Overview of Simulations of Quantum Systems

Croucher ASI, Hong Kong, December 7 2005

Roberto Car, Princeton University

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

/I I IM R R

{ }({ }) [{ },{ }]iI I iR Min E R

0

1( ) , ( )

obs

obs

A A R t R t dt A

• 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

1

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({ }){ }exp

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B

Rd R

k TF Zk T Z

Rd R

k T

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A

F F F d

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Adiabatic switching (Watanabe and Reinhardt):

Calculated melting line – Clapeyron slopes are in red

The complete phase diagram of Carbon

X. Wang, S. Scandolo, R.C, PRL 95 (2005)

Phase diagram of carbon from DFT simulations

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)

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

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

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

,dS

i H S Sdt

C

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

dIG

dV

2

ii

eG T

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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