Observation functional An exact generalization of DFT

2. Observation functional An exact generalization of DFT Philippe CHOMAZ - GANIL States, observables, observations Variational principles Generalized mean-Field Hartree-Fock Hierarchies and fluctuations Exact generalized density functional Exact generalized Kohn-Sham Eq. 1

3. A) States, Observables and Observations Many-body wave function Hilbert or Fock space • States • Observables • Observation 31

4. A) States, Observables and Observations Many-body wave function Hilbert or Fock space Density matrix Liouville space • States • Observables • Observation Scalar product in matrix space 31

5. B) Variational principles • Static • Dynamics Schrödinger equation Extremum of the action I 31

6. B) Variational principles Zero Temperature minimum energy E • Static • Dynamics Finite T minimum free energy Entropy Liouville equation Schrödinger equation Balian and Vénéroni double principle Extremum of the action I Observables backward from t1 Density forward from t0 31

7. C) Generalized mean-field • Coherent states • Generalized density • Extremum action Group transformation Lie Algebra Group parameters Mean-field <=> Ehrenfest 31

8. Maximum entropy trial state With the constraints Constrained entropy Lagrange multipliers C) Generalized mean-field Trial observables • Coherent states • Generalized density • Extremum action Group transformation Lie Algebra Group parameters Mean-field <=> Ehrenfest 31

9. D) Hartree Fock One-body observables One-body density • Lie algebra • Observation • Trial states • Hamiltonian • Independent particles • Mean Field Independent particle state Thouless theorem (Slaters) 31

10. E) Hierarchies and fluctuations Close the Lie Algebra including A an H • Exact dynamics • Hierarchy • Projections <A> • Minimum entropy • Correlation • MF Langevin • Mean-Field Coupled equations 31

11. F) Exact generalized Density functional Or • Exact State • Exact Observations • Exact E functional • Min in a subspace • Constrained energy <=> external field Generalized density 31

12. G) Exact Generalized Kohn-Sham Eq. • Exact E functional • For a set of observations • Exact ground state E => exact densities  • Variation • Equivalent to mean-field Eq. with Lie algebra including Al , {Al , A’m } Generalized density Exact E and  in an external field U=zlAl 31

13. G) Exact Generalized Kohn-Sham Eq. • Remarks • Exact for E and all observations <Al> =lincluded in E[] • Easy to go from a set of Al to a reduced set A’l => E’[‘]=min‘=cst E[] 31

14. H) Density functional theory : LDA ^ • The only information needed is the energy => functionals of r • Local density approximation • Energy density functional • Local densities matter ,kinetic ,current • Mean field 35

15. H) LDA: Skyrme case • Standard case few densities • Matterisoscalar isovector • kineticisoscalar isovector • Spin isoscalar isovector • Energy functional • Mean-field q=(n,p) 36

16. Skyrme parameters H) LDA: Skyrme case • Standard case few densities • Matterisoscalar isovector • kineticisoscalar isovector • Spin isoscalar isovector • Energy functional • Mean-field q=(n,p) 36