(r1)ϕ2(r2)ϕ3(r3)...ϕN (rN ) .
Total energy:
ΨH | ˆH|ΨH =
N
i
d3
r ϕ∗
i (r) −
2
2m
2
+ ˆVion(r) ϕi(r) +
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 20/64
ENFMC

96. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree’s Method (1928)
Wave function
ΨH (r) = ϕ1(r1)ϕ2(r2)ϕ3(r3)...ϕN (rN ) .
Total energy:
ΨH | ˆH|ΨH =
N
i
d3
r ϕ∗
i (r) −
2
2m
2
+ ˆVion(r) ϕi(r) +
+
e2
2
N
i
N
j
d3
r d3
r
|ϕi(r)|2|ϕj(r )|2
|r − r |
.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 20/64
ENFMC

97. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree’s Method (1928)
Wave function
ΨH (r) = ϕ1(r1)ϕ2(r2)ϕ3(r3)...ϕN (rN ) .
Total energy:
ΨH | ˆH|ΨH =
N
i
d3
r ϕ∗
i (r) −
2
2m
2
+ ˆVion(r) ϕi(r) +
+
e2
2
N
i
N
j
d3
r d3
r
|ϕi(r)|2|ϕj(r )|2
|r − r |
.
“Hartree functional”: n(r) =
N
i
|ϕi(r)|2
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 20/64
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98. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree’s Method (1928)
Wave function
ΨH (r) = ϕ1(r1)ϕ2(r2)ϕ3(r3)...ϕN (rN ) .
Total energy:
ΨH | ˆH|ΨH =
N
i
d3
r ϕ∗
i (r) −
2
2m
2
+ ˆVion(r) ϕi(r) +
+
e2
2
N
i
N
j
d3
r d3
r
|ϕi(r)|2|ϕj(r )|2
|r − r |
.
“Hartree functional”: n(r) =
N
i
|ϕi(r)|2
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 20/64
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99. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree’s Method (1928)
Wave function
ΨH (r) = ϕ1(r1)ϕ2(r2)ϕ3(r3)...ϕN (rN ) .
Total energy:
ΨH | ˆH|ΨH =
N
i
d3
r ϕ∗
i (r) −
2
2m
2
+ ˆVion(r) ϕi(r) +
+
e2
2
N
i
N
j
d3
r d3
r
|ϕi(r)|2|ϕj(r )|2
|r − r |
.
“Hartree functional”: n(r) =
N
i
|ϕi(r)|2
ΨH | ˆU|ΨH =
e2
2
d3
r d3
r
n(r)n(r )
|r − r |
= UH [n] .
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 20/64
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100. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree-Fock Method (1930)
John Slater, Vladimir Fock
Fermionic wave function: antisimetrized product
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 21/64
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101. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree-Fock Method (1930)
John Slater, Vladimir Fock
Fermionic wave function: antisimetrized product
Ψ(x1, x2, ..., xi, ..., xj, ..., xN ) =
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 21/64
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102. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree-Fock Method (1930)
John Slater, Vladimir Fock
Fermionic wave function: antisimetrized product
Ψ(x1, x2, ..., xi, ..., xj, ..., xN ) = − Ψ(x1, x2, ..., xj, ..., xi, ..., xN ) ,
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 21/64
ENFMC

103. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree-Fock Method (1930)
John Slater, Vladimir Fock
Fermionic wave function: antisimetrized product
Ψ(x1, x2, ..., xi, ..., xj, ..., xN ) = − Ψ(x1, x2, ..., xj, ..., xi, ..., xN ) ,
where x = (r, σ).
Slater determinant
ΨHF (r) =
1
√
N!
ϕ1(x1) ϕ1(x2) · · · ϕ1(xN )
ϕ2(x1) ϕ2(x2) · · · ϕ2(xN )
...
...
...
...
ϕN (x1) ϕN (x2) · · · ϕN (xN )
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 21/64
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104. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Example
For two orbitals:
Slater determinant:
ΨHF
(x1, x2) =
1
√
2
ϕ1(x1) ϕ1(x2)
ϕ2(x1) ϕ2(x2)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 22/64
ENFMC

105. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Example
For two orbitals:
Slater determinant:
ΨHF
(x1, x2) =
1
√
2
ϕ1(x1) ϕ1(x2)
ϕ2(x1) ϕ2(x2)
=
1
√
2
[ϕ1(x1)ϕ2(x2) − ϕ1(x2)ϕ2(x1)]
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 22/64
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106. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Example
For two orbitals:
Slater determinant:
ΨHF
(x1, x2) =
1
√
2
ϕ1(x1) ϕ1(x2)
ϕ2(x1) ϕ2(x2)
=
1
√
2
[ϕ1(x1)ϕ2(x2) − ϕ1(x2)ϕ2(x1)]
ΨHF | ˆH|ΨHF = 1/2 dx1 dx2·
[ϕ∗
1(x1)ϕ∗
2(x2) − ϕ∗
1(x2)ϕ∗
2(x1)] Σˆhi + ˆU [ϕ1(x1)ϕ2(x2) − ϕ1(x2)ϕ2(x1)]
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 22/64
ENFMC

107. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree-Fock method
Total energy:
ΨHF | ˆH|ΨHF =
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 23/64
ENFMC

108. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree-Fock method
Total energy:
ΨHF | ˆH|ΨHF =
i σ
d3
r ϕ∗
iσ(r) −
2
2m
2
+ ˆVext(r) ϕiσ(r) +
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 23/64
ENFMC

109. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree-Fock method
Total energy:
ΨHF | ˆH|ΨHF =
i σ
d3
r ϕ∗
iσ(r) −
2
2m
2
+ ˆVext(r) ϕiσ(r) +
+
e2
2 i,j σi ,σj
d3
r d3
r
|ϕiσi
(r)|2
|ϕjσj
(r )|2
|r − r |
−
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 23/64
ENFMC

110. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree-Fock method
Total energy:
ΨHF | ˆH|ΨHF =
i σ
d3
r ϕ∗
iσ(r) −
2
2m
2
+ ˆVext(r) ϕiσ(r) +
+
e2
2 i,j σi ,σj
d3
r d3
r
|ϕiσi
(r)|2
|ϕjσj
(r )|2
|r − r |
−
−
e2
2 i,j σ
d3
r d3
r
ϕ∗
iσ(r)ϕ∗
jσ(r )ϕiσ(r )ϕjσ(r)
|r − r |
= T + Vext + UH + Ex
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 23/64
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111. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree-Fock method
Total energy:
ΨHF | ˆH|ΨHF =
i σ
d3
r ϕ∗
iσ(r) −
2
2m
2
+ ˆVext(r) ϕiσ(r) +
+
e2
2 i,j σi ,σj
d3
r d3
r
|ϕiσi
(r)|2
|ϕjσj
(r )|2
|r − r |
−
−
e2
2 i,j σ
d3
r d3
r
ϕ∗
iσ(r)ϕ∗
jσ(r )ϕiσ(r )ϕjσ(r)
|r − r |
= T + Vext + UH + Ex
Coulomb energy, direct and indirect:
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 23/64
ENFMC

112. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree-Fock method
Total energy:
ΨHF | ˆH|ΨHF =
i σ
d3
r ϕ∗
iσ(r) −
2
2m
2
+ ˆVext(r) ϕiσ(r) +
+
e2
2 i,j σi ,σj
d3
r d3
r
|ϕiσi
(r)|2
|ϕjσj
(r )|2
|r − r |
−
−
e2
2 i,j σ
d3
r d3
r
ϕ∗
iσ(r)ϕ∗
jσ(r )ϕiσ(r )ϕjσ(r)
|r − r |
= T + Vext + UH + Ex
Coulomb energy, direct and indirect:
ΨH | ˆU|ΨH = UH > 0
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 23/64
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113. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hartree-Fock method
Total energy:
ΨHF | ˆH|ΨHF =
i σ
d3
r ϕ∗
iσ(r) −
2
2m
2
+ ˆVext(r) ϕiσ(r) +
+
e2
2 i,j σi ,σj
d3
r d3
r
|ϕiσi
(r)|2
|ϕjσj
(r )|2
|r − r |
−
−
e2
2 i,j σ
d3
r d3
r
ϕ∗
iσ(r)ϕ∗
jσ(r )ϕiσ(r )ϕjσ(r)
|r − r |
= T + Vext + UH + Ex
Coulomb energy, direct and indirect:
ΨH | ˆU|ΨH = UH > 0
ΨHF | ˆU|ΨHF = UH + Ex (Ex < 0)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 23/64
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114. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Exchange energy (Fock)
Antisimetrized wave function: Pauli principle pushes same
spin electrons apart,
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 24/64
ENFMC

115. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Exchange energy (Fock)
Antisimetrized wave function: Pauli principle pushes same
spin electrons apart, reducing the Coulomb repulsion
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 24/64
ENFMC

116. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Exchange energy (Fock)
Antisimetrized wave function: Pauli principle pushes same
spin electrons apart, reducing the Coulomb repulsion
Exchange contribution Ex < 0
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 24/64
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117. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Exchange energy (Fock)
Antisimetrized wave function: Pauli principle pushes same
spin electrons apart, reducing the Coulomb repulsion
Exchange contribution Ex < 0
Fock exchange depends on the orbitals:
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 24/64
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118. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Exchange energy (Fock)
Antisimetrized wave function: Pauli principle pushes same
spin electrons apart, reducing the Coulomb repulsion
Exchange contribution Ex < 0
Fock exchange depends on the orbitals:
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 24/64
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119. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Exchange energy (Fock)
Antisimetrized wave function: Pauli principle pushes same
spin electrons apart, reducing the Coulomb repulsion
Exchange contribution Ex < 0
Fock exchange depends on the orbitals:
Ex[ϕ] = −
e2
2 i,j σ
d3
r d3
r
ϕ∗
iσ(r)ϕ∗
jσ(r )ϕiσ(r )ϕjσ(r)
|r − r |
(1)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 24/64
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120. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Exchange energy (Fock)
Antisimetrized wave function: Pauli principle pushes same
spin electrons apart, reducing the Coulomb repulsion
Exchange contribution Ex < 0
Fock exchange depends on the orbitals:
Ex[ϕ] = −
e2
2 i,j σ
d3
r d3
r
ϕ∗
iσ(r)ϕ∗
jσ(r )ϕiσ(r )ϕjσ(r)
|r − r |
(1)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 24/64
ENFMC

121. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Correlation energy
Hartree-Fock: a single Slater determinant
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 25/64
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122. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Correlation energy
Hartree-Fock: a single Slater determinant
The real electronic repulsion is more correlated than the
mean-field description
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 25/64
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123. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Correlation energy
Hartree-Fock: a single Slater determinant
The real electronic repulsion is more correlated than the
mean-field description
In quantum chemistry and many-body perturbation theory we
call “correlation” the contributions beyond a single Slater
determinant
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 25/64
ENFMC

124. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Correlation energy
Hartree-Fock: a single Slater determinant
The real electronic repulsion is more correlated than the
mean-field description
In quantum chemistry and many-body perturbation theory we
call “correlation” the contributions beyond a single Slater
determinant
Coulomb energy: Hartree + Exchange + Correlation
Ψ| ˆU|Ψ = EH + Ex + Ec
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 25/64
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125. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Correlation energy
Hartree-Fock: a single Slater determinant
The real electronic repulsion is more correlated than the
mean-field description
In quantum chemistry and many-body perturbation theory we
call “correlation” the contributions beyond a single Slater
determinant
Coulomb energy: Hartree + Exchange + Correlation
Ψ| ˆU|Ψ = EH + Ex + Ec
Tipically in atoms and molecules, Ec ≈ 0.1Ex...
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 25/64
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126. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Outline
1 Electronic structure
2 Hartree and Hartree-Fock methods
3 Thomas-Fermi model
4 Hohenberg-Kohn Theorem
5 Kohn-Sham Scheme
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 25/64
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127. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Thomas-Fermi model (1927)
In 1927, Llewellyn Thomas and Enrico
Fermi proposed independently a method
based on semiclassical and statistical
ideas to determine the ground-state of
many-electron atoms. The N electrons
are treated as a Fermi gas on its ground
state, confined by an effective potential
that goes to zero in the infinty.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 26/64
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128. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Thomas-Fermi model (1927)
In 1927, Llewellyn Thomas and Enrico
Fermi proposed independently a method
based on semiclassical and statistical
ideas to determine the ground-state of
many-electron atoms. The N electrons
are treated as a Fermi gas on its ground
state, confined by an effective potential
that goes to zero in the infinty.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 26/64
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129. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Thomas-Fermi model
The total energy E in the TF model is the sum of the Fermi gas
energy
EF =
2
2m
3π2
n
2/3
with the effective potential
Veff (r) = Vext(r) + e2
d3
r
n(r )
|r − r |
.
E = EF + Veff
The density can be written in terms of these contributions
n(r) =
1
3π2
2m
2
3/2
(E − Vef (r))3/2
,
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 27/64
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130. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Thomas-Fermi model
With Poisson’s equation,
2
ϕ(r) = 4πe n(r) , ϕ(r) = −Vef (r)/e
we can obtain a self-consistent solution, iterating the TF density
and the Poisson’s equation density.
n(r) =
1
3π2
2m
2
3/2
(E − Vef (r))3/2
,
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 28/64
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131. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Thomas-Fermi model
Ground-state energy of atoms via Fermi gas
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 29/64
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132. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Thomas-Fermi model
Ground-state energy of atoms via Fermi gas
Approximates the kinetic energy of atoms by use of the
density of the uniform electron gas
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 29/64
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133. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Thomas-Fermi model
Ground-state energy of atoms via Fermi gas
Approximates the kinetic energy of atoms by use of the
density of the uniform electron gas
Rewriting the total energy as a functional of the density
E = TTF [n] + d3
r n(r)vext(r) +
e2
2
d3
rd3
r
n(r)n(r )
|r − r |
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 29/64
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134. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Thomas-Fermi model
Ground-state energy of atoms via Fermi gas
Approximates the kinetic energy of atoms by use of the
density of the uniform electron gas
Rewriting the total energy as a functional of the density
E = TTF [n] + d3
r n(r)vext(r) +
e2
2
d3
rd3
r
n(r)n(r )
|r − r |
we obtain the Thomas-Fermi approximation to the kinetic energy,
TTF [n] = ts(n(r))n(r)d3
r =
3
10
2
m
3π2 2/3
d3
r n5/3
(r)
known also as the first local density approximation (LDA).
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 29/64
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135. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Thomas-Fermi model
Qualitative trends of total energies
No chemical binding, and exact only in the Z → ∞ limit
Fermi energy sphere is purely kinetic
Absence of quantum correlations (xc)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 30/64
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136. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Thomas-Fermi model
Qualitative trends of total energies
No chemical binding, and exact only in the Z → ∞ limit
Fermi energy sphere is purely kinetic
Absence of quantum correlations (xc)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 30/64
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137. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Dirac’s approximation (1929)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 31/64
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138. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Dirac’s approximation (1929)
Dirac derives the exchange energy density of the electron gas
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 31/64
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139. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Dirac’s approximation (1929)
Dirac derives the exchange energy density of the electron gas
ELDA
x [n] = −Axe2/3
d3
rn4/3
(r) ,
Thomas-Fermi-Dirac model
E ≈ ETFD
[n] = TLDA
s [n] + UH [n] + ELDA
x + V [n] .
First density functionals
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 31/64
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140. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Timeline
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 32/64
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141. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Remark
“In the 20s and early 30s there was a flush of
successes in establishing the ability of quantum
mechanics to describe the simplest molecules
accurately:
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 33/64
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142. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Remark
“In the 20s and early 30s there was a flush of
successes in establishing the ability of quantum
mechanics to describe the simplest molecules
accurately:
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 33/64
ENFMC

143. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Remark
“In the 20s and early 30s there was a flush of
successes in establishing the ability of quantum
mechanics to describe the simplest molecules
accurately: the Born-Oppenheimer approximation,
the nature of chemical bonding, and the
fundamentals of molecular spectroscopy.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 33/64
ENFMC

144. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Remark
“In the 20s and early 30s there was a flush of
successes in establishing the ability of quantum
mechanics to describe the simplest molecules
accurately: the Born-Oppenheimer approximation,
the nature of chemical bonding, and the
fundamentals of molecular spectroscopy. But then
the quantitative theory of molecular structure, which
we call quantum chemistry, was stymied, by the
difficulty of solving the Schr¨odinger equation for
molecules.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 33/64
ENFMC

145. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Remark
“In the 20s and early 30s there was a flush of
successes in establishing the ability of quantum
mechanics to describe the simplest molecules
accurately: the Born-Oppenheimer approximation,
the nature of chemical bonding, and the
fundamentals of molecular spectroscopy. But then
the quantitative theory of molecular structure, which
we call quantum chemistry, was stymied, by the
difficulty of solving the Schr¨odinger equation for
molecules. The senior chemical physicists of the 30s
pronounced the problem unsolvable.”
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 33/64
ENFMC

146. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Remark
“In the 20s and early 30s there was a flush of
successes in establishing the ability of quantum
mechanics to describe the simplest molecules
accurately: the Born-Oppenheimer approximation,
the nature of chemical bonding, and the
fundamentals of molecular spectroscopy. But then
the quantitative theory of molecular structure, which
we call quantum chemistry, was stymied, by the
difficulty of solving the Schr¨odinger equation for
molecules. The senior chemical physicists of the 30s
pronounced the problem unsolvable.” But the younger
theoreticians in the period coming out of WWII
thought otherwise. (...) It would not be as easy as
handling an infinite periodic solid, but a number of us
set to work.”
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 33/64
ENFMC

147. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Remark
“In the 20s and early 30s there was a flush of
successes in establishing the ability of quantum
mechanics to describe the simplest molecules
accurately: the Born-Oppenheimer approximation,
the nature of chemical bonding, and the
fundamentals of molecular spectroscopy. But then
the quantitative theory of molecular structure, which
we call quantum chemistry, was stymied, by the
difficulty of solving the Schr¨odinger equation for
molecules. The senior chemical physicists of the 30s
pronounced the problem unsolvable.” But the younger
theoreticians in the period coming out of WWII
thought otherwise. (...) It would not be as easy as
handling an infinite periodic solid, but a number of us
set to work.” Robert Parr
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 33/64
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148. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Timeline
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 34/64
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149. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Timeline
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 34/64
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150. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Timeline
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 34/64
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151. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Warning
“Density-functional theory (DFT) is a
subtle, seductive, provocative business.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 35/64
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152. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Warning
“Density-functional theory (DFT) is a
subtle, seductive, provocative business.”
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 35/64
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153. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Warning
“Density-functional theory (DFT) is a
subtle, seductive, provocative business.
Its basic premise, that all the intricate
motions and pair correlations in a
many-electron system
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 35/64
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154. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Warning
“Density-functional theory (DFT) is a
subtle, seductive, provocative business.
Its basic premise, that all the intricate
motions and pair correlations in a
many-electron system are somehow
contained in the total electron density
alone,
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 35/64
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155. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Warning
“Density-functional theory (DFT) is a
subtle, seductive, provocative business.
Its basic premise, that all the intricate
motions and pair correlations in a
many-electron system are somehow
contained in the total electron density
alone, is so compelling it can drive one
mad.”
Axel D. Becke
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 35/64
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156. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Outline
1 Electronic structure
2 Hartree and Hartree-Fock methods
3 Thomas-Fermi model
4 Hohenberg-Kohn Theorem
5 Kohn-Sham Scheme
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 35/64
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157. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hohenberg-Kohn theorem (1964)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 35/64
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158. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hohenberg-Kohn theorem (1964)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 34/64
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159. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
“There seems to be a misguided belief
that a one-particle density can determine
the exact N-body ground state.”
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 35/64
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160. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
“There seems to be a misguided belief
that a one-particle density can determine
the exact N-body ground state.”
Criticism on the Hohenberg-Kohn theorem, (1980).
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 35/64
ENFMC

161. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
“There seems to be a misguided belief
that a one-particle density can determine
the exact N-body ground state.”
n(r) Ψ(r1, r2, ..., rN )
Criticism on the Hohenberg-Kohn theorem, (1980).
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 35/64
ENFMC

162. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
“There seems to be a misguided belief
that a one-particle density can determine
the exact N-body ground state.”
n(r) · · · · · · · · · · · · Ψ(r1, r2, ..., rN )
Criticism on the Hohenberg-Kohn theorem, (1980).
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 35/64
ENFMC

163. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
An N-electron system has a complex multi-dimensional
wavefunction Ψ(r1, r2, ..., rN ) that depends on the
coordinates of all of its electrons
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 36/64
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164. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
An N-electron system has a complex multi-dimensional
wavefunction Ψ(r1, r2, ..., rN ) that depends on the
coordinates of all of its electrons
Integrating out N-1 degrees of freedom of |Ψ|2, we obtain the
probability of finding one electron in the volume element d3r
at r
n(r) = N d3
r2 · · · d3
rN |Ψ(r1, r2, ..., rN )|2
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 36/64
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165. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
An N-electron system has a complex multi-dimensional
wavefunction Ψ(r1, r2, ..., rN ) that depends on the
coordinates of all of its electrons
Integrating out N-1 degrees of freedom of |Ψ|2, we obtain the
probability of finding one electron in the volume element d3r
at r
n(r) = N d3
r2 · · · d3
rN |Ψ(r1, r2, ..., rN )|2
⇒ Can we eliminate all reference to N-electron wavefunction,
working entirely in terms of 1-electron density?
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 36/64
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166. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
The 1964-65 papers
A 40-year-old researcher doing a sabbatical in Paris, interested
in the electronic structure of alloys, publishes two seminal
papers.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 37/64
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167. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
The 1964-65 papers
A 40-year-old researcher doing a sabbatical in Paris, interested
in the electronic structure of alloys, publishes two seminal
papers.
In the first, he prooves by reductio ad absurdum, that the
relation
Ψ(r1, r2, ..., rN ) ⇒ n(r)
can be inverted:
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 37/64
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168. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
The 1964-65 papers
A 40-year-old researcher doing a sabbatical in Paris, interested
in the electronic structure of alloys, publishes two seminal
papers.
In the first, he prooves by reductio ad absurdum, that the
relation
Ψ(r1, r2, ..., rN ) ⇒ n(r)
can be inverted:
Given a ground-state density it is possible, in principle, to
calculate the corresponding ground-state wave function.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 37/64
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169. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
The 1964 paper
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 38/64
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170. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
General formulation: density as basic variable
“We shall be considering a collection of arbitrary number of electrons,
enclosed in a large box and moving under the influence of an external
potential v(r) and the mutual Coulomb repulsion.”
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 39/64
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171. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
General formulation: density as basic variable
“We shall be considering a collection of arbitrary number of electrons,
enclosed in a large box and moving under the influence of an external
potential v(r) and the mutual Coulomb repulsion.”
The Hamiltonian has the form H = T + V + U, where
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 39/64
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172. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
General formulation: density as basic variable
“We shall be considering a collection of arbitrary number of electrons,
enclosed in a large box and moving under the influence of an external
potential v(r) and the mutual Coulomb repulsion.”
The Hamiltonian has the form H = T + V + U, where
T = −
1
2
ψ∗
(r) 2
ψ(r)d3
r (2)
V = v(r)ψ∗
(r)ψ(r)d3
r (3)
U =
1
2
1
|r − r |
ψ∗
(r)ψ∗
(r )ψ(r )ψ(r)d3
r d3
r (4)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 39/64
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173. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
General formulation: density as basic variable
“We shall be considering a collection of arbitrary number of electrons,
enclosed in a large box and moving under the influence of an external
potential v(r) and the mutual Coulomb repulsion.”
The Hamiltonian has the form H = T + V + U, where
T = −
1
2
ψ∗
(r) 2
ψ(r)d3
r (2)
V = v(r)ψ∗
(r)ψ(r)d3
r (3)
U =
1
2
1
|r − r |
ψ∗
(r)ψ∗
(r )ψ(r )ψ(r)d3
r d3
r (4)
We consider T and U “universal” terms, whereas the external potential
v(r) determines the specificities of the system/Hamiltonian.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 39/64
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174. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Phys. Rev. 136 B864 (1964).
We shall in all that follows assume for simplicity that we are only dealing
with situations in which the ground state is nondegenerate.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 40/64
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175. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Phys. Rev. 136 B864 (1964).
We shall in all that follows assume for simplicity that we are only dealing
with situations in which the ground state is nondegenerate.
We denote the electronic density in the ground state Ψ by n(r),
n(r) = Ψ|ˆn|Ψ = Ψ|ψ ψ|Ψ (5)
which is clearly a functional of v(r) (v → H → Ψ → n).
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 40/64
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176. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Phys. Rev. 136 B864 (1964).
We shall in all that follows assume for simplicity that we are only dealing
with situations in which the ground state is nondegenerate.
We denote the electronic density in the ground state Ψ by n(r),
n(r) = Ψ|ˆn|Ψ = Ψ|ψ ψ|Ψ (5)
which is clearly a functional of v(r) (v → H → Ψ → n).
We shall now show that conversely v(r) is a unique functional of n(r),
apart from a trivial additive constant.”
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 40/64
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177. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Phys. Rev. 136 B864 (1964).
The proof proceeds by reductio ad absurdum.
Assume that another potential v (r), different by more than a constant,
with ground state Ψ gives rise to the same density n(r).
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 41/64
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178. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Phys. Rev. 136 B864 (1964).
The proof proceeds by reductio ad absurdum.
Assume that another potential v (r), different by more than a constant,
with ground state Ψ gives rise to the same density n(r).
Now clearly Ψ cannot be equal to Ψ since they satisfy different
Schr¨odinger equations:
H |Ψ = (T + U + V ) |Ψ = E |Ψ
H |Ψ = (T + U + V ) |Ψ = E |Ψ .
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 41/64
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179. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Phys. Rev. 136 B864 (1964).
The proof proceeds by reductio ad absurdum.
Assume that another potential v (r), different by more than a constant,
with ground state Ψ gives rise to the same density n(r).
Now clearly Ψ cannot be equal to Ψ since they satisfy different
Schr¨odinger equations:
H |Ψ = (T + U + V ) |Ψ = E |Ψ
H |Ψ = (T + U + V ) |Ψ = E |Ψ .
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 41/64
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180. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hence, if we denote the Hamiltonian and ground-state energies
associated with Ψ and Ψ by H, H and E, E ,
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 42/64
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181. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hence, if we denote the Hamiltonian and ground-state energies
associated with Ψ and Ψ by H, H and E, E ,
we have by the minimal property of the ground state,
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 42/64
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182. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hence, if we denote the Hamiltonian and ground-state energies
associated with Ψ and Ψ by H, H and E, E ,
we have by the minimal property of the ground state,
E = Ψ |H |Ψ
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 42/64
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183. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hence, if we denote the Hamiltonian and ground-state energies
associated with Ψ and Ψ by H, H and E, E ,
we have by the minimal property of the ground state,
E = Ψ |H |Ψ < Ψ|H |Ψ
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 42/64
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184. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hence, if we denote the Hamiltonian and ground-state energies
associated with Ψ and Ψ by H, H and E, E ,
we have by the minimal property of the ground state,
E = Ψ |H |Ψ < Ψ|H |Ψ
< Ψ|T + U + V |Ψ
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 42/64
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185. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hence, if we denote the Hamiltonian and ground-state energies
associated with Ψ and Ψ by H, H and E, E ,
we have by the minimal property of the ground state,
E = Ψ |H |Ψ < Ψ|H |Ψ
< Ψ|T + U + V |Ψ
< Ψ|T + U + V − V + V |Ψ
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 42/64
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186. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hence, if we denote the Hamiltonian and ground-state energies
associated with Ψ and Ψ by H, H and E, E ,
we have by the minimal property of the ground state,
E = Ψ |H |Ψ < Ψ|H |Ψ
< Ψ|T + U + V |Ψ
< Ψ|T + U + V − V + V |Ψ
< Ψ|H − V + V |Ψ
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 42/64
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187. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hence, if we denote the Hamiltonian and ground-state energies
associated with Ψ and Ψ by H, H and E, E ,
we have by the minimal property of the ground state,
E = Ψ |H |Ψ < Ψ|H |Ψ
< Ψ|T + U + V |Ψ
< Ψ|T + U + V − V + V |Ψ
< Ψ|H − V + V |Ψ
E < E + Ψ|V − V |Ψ ,
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 42/64
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188. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hence, if we denote the Hamiltonian and ground-state energies
associated with Ψ and Ψ by H, H and E, E ,
we have by the minimal property of the ground state,
E = Ψ |H |Ψ < Ψ|H |Ψ
< Ψ|T + U + V |Ψ
< Ψ|T + U + V − V + V |Ψ
< Ψ|H − V + V |Ψ
E < E + Ψ|V − V |Ψ ,
so that
E < E + d3
r n(r) [v (r) − v(r)] .
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 42/64
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189. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
...
E < E + d3
r n(r) [v (r) − v(r)] . (6)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 43/64
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190. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
...
E < E + d3
r n(r) [v (r) − v(r)] . (6)
Interchanging primed and unprimed quantities, we find in exactly the
same way that
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 43/64
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191. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
...
E < E + d3
r n(r) [v (r) − v(r)] . (6)
Interchanging primed and unprimed quantities, we find in exactly the
same way that
E < E + d3
r n(r) [v(r) − v (r)] . (7)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 43/64
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192. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
...
E < E + d3
r n(r) [v (r) − v(r)] . (6)
Interchanging primed and unprimed quantities, we find in exactly the
same way that
E < E + d3
r n(r) [v(r) − v (r)] . (7)
Addition of 6 and 7 leads to the inconsistency
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 43/64
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193. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
...
E < E + d3
r n(r) [v (r) − v(r)] . (6)
Interchanging primed and unprimed quantities, we find in exactly the
same way that
E < E + d3
r n(r) [v(r) − v (r)] . (7)
Addition of 6 and 7 leads to the inconsistency
E + E < E + E . (8)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 43/64
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194. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
...
E < E + d3
r n(r) [v (r) − v(r)] . (6)
Interchanging primed and unprimed quantities, we find in exactly the
same way that
E < E + d3
r n(r) [v(r) − v (r)] . (7)
Addition of 6 and 7 leads to the inconsistency
E + E < E + E . (8)
Thus v(r) is (to within a constant) a unique functional of n(r).
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 43/64
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195. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
...
E < E + d3
r n(r) [v (r) − v(r)] . (6)
Interchanging primed and unprimed quantities, we find in exactly the
same way that
E < E + d3
r n(r) [v(r) − v (r)] . (7)
Addition of 6 and 7 leads to the inconsistency
E + E < E + E . (8)
Thus v(r) is (to within a constant) a unique functional of n(r).
Since, in turn, v(r) fixes H we see that the full many-particle ground
state is a unique functional of n(r).
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 43/64
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196. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
The original proof
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 44/64
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197. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hohenberg-Kohn theorem
Since n(r) determines v(r), it gives us the full Hamiltonian.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 45/64
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198. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hohenberg-Kohn theorem
Since n(r) determines v(r), it gives us the full Hamiltonian.
Hence n(r) determines implicitly all properties derivable from H through
the solution of the Schr¨odinger equation, such as:
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 45/64
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199. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hohenberg-Kohn theorem
Since n(r) determines v(r), it gives us the full Hamiltonian.
Hence n(r) determines implicitly all properties derivable from H through
the solution of the Schr¨odinger equation, such as:
the many-body eigenstates Ψ0(r1, ..., rN ), Ψ1(r1, ..., rN ), ...
the Green’s functions G(r1t1; ...; rN tN )
the response functions χ(r, r , ω)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 45/64
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200. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Hohenberg-Kohn theorem
Since n(r) determines v(r), it gives us the full Hamiltonian.
Hence n(r) determines implicitly all properties derivable from H through
the solution of the Schr¨odinger equation, such as:
the many-body eigenstates Ψ0(r1, ..., rN ), Ψ1(r1, ..., rN ), ...
the Green’s functions G(r1t1; ...; rN tN )
the response functions χ(r, r , ω)
all observables Ψ[n]| ˆO |Ψ[n] = O[n]
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 45/64
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201. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
“Finally it occurred to me that for a single particle there is
an explicit elementary relation between the potential v(r)
and the density, n(r), of the groundstate. Taken together,
these provided strong support for the conjective that the
density n(r) completely determines the external potential
v(r). This would imply that n(r) which integrates to N, the
total number of electrons, also determines the total
Hamiltonian H and hence all properties derivable from H
and N, e.g. the wavefunction of the 17th excited state! (...)
Could this be true? And how could it be decided? Could
two different potentials, v1(r) and v2(r), with associated
different groundstates Ψ1(r1, ...rN ) and Ψ2(r1, ...rN ) give
rise to the same density distribution? It turned out that a
simple 3-line argument, using my beloved Rayleigh Ritz
variational principle, confirmed the conjecture. It seemed
such a remarkable result that I did not trust myself.”
W.Kohn
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 46/64
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202. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
“Finally it occurred to me that for a single particle there is
an explicit elementary relation between the potential v(r)
and the density, n(r), of the groundstate. Taken together,
these provided strong support for the conjective that the
density n(r) completely determines the external potential
v(r). This would imply that n(r) which integrates to N, the
total number of electrons, also determines the total
Hamiltonian H and hence all properties derivable from H
and N, e.g. the wavefunction of the 17th excited state! (...)
Could this be true? And how could it be decided? Could
two different potentials, v1(r) and v2(r), with associated
different groundstates Ψ1(r1, ...rN ) and Ψ2(r1, ...rN ) give
rise to the same density distribution? It turned out that a
simple 3-line argument, using my beloved Rayleigh Ritz
variational principle, confirmed the conjecture. It seemed
such a remarkable result that I did not trust myself.”
W.Kohn
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 46/64
ENFMC

203. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
“Finally it occurred to me that for a single particle there is
an explicit elementary relation between the potential v(r)
and the density, n(r), of the groundstate. Taken together,
these provided strong support for the conjective that the
density n(r) completely determines the external potential
v(r). This would imply that n(r) which integrates to N, the
total number of electrons, also determines the total
Hamiltonian H and hence all properties derivable from H
and N, e.g. the wavefunction of the 17th excited state! (...)
Could this be true? And how could it be decided? Could
two different potentials, v1(r) and v2(r), with associated
different groundstates Ψ1(r1, ...rN ) and Ψ2(r1, ...rN ) give
rise to the same density distribution? It turned out that a
simple 3-line argument, using my beloved Rayleigh Ritz
variational principle, confirmed the conjecture. It seemed
such a remarkable result that I did not trust myself.”
W.Kohn
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 46/64
ENFMC

204. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
“Finally it occurred to me that for a single particle there is
an explicit elementary relation between the potential v(r)
and the density, n(r), of the groundstate. Taken together,
these provided strong support for the conjective that the
density n(r) completely determines the external potential
v(r). This would imply that n(r) which integrates to N, the
total number of electrons, also determines the total
Hamiltonian H and hence all properties derivable from H
and N, e.g. the wavefunction of the 17th excited state! (...)
Could this be true? And how could it be decided? Could
two different potentials, v1(r) and v2(r), with associated
different groundstates Ψ1(r1, ...rN ) and Ψ2(r1, ...rN ) give
rise to the same density distribution? It turned out that a
simple 3-line argument, using my beloved Rayleigh Ritz
variational principle, confirmed the conjecture. It seemed
such a remarkable result that I did not trust myself.”
W.Kohn
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 46/64
ENFMC

205. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
“Finally it occurred to me that for a single particle there is
an explicit elementary relation between the potential v(r)
and the density, n(r), of the groundstate. Taken together,
these provided strong support for the conjective that the
density n(r) completely determines the external potential
v(r). This would imply that n(r) which integrates to N, the
total number of electrons, also determines the total
Hamiltonian H and hence all properties derivable from H
and N, e.g. the wavefunction of the 17th excited state! (...)
Could this be true? And how could it be decided? Could
two different potentials, v1(r) and v2(r), with associated
different groundstates Ψ1(r1, ...rN ) and Ψ2(r1, ...rN ) give
rise to the same density distribution? It turned out that a
simple 3-line argument, using my beloved Rayleigh Ritz
variational principle, confirmed the conjecture. It seemed
such a remarkable result that I did not trust myself.”
W.Kohn
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 46/64
ENFMC

206. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
“Finally it occurred to me that for a single particle there is
an explicit elementary relation between the potential v(r)
and the density, n(r), of the groundstate. Taken together,
these provided strong support for the conjective that the
density n(r) completely determines the external potential
v(r). This would imply that n(r) which integrates to N, the
total number of electrons, also determines the total
Hamiltonian H and hence all properties derivable from H
and N, e.g. the wavefunction of the 17th excited state! (...)
Could this be true? And how could it be decided? Could
two different potentials, v1(r) and v2(r), with associated
different groundstates Ψ1(r1, ...rN ) and Ψ2(r1, ...rN ) give
rise to the same density distribution? It turned out that a
simple 3-line argument, using my beloved Rayleigh Ritz
variational principle, confirmed the conjecture. It seemed
such a remarkable result that I did not trust myself.”
W.Kohn
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 46/64
ENFMC

207. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
“Finally it occurred to me that for a single particle there is
an explicit elementary relation between the potential v(r)
and the density, n(r), of the groundstate. Taken together,
these provided strong support for the conjective that the
density n(r) completely determines the external potential
v(r). This would imply that n(r) which integrates to N, the
total number of electrons, also determines the total
Hamiltonian H and hence all properties derivable from H
and N, e.g. the wavefunction of the 17th excited state! (...)
Could this be true? And how could it be decided? Could
two different potentials, v1(r) and v2(r), with associated
different groundstates Ψ1(r1, ...rN ) and Ψ2(r1, ...rN ) give
rise to the same density distribution? It turned out that a
simple 3-line argument, using my beloved Rayleigh Ritz
variational principle, confirmed the conjecture. It seemed
such a remarkable result that I did not trust myself.”
W.Kohn
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 46/64
ENFMC

208. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
“Finally it occurred to me that for a single particle there is
an explicit elementary relation between the potential v(r)
and the density, n(r), of the groundstate. Taken together,
these provided strong support for the conjective that the
density n(r) completely determines the external potential
v(r). This would imply that n(r) which integrates to N, the
total number of electrons, also determines the total
Hamiltonian H and hence all properties derivable from H
and N, e.g. the wavefunction of the 17th excited state! (...)
Could this be true? And how could it be decided? Could
two different potentials, v1(r) and v2(r), with associated
different groundstates Ψ1(r1, ...rN ) and Ψ2(r1, ...rN ) give
rise to the same density distribution? It turned out that a
simple 3-line argument, using my beloved Rayleigh Ritz
variational principle, confirmed the conjecture. It seemed
such a remarkable result that I did not trust myself.”
W.Kohn
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 46/64
ENFMC

209. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Parenthesis
“The approach Hohenberg and Kohn took (...) reflects the
way Kohn chose to frame the final published paper. There
is no mention of the alloy problem or even of any desire to
re-formulate the electronic structure problem for solids.
Instead, the title of the HK paper is simply
“Inhomogeneous electron gas” and the first line of the
abstract announces that “this paper deals with the ground
state of an interacting electron gas in an external potential
v(r).” The Introduction goes on (...) HK then remind the
reader about the Thomas-Fermi method.”
Andrew Zangwill, arxiv:1403:5164
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 47/64
ENFMC

210. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Parenthesis
“The approach Hohenberg and Kohn took (...) reflects the
way Kohn chose to frame the final published paper. There
is no mention of the alloy problem or even of any desire to
re-formulate the electronic structure problem for solids.
Instead, the title of the HK paper is simply
“Inhomogeneous electron gas” and the first line of the
abstract announces that “this paper deals with the ground
state of an interacting electron gas in an external potential
v(r).” The Introduction goes on (...) HK then remind the
reader about the Thomas-Fermi method.”
Andrew Zangwill, arxiv:1403:5164
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 47/64
ENFMC

211. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Parenthesis
“The approach Hohenberg and Kohn took (...) reflects the
way Kohn chose to frame the final published paper. There
is no mention of the alloy problem or even of any desire to
re-formulate the electronic structure problem for solids.
Instead, the title of the HK paper is simply
“Inhomogeneous electron gas” and the first line of the
abstract announces that “this paper deals with the ground
state of an interacting electron gas in an external potential
v(r).” The Introduction goes on (...) HK then remind the
reader about the Thomas-Fermi method.”
Andrew Zangwill, arxiv:1403:5164
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 47/64
ENFMC

212. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Parenthesis
“The approach Hohenberg and Kohn took (...) reflects the
way Kohn chose to frame the final published paper. There
is no mention of the alloy problem or even of any desire to
re-formulate the electronic structure problem for solids.
Instead, the title of the HK paper is simply
“Inhomogeneous electron gas” and the first line of the
abstract announces that “this paper deals with the ground
state of an interacting electron gas in an external potential
v(r).” The Introduction goes on (...) HK then remind the
reader about the Thomas-Fermi method.”
Andrew Zangwill, arxiv:1403:5164
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 47/64
ENFMC

213. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Parenthesis
“The approach Hohenberg and Kohn took (...) reflects the
way Kohn chose to frame the final published paper. There
is no mention of the alloy problem or even of any desire to
re-formulate the electronic structure problem for solids.
Instead, the title of the HK paper is simply
“Inhomogeneous electron gas” and the first line of the
abstract announces that “this paper deals with the ground
state of an interacting electron gas in an external potential
v(r).” The Introduction goes on (...) HK then remind the
reader about the Thomas-Fermi method.”
Andrew Zangwill, arxiv:1403:5164
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 47/64
ENFMC

214. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Parenthesis
“The approach Hohenberg and Kohn took (...) reflects the
way Kohn chose to frame the final published paper. There
is no mention of the alloy problem or even of any desire to
re-formulate the electronic structure problem for solids.
Instead, the title of the HK paper is simply
“Inhomogeneous electron gas” and the first line of the
abstract announces that “this paper deals with the ground
state of an interacting electron gas in an external potential
v(r).” The Introduction goes on (...) HK then remind the
reader about the Thomas-Fermi method”
Andrew Zangwill, arxiv:1403:5164
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 47/64
ENFMC

215. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Outline
1 Electronic structure
2 Hartree and Hartree-Fock methods
3 Thomas-Fermi model
4 Hohenberg-Kohn Theorem
5 Kohn-Sham Scheme
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 47/64
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216. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Kohn Sham (1965)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 47/64
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217. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
Kohn Sham (1965)
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 46/64
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218. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
We have seen that from a 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.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 47/64
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219. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
We have seen that from a 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.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 47/64
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220. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
1964 paper’s concerns
HK mentions an universal functional (F[n] = T[n] + U[n])
and recognize the necessity to determine it
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 48/64
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221. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
1964 paper’s concerns
They distinguish the Hartree term (classical Coulomb energy)
and separate it from the functional to be approximated
HK knew that the Thomas-Fermi model follows from (15) by
approximating the kinetic energy
E = d3
r n(r)vext(r) +
1
2
d3
rd3
r
n(r)n(r )
|r − r |
+ TTF [n]
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 49/64
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222. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
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.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 50/64
ENFMC

223. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
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.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 50/64
ENFMC

224. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
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.
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 50/64
ENFMC

225. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
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 50/64
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226. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
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 51/64
ENFMC

227. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
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 51/64
ENFMC

228. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
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 51/64
ENFMC

229. Motivation ElStr Hartree-Fock Thomas-Fermi Hohenberg-Kohn Kohn-Sham
First Kohn and Sham observed that in the Hartree method,
each electron moves independently in an effective potential
which does not recognize the individual identity of the other
electrons.
−
2
2m
2
+ ˆveff (r) ϕi(r) = iϕi(r) ,
The kinetic energy for independent (non-interacting) electrons
is:
TS [n] =
N
i=1
ϕ∗
(r) −
2
2m
2
ϕ(r)d3
r
(S: single-particle)
OBS: The true kinetic energy of an interacting system is not
TS , we miss a term that describes the correlated motion
Mariana M. Odashima Introduction to density functional theory XXXVIII ENFMC Foz do Iguac¸u 52/64
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