A Graph-Theoretical Kinetic Monte Carlo Framework for
on-Lattice Chemical Kinetics
Michail Stamatakis and Dionisios G. Vlachos
Department of Chemical Engineering, University of Delaware, Newark, DE 19716, USA
SUPPLEMENTARY MATERIAL
1. Rate Expressions for Elementary Steps Involving Gas Species
For a surface reaction of the type X* + Y*  Z*, the partition functions of all species have
only vibrational components, and thus, the rate is calculated as:
kSurface 
 E  
q vib
k T
 B  exp  

q vib,X  q vib,Y
h
 kB  T 
(1)
The vibrational partition function of species X is expressed as a product of contributions from
each vibrational mode of that species:
q vib,X
 h  k 
exp  

 2  kB  T 
 
 h  k 
k
1  exp  

 kB  T 
(2)
and similar equations hold for species Y and Z.
For an Eley-Riedel reaction of the type X(gas) + Y*  Z*, the rate is given as:
k fwd
Eley  Riedel  k TST  N X 
 E  
k T
Q
 B  exp  
  NX
QXgas  QY
h
 kB  T 
(3)
where NX is the number of X molecules in the gas phase, Ast the surface area of the site(s) over
which the reaction takes place, and Auc the area of the surface unit cell. The partition functions
QY and Q have only vibrational components and can be computed from equations similar to (2).
The partition function for X(gas) can be written as:
QXgas  q vib,Xgas  q rot,Xgas  q trans3D,Xgas
(4)
The distance of X(gas) from the surface can be used as a reaction coordinate, thereby motivating
the decoupling of the z component of the translational partition function as follows:
q trans3D,Xgas  q trans2D,Xgas  l z 
2   mX  k B  T
h
(5)
where l z the length of the gas-phase in the z-direction and the 2D translational partition function
over a unit cell of the surface is:
-1-
q trans2D,Xgas = A st 
2    mX  k B  T
h2
(6)
where Ast the effective area of the site(s) where the reaction takes place. In equation (4), the
rotational component for a linear molecule is:
8  2  I  k B  T
  h2
q rot,Xgas 
(7)
where I is the moment of inertia and  the symmetry number of the molecule. For molecules
with non-linear geometry:
3
1
  Ia  Ib  Ic  2   8  2  k B  T  2
q rot,Xgas 



h2


(8)
By substituting equations (4, 5) into (3) we obtain:
 E  
q vib
NX  k B  T

 exp  

q vib,Xgas  q rot,Xgas  q trans2D,Xgas  q Y,vib l z  2   mX  k B  T
 kB  T 
k fwd
Eley  Riedel 
(9)
Further, from the ideal gas law pXV = NXkBT and noting that V = l zAuc we end up with:
k
fwd
Eley  Riedel
 E  
q vib
pX  Ast


 exp  

q vib,Xgas  q rot,Xgas  q trans2D,Xgas  q Y,vib 2   mX  k B  T
 kB  T 
(10)
For the reverse reaction:
 E   E rxn 
q vib k B  T

 exp  

q Z,vib
h
kB  T


k rev
Eley  Riedel 
(11)
where Erxn is the reaction energy changeErxn > 0 for exothermic, < 0 for endothermic).
An activated adsorption elementary step denoted as X(gas) + *  X*, can be treated similarly:
k
fwd
Activated
Adsorption
 E  
q vib
p X  Ast


 exp  

q X,vib  q X,rot  q X,trans2D 2   mX  k B  T
 kB  T 
(12)
and for the desorption:
k rev
Activated
Adsorption

 E   Eads 
q vib k B  T

 exp  

q X,vib
h
kB  T


(13)
For non-activated (endothermic) adsorption one assumes an early 2D gas-like transition state
and thus the rates for adsorption and desorption simplify to:
k fwd
Non  Activated 
Adsorption
k rev
Non  Activated 
Adsorption
p X  Ast
2   m X  k B  T
(14)
q vib,Xgas  q rot,Xgas  q trans2D,Xgas k  T
 E 
 B  exp  ads 
q vib,X
h
 kB  T 
-2-
(15)
2. Comparison of the Computational Times of Graph-Theoretical
KMC and a Simpler KMC Approach
The table below provides the CPU times for three different runs for each KMC framework,
namely the graph-theoretical KMC (GT-KMC) and a KMC approach capable of simulating 1and 2-site processes only (12-KMC). All times are reported in seconds. The ZGB system (Table
3 in main text) was simulated for PA = PB2 = 0.5 and tfinal = 100. The lattice size is given in
number of sites in x direction by number of sites in the y direction, thus the 100  100 lattice has
10000 sites. The lattice coordination number was equal to 4 in all simulations.
Lattice
Size
GT-KMC
run 1
run 2
run 3
12-KMC
tavgGTKMC
run 1
run 2
run 3 tavg12-KMC
| Δt avg |
| Δt avg |
t avg12-KMC
25  25
22.4
22.4
23.5
22.7
16.9
16.5
16.3
16.6
6.2
37%
35  35
44.0
44.4
46.1
44.8
35.0
32.2
32.1
33.1
11.7
35%
50  50
91.6
92.1
95.0
92.9
73.7
68.5
68.2
70.1
22.8
32%
71  71
184.2
186.1
184.4
184.9
155.3
142.6
142.1
146.7
38.2
26%
100  100
363.5
363.9
363.8
363.8
308.6
295.8
295.3
299.9
63.9
21%
141  141
727.3
727.7
728.3
727.8
648.2
623.2
623.0
631.4
96.3
15%
200  200
1485.5
1486.4
1489.4
1487.1
1389.0
1330.6
1327.7
1349.1
138.0
10%
283  283
3049.7
3050.9
3056.8
3052.5
2894.8
2811.9
2808.0
2838.2
214.2
8%
400  400
6210.3
6213.4
6220.6
6214.8
6038.9
5855.3
5913.8
5936.0
278.8
5%
12686.0 12250.0 12225.0 12387.0
421.3
3%
566  566
12658.0 12599.0 13168.0 12808.3
-3-
3. Water-Gas Shift Elementary Reaction Patterns
In the following graphs, different site types are denoted as follows:
top
CO adsorption
CO(gas)
CO*
*
O2 dissociative adsorption
O2(gas)
*
*
O*
O*
*
*
H2 dissociative adsorption
H2(gas)
*
*
H*
H*
*
*
CO2 dissociative adsorption
CO2(gas)
*
*
O*
CO*
*
*
H2O adsorption
H2O(gas)
*
H2O*
H2O decomposition
H2O*
*
OH*
H*
-4-
fcc hollow
bridge
OH decomposition
*
*
O*
H*
OH*
*
OH-OH disproportionation
*
OH*
O*
OH*
*
H2O*
*
*
CO-OH disproportionation
H2O*
CO*
*
tCOOH*
COOH decomposition
*
tCOOH*
*
H*
CO2(gas)
tCOOH-O disproportionation
tCOOH*
*
O*
OH*
*
*
*
*
CO2(gas)
tCOOH-OH disproportionation
CO2(gas)
tCOOH*
OH*
*
H2O*
HCO formation
H*
CO*
*
*
*
*
O*
CH*
HCO**
-5-
HCOO formation
O*
O*
*
*
CH*
O*
O*
CH*
HCO**
HCOO***
HCOO decomposition
*
O*
CO2(gas)
H*
O*
CH*
*
*
*
HCOO***
HCOO-O disproportionation
O*
O*
CO2(gas)
*
O*
CH*
*
OH*
*
HCOO***
HCOO-OH disproportionation
OH*
H2O*
*
O*
CH*
CO2(gas)
*
O*
*
*
HCOO***
-6-
*
4. Data Used in the Rate Constant Calculations
Table S3.1: Total energies ETotal, moments of inertia IA or product thereof (IAIBIC) for nonlinear molecules,
symmetry numbers s, and vibrational frequencies  for gas phase species.
Gas Species
CO
CO2
O2
H2
Gas Species
H2O
Linear Molecules
IA (10−47kgm2)
14.568
71.4988
19.471
0.47203
Non-Linear Molecules
ETotal (eV)
IAIBIC (10−141kg3m6)
-467.4604
5.8577
ETotal (eV)
-588.9694
-1025.4623
-867.2017
-31.6629

1
2
2
2
 ν (cm−1)
2143
667, 1333, 2349
1580
4190

2
 ν (cm−1)
1595, 3657, 3756
Moments of inertia were read from Ott, J. B., and Boerio-Goates, J. (2000). “Chemical thermodynamics: advanced
applications”. Chemical Thermodynamics, ed. J. Boerio-Goates. Vol. 2. London: Academic Press.
Vibrational frequencies of polyatomics (CO2, H2O) were read from Pople, J. A., H. B. Schlegel, et al. (1981).
“Molecular-Orbital Studies of Vibrational Frequencies”. International Journal of Quantum Chemistry: 269-278.
Table S3.2: Total energies ETotal, and vibrational frequencies  for surface species and transition states. The free
slab energy is Eslab = −11448.22 eV. Vibrational frequencies > 200 cm−1 are reported. The numbers under the
transition states column correspond to the elementary steps of Table 2 in the main manuscript. Adsorption steps
having early gas-species-like transition states are not mentioned.
Surface Species
CO*
O*
H2O*
OH*
H*
HCO**
HCOO***
COOH*
Transition State
3
4
6
7
8
9
10
11
12
13
14
15
16
17
ETotal (eV)
-12039.09
-11882.98
-11916.04
-11899.15
-11464.54
-12054.46
-12489.87
-12490.26
ETotal (eV)
-11479.95
-12472.44
-11915.26
-11898.21
-12350.53
-12489.54
-12489.40
-12924.54
-12941.08
-12054.10
-12488.19
-12488.83
-12922.88
-12939.78
 ν (cm−1)
367, 374, 455, 2010
366, 394, 467
241, 454, 554, 1559, 3483, 3606
494, 855, 3576
692, 701, 986
235, 333, 377, 535, 718, 1129, 1284, 2907
287, 322, 336, 733, 904, 1264, 1288, 1535, 2995
267, 308, 434, 572, 655, 1076, 1206, 1675, 3341
 ν (cm−1)
305, 310, 1601, 2255
290, 350, 370, 422, 545, 1896
254, 282, 493, 524, 870, 1866, 3552
260, 410, 477, 1796
268, 432, 565, 578, 724, 839, 1353, 1454, 3551
310, 386, 409, 512, 553, 839, 1926, 3545
275, 319, 555, 576, 1157, 1804, 1958
235, 264, 324, 448, 568, 630, 1048, 1120, 1174, 1764, 2234
267, 308, 434, 494, 572, 655, 855, 1076, 1206, 1675, 3341, 3576
213, 288, 315, 624, 977, 1597, 2053
285, 305, 412, 475, 555, 818, 1057, 1488, 2975
207, 370, 608, 895, 1030, 1240, 1532, 1644
235, 264, 324, 448, 568, 630, 1048, 1120, 1174, 1764, 2234
201, 225, 375, 473, 652, 674, 963, 977, 1103, 1292, 1621, 1959, 3052
-7-
5. Effect of Diffusion
To evaluate the effect of diffusion, the hopping of CO and H species was accounted for, since
these are the two most abundant species on the surface:
CO diffusion
CO*
*
CO*
*
*
*
H diffusion
H*
*
H*
*
*
*
These two processes were assumed to be fast compared to the adsorption-desorption dynamics of
the two species and thus the kinetic constants were taken such that the frequency of diffusion
events is at least two orders of magnitude higher frequency of adsorption-desorption events. A
comparison of the KMC results in the absence and presence of diffusion are shown below.
Without Diffusion
With Diffusion
Coverage fractions
0.8
0.6
0.4
0.2
H*
CO*
1
Coverage fractions
H*
CO*
1
0
0.8
0.6
0.4
0.2
0
0
1
2
3
Time (s)
4
5
-3
x 10
0
1
2
3
Time (s)
4
5
-3
x 10
The figure above shows the coverages for the parameter set of Figure 9a in the main manuscript.
Including diffusion in the reaction mechanism does not affect the CO coverage, or the H2O
coverage which remains low (not shown). On the other hand, diffusion affects the H coverage, as
evidenced by the higher values of the latter when this process is present. This effect can be
explained by the fact that H2 desorbs dissociatively: in the absence of diffusion the two hydrogen
adatoms produced by water and carboxyl decomposition remain within the same neighborhood
and are ready to desorb. However, in the presence of diffusion, the two adatoms no longer
remain localized, and thus, H desorption is limited by how fast an adatom can diffuse and find a
-8-
“partner” with which it can desorb associatively. Since the process of desorption is hindered by
this effect, higher H2 coverages are observed.
Without Diffusion
With Diffusion
CO_adsorption
CO_diffusion
H2_dissociative_adsorption
H_diffusion
H2O_adsorption
H2O_decomposition
CO_OH_disproportionation
COOH_decomposition
HCO_formation
CO_adsorption
H2_dissociative_adsorption
H2O_adsorption
H2O_decomposition
CO_OH_disproportionation
COOH_decomposition
fwd
rev
HCO_formation
0
10
2
4
10
10
6
8
10
10
10
fwd
rev
0
10
2
10
Event frequency (s-1)
4
10
10
6
8
10
10
10
10
Event frequency (s-1)
The figure above shows the histograms for the reaction statistics for the parameter set of Figure
9d. Diffusion of CO and H was assumed to be fast compared to the adsorption-desorption
dynamics of these species. Thus, the frequency of diffusion events was at least 2 orders or
magnitude higher than that of adsorption or desorption events. The frequencies of events other
than diffusion are similar in the two simulations. Note that HCO formation occurs more
frequently when diffusion is present due to the higher H coverage on the lattice.
Without Diffusion
10
10
10
2
E =
12.0
a
kcal
/m
1
10
ol
E
a
=
24
.3
0
kc
al/
mo
TOF (s -1)
TOF (s -1)
10
With Diffusion
l
-1
1.5
1.6
1.7
1.8
1.9
2
10
10
10
2.1
-1
2
1
E =
a 16
.1 k
cal/
mo
l
E
a
=
28
0
.7
k
ca
l/
m
ol
-1
1.5
1.6
1.7
1.8
1.9
2
2.1
-1
1000/T (K )
1000/T (K )
The figure above shows comparisons of the Arrhenius plots. The qualitative characteristics of the
graphs are the same, and the apparent activation energies are also similar (within 5 kcal/mol).
-9-
6. Error Estimates for the Arrhenius Plot
Given that NH2 molecules have been produced over the time course of the simulation, t sim = tfin −
tini from a total number of sites SL, we need to estimate the rate per monolayer at which H2 is
produced, assuming that the production of each H2 molecules follows a Poisson process. Thus,
the waiting times for the production of each molecule are exponentially distributed (memoryless
process):
i ~ Exp  r 
(16)
and thus the overall number of H2 molecules produced follows a Poisson distribution:
N H2 ~ P   
 NH 2  e
P  N H2 |   
N H2 !

where   r  SL  t sim
(17)
We use Bayesian inference to obtain a distribution for  from which the expectation ̂ , and a
confidence interval (CI) can be deduced. In this approach, we postulate a prior distribution for ,
that incorporates our existing knowledge on the value of this parameter. Then, using the
information about NH2 we find a posterior distribution from which ̂ and the CI are obtained.
P   | N H2   c  P  N H2 |    P   
(18)
where c is a constant. If the prior P() and the posterior P(|NH2) belong to the same family of
distributions, then the latter family is referred to as the conjugate prior of P(NH2|). For the
Poisson distribution the conjugate prior is the Gamma distribution:
 ~ Γ  ,  

P  

  1  e 


 
(19)
where () is the gamma function:

  z    t z 1  e  t dt
(20)
0
Therefore, if one uses Γ (,) as the prior distribution, then given n samples for NH2 the posterior
distribution will be:
 | N H1 2 , N H22 ,..., N Hn2 ~ Γ    N Total
H2 ,   n 
 
 
n
i
where N Total
H2   NH2
 
 
(21)
i 1
From the above equation, it turns out that if the prior gamma distribution has   0 and   0,
the expectation of the rate is identical the maximum likelihood estimator:
1
ˆ   N Total
H2
n
 
 
 
given N H12 , N H22 ,..., N Hn2
(22)
Further, the 95% CI can be found by solving the following equations:
  N Total
H 2 ,  min 
  N Total
H2 
 0.05
and
  N Total
H 2 ,  max 
  N Total
H2 
-10-
 0.95
(23)
where (k,x) is the lower incomplete gamma function:
x
  z, x    t z 1  e  t dt
(24)
0
Thus, the 95% CI for the rate r can be found by scaling the CI for  by SLtsim.
-11-