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

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

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

Non-uniform system thermodynamics - quantum Grand potential - quantum temperature β = 1/K B T local chemical potential pair potential

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?

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

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

Effective temperature – classical virial equation Effective local chemical potential

Uniform Fermi Fluid HNC Quantum input for uniform ideal Fermi gas

r -2 tail Effective interaction potential Effective temperature

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

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

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

Prediction: Pair Correlation function r s = 5

T=0

Local field corrections T=0 r0=5

Model for the effective potential

Application to Charges in a harmonic Trap HNC OCP direct correlations of the Jellium model ( classical – no quantum effects)

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

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

Shells from diffraction (r s = 5) for Kelbg potential

Origin of classical shell structure – Coulomb correlations

Degeneracy effects

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 ??