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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