Roberto CarPrinceton University

Berlin, July 1st 2009

Challenges for �rst-principlesmolecular simulations

‘The fundamental laws necessary to the mathematicaltreatment of large parts of physics and the whole of chemistryare thus fully known, and the difficulty lies only in the fact thatapplication of these laws leads to equations that are toocomplex to be solved’

P.A.M Dirac, Proc. Roy. Soc. A 123, 714 (1929)

Why, 80 years later and more than 50 years after theadvent of digital computers, the difficulty identified byDirac is still there, in spite of substantial progress?

At the molecular scale the fundamental laws of physicsare known

The exponential barrier

requires data

requires data

This program can succeed if we can: (a) map the N-quantum particle problemonto an equivalent, fictitious, non-interacting quantum system, or/and (b) mapan interacting quantum problem onto an equivalent, fictitious, classical one

N quantum particles:

N classical particles:

R1,P1;R2,P2;...;R

N,PN

To predict the behavior of solids and liquids by solving the basic equations ona (classical) computer the exponential barrier needs to be overcome

Examples

Born-Oppenheimer (BO) separation:

+ Classical nuclei:

+ Electronic Density Functional Theory:

�(R1,R2,...,R

N)� �(R

1�R

1)�(R

2�R

2)...�(R

N�R

N)

E[{R}]

Ab-initio Molecular Dynamics maps the system onto an effective classicalsystem and does overcome the exponential barrier

Quantum nuclear dynamics, even within BO, does not overcome theexponential barrier:

�(R1,R2,...,R

N)� �

1(R1)�2(R2)...�

N(RN)

Even with non-overlapping � 's E[{R}] requires MN data

at each nuclear configurationSimulating quantum nuclei is a challenge

Gauging the importance of quantum nuclei: isotopeeffects

Much larger isotope effects are seen in phase transitions where tunnelingplay a role: e.g. Tc for the ferroelectric to paraelectric transition in KPD(KH2PO4) increases by ~ 90K when protons are substituted by deuterons

Why does QM lead to mass dependence

Configurations are weighted by a Boltzmann factor

( )exp / BE k T�

E include kinetic (depending on mass and momentum)and potential (depending on position) contributions.

In classical mechanics kinetic and potential contributionsto the Boltzmann weight factorize, leading to a partitionfunction that is the product of an ideal and an excess part.In quantum mechanics they do not (uncertainty principle).

Nuclear quantum effects in simulations

If we are only interested in static, rather than in dynamic, equilibriumproperties the exponential barrier of quantum mechanics can again bebypassed

This is because quantumstatistical mechanics at finitetemperature can be mappedonto classical statisticalmechanics via the FeynmanPath Integral concept

Close Feynman paths result from HZ Tr e ��� �= � �

Sampling the Feynman paths

A = drA(r)�(r,r)�

By repeatedly inserting the completeness relation

The important configurations of the classicalring (open) polymers (necklaces) can besampled with molecular dynamics or withMonte Carlo

The PI-AIMD technique was introduced by Marx and Parrinello (1994)

Converged PIMD simulations for water and ice

Some details:–32 polymer beads–64 molecules (liquid water)– 8 (valence) electrons permolecule– 96 molecules (hexagonal ice)– valence electrons and nuclei arereplicated 32 times (potentialenergy surface from DFT as instandard ab-initio MD)

The simulation is possible on massively parallelcomputers like the IBM BG/L platform

Comparison with experimental diffraction (elastic scattering) data

Quantum water has more brokenbonds consistent with the highermelting point of heavy water

Exp: x-ray diffraction data (T. Head-Gordon et al.)

From J Morrone and RC, PRL (2008)

Can semi-classical approximations be used in nontunneling situations?

Landau and Lifshitz suggest

The momentum distribution is theFourier transform of the end-to-enddistribution n(x)

f (x) = ��

�xlogn(x) n(x) = dr�(r)�(r + x)�

For a particle in the ground state �(r)

From L Lin, J Morrone, RC, M Parrinello (in preparation)

J Morrone RC,PRL 2008

Ice Ih

Does zero-point motion matter?

Difference spectrum:

Th vs Exp

Water at two temperatures

Water: Th vs Exp

Ice: Th vs Exp

From W Chen, X Wu, RC (2009)

1s x-ray absorption spectra:

in the liquid calculations report anaverage of several local configurations

in ice only a single configuration (theclassical DFT T=0 configuration) hasbeen used in the calculation, i.e. zero-point motion was not included

The calculations were based on anapproximated BSE-GW treatment

Equilibrium and Non-equilibrium

• Statics is a special case of dynamics: classical trajectories generated byNewton’s equations describe thermodynamic equilibrium, approach toequilibrium, and even states very far from equilibrium, of a system ofclassical particles

• One suspects that the same should be true for a system of quantumparticles but solutions of the time dependent problem for a many-bodyquantum system are missing because of the exponential barrier: thisdifficulty could be circumvented (in some cases) when dealing withthermodynamic equilibrium

• Can we find approximated dynamics that describe the approach toequilibrium of a quantum system and do not suffer (at least in some simplebut non-trivial cases) of the exponential barrier?

The answer is yes, renouncing to the short time details of the dynamics,quantum master equations can be formulated that parallel the Fokker-Planck equation in the classical case; these equations can be solvedwithin suitable mean-field approximations

These approximated dynamical approaches are called kineticapproaches

One example is the Boltzmann transport equation which iswidely used to model electron transport in real devices, from themacroscale down to the microscale. The approach is veryrobust but somewhere down toward the nanoscale it must fail

Kinetics:on a coarse time scale dynamics

is approximated by kinetics

field collisions

df f fdt t t

� �� � � �= +� � � �

� �� � � �

Steady State:

field collisions

f ft t

� �� � � �= �� � � �

� �� �

( , ; )f f x p t� is a classical probability distributionHere

Example: Boltzmann’s kinetic equation

Quantum kinetic formulation

For devices of dimension comparable to the electron wavelength,the semi-classical Boltzmann equation should be replaced by aquantum-mechanical Liouville-Master equation for the reduceddensity operator describing a quantum system coupled to a bath

f S�

[ ] [ ],dS i H S Sdt

= � + C

[ ] [ ],i H S S= CSteady State

Important approximations

• interaction with the bath to 2nd order in perturbation theory

• the harmonic bath is in thermal equilibrium

• Markov approximation, i.e. coarse-graining in time so that onthe time scale of system damping electron-phonon scatteringprocesses are treated as instantaneous processes via Fermi’sgolden rule

c Rt� �� �

A master equation can be established for a Kohn-Shamsystem by extending time-dependent density functionaltheory to dissipative systems (K Burke, RC, R Gebauer,PRL 2005)

This allows us to study the evolution of a reducedsingle-particle density matrix, eliminating theexponential barrier for the electrons. The approach isnon-trivial because it includes quantum dissipativeprocesses (quantum inelastic scattering with a phononbath)

Open and Closed BoundaryConditions

A ring geometry, which allows current flow with close boundaries, can berealized with a proper choice of the gauge

,�= ��E x� = � �E

,Ac t�

= ��

E A c t= � E

The v-gauge corresponds to a ring geometry in which anelectric current is induced by a magnetic flux

x-gauge

v-gauge

The electrons are then subject to a steady electromotiveforce. Coupling to a heat bath prevent them fromaccelerating indefinitely

�Sn,m =�i Hn, pE (t)Sp,m� Sn, p H p,m

E (t)( )p�

+ �n,m� Sn,m( ) �n, p +�m, p( )Sp, p � Sn,m � p,n +� p,m( ) 1� Sp, p( )p�

p�

The Liouville-Master equation

Here:( )

2

0( ) ( ) [ ]2 HXC

p tH t U x V n�

�= + +

EE

In the numerical implementation the electric field is systematically“gauged” away to avoid indefinite “growth” of the Hamiltonianwith time. The result is that the effect of the electric field istransferred from the Hamiltonian to the density matrix

R. Gebauer and R.C. (PRB 2004)

�Sn,m =�i Hn, pE (t)Sp,m� Sn, p H p,m

E (t)( )p�

+ �n,m� Sn,m( ) �n, p +�m, p( )Sp, p � Sn,m � p,n +� p,m( ) 1� Sp, p( )p�

p�

The effect of the bath contribution

( )2

, 2

( ) 1 ( ) if

( ) ( ) if

e ph mn mn n m

n m

e ph nm nm n m

n V m n e e

n V m n e e

� � �

� � �

�

�

� + <�� = �

� >�

1( )1Tn

e�� =

�

Because of the collision term the electrons take a Fermi-Diracdistribution in absence of applied electromotive force

A simple application: a 1D DBRTS(Double Barrier Resonant Tunneling System)

Although calculations are performed in the velocity gauge, for illustrative purposes werepresent on the right panel the external field as a scalar potential varying linearlyalong the wire (position gauge).

On the coarse grained time scale of Markovian dynamics the physical current is

j(x;t) � jH (x;t) + jC(x;t)

[ ] [ ],dS i H S Sdt

= � + C( ) ( ) ( )H

d r j r j rdt�

= �� � �� �C

Generalized continuity equation (R. Gebauer and RC, PRL 2004)

The collision current is a quantum effect: collisions that change momentum also changeposition (density distribution).

I-V characteristics

Intrinsic bistability results fromcharging the resonant level

The effect is more pronouncedat small dissipative coupling

hysteresis effects were first observed inGaAs-AlAs heterostructures byGoldman, Tsui, and Cunningham, PRL(1987)

From R Gebauer and RC, PRB (2004)

Pseudopotential plane wavecalculations on molecular junctions

within DFT-GGA

From S. Piccinin, R. Gebauer, K. Burke, R.C., in preparation

A 3-atom gold wire

Visualization of the electronic current flow

Calculations using pseudopotentials andplane waves in a supercell geometry

Potential drop in the position gauge

Issues: current contribution due to the bath;effect of dissipation on current characteristics

At sufficiently large dissipative coupling theconductance measured across the “ballistic”junction shows saturation, becoming“independent” of the dissipative coupling

Conductance fluctuationswith the number of atoms inthe wire (as found in NEGFcalculations, but out ofphase with experiment).The conductance in ourcalculation decreases withthe number of atoms in thewire: an ohmic proximityeffect.

Experiments: Smit et al. PRL(2003)

Realistic calculations with master equation

• Large systems needed for realistic el-ph scattering lengths, hybridapproaches may facilitate calculations but still large systems needed:schemes more approximated than full DFT would be useful, like e.g.schemes used successfully in the study of nanodots (Zunger)

• Bridging size and time scales: when quantum master equationreproduces semi-classical Boltzmann

• Small scale constrictions (e.g. single molecules): ballistic regime. Inthis regime equilibration is achieved without dissipation: it is a fastelectronic interference effect (Gross et al). The time scale is muchshorter than that of dissipative effects. In this regime, in absence ofnon-linear resonance effects, linear response is adequate, and Jouleheating is negligible (Kohn&Luttinger, 1957). However, at sufficientlylarge bias Joule heating should become important.

Coupled electron-ion system

• Both electrons and ions should be treated quantum-mechanically toenforce detailed balance (need spontaneous emission) which is essentialto equilibration; however the harmonic approximation (even a verysimplified one) can be used for the ion dynamics; relaxation to equilibriumis a slow process.

• Ehrenfest dynamics in which the ions are treated classically cannot leadto equilibration (strictly speaking it is only valid when the electrons stay inthe ground state); however when the electrons are excited it can describecorrectly short time dynamics without coherence loss.

• Time dependent density functional theory treats the external guidingfield classically thus it is not adequate in principle to describe relaxationto equilibrium. It is well suited to deal with intense laser fields (for whichspontaneous processes are negligible compared to stimulatedprocesses).

• Is it really true that no relaxation to equilibrium can occur within TDFT:what about purely electronic dissipative processes?

The first-principle issue

“Don’t forget that the reason a physicist can really calculate from first principlesis that he chooses only simple problems. He never solves a problem with 42 oreven 6 electrons in it. So far, he has been able to calculate reasonablyaccurately only the hydrogen atom and the helium atom”R.P. Feynman, The Feynman lectures on physics, 3rd printing (1969)

Strictly speaking he is of course right. Even though we can calculatenowadays fairly complicated systems with 1000s electrons and almost nofree parameters, we still need to rely on physical approximations. Differentphenomena may need different approximations: density functional methodsalbeit powerful are not the dream machine that solves all problems.

... The situation may change though when quantum computers wouldbecome reality because then the exponential barrier of quantum dynamicswould be leveled down...