Transcript Lecture 6

Thermal Behavior – I
Chemical Processes & Transition State Theory
[based on Chapter 6, Sholl & Steckel]
• From zero K to warmer situations!
• Kinetics of processes
– Example: Atomic diffusion on surfaces
– Transition state theory
– Determining transition states (or saddle-points) numerically
• The nudged elastic band method
– Connecting atomic level processes to overall dynamics
• The kinetic Monte Carlo (kMC) method
• Case studies
– Catalytic NO decomposition
– Catalytic CO oxidation
– Other catalytic reactions
Key Dates/Lectures
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Oct 12 – Lecture
Oct 19 – No class
Oct 26 – Midterm Exam
Nov 2 – Lecture
Nov 9 – Lecture
Nov 16 – Guest Lectures
Nov 30 – MRS week – no class
Dec 7?? – In-class term paper presentations
Term Paper/Presentation
• Choose topic close to your research.
• Cast the term paper/presentation like a “proposal” 
identify problem, provide background, discuss past DFT
work, and identify open issues & future work
• DFT has to be a necessary component of term paper.
• Literature search: Phys. Rev. B, Phys. Rev. Lett., Appl.
Phys. Lett., J. Phys. Chem., Nano Letters, etc., within the
last 10 years.
From zero K to warmer
situations!
• All calculations considered so far deal with the “ground
state”, meaning at absolute zero temperature
• Does this mean that all such results are meaningless?
• Not really, as these results correspond to the “internal
energy” or “enthalpy”
• Many methods are available to incorporate temperature
– Through direct inclusion of entropic contributions
– Through “Molecular Dynamics”
– Through transition state theory & kinetic Monte Carlo
Example: Surface diffusion
• Lets first consider surface adsorption, a necessary first step
underlying many processes (e.g., crystal growth, catalysis)
• Specific example: Ag atom on Cu(100) surface
• Three distinct adsorption sites; what is the nature of each of these
sites? (i.e., are they minima, maxima, etc.)
Potential energy surface
• Hollow site: global minimum
• Bridge site: 1st order saddle point
• On top site: 2nd order saddle point
Potential energy surface (PES)
• PES computed using DFT at zero K
• System is “dynamic” on this PES, and the
kinetic energy determines the temperature
• Thus, zero K computations are very useful
and relevant!
1-dimensional PES
• Only the lowest energy transition
state (or “barrier”) will matter
1-dimensional transition state theory
Rate of transition from A to B
kA  B

 E t  E A 
  exp

kT


Probability of system
being at transition state
Vibrational frequency at A
~ 1012-1013 Hz
In this 1-d example, the minimum is characterized by a real
frequency, while the transition state displays a imaginary
frequency (why?) – this is a signature of a transition state …
What about in a 3-d system containing N atoms?
3-d transition state theory
• One has to perform “normal mode analysis” to determine the normal
mode frequencies of the equilibrium and the transition states
• What are normal modes?
• A system with N atoms will display 3N real frequencies at
equilibrium and 3N-1 real frequencies at a 1st order saddle point
• The A  B rate is given by
kA  B
 E t  E A 
1   2  ...  3N
 t
exp

t
t
1   2  ...  3N 1
kT 

• Still, the first fraction works out to ~ 1012-1013 Hz, and hence the rate
of the process is dominated by DE

• The above expression applies to any process (not just site-to-site
hopping of adsorbates) as long as starting and ending equilibrium
situations, and transition states can be defined
Two questions
• To calculate rates of elementary processes, we need the
activation barrier (i.e., energy difference between
transition state and initial equilibrium situation). How do
we determine the barriers?
• Even after the barriers for all (or most) possible
elementary steps are determined, how do we “assemble”
all this information to determine overall macroscopic
experimental quantities such as “turn over frequency”
(TOF) or conversion efficiency?
Determining Extrema
• Transition state is a maximum along one “direction” in phase space,
but a minimum along other orthogonal directions
• First, let us review how minima are found numerically: consider a
function (Energy, E, which is a function of several coordinates) that
needs to be minimized; let us suppose that methods are available to
compute the function value (DFT energy) and its first derivatives
(Hellmann-Feynman forces) at a chosen set of coordinates
• The most obvious choice for minimizing the function numerically is
the steepest descent method
• A much better choice is some flavor of the conjugate gradient
method
Steepest Descent (SD)
• Move along the negative gradient of the function till you
reach a minimum (the “line search”)
• Then find the gradient again, and commence another
line search, etc.
• This can take any number of line searches even if we
are in the quadratic region
Conjugate Gradient (CG)
•
•
•
•
First search direction is identical to the steepest descent (SD) method
Subsequent search directions are linear combinations of the new gradient
direction and the previous gradient direction (called “conjugate” directions)
This is done to account for the fact that the new SD gradient direction
contains an “already searched direction” component
For a N-dimensional function in the quadratic region, the CG algorithm
takes exactly N line minimizations to locate the minimum
Transition State
• Finding transition states is an entirely new ball game, as the point of
interest is a maximum and minima simultaneously
• Starting with an initial guess (as is always the case), attempting to
find a point with with zero (or negligible) gradient will inevitably take
us to one of the local minima, rather than the transition state!
• Remember: criterion satisfied by transition state: first derivative is
zero, and second derivative is negative along one dimension and
positive along all other dimensions
• A few methods are available, the most popular one in electronic
structure calculations being the “nudged elastic band” method
Nudged Elastic Band Method
•
•
•
Much more intensive than conjugate gradient, and works very differently
Involves multiple configurations along the reaction coordinate separated by
fictitious “springs” to keep the configurations from “falling” into a local
minimum
A snapshot:
O Interstitial Migration
Si:HfO2 Interfaces
Si
HfO2
Interfacial segregation:
Hf Thermodynamic driving
force (implied by
O decreasing formation
energy as interface is
approached)
Kinetic driving force, and
O penetration into Si
(implied by decreasing
migration barriers as
interface is approached)
Excess O interstitials
lead to the formation
of SiOx at the interface
C. Tang & R. Ramprasad, Phys. Rev. B 75, 241302 (2007)
Point Defect Migration
Amorphous HfO2
O
Hf
O vacancy
O interstitial
Hf vacancy
VO2+ most mobile in a-HfO2
C. Tang & R. Ramprasad, Phys. Rev. B (in print)
Now What ….
• We have the frequencies, we have the
barriers, and hence we have the rates …
• How do we put it all together?
• We roll the dice!
– The kinetic Monte Carlo (kMC) method
– “kinetic” because rates and temperatures are
involved, and “Monte Carlo” because
elementary processes are stochastic
The kinetic Monte Carlo (kMC) Method
• The idea behind this method is straightforward: If we know the rates
for all processes that can occur given the current configuration of
our atoms, we can choose an event in a random way that is
consistent with these rates. By repeating this process, the system’s
time evolution can be simulated
The kMC Algorithm
• Consider a Pd-alloyed Cu surface, with a predetermined number of
Ag atoms randomly adsorbed on this surface
• The dynamical evolution of this system may be modeled using the
following algorithm:
• The output of a kMC simulation is typically: surface coverage at
given (T, P) conditions; rate of formation of various competing
products via competing mechanisms
• kMC will be contrasted with “Molecular Dynamics (MD)” in the last
lecture
Case Studies of Catalysis
• Although a reaction/process may be allowed by
thermodynamics, it may be slow at low temperatures due
to large barriers
• Catalyst: A magical substance that speeds up chemical
reactions without (of course) altering the overall
thermodynamics
• Large barriers may be due to steric or bond breakage
reasons, or because the process may be “forbidden”
H2 + D2  2HD
• Intuitively, we may expect this reaction to occur
as follows:
H
D
H
D
H
D
H
H
H
D
D
D
• But this almost never happens, and H and D
atomic intermediates are generally found during
the course of the reaction. Why?
Orbital Symmetry Considerations
H
H
D
D
H
H
D
D
Ground state of reactants/products correlates with excited
state of products/reactants, and hence, high barrier!
Catalytic decomposition of NO
• NO is one of the harmful
effluents of automotive exhaust
• Need to accomplish: 2NO 
N2 + O2; although reaction is
thermodynamically downhill, it
has a huge barrier (because it
is symmetry-forbidden: next
slide)
• Mechanism of catalytic
decomposition of NO using a
Cu-exchanged zeolite catalyst
2NO
N2 + O2
Gas Phase Reactions
2NO  N2 + O2
(Symmetry Forbidden)
2NO  N2O + O  N2 + O2
(Symmetry Allowed)
Ramprasad, et al, J. Phys. Chem. B (1997)
Ramprasad, PhD Thesis
Catalytic decomposition of NO
• Various modes of interaction of NO with Cu-exchanged
zeolites
• Mechanism of NO decomposition is many step process
2NO
N2 + O2
Ramprasad, PhD Thesis
Schneider et al, J. Phys. Chem. B (1998)
Catalytic NO Decomposition
Correlation Diagram
Catalytic CO Oxidation
• One of the most studied catalytic reactions on metal and metal oxide
surfaces
• Spawned careful surface science work (cf. Ertl’s work from the
1960s)
• Essential steps:
– Adsorption of CO (generally unactivated, meaning negligible activation
barrier)
– Dissociative adsorption of O2 (may be activated)
– Surface diffusion of CO and O (generally with a large barrier)
– Reaction of CO and O to form weakly bound CO2 (may be activated)
– CO2 desorption (may be activated)
•
•
•
RuO2(110) surface has 2 types of
adsorption sites: coordinatively
unsaturated site (CUS) & bridge
site, forming alternating rows
Energies and barriers for all
elementary steps (previous slide)
were computed, and used in a
kMC simulation
Results: Surface phase diagram of
RuO2, and CO2 conversion
efficiency
CO Oxidation on RuO2 surfaces
• Surface phase diagram and turn over frequency (TOF) for CO2
kMC Simulation
• Barrier for COcus + Ocus  CO2 was lowest
• If this was the only operative reaction, it should have resulted in a
rate of CO2 production proportional to  (1- ), where  is the
coverage of O on the cus sites; but this was not the case
• kMC simulation clarified this …
Catalyst Design from First Principles