Rare events: classical and quantum

1
Rare events classical and quantum
Roberto Car, Princeton University
Croucher ASI, Hong Kong, Dec. 9 2005
2
Reaction Pathways
FPMD simulations are currently limited to time
scales of tens of ps Most chemical reactions are
activated processes that occur on longer time
scales and are not accessible in direct FPMD
simulations (and would not be accessible even in
classical MD simulations). Identifying reaction
pathways is central to the study of chemical
reactions. The string method for reaction
pathways (W. E et al (PRB 66 (2002))) can be
easily combined with FPMD
3
String Method (at T0)
A Minimum Energy Path
connecting two end points
satisfies
A longitudinal constraint, requiring only uniform
stretching, is imposed by Lagrange multipliers
This is easily solved by Damped Molecular
Dynamics using the SHAKE procedure for the
Lagrange multipliers
4
Damped Molecular Dynamics of a String
In this way an initial trial path is locally
optimized to get a MEP
This is closely related to the NEB method by H.
Jonsson and co. the latter can be seen as a
string method in which a constraint is imposed by
a penalty function (rather than a Lagrange
multiplier)
5
First Principles String Molecular Dynamics
Y. Kanai, A. Tilocca, A. Selloni and R.C., JPC
(2004)
6
Acetylene interacting with a partially
hydrogenated Si(111) surface reaction pathways
from string damped molecular dynamics
A surface chain reaction
Takeuchi, Kanai, Selloni JACS (2004)
7
Influence of xc functional PBE (GGA) vs. TPSS2
(meta-GGA)
Reaction Energy (eV)
H2Si(100) H2 Si2H 4
PBE 1.94 2.11
TPSS 2.18 2.37
B3LYP __ 2.25
QMC 2.40 0.15 _

• ? DFT-GGA underestimates the
• barriers for these reactions 3,4.
• ? Barriers as well as reaction energies improve
  using meta-GGA (TPSS).
• ? There are, however, situations where neither
  B3LYP nor TPPS work well (e.g. a proton transfer
  reaction in a H-bond)

Reaction Barriers (eV)

Intra Inter 1,2 1,1
PBE 0.40 0.24 0.98 0.50
TPSS 0.56 0.40 1.27 0.67
B3LYP _ _ 1.31 0.60
QMC 0.66 0.15 0.54 0.09 _ _
H2Si(100)
H2Si2H4
8
Long time evolution due to activated processes
coarse grained dynamics by kMC
Activation energies an reaction pathways
identified by the string method provide the input
data for kinetic Monte Carlo simulations (kMC).
This multi-scale approach allows us to model
long-time micro-structural evolution (i.e.
processes that occur on time scales of minutes or
even hours and are completely outside the realm
of MD simulations.
9
kinetic Monte
Carlo Continuous atomic dynamics is replaced by
a Markov process consisting of a succession of
hops with rates ri, which must be known in
advance
10
Example Oxygen Diffusion in YSZ

• YSZ has a fluorite structure with oxygen in
  tetrahedral sites
• Oxygen diffuses primarily in lt100gt directions
  across lt110gt cation edges
• Molecular Dynamics (MD) and Monte Carlo
  simulations suggest that the cations on the lt110gt
  edge determine the oxygen ion diffusion barrier
• Oxygen diffusivity determined by set of lt110gt
  edges traversed (can be Zr-Zr, Zr-Y, Y-Y)

a 5.629 Å
11
Kinetic Monte Carlo Simulation

• Random (frozen) fcc cation lattice with Y and Zr
  according to bulk concentration
• Oxygen ions and vacancies distributed on
  tetrahedral sites according to Y2O3 concentration
• Oxygen vacancies hop to new sites using rates
  determined from first-principles calculations
  (repeat 109 times)
• A periodic cell with 1,000,000 oxygen ions and
  500,000 cations is employed
• Repeat over a range of Y and oxygen vacancy
  concentrations

Oxygen vacancy in Cation lattice
12
Calculated Results Oxygen Diffusivity
13
What can we do if we only know the starting point
but not the end point of a reaction?

• Metadynamics (Laio and Parrinello (2002)) gives a
  viable strategy, provided we know the important
  reaction coordinates (collective variables)
• In this approach the microscopic dynamics is
  biased by a coarse grained (in the space of the
  order parameters) history dependent dynamics

14
Cope Rearrangement
?
?
?
1,5-hexadiene
15
cope rearrangement of 1,5-hexadiene
16
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.
17
Boltzmanns equation, the standard approach for
bulk transport, includes kinetics and dissipation
Steady State
is a classical probability distribution
18
Quantum formulation
When the dimensions of a device are comparable to
the electron wavelength, the semi-classical
Boltzmann equation should be replaced by a
quantum-mechanical Liouville-Master equation for
the reduced density operator describing a quantum
system coupled to a heat bath
Steady State
19
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)
20

x-gauge
v-gauge
The v-gauge corresponds to a ring geometry in
which an electric current is induced by a
magnetic flux
The electrons are subject to a steady
electromotive force coupling to a heat bath
prevent them from accelerating indefinitely
21
The Liouville-Master equation
Here
The collision term gives a Fermi-Dirac
distribution to the electrons in absence of
applied electromotive force In the numerical
implementation the electric field is
systematically gauged away to avoid indefinite
growth of the Hamiltonian with time
22
Benzene dithiol between gold electrodes
Atomic point contact (Gold on gold)
23
Results for an applied bias of 1eV
Gebauer, Piccinin, RC ChemPhysChem 2005
24
I-V characteristics
Quantitatively similar results to S. Ke, H.U.
Baranger, W. Yang, JACS (2004)
25
Steady state electron current flux through an
atomic point contact (S. Piccinin, R. Gebauer,
R.C., to be published)
26
Quantum tunneling through a molecular contact
Landauer formula
27
Carbon nanotube suspended between two gold
electrodes
A self-consistent tight binding calculation
28
I-V characteristics CNT on gold
Tight-binding calculations using self-consistent
master equation, including nanotube, contacts and
gold electrodes
Experiment from Tao, Kane, and Dekker PRL 84,
2941 (2000)