Sandipan Dutta and James Dufty Department of Physics, University of Florida

1. Classical Representation of Quantum Systems Sandipan Dutta and James Dufty Department of Physics, University of Florida Work supported under US DOE Grants DE-SC0002139 and DE-FG02-07ER54946

2. Overview • Objective – exploit classical methods for to describe correlations in quantum systems. • Method – map quantum system thermodynamics onto a representative classical system. • Applications – Explicitly build the representative systems for ideal Fermi gas and weakly coupled jellium: interaction potential local chemical potential temperature. • Application - shell structure of confined charges.

3. Can it work? Dharma-wardana and Perrot, PRL 84, 959 (2000); see review Dharma-wardana (2011), Arxiv: 1103 6070v1 ideal gas potential Deutch wavelength fit to T=0 xc energy Implement classical stat mech via HNC - examples

4. Non-uniform system thermodynamics - quantum Grand potential - quantum temperature β = 1/KB T local chemical potential pair potential

5. Non-uniform system thermodynamics - classical Grand potential - classical effective temperature effective local chemical potential effective pair potential Problem: how to define classical parameters to impose equivalence of thermodynamics and structure?

6. Definition of classical / quantum equivalence Solve for the unknown parameters in

7. Solution of the thermodynamic parameters Effective interaction potential– HNC equation Ornstein-Zernike equation

8. Effective local chemical potential Effective temperature – classical virial equation

9. Uniform Fermi Fluid HNC Quantum input for uniform ideal Fermi gas

10. Effective temperature r -2 tail Effective interaction potential

11. Predictions from the map- Internal Energy By definition: By calculation: exact

12. Representative system for Jellium in weak coupling Coulomb effects exchange effects Weak coupling limit: Proposed approximate classical jellium potential

13. Some properties of the RPA classical potential perfect screening sum rule Large r: Comparison with Dharma-wardana ( low density – diffraction only )

14. Prediction: Pair Correlation function rs = 5

15. T=0

17. Local field corrections T=0 r0=5

18. Model for the effective potential

19. Application to Charges in a harmonic Trap HNC OCPdirectcorrelations of the Jellium model ( classical – no quantum effects)

20. Lowest order map – inhomogeneous ideal Fermi gas LDA (Thomas-Fermi)

21. Quantum effects on shell formation in mean field limit . mean field Classical limit, Coulomb no shell structure at any coupling strength Diffraction

22. Shells from diffraction (rs = 5) for Kelbg potential

23. Origin of classical shell structure – Coulomb correlations

24. Degeneracy effects

25. Summary • Quantum – Classical map defined for thermodynamics and structure • Implementation of map with two exact limits • Application to jellium via HNC integral equation - in progress (need finite T simulation data for benchmark!) • Application to shell structure for charges in trap • Extension to orbital free density functional theory ??