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## Overview of Simulations of Quantum Systems

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**Overview of Simulations of Quantum Systems**Roberto Car, Princeton University Croucher ASI, Hong Kong, December 7 2005**Foreword**This is a vast subject. I will only be able to give a very partial overview of the field from a very personal perspective: the perspective of Molecular Dynamics simulations based on Density Functional Theory.**First Principles Molecular Dynamics**• Classical atomic trajectories • Interatomic potential from the quantum-mechanical ground-state of the electrons • Equilibrium properties (static and dynamic) as temporal averages**This approach opened the way to first-principles studies of**liquids and has greatly contributed to the application of DFT to complex material and molecular systems. • Three illustrative examples from my own work: (a) the phase diagram of C, (b) the IR absorption of liquid water (a dynamical property), (c) solvent mediated force between two methane molecules in water (hydrophobic effect)**Free energy by thermodynamic integration**Adiabatic switching (Watanabe and Reinhardt):**Phase diagram of carbon from DFT simulations**Calculated melting line – Clapeyron slopes are in red The complete phase diagram of Carbon X. Wang, S. Scandolo, R.C, PRL 95 (2005)**Dynamic response of water to an electric field: IR**spectroscopy Within linear response theory the infrared absorption coefficient derives from the fluctuations of the cell dipole moment M = i i The modes at ~ 185 cm-1 which are associated to hindered translations of the water molecules M. Sharma, R. Resta, R.C., PRL 95 (2005)**Rigid translations of the central molecule are hindered by**the H-bonds that a molecule forms with its neighbors, which define a (distorted) local tetrahedral cage Translations of a rigid dipole do not couple to uniform electric fields. Hence the origin of the IR feature at ~185 cm-1 must be electronic. It has been attributed (Madden and Impey, CPL 1986) to an induced molecular dipole, a consequence of the dynamic polarizability of the water molecule (induced intramolecular dipole). First principles molecular dynamics simulations do not support this interpretation but show that the effect is mainly intermolecular**Two methanes (CH4) in water attract each other by**hydrophobic effect J-L Li, RC, C. Tang, NS Wingreen, (2005)**The average effective force is obtained from the average**force of constraint that keeps the two molecules at fixed distance. The potential of mean force is obtained by integrating the average effective force**Issues**• Is DFT bonding good enough? • Multiple time and size scales: coarse graining • Quantum effects in nuclear motion? • Non equilibrium quantum processes: electron transport**Is DFT good enough?**Melting temperature from different DFT approximations for Si: LDA: 1350+/-50 K (O.Sugino, RC (1995)), GGA: 1492+/-50 K (D. Alfe, M. Gillan (2003)), MetaGGA (TPSS): 1700+/-50 K (X.Wang, S.Scandolo, RC (to be published)). EXP: 1687 K**Chemical reactions**In all these reactions we observe a systematic improvement in the barrier going from LDA to GGA to METAGGA with B3LYP being closer to experiment or accurate quantum chemical calculations Y.Kanai, X. Wang, A. Selloni, RC (2005)**Quantum nuclei: thermal equilibrium properties**• First Principles Path Integral Molecular Dynamics (M. Parrinello and collaborators) • Here I just mention a recent extension of the scheme (D. Sebastiani and RC) to compute the proton momentum distribution (which can be compared with Compton neutron scattering experiments)**Modeling quantum systems in non-equilibrium situations:**Molecular Electronics: We are interested in the steady state current. The relaxation time to achieve stationary conditions is large compared to the timescales of both electron dynamics and lattice dynamics. This makes a kinetic approach possible.**A scheme introduced by R. Gebauer and RC allows to deal with**an electron flux in a close circuit. (PRL 2004, PRB2004) Kinetic approach: master equation The single-particle Kohn-Sham approach is generalized to dissipative quantum system (Burke, Gebauer, RC, PRL 2005)**Benzene dithiol between gold electrodes**Atomic point contact (Gold on gold)**Steady state electron current flux through an atomic point**contact (S. Piccinin, R. Gebauer, R.C., to be published)**Quantum tunneling through a molecular contact**Landauer formula**Conclusions**• DFT based quantum simulation remains a very active area • A number of challenging issues exist (functionals, large and complex systems, rare events, quantum effects (equilibrium, non equilibrium) • Coarse graining in space and time would open new perspectives