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Regional School on Physics at the Nanoscale:Regional School on Physics at the Nanoscale:Theoretical and computational aspectsTheoretical and computational aspects

1414--25 December 2009, Hanoi, Vietnam25 December 2009, Hanoi, Vietnam

Directors:

B. AltshulerM. Kiselev V. KravtsovN.V. Lien S. Scandolo

Local organizer:

N.H. Quang

Topics:

> Introduction to Nanophysics> Physics of Carbon Nanotubes> Physics of graphene and

graphene nanoribbons> Mesosocopic systems> Quantum dots> Nanowires, Nanoclusters and

Nanostructures on surfaces> Computational approaches

based on:. Density Functional Theory. Molecular dynamics

The Abdus Salam International Centre for Theoretical Physics(under UNESCO / IAEA), Trieste, Italy

ICTP’s mission:To foster the growth of advancedstudies and research in physicsand mathematics, especiallyamong researchers fromdeveloping countries

Programs:- Conferences / Schools throughout the year- Associates (our faculty “in the field”)- Visitors (junior, STEP, TRIL, etc)- Diploma Course

www.ictp.it

Santiago 2009

Cape Town 2007 & 2010

Abuja 2007

Esfahan 2005

Bangalore 2006Hanoi 2009

Beijing 2006

Addis Ababa 2008

Trieste 2004

Recent ICTP Schools on theoretical nano / materials science

Sao Paulo 2011

65 participants:

Bangladesh

India

Indonesia

Lao People's Democratic Republic

Malaysia

Nepal

New Zealand

Pakistan

People's Republic of China

Philippines

Sri Lanka

Thailand

Vietnam

Republic of Korea

Small is different

New science

Example: CdS nanoparticles of different size have different colors

New technologies

Example: Ability to deposit objectson surfaces with nanoscale precision

Is Ohm’s law still valid in carbon nanotubes?

1 nm

0.5 nm

Does a water nanodrop behave like liquid water?

Regional School on Physics at the Nanoscale:Regional School on Physics at the Nanoscale:Theoretical and computational aspectsTheoretical and computational aspects

1414--25 December 2009, Hanoi, Vietnam25 December 2009, Hanoi, Vietnam

Directors:

B. AltshulerM. Kiselev V. KravtsovN.V. Lien S. Scandolo

Local organizer:

N.H. Quang

Topics:

> Introduction to Nanophysics> Physics of Carbon Nanotubes> Physics of graphene and

graphene nanoribbons> Mesosocopic systems> Quantum dots> Nanowires, Nanoclusters and

Nanostructures on surfaces> Computational approaches

based on:. Density Functional Theory. Molecular dynamics

Ab-initio simulations in the nanosciences:(1) Methods

Sandro Scandolo

The Abdus SalamInternational Center forTheoretical Physics (ICTP)Trieste, Italy

www.ictp.it

Regional School on Science at the Nanoscale, Hanoi December 2009

Molecular Dynamics

)(21)()()()1( 32 tOt

MFttvtRttR

I

IIII ∆+∆+∆+=∆+

)(21)()()()2( 32 tOt

MFttvtRttR

I

IIII ∆−∆+∆−=∆−

)()(2)()()2()1( 42 tOtMFtRttRttR

I

IIII ∆+∆+=∆−+∆++

)(txI

tt∆

)()()(2)( 42 tOtMFttRtRttR

I

IIII ∆+∆+∆−−=∆+

II

I Fdt

RdM =2

2

Verlet’s algorithm

How to integrate Newton’s equations

Initialize: select starting atomic positions and velocities as close aspossible to thermal equilibrium

Integrate: compute all forces and determine new positions using Verlet’s algorithm

Equilibrate: let the system loose memory of the initial configurations,i.e. let it reach thermal equilibrium

Average: accumulate averages of observables of interest (A)

∫∫

−

−=

PdRdE

PdRdEPRAA rr

rrrr

)exp(

)exp(),(

β

β∫=T

dttPtRAT

A0

))(),((1 rr

Average in Molecular Dynamics = Average in Statistical Mechanics

A typical molecular dynamics protocol

Sum of pair interactions?

How to treat chemical complexity?

Where are the electrons?

Molecular Dynamics

II

II dR

RdEFdt

RdM })({2

2

−==

)(21})({

,

)2(∑ −=JI

JI RRRE φ

d

)2(φ

JI RRd −=

A very simple choice:

But the interaction betweentwo atoms is partially mediated by the presence of a third atom…

∑

∑

+

+=

KJI

JI

KJI

JIRE

,,

)3(

,

)2(

...),,(61

),(21})({

φ

φ

I

J

K

Interatomic potentials

“Empirical” potentials

Starting point: Pair potentials (Lennard-Jones, Born-Mayer, Coulomb, etc)

+ three-body corrections

+ density dependent terms (embedded atom models)

+ atomic distorsion terms (includes polarization)

+ charge transfer terms

...,,426

012

0 etceR

ZeRR

RR Rαεε −+−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛

JI RRR −=

I

J

K

[ ] KJIVKJI ˆ3/1cos),,( 0)3( =+= ααφ

Parameters are determined from “empirical” data, such as experimental EOS, vibrations, phase diagrams, dynamical properties, etc.

Quantum simulations: The “standard model”

Hψ = Eψ

Schroedinger equationfor electrons

“Molecular dynamics”for atoms

Ma = F = -dE/dR

e--e- interactions: Density Functional Theory

e--nuclei interactions:Pseudopotentials

“Ab-initio” molecular dynamics = Classical molecular dynamics in potential energy surface generated by thelectrons in their quantum ground state

R. Cohen

Electrons respond much faster thannuclei to external forces, because of their lighter mass, therefore:

Electrons are always in their instantaneous quantum mechanicalground state, for each given {R}.

Consequence:Every MD step requires the calculation of the QM ground state of the electrons

Where are the electrons? The adiabatic approximation

II

II dR

RdEFdt

RdM })({2

2

−==

The electronic Hamiltonian Hedepends parametrically on thenuclear positions {R}

Nuclei

e-

e-

e-

e-

e-e-

{ }( ) { }( ) eee RHRE ψψ=

Density functional theory

Electron-electron (many-body)interaction

))(('')'(2

iexci

e

j ji

rVrdrrr

rre rr

rr

r

rr ρρ +−

≡− ∫∑

Density-functional theory [W. Kohn et al. 1964-1965] states that the e-e interaction can be written as a one-electron “effective” term:

However, the exact functional form of Vxc is not (yet) known.Current approximations to the exact Vxc go under different names (LDA, BLYP, BP, GGA, “hybrid”, et al). While GGA and hybrid functionals provide slightly better results than other approximations, the choice of the Vxc is often made in such a way to improve agreement with exps (“ab fine”??).

The pseudopotential approximation

+ Core states are irrelevant

+ Valence states have nodes asa consequence of orthogonalitywith core states

The 1/r Coulomb potential can be replaced by a suitably constructed effective potential, or pseudopotential

Electron-ioninteraction

Valencestate

Core state

I

e

II

II dR

RHddR

RdEFdt

RdM 002

2 })({})({ ψψ−=−==

“Ab-initio” forces: the Hellmann-Feynman theorem

00

0000

000

000

000

000

00

})({

})({})({

})({})({})({

})({})({})({})({

ψψ

ψψψψ

ψψψψψψ

ψψψψψψψψ

I

e

II

e

III

e

Iee

II

e

I

e

dRRdH

dRd

REdR

RdH

dRdRE

dRdRE

dRRdH

dRdRHRH

dRd

dRRdH

dRRHd

−=

−−=

−−−=

−−−=−

Forces can be calculated without recalculating the ground state wave function for small atomic displacements

How to keep electron in the ground state (1)

II

II dR

RdEFdt

RdM })({2

2

−==

00 })({})({ ψψ RHRE e=

jeiji

jie HccH ϕϕψψ ∑=,

*

ii

ic ϕψ ∑=

If we expand wavefunctions in a basis set

finding the ground state is equivalent tominimizing a quadratic form in the {c}’s(variational principle). So, standard minimizationschemes can be used, e.g. steepest descent:

ψψψδψδψ ee HH −=−=

•

How to keep electron in the ground state (2)

II

II dR

RdEFdt

RdM })({2

2

−==

00 })({})({ ψψ RHRE e=

ψψ })({RHe−=•

{R}

ψψ })({RHe−=••

III dR

RdERM })({−=••

The Car-Parrinello algorithm

down to the minimumfor each {R}

???

Born-Oppenheimer –versus- Car-Parrinello AIMD

Born-Oppenheimer Car-Parrinello

Born-Oppenheimer –versus- Car-Parrinello forces

from M Sprik

Born-Oppenheimer –versus- Car-Parrinello: summary

• R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985)

• M. Payne, M. Teter, D. Allan, T. Arias, J. Joannopoulos, Rev. Mod. Phys. 64, 1045 (1992).

• D. Marx, J. Hutter, "Ab Initio Molecular Dynamics: Theory and Implementation", in "Modern Methods and Algorithms of Quantum Chemistry" (p. 301-449), Editor: J. Grotendorst, (NIC, FZ Jülich 2000)

• http://www.theochem.ruhr-uni-bochum.de/research/marx/cprev.en.html

• R. Rousseau and S. Scandolo, “Car-Parrinello Molecular Dynamics”, in “Encyclopedia of Condensed Matter Physics”, edited by G. Bassani, G. Liedl, and P. Wyder, Elsevier, Amsterdam (2005)

Ab-initio Molecular Dynamics: bibliography

Quantum ESPRESSO is an open-source suite of computer codes for electronic-structurecalculations and materials modeling. It is based on density-functional theory, plane waves, and pseudopotentials.

Features include: structural optimizations, dielectric and Raman tensorsphonons infrared spectraelastic constants NMR spectraab-initio molecular dynamics etc…

“First-principles codes for computational crystallography in the Quantum-ESPRESSO package”S. Scandolo et al., Z. Kristallogr. 220, 574 (2005)

www.quantum-espresso.org

Asian-Pacific School, Beijing, July 2004