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Observation functional An exact generalization of DFT

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Presentation on theme: "Observation functional An exact generalization of DFT"— Presentation transcript:

1

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 31

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

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

6 B) Variational principles
Static Dynamics Zero Temperature minimum energy E Finite T minimum free energy Entropy Schrödinger equation Liouville 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 C) Generalized mean-field
Coherent states Generalized density Extremum action Trial observables Group transformation Maximum entropy trial state With the constraints Constrained entropy Lagrange multipliers Lie Algebra Group parameters Mean-field <=> Ehrenfest 31

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

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

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

12 Exact E and  in an external field U=zlAl
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 Energy functional
Matter isoscalar isovector kinetic isoscalar isovector Spin isoscalar isovector Energy functional Mean-field q=(n,p) 36

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

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