Database of atomistic reaction mechanisms with application to kinetic Monte Carlo.

Abstract

Kinetic Monte Carlo is a method used to model the state-to-state kinetics of atomic systems when all reaction mechanisms and rates are known a priori. Adaptive versions of this algorithm use saddle searches from each visited state so that unexpected and complex reaction mechanisms can also be included. Here, we describe how calculated reaction mechanisms can be stored concisely in a kinetic database and subsequently reused to reduce the computational cost of such simulations. As all accessible reaction mechanisms available in a system are contained in the database, the cost of the adaptive algorithm is reduced towards that of standard kinetic Monte Carlo.

Full text

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THE JOURNAL OF CHEMICAL PHYSICS 137, 014105 (2012)
Database of atomistic reaction mechanisms with application
to kinetic Monte Carlo
Rye Terrell, Matthew Welborn, Samuel T. Chill, and Graeme Henkelmana)
Department of Chemistry and Biochemistry and the Institute for Computational Engineering and Sciences,
The University of Texas at Austin, Austin, Texas 78712-0165, USA
(Received 16 May 2012; accepted 8 June 2012; published online 3 July 2012)
Kinetic Monte Carlo is a method used to model the state-to-state kinetics of atomic systems when
all reaction mechanisms and rates are known a priori. Adaptive versions of this algorithm use sad-
dle searches from each visited state so that unexpected and complex reaction mechanisms can also
be included. Here, we describe how calculated reaction mechanisms can be stored concisely in a
kinetic database and subsequently reused to reduce the computational cost of such simulations. As
all accessible reaction mechanisms available in a system are contained in the database, the cost of
the adaptive algorithm is reduced towards that of standard kinetic Monte Carlo. © 2012 American
Institute of Physics. [http://dx.doi.org/10.1063/1.4730746]
I. INTRODUCTION
The interesting kinetics of many chemical and material
systems are governed by rare events. Chemical reactions and
diffusion in solids, for example, typically occur on time scales
of milliseconds or longer. Standard molecular dynamics algo-
rithms are limited by the femtosecond time scale of atomic vi-
brations and are not suitable for directly modeling rare events.
A. Kinetic Monte Carlo
Fortunately, in many rare event systems, there is a natu-
ral separation of time scales between fast vibrations within
stable states and slow kinetics between states. If the states
can be characterized, as well as the transition times between
them, the kinetic Monte Carlo (KMC) algorithm is suitable
to model the state-to-state kinetics of the system.1,2In the
case of Markovian dynamics, the transition times between
two adjacent states are characterized by a rate constant. For
each state visited along a KMC trajectory, an exit process i is
chosen stochastically with a probability, ?i/?tot, proportional
to its rate, ?i, where ?tot=??jis the escape rate to any
transition by ?t = −ln(µ)/?tot, where µ is a random number
distributed uniformly on (0, 1].
In order to calculate a KMC trajectory, the mechanism
and rate of every processes that might occur during the course
of the simulation must be known a priori. For this reason, the
applicability of KMC is limited to the simplest of chemical
and material systems.
product state. The simulation time is incremented after each
B. Adaptive kinetic Monte Carlo
The adaptive kinetic Monte Carlo (AKMC) algorithm re-
laxes the requirement to know all reactive events a priori by
determining the exit processes and rate for each state that is
a)henkelman@mail.utexas.edu.
visited during the the simulation.3Using a min-mode follow-
ing algorithm, such as the dimer method,4saddle points are
found along minimum energy pathways exiting the current
state. The rate for leaving the current state can then be cal-
culated with harmonic transition state theory.5,6Similar ap-
proaches have been used in the hybrid eigenvector-following
method7to evaluate rates in reaction networks8and, more
recently, in the kinetic activation-relaxation technique.9The
AKMC method has also been referred to as on-the-fly and
off-lattice KMC.10
For an accurate AKMC simulation, one needs to be con-
fident that all energetically relevant processes leading out of
each state visited are included in the rate table for that state.
A statistical confidence parameter has been defined,
C = 1 − 1/(αNr),
(1)
which is a function of the number of sequential searches Nr
that find previously discovered processes that are also ener-
getically relevant. The constant α is the relative probability
of finding the least likely process as compared to the most
likely.11The criterion for a process to be energetically rele-
vant is for it to have a barrier within a specified energy win-
dow of the lowest barrier process. Only these processes are
relevant because the chance of selecting a process with a bar-
rier of 30 kT higher than the lowest barrier process is negli-
gible (ca. e−30). To have confidence in an AKMC simulation,
saddle searches are performed in each new state visited un-
til C reaches a specified value, at which point a KMC step is
taken to the next state.
The advantage of AKMC over KMC is that reaction
mechanisms can be revealed during the simulation. The dis-
advantage is that AKMC requires more computational re-
sources to search for processes in each new state visited. The
computational cost of the saddle searches is particularly ev-
ident when energies and forces are evaluated with an elec-
tronic structure method, such as density functional theory
(DFT). The number and cost of these saddle searches can
be significantly reduced—particularly in the case of large
0021-9606/2012/137(1)/014105/7/$30.00© 2012 American Institute of Physics
137, 014105-1

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014105-2 Terrell et al.J. Chem. Phys. 137, 014105 (2012)
systems—using the method of saddle point recycling.11Since
the method of saddle point recycling is related to that of the
kinetic database, it is briefly reviewed here.
C. Saddle point recycling
Suppose that from a reactant state R, the set of saddles
and their corresponding products, Siand Pi, have been de-
termined. Saddles are usually found by making local random
displacements from R and converging to nearby saddle points
with a min-mode following algorithm. Assuming that a spec-
ified confidence in the rate table is reached, one of these pro-
cesses, j, is selected by the KMC algorithm and the system is
moved to the corresponding product state, Pj. It is now possi-
ble to check if any of the processes that were available to the
system in the original state R are still available to the system
in the new state R?= Pj. Suggestions of new saddle point ge-
ometries can be made by applying the vector between R and
each connecting saddle Pito the current state,
S?
i= R?+ (Si− R).
(2)
Another approach, used in Ref. 11, is to set the position of
any atom in S?
cantly between R and R?and in the position of R?otherwise.
In either scheme, the min-mode following algorithm is used
to re-converge the recycled saddle point approximations, S?
Processes which are unaffected by motions of atoms in the
selected process j will converge rapidly in the new state, and
usually cost far fewer force evaluations than a typical random
search. When a conflict occurs, a recycled saddle geometry,
S?
can make a qualitative difference for the cost of AKMC sim-
ulations. In order to reach confidence in state R?, only sad-
dle searches with initial displacements in the neighborhood
of atoms that moved significantly from state R to R?are re-
quired. If this region is local, the overall computational cost—
measured in terms of the number of force evaluations—does
not increase with the total system size.11
Recycling is a fast, simple method that works well in
many situations, but it has two limitations. First, recycling
cannot make reliable suggestions for the region around the
atoms that moved significantly from state R to state R?. Sec-
ond, processes are only recycled from the previous state of
the simulation; the method cannot take advantage of process
information acquired at any earlier state. An illustration of
these limitations can be seen in the example of a heptamer
island diffusing on a (111) surface (see Fig. 1). The interac-
tion potential is taken to be of the Morse double exponential
form, using parameters to match bulk Pt.12At room tempera-
ture, the heptamer diffuses by sliding as a unit in one of three
directions from FCC to HCP hollow sites. After the slide oc-
curs, it is energetically unfavorable to slide again in the same
direction because the atoms in the island would move from
(favorable) hollow sites to (unfavorable) top sites. The recy-
cling algorithm then fails to predict saddles for two reasons:
all processes available in the previous state contain the same
set of moving atoms as the selected process, and cannot occur
in the current state because of the symmetry of the substrate.
ito the saddle geometry Siif it moved signifi-
i.
i, may not converge, and the process is discarded. Recycling
FIG. 1. A heptamer island slides on a (111) surface. At room temperature,
the island slides as a unit, but not in the same direction consecutively.
D. Kinetic database
The kinetic database (KDB) is developed as a general-
ization of the process recycling method that does not suffer
from the two problems mentioned. In our implementation,
processes are added to the KDB using a minimal represen-
tation that includes only moving atoms and their immediate
environment. The KDB can be queried with a new configura-
tion to provide suggestions of available saddle point geome-
tries. These suggestions are used to speed up AKMC simula-
tions by reducing the number of random searches needed to
reach confidence. Further, if a KDB is sufficiently populated
and trusted, it can be used to provide processes and rates for a
KMC simulation withno need to perform random searches. In
this way, when the kinetic events of the system are known, we
can reduce the computational cost of AKMC towards that of
KMC. Finally, the KDB is very general so that it can be used
to suggest reaction mechanisms from any interesting configu-
ration under investigation, by hand, or as part of an algorithm
for sampling or modeling the kinetics of atomic motion.
E. Previous work
The KDB is not the first method to store configurational
and kinetic information for use in a KMC simulation; there
have been a number of such efforts along these lines. One ap-
proach, called self-learning KMC (SL-KMC),13uses a two-
dimensional occupancy matrix to define environments in the
neighborhood of diffusing atoms on a crystal surface. When
an atom is found to be in a known environment, the possible
diffusion events can then be looked up in a stored table. This
method can be valuable for short-ranged interactions between
species on a two-dimensional lattice, but as the interaction
length increases (even to second neighbors), the number of
possible environments makes the approach intractable. A re-
cent extension to three dimensions has a similar scaling with
interaction range.14
Another interesting idea, put forward by El-Mellouhi
et al.,9is to store the local environments around active
atoms in terms of their bonding topology. An algorithm
called NAUTY (Ref. 15) is used to generate a hash code of
the bonding graph, allowing for a very efficient compari-
son of different topologies by simple string matching. It is
the binary definition of a bond, based upon the distance be-
tweenneighbors,whichenablesthisconcisedescriptionofthe

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014105-3Terrell et al. J. Chem. Phys. 137, 014105 (2012)
environment around an atom. The network picture of bond-
ing is appropriate for a crystalline solid, such as Si, but it has
limitations when there is a continuous distribution of bond
lengths between atoms, such as in disordered materials. Then,
many topologies are required to accurately describe the range
of events available to the system. Furthermore, if a sufficient
number of neighbors are included in the bonding topology to
accurately define the environment of an atom, the number of
topologies continues to grow in the course of a simulation,
and there is a diminishing chance of revisiting topologies.16
Recently, another approach called local environment
KMC (LE-KMC) was reported.17In LE-KMC, local environ-
ments are stored as a set of positions between a central atom
and its neighbors, up to a radial cutoff. As in SL-KMC, an en-
vironmentismatchedtoaqueryconfiguration whenallneigh-
bors are present at the specified relative position. Off-lattice
atoms are allowed in LE-KMC by using a histogram of dis-
tances, instead of only those which correspond to fixed lattice
sites. The limitations of the two methods, however, are simi-
lar. In both cases, a large environment region, which provides
the most accurate rates, also leads to an exponentially large
number of possible environments. LE-KMC was shown to be
effective for modeling diffusion on a crystalline surface, but
it remains to be seen if it can work in amorphous materials, or
even when a significant number of defects are present.
TheKDBpresentedherehasmanysimilaritiestoexisting
methods. The goals are the same: to store kinetic events cal-
culated previously so they can be used in current simulations.
The method details, however, are different both in philoso-
phy and implementation. As described next, our strategy is to
store a minimal representation of a kinetic event so that the
number of events in the database remains tractable. Allow-
ing for the rotation and translation of configurations in our
database when matching a query configuration further mini-
mizes duplicate entries. Our aim is not to quantify an envi-
ronment well-enough for the database to provide a reliable
energy barrier or rate. Rather, the KDB provides only esti-
mates of reactant, saddle, and product geometries, which are
consistent with a query configuration. A subsequent optimiza-
tion step is necessary to verify that the process is valid and to
determine the activation barrier. So, while the KDB can pro-
vide reaction mechanisms for a KMC simulation, it does not
replace the calculation of the rate table. Finally, it is designed
to be quite general so that it can be used for the determination
of reaction mechanisms outside of KMC simulations.
II. METHOD
A. Populating the kinetic database
When a new reaction mechanism is found, such as in the
course of an AKMC simulation, the process is described by
the atomic configurations of the reactant, saddle, and prod-
uct states. The subset of atoms in these configurations which
are local to the process are used to populate the KDB. Atoms
which move significantly during the process, as well as their
neighbors, are considered local to the process. More specifi-
cally, atom i is determined to have moved significantly if
max(|ri− si|,|si− pi|,|ri− pi|) ≥ dd,
(3)
(a)
(b)
FIG. 2. (a) A crystal surface (light grey) with adatoms (dark grey) in which
one adatom (orange) undergoes a hopping process and is identified as a mov-
ing atom. This adatom and its neighbors are determined to be local to the
process. (b) The coordinates of the local atoms are extracted and stored in the
database as reactant, saddle, and product configurations.
where ri, si, and piare the positions of atom i in the reac-
tant R, saddle S, and product P, configurations, respectively,
and ddis a cutoff distance. In test cases of adatom diffusion,
0.7 Å was found to be an effective value. Atom j is determined
to be a neighbor of atom i if
min(rij,sij,pij) < dn,
(4)
where rij= |ri− rj| and the neighbor bond length dn= σ(ci
+ cj) is taken to be the sum of the covalent radii ciand cjfor
the two atoms multiplied by a scaling factor σ.18Our value
of σ = 1.2 allows for a neighbor be 20% beyond the covalent
bond length.
The atoms which are local to the process are inserted
into the database in the reactant, saddle, and product states.
An example of this extraction is illustrated in Fig. 2(a). The
hopping adatom is determined to have moved significantly by
Eq. (3), and its neighboring atoms are determined by Eq. (4).
The minimal representation of the process, which is inserted
into the database, is shown in Fig. 2(b).
B. Querying the kinetic database
The KDB is typically queried with a minimum energy
configuration (query configuration) for which saddle point
suggestions are desired. In broad strokes, the query algorithm
searchesthequeryconfigurationforanylocalconfigurationof
atoms that match the reactant or product configuration of any
process in the KDB. Once a list of candidate processes has
been determined, the saddle points in the KDB are mapped to
the query configuration to generate saddle point suggestions.
Specifically, the query algorithm first selects those KDB
entries that contain at least as many atoms, by type, as the
query configuration. For each candidate KDB entry, a set of
mappings are generated between atoms from the KDB con-
figuration to atoms in the query configuration. Each mapping
is a transformation of the KDB configuration to a location in
the query configuration where the local geometry closely re-
sembles that of the KDB configuration.
The first step of the mapping procedure identifies atoms
in the query configuration with the same number and types
of neighbors as the mobile atoms in the KDB process. This
is illustrated in Fig. 3, where the query configuration is a

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014105-4Terrell et al.J. Chem. Phys. 137, 014105 (2012)
database
configuration
1
2
3
4
5
6
7
mapping algorithm
step 2
suggested
saddle
matches within the query configuration
step 1
step 3
final successful
mappings
partial mappings
at each step
FIG. 3. The mapping algorithm identifies locations within the query config-
uration that match the geometry of a minimum configuration in the database.
The algorithm generates a list of mappings from the database configuration
atoms to their counterparts in the query configuration. The resulting map-
pings are used to suggest saddle points for the query configuration (final
column).
crystal substrate with supported adatom clusters. For simplic-
ity, only the adatoms are considered in the matching algo-
rithm; an extension to include the substrate atoms does not
change the nature of the algorithm. The top row in Fig. 3
shows the reactant configuration of the KDB entry under con-
sideration (the database configuration). The entire KDB en-
try is shown in Fig. 2. In each step of the query algorithm,
a different atom from the KDB configuration is matched to
an atom in the query configuration. In Fig. 3, the atoms are
matched in steps 1−3, in the sequence red, green, and then
blue.
In the first step, the (red) mobile atom is selected from the
KDB configuration and is mapped to atoms in the query con-
figuration that share the same number and type of neighbors
(one adatom neighbor, in this case). All seven matching atoms
are colored red in the seven rows of matching configurations.
After the initial set of mappings have been found in step
1, the algorithm selects a second atom from the unmapped
set of atoms in the KDB configuration (step 2, first row,
green atom) and calculates the distance from that atom to all
mapped atoms in the KDB configuration (at this step, only the
red atom). The algorithm iterates over partial mappings found
in step 1 and finds atoms in the query configuration where
the distance from that atom and the already-mapped atoms
for that mapping match the distances determined for the KDB
configuration. The distances are considered to match when
??r(k)
atom i and the mapped atoms j, in the KDB (k) and query
(q) configurations. The cutoff distance dcis taken to be 0.3 Å,
which we have found to be a robust value for a variety of solid
state systems.
Matchingatomsareusedtogenerateanewsetofpossible
mappings from the previous step. This process is illustrated in
the second column of Fig. 3 (step 2), where the green atom of
the KDB configuration has been mapped to the green atoms
of the query configuration. In this case, all mappings at step 1
result in a single mapping at step 2, but in general, there can
be more or fewer mappings in each subsequent step.
The matching procedure is repeated until all atoms of
the KDB configuration have been mapped to a corresponding
atom within the query configuration. In Fig. 3, the procedure
is complete in step 3 of the algorithm, where the third and fi-
nal atom of the KDB configuration (blue) has been mapped to
atoms (blue) in the query configuration. Note that in step 2, a
single partial mapping has branched into two in the final step,
and that the mappings in rows 6 and 7 have been rejected
because no atom was found whose distances to the mapped
atoms matched the distances found for the third (blue) KDB
configuration atom.
After a set of mappings from the atoms in the KDB mini-
mum energy configuration to atoms in the query configuration
are found, an affine transformation ?(R) = R?is used so that
the transformed position of atom i is
ij− r(q)
ij
??< dc,
(5)
where rij is the distance between the selected (unmapped)
r?
i= ?rotri+ ?trans,
(6)
where ?rotis a 3 × 3 rotation matrix and ?transis a translation
vector, chosen to minimize
N(k)
?
i
??r?(k)
i
− r(q)
i
??2.
(7)
Here, r?(k)
formed KDB and query configurations, and N(k)is the num-
ber of atoms in the KDB configuration. There are a num-
ber of algorithms to determine the optimal rotation to min-
imize the root-mean-square deviation (RMSD) between the
i
and r(q)
i
are the positions of atom i in the trans-

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014105-5Terrell et al.J. Chem. Phys. 137, 014105 (2012)
mapped atoms. While the Kabsch algorithm19is a common
choice, we used a quaternion-based approach that avoids
rotoinversions.20
We apply the transformation to the KDB configuration
and calculate a score for the match,
max??r?(k)
in the KDB and query configurations. If this distance is less
than a desired tolerance, the saddle configuration of the KDB
entry S is used to make a saddle point suggestion by applying
? to S. The saddle suggestions for the query configuration in
Fig. 3 are shown in the final column.
i
− r(q)
i
??,
(8)
which is the greatest distance between any two mapped atoms
III. RESULTS
A. Performance of the KDB
The accuracy and performance of the KDB was com-
pared to the recycling algorithm using the model system
of Pt heptamer island formation and diffusion on a Pt(111)
surface.12The AKMC algorithm without recycling or the
KDB was applied to the starting configuration A in Fig. 4(a).
For this simulation, we used a confidence parameter of 0.99,
which corresponds to a confidence requirement of 100 ran-
dom saddle searches without finding a new saddle (assuming
unity α in Eq. (1)). In the initial configuration (A), adatoms
are scattered randomly on the surface. Over the course of the
simulation the atoms form clusters (B), then a compact island
(D), and finally the stable heptamer (E) after 0.5 µs.
A comparison of saddle point recycling and the KDB al-
gorithm with standard AKMC are shown in Fig. 4(b). The
number of saddle searches required to reach confidence in
each state with the standard AKMC algorithm is given by the
green line.
The simulation was then run again with the recycling al-
gorithm. The same confidence parameter and state-to-state
trajectory was imposed upon the simulation to ensure a fair
comparison to standard AKMC. The red lines show that the
recycling algorithm successfully predicts saddles and reduces
the average number of saddle searches required to reach con-
fidence. Note that as the island forms, the number of relevant
processes that can occur outside of the region of the most
recent process tends to decrease and the fraction of saddles
found with the recycling algorithm along with it.
The performance of the KDB was measured in the same
way (blue lines). The KDB predicts, on average, a greater
percentage of saddles than the recycling algorithm. The ad-
vantage of the KDB becomes obvious as the simulation pro-
gresses, in spite of the increasing proximity of processes.
After the heptamer island forms, it diffuses via sliding along
the surface. After this process is stored in the KDB, it per-
forms perfectly in the diffusive part of the trajectory, where
the recycling algorithm performs poorly.
B. Bridging the gap to KMC
If the KDB is sufficiently populated, it is possible to run
an AKMC simulation using only the suggestions from the
database, supplemented by no additional saddle searches. In
this simulation mode, for each new state reached, the KDB
is queried for suggestions, those suggestions are then refined
and, if valid, are used to populate the rate table for that state.
To test the efficacy of this technique, we created 128
AKMC simulation trajectories of Pt island ripening, as de-
scribed in the previous section. Each simulation was started
with isolated adatoms and run until they coalesced into the
compact Pt heptamer island. The KDB was populated using
the information from two of these trajectories. Then, 128 new
AKMC trajectories were simulated using only the saddle sug-
gestions from the KDB. The center panel of Fig. 5 is a plot
of the elapsed time as a function of the AKMC step for each
of the standard AKMC simulations (red) and the KDB-only
simulations (blue). Most trajectories follow a path of fast mo-
tion of surface atoms (e.g., bottom left panel) coalescing into
a asymmetrical heptamer island and then undergoing some
slow processes (e.g., middle left panel) rearranging into the
symmetrical heptamer island. The primary line in the figure
corresponds to the fast diffusion processes, while the vertical
lines are the slow island rearrangement processes. The odds
of reaching the symmetric island through the diffusion of sin-
gle atoms and smaller islands is low as compared to forming
an asymmetric island that must rearrange to form the low-
est energy compact structure, which can be seen by the small
fraction of trajectories that end on the primary line.
The AKMC simulations found the compact island on
time scales between 10−11than 10−4s; the KDB-only ap-
proach found a similar distribution. There was, however, one
100
200
300
400
saddle searches
ABCDE
010203040
0
20
40
60
80
100
AKMC simulation state
saddles predicted (%)
recycling
KDB
AKMC
(a)
(b)
0 ps
6.1 ps
56 ps
482 ns
494 ns
FIG. 4. (a) AKMC trajectory of a heptamer island formation process on a
crystal surface. (b) At least 100 saddle searches are required (Nrin Eq. (1))
to reach the convergence criterion C = 0.99; more are required if new saddles
are discovered. The method of saddle point recycling reduces this number,
and the KDB is even more effective, particularly after a compact island forms
(state E) and slides diffusively by a known mechanism.

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014105-6Terrell et al.J. Chem. Phys. 137, 014105 (2012)
Count
0204060
AKMC step
102
Reactant
Product
103
100
101
104
105
1.05 s-1
1.36 x1011 s-1
9.08 x107 s-1
10-4
time (s)
10-5
10-6
10-7
10-8
10-9
10-10
10-11
10-12
10-13
0.53 s
FIG. 5. In the center panel, the simulation time is shown over the course of 128 AKMC (red) and AKMC+KDB (blue) simulations. Each simulation is run
until the lowest energy compact heptamer island, shown in Fig. 1, is reached. The KDB was populated with all processes discovered by only two of the
AKMC simulations. The AKMC+KDB simulations were then run using only saddle points suggested from the KDB, with no additional saddle searches. The
comparable distribution of the island formation times is shown in the right panel. On the left, three events are shown. The lower panel shows a typical single-
atom fast diffusion process, which forms the primary initial path of trajectories. The middle is an example of a slower concerted mechanism involving a pivot of
the cluster between hcp and fcc hollow sites. On top is a slow concerted sliding mechanism, which was followed in an AKMC+KDB; an unlikely event caused
by the incomplete population of the KDB. The middle mechanism, which is much faster, can occur from the same state. Had that trajectory also been used to
populate the KDB, the slow mechanism would almost certainly have be avoided.
outlier in the KDB-only distribution with an island formation
time of 0.53 s. This outlier is due to the simulation trajec-
tory reaching a local configuration for which there was in-
complete information in the KDB, and subsequently select-
ing the low rate process shown in the top left panel of Fig. 5.
The KDB-only trajectory failed (in terms of accuracy) in
this case because neither of the trajectories that were used
to populate the KDB passed through a state that had a pro-
cess like the one shown in the middle left panel, with a much
faster rate. The accuracy of simulations that use the KDB-
only approach is dependent on how complete the KDB is
with respect to the trajectory of the simulation. In the limit
of a complete database, the KDB distribution of island for-
mation times is indistinguishable from that of the AKMC
simulations.
The computational cost for the 128 KDB-only simula-
tions with two seed trajectories was 2.1% that of the original
128 simulations, measured in terms of number of force eval-
uations including the force evaluations from the two AKMC
simulationsusedtopopulatetheKDB.ThecostofeachKBD-
only trajectory was, on average, 1.4% the cost of a standard
AKMC trajectory.
C. Application to an off-lattice DFT system
To demonstrate the applicability of the KDB algorithm
on a somewhat more complex system, we modeled the kinet-
ics of a bulk Si lattice at 500 K with two B atoms initially
sharing one interstitial site, a so-called B2I cluster. Boron is
commonly used as a dopant for p-type Si. Due to the scal-
ing relations of semiconductor devices, smaller devices re-
quire a higher concentration of dopants. At very high con-
centrations, the dopant atoms can coalesce into inactive clus-
ters. The mechanism of B-cluster break-up and diffusion is
important for understanding the fabrication of nanoscale Si
devices.21
In our simulation of B2I, the energy of the system and
the forces on the atoms were calculated with DFT using the
Vienna ab initio simulation package.22For our calculations,
we used the PW-91 exchange-correlation functional,23a plane
wave cutoff of 200 eV, and a 2×2 ×2 Monkhorst-Pack
k-point mesh.24States at the Fermi level were smeared by a
Gaussian width of 0.05 eV. A Si supercell, initially contain-
ing 64 Si atoms, was relaxed in a cubic box of side length
10.914 Å. One Si atom was removed and replaced with a
pair of B atoms; the geometry was relaxed to create the initial
state. Geometries were considered converged when the force
on all atoms dropped below 0.01 eV/Å.
For each simulation state, a confidence parameter of 0.95
was used. First, KDB-suggested saddles were converged, fol-
lowed by the required number of random searches to reach
confidence. The number of successful KDB predictions as a
fraction of total saddles found in each state is shown in Fig. 6.
In the initial state, the KDB has not yet learned anything about
the system, so it predicts no saddles. State 2 is identical to
the initial state by a rotation, and the KDB predicts all rele-
vant saddles. State 10 has a local configuration that is unique
among those seen so far, resulting in the identification of only
the reverse process being identified by the KDB. In state 20,
the B dimer is separated by a Si atom. This state also contains
local configurations unique among those visited, so the KDB
is unable to predict most of the available saddles. As the sim-
ulation progresses, however, the fraction of saddles predicted
by the KDB is very high. Overall, the KDB dramatically
improves computational efficiency, allowing for a 32-state

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014105-7Terrell et al.J. Chem. Phys. 137, 014105 (2012)
saddles predicted (%)
0
20
40
60
80
100
AKMC simulation state
0102030
0 s3.7x10-9 s 1.4x101 s8.3x102 s1.5x103 s
2
FIG. 6. The fraction of saddles successfully predicted with the kinetic
database used during an AKMC simulation of bulk Si with two B atoms
sharing a Si site, demonstrating the KDB’s ability to deal with a complex,
off-lattice system with energies and forces calculated with DFT.
simulation showing the breakup of the B2I cluster on a time
scale of an hour at 500 K.
IV. CONCLUSION
A database of kinetic events has been introduced and
used to accelerate the AKMC algorithm by reducing the num-
ber of saddle searches needed to reach confidence in the rate
table from most states. We have shown how a well-populated
KDB can be used as the basis of a KMC simulation without
the need for any additional saddle point searches, and there-
fore requiring only a fraction of the computational effort. The
KDB is a very general tool for predicting available saddles
to a query configuration; it can be used for off-lattice simula-
tions and in combination with DFT. Software for the KDB is
available at http://theory.cm.utexas.edu/kdb.
ACKNOWLEDGMENTS
The work was supported by the National Science Foun-
dation (Award No. CHE-1152342). The authors are grateful
for allocations of computing resources at the Texas Advanced
Computing Center, and for donations of computer time from
participants of the EON distributed computing project.
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