1. DFT, 2007

2. Observation functional An exact generalization of DFT  States, observables, observations  Variational principles  Generalized density functional  Exact generalized Kohn-Sham Eq.  Finite temperature extension  Skyrme from Relativistic MF Ph. Chomaz, C. Ducoin, K. Hasnaoui - GANIL-Caen E. Khan, J. Margueron - IPN-Orsay C. Providencia, D.P.Menezes, L. Brito, and Ph. Chomaz, PRC2007

3. DFT, 2007

4. A) States, Observables and Observations

5. DFT, 2007 States Observables Observation A) States, Observables and Observations Many-body wave function Hilbert or Fock space

6. DFT, 2007 States Observables Observation A) States, Observables and Observations Many-body wave function Hilbert or Fock space Density matrix Liouville space

7. DFT, 2007 States Observables Observation A) States, Observables and Observations Many-body wave function Hilbert or Fock space Density matrix Liouville space Scalar product in matrix space Generalized density

8. DFT, 2007

9. Static Dynamics B) Variational principles Schr ö dinger equation Extremum of the action I Balian and Vénéroni double principle Liouville equation Observables backward from t 1 Density forward from t 0

10. DFT, 2007 Static Dynamics B) Variational principles Zero Temperature minimum energy E Finite T minimum free energy Entropy Schr ö dinger equation Extremum of the action I Balian and Vénéroni double principle Liouville equation Observables backward from t 1 Density forward from t 0

11. DFT, 2007 Static Dynamics B) Variational principles Zero Temperature minimum energy E Finite T minimum free energy Entropy Schr ö dinger equation Extremum of the action I Balian and Vénéroni double principle Liouville equation Observables backward from t 1 Density forward from t 0

12. DFT, 2007

13. C) Exact generalized Density functional

14. DFT, 2007 Observations E functional C) Exact generalized Density functional Generalized density

15. DFT, 2007 Observations E functional  Min in a subspace with the density constraint C) Exact generalized Density functional Generalized density

16. DFT, 2007 Observations E functional  Min in a subspace with the density constraint Theorem:  Two steps minimization  Exact ground state C) Exact generalized Density functional Generalized density Or

17. DFT, 2007 Observations E functional  Min in a subspace with the density constraint Theorem:  Two steps minimization  Exact ground state C) Exact generalized Density functional Generalized density Or Demonstration:

18. DFT, 2007 E functional  Min in a subspace with the density constraint C) Generalized DFT

19. DFT, 2007 E functional  Min in a subspace with the density constraint Lagrange multiplier:  Min with no constraint C) Generalized DFT

20. DFT, 2007 E functional  Min in a subspace with the density constraint Lagrange multiplier:  Min with no constraint  External field C) Generalized DFT and external field

21. DFT, 2007 E functional  Min in a subspace with the density constraint Lagrange multiplier:  Min with no constraint  External field  Legendre transform C) Generalized DFT and external field

22. DFT, 2007 E functional  Min in a subspace with the density constraint Lagrange multiplier:  Min with no constraint  External field  Legendre transform C) Generalized DFT and external field  Existence / analyticity No degeneracy / no phase trans.  Invertibility

23. DFT, 2007 E functional  Min in a subspace with the density constraint Lagrange multiplier:  Min with no constraint  External field  Legendre transform C) Generalized DFT and external field  Existence / analyticity No degeneracy / no phase trans.  Invertibility Cf. Höhenberg-Kohn theorem

24. DFT, 2007 E functional  Min in a subspace with the density constraint Lagrange multiplier:  Min with no constraint  External field  Legendre transform C) Generalized DFT and external field  Existence / analyticity No degeneracy / no phase trans.  Invertibility

25. DFT, 2007

26. Exact E functional  For a set of observations D) Exact Generalized Kohn-Sham Eq. Generalized density

27. DFT, 2007 Exact E functional  For a set of observations Exact ground state E  => exact densities  D) Exact Generalized Kohn-Sham Eq. Generalized density

28. DFT, 2007 Exact E functional  For a set of observations Exact ground state E  => exact densities  Variation  Equivalent to MF Equation with Lie algebra including A D) Exact Generalized Kohn-Sham Eq. Generalized density

29. DFT, 2007 Remarks  Exact for E and all observations =  l  included in E[  ]  Easy to go from a set of A l to a reduced set A’ l => E’[  ‘]=min  ‘=cst E[  ] D) Exact Generalized Kohn-Sham Eq.

30. DFT, 2007 Lie algebra Observation Trial states Hamiltonian Independent particles Mean Field E) Hartree Fock Kohn-Sham One-body density One-body observables Thouless theorem (Slaters) Independent particle state

31. DFT, 2007

32. One body, one body density E) Hartree Fock Kohn-Sham

33. DFT, 2007 One body, one body density => Mean field E) Hartree Fock Kohn-Sham

34. DFT, 2007 One body, one body density => Mean field Local density approximation  Energy density functional  Local densities matter, kinetic, current  Mean field E) Hartree Fock Kohn-Sham : LDA

35. DFT, 2007 One body, one body density => Mean field Local density approximation  Energy density functional  Local densities matter, kinetic, current  Mean field E) Hartree Fock Kohn-Sham : LDA

36. DFT, 2007 E) LDA: Skyrme case 36 Standard case few densities  Matter isoscalar isovector  kinetic isoscalar isovector  Spin isoscalar isovector Energy functional Mean-field q=(n,p)

37. DFT, 2007 Standard case few densities  Matter isoscalar isovector  kinetic isoscalar isovector  Spin isoscalar isovector Energy functional Mean-field q=(n,p) E) LDA: Skyrme case 36 Skyrme parameters

38. DFT, 2007

39. Maximum entropy Under energy constraint  Free energy Equilibrium  Partition sum Equation of states  Legendre transform  Free energy 1st order Phase transition F) Finite temperature

40. DFT, 2007 Minimum entropy Observables Generalized densities Free energy functional Lagrange “external field” Minimum under constraints  Extrenal field  Max entropy states  Partition sum  EOS  Legendre transform F) Free energy functional

41. DFT, 2007 Minimum entropy Observables Generalized densities Free energy functional Lagrange “external field” Minimum under constraints  Extrenal field  Max entropy states  Partition sum  EOS  Legendre transform F) Free energy functional

42. DFT, 2007 Thermo with external field Partition sum  dependent free energy Free energy functional Link with density  Legendre Existence /analyticity Curvature = fluctuation => Unicity F) Free energy functional No phase trans. Convex free energy

43. DFT, 2007 Thermo with external field Partition sum  dependent free energy Free energy functional Link with density  Legendre Existence /analyticity Curvature = fluctuation => Unicity F) Free energy functional No phase trans. Convex free energy

44. DFT, 2007 Max Free energy functional Assume a Lie algebra  Trial density  Density  Entropy, S variation Mean field  Energy, E variation  Free energy variation Generalized Kohn-Sham F) Finite T Generalized Kohn-Sham Eq.

45. DFT, 2007 Max Free energy functional Assume a Lie algebra  Trial density  Density  Entropy, S variation Mean field  Energy, E variation  Free energy variation Generalized Kohn-Sham F) Finite T Generalized Kohn-Sham Eq.

46. DFT, 2007

47. Free energy observation-functional Thermo with external field  Partition sum   dependent free energy Free energy funct. = Legendre tr. Unique density Problem with phase transition General theorems and methods Clarification of comparisons between functionals Systematic method to reduce information (eg subset of A or  ) Inclusion of correlations (extension of A, eg fluctuations) G) Conclusion

48. DFT, 2007

49. Skyrme functional  Isospin part  Kinetic  as  -functional (Matter properties LDA) H) DFT: Skyrme versus Relat. MF C. Providencia, D.P.Menezes, L. Brito, and Ph. Chomaz, PRC2007

50. DFT, 2007 Skyrme functional  Isospin part  Kinetic  as  -functional (Matter properties LDA) H) DFT: Skyrme versus Relat. MF C. Providencia, D.P.Menezes, L. Brito, and Ph. Chomaz, PRC2007 Relat. M.F. (Walecka)  Nucleon fields  Meson fields  pot.  mass

51. DFT, 2007 Equilibrium Meson field Densities H) Equilibrium Relativistic MF C. Providencia, D.P.Menezes, L. Brito, and Ph. Chomaz, PRC2007  pot.  mass  Barions  Sacalar  Kinetic

52. DFT, 2007 Energy functional Densities H) Equilibrium Relativistic MF C. Providencia, D.P.Menezes, L. Brito, and Ph. Chomaz, PRC2007  Barions  Sacalar  Kinetic

53. DFT, 2007 H) Non-Relat. Approxim => Skyrme C. Providencia, D.P.Menezes, L. Brito, and Ph. Chomaz, PRC2007  Scalar  Kinetic Non relat. approx. Densities

54. DFT, 2007 H) Non-Relat. Approxim => Skyrme C. Providencia, D.P.Menezes, L. Brito, and Ph. Chomaz, PRC2007  Scalar  Kinetic  Relativistic  Skyrme Non relat. approx. Densities Energy functional

55. DFT, 2007 H) Non-Relat. Approxim => Skyrme C. Providencia, D.P.Menezes, L. Brito, and Ph. Chomaz, PRC2007  Scalar  Kinetic  Relativistic  Skyrme Non relat. approx. Densities Energy functional

56. DFT, 2007 Comparison of functionals H) Non-Relat. Approxim => Skyrme C. Providencia, D.P.Menezes, L. Brito, and Ph. Chomaz, PRC2007

57. DFT, 2007 Comparison of functionals H) Non-Relat. Approxim => Skyrme C. Providencia, D.P.Menezes, L. Brito, and Ph. Chomaz, PRC2007 fit

58. DFT, 2007