Supporting Information for
Predictive Beyond-Mean-Field Rate Equations for Multisite Lattice-Gas
Models and for Catalytic Surface Reactions: CO-Oxidation on Pd(100)
Da-Jiang Liu,1 Federico Zahariev,1 Mark S. Gordon,1,2 and James W. Evans1,3,4
Laboratory – USDOE, Iowa State University, Ames, Iowa 50011
of Chemistry, Iowa State University, Ames, Iowa 50011
3Department of Mathematics, Iowa State University, Ames, Iowa 50011
4Department of Physics & Astronomy, Iowa State University, Ames, Iowa 50011
1Ames
2Department
Journal of Physical Chemistry C (Special Issue in honor of M.S. Gordon)
S1. Cluster geometries for analysis of CO adsorption at br sites on Pd(100) facets
We choose clusters geometries such that, even for relatively small sizes, the top
facet corresponds to a fcc(100) surface and includes a central br site (which is
separated as far as possible from the facet edge). The smallest such cluster has 8
atoms with 6 atoms in the top layer and 2 atoms beneath. The next largest cluster has
20 atoms with 12 in the top layer, 6 in the middle layer, and 2 in the bottom layer. See
Figure S1. The general cluster in this sequence has Lx(L+1) atoms in the top layer,
(L-1)xL in the next underlying layer, (L-2)x(L-1) in the next, etc., and 2x1 in the bottom
layer for a total of L(L+1)(L+2)/3 atoms (for L  2).
Figure S1. Schematic of geometries of 8-atom, 20-atom,… Pd clusters used to assess the CO
adsorption energy at the central bridge site.
1
S2. Further results from plane-wave DFT VASP analysis.
As indicated in Sec.3.3, plane-wave DFT calculations were performed using the
VASP code (version 5.3.5) with the projector augmented-wave (PAW) method. The
energy cutoff for plane-wave basis set is 400 eV. For periodic slab calculations
modeling the fcc(100) surface, the Methfessel-Paxton method is employed with  = 0.2
eV. The slabs are chosen to be separated by 1.2 nm in the direction orthogonal to the
fcc(100) surface, and the computational supercell vectors correspond to various
multiples of those for the primitive unit cell of the (100) surface. For cluster calculations,
Gaussian smearing with  = 0.2 eV is also used as the default. Also for these cluster
calculations in VASP, each cluster is contained in a supercell which is repeated with a
rectangular periodicity. The size of the supercell is so chosen that the closest separation
between the edges of periodic images of the cluster is at least 1.2 nm. Since the
periodicity has no physical meaning, a 1x1 or -point only k-point grid is used. Unless
stated otherwise, all calculations have paired spins (corresponding to the singlet state).
Results are shown in Table SI from periodic plane-wave DFT analysis using slab
geometries for the PBE functional for the CO adsorption energy on Pd(100) at various
sites. Results are obtained by averaging over slab thickness of 7-12 layers to minimize
quantum size effects. It is clear that lateral CO adspecies interactions have minimal
effect on estimates of adsorption energy even for the smallest p(2x2) unit cell. Indeed
lateral interactions between CO on br sites separated by d = 2a were estimated in Ref.6
to be only 0.020 eV, and interactions for larger d are significantly smaller.
p(2x2)
(55)R26.6
(2222)R45
CO coverage
0.25 ML
0.20 ML
0.125 ML
br
-1.902
-1.933
-1.942
4fh
-1.842
-1.861
-1.882
top
-1.475
-1.489
-1.514
Table S1. PBE estimates of CO adsorption energies, EaCO (in eV), on Pd(100) at various sites
from plane-wave DFT VASP analysis using slab geometries.
We have also performed limited slab calculations using the hybrid HSE06
functional to assess the CO adsorption energy at br sites. Given the expense of such
analysis in VASP, we restrict attention to thin slabs using a small c(2x2) unit cell with a
(4x4) q-point grid for the exact exchange versus the full 8x8 k-point grid, and use frozen
PBE optimized geometries. Results shown in Table S2 varying slab thickness, m (in
atomic layers), indicate very similar adsorption energy estimates for HSE06 and PBE.
m
1
2
3
4
5
HSE06
-2.583
-2.072
-1.927
-2.104
-1.886
PBE
-2.497
-1.962
-1.896
-1.915
-1.901
Table S2. HSE06 (versus PBE) estimates of CO adsorption energies (in eV) at br sites on
Pd(100).
2
Extensive results for plane-wave VASP results for CO adsorption energies at
various adsorption sites on Pd clusters of various sizes are shown in Table S3. The
cluster geometry for analysis of adsorption energy at br sites is described in S1. For
adsorption at 4fh sites, the series of clusters considered have the following structure:
2x2 atoms in the top layer, and 1 beneath [size 5 atoms]; 4x4 atoms in the top layer,
3x3 in the next layer, then 2x2, then 1 [size 30 atoms]; etc., and in the general case
(2L)x(2L) in the top layer, (2L-1)x(2L-1) in the next layer, etc. [size 2L(2L+1)(4L+1)/6
atoms]. For these clusters, the adsorbed CO can be and is placed at a 4fh site exactly
in the center of the top layer. For adsorption at top sites, the series of clusters
considered have the following structure: 3x3 atoms in the top layer, 2x2 in the next, and
1 atom beneath [size 14 atoms]; 5x5 in the top layer, 4x4 in the next, etc. [size 55
atoms]; etc., and in the general case (2L+1)x(2L+1) atoms in the top layer, (2L)x(2L) in
the next, etc. [size (2L+1)(2L+2)(4L+3)/6 atoms]. For these clusters, the adsorbed CO
can be and is placed at a top site exactly in the center of the top layer.
size
8
20
40
70
112
168
240
330
br
-2.042
-2.138
-2.014
-1.870
-2.028
-1.905
-1.901
-2.002
size
5
30
91
204
385
4fh
-2.462
-2.190
-1.737
-1.758
-1.836
size
14
55
140
285
top
-1.638
-1.496
-1.497
-1.523
Table S3. PBE estimates of CO adsorption energies (in eV) on Pd(100) at various sites from
plane-wave DFT VASP analysis using clusters for various sizes (in atoms).
We have also performed analysis of CO adsorption at br sites on Pd clusters
using the hybrid HSE06 functional. The results shown in Table S4 are similar to those
obtained from the PBE functional.
size
8
20
40
HSE06
-1.988
-1.961
-2.091
PBE
-2.042
-2.138
-2.014
Table S4. HSE06 (versus PBE) estimates of CO adsorption energies (in eV) at br sites on an 8atom Pd(100) cluster.
3
S3. Comparison of various predictions for the CO adsorption energy at br sites
for an 8 atom Pd cluster from analysis using a localized atomic orbital basis.
We use a LANL2DZ basis set with effective core potentials (ECP) for Pd, this
choice being effectively utilized for other transition metal clusters, and 6-311++G(d,p)
basis set for CO. Various levels of theory are considered including Hartree Fock (HF)
and CR-CC(2,3) are performed using GAMESS, and CR-CC(2,3) calculations are
performed using frozen core electrons. DFT calculations for PBE and PBE0 functionals
are performed using NWChem. In all cases, the cluster geometry is fixed to that
obtained from the PBE VASP analysis, and the analysis is performed for paired
electrons (i.e., for the singlet state). Table S5 presents results for these different levels
of theory for the total energy of the CO+8Pd system, the 8Pd system, and an isolated
CO. The binding energy for CO is obtained from subtracting the energy of the
(separated) 8Pd cluster and the CO from that for the CO+8Pd system. Table S6
presents the results for the CO singlet-triplet energy split for these same levels of
theory.
HF
PBE
PBE0
CR-CC(2,3)
CO+8Pd
-1119.488284 H
-1127.257645 H
-1127.006634 H
-1121.078103 H
8Pd
-1006.838287 H
-1013.963596 H
-1013.720598 H
-1007.944710 H
CO
-112.767247 H
-113.221434 H
-113.216431 H
-113.096390 H
Binding
-3.19 eV
-1.98 eV
-1.89 eV
-1.01 eV
Table S5. Total energy of the CO+8Pd system, the 8Pd system, and an isolated CO, from
which the adsorption energy of CO at a br site (Binding) is determined.
S
T
ET-S
HF
-112.767247 H
-112.568161 H
5.4174 eV
PBE
-113.221434 H
-113.004427 H
5.9051 eV
PBE0
-113.216431 H
-112.999817 H
5.8944 eV
CR-CC(2,3)
-113.096390 H
-112.868289 H
6.2069 eV
Table S6. Energy of CO singlet and triplet states from which the singlet-triplet energy split is
determined.
4
S4. Unrestricted spin analysis of the cluster size-dependence of CO adsorption
based on a localized atomic orbital analysis for PBE and PBE0 in NWChem.
Below, we present results for the total energy of an isolated CO (in Table S7) and
of the CO+nPd system and the nPd system (for n = 8 in Table S8, and for n = 20 in
Table S9), for PBE and PBE0 functionals using the localized atomic basis described in
S3. Analysis is performed for unpaired spins with various spin multiplicities. Energies in
the Tables are given in Hartree (H) and we use 1 Hartree = 27.2114 eV to convert from
Hartree (H) to eV.
CO (singlet):
PBE
PBE0
-113.2214 H
-113.2164 H
Table S7. Ground state energy in Hartree for CO (corresponding to the singlet state).
CO+8Pd:
Hartree
PBE
PBE0
M=1
-1127.2576
-1127.0066
M=3
-1127.258
-1127.009
M=5
-1127.2487
-1127.0084
M=7
-1127.2231
-1126.9673
M=9
-1127.1720
-1126.9078
M = 11
-1127.1006
-1126.8406
8Pd:
Hartree
PBE
PBE0
M=1
-1013.9635
-1013.7206
M=3
-1013.9707
-1013.7321
M=5
-1013.980
-1013.745
M=7
-1013.9499
-1013.7004
M=9
-1013.9062
-1013.6540
M = 11
-1013.8272
-1013.5831
Table S8. Energies in Hartree for various spin multiplicities (M) for CO+8Pd and for 8Pd from
unrestricted spin (open shell) calculations. The M-value and energies for the ground state are
indicated in bold (and blue color).
CO+20Pd:
Hartree
PBE
PBE0
Hartree
PBE
PBE0
20Pd:
Hartree
PBE
PBE0
Hartree
PBE
PBE0
M=1
-2648.5449
-2647.8517
M=3
-2648.5484
-2647.8679
M = 13
2648.5406
-2647.8824
M=1
-2535.2546
-2534.5759
M=3
-2535.2528
-2534.5848
M = 13
-2535.2632
-2534.5988
M=5
-2648.5505
-2647.8820
M=7
-2648.5488
-2647.8934
M = 15
-2648.5061
-2647.8279
M=5
-2535.2595
-2534.5914
M=7
-2535.267
-2534.6037
M = 15
-2535.2366
-2534.5820
M=9
-2648.5585
-2647.906
M = 11
-2648.562
-2647.9011
M = 17
2648.4651
-2647.793
M=9
-2535.2644
-2534.6125
M = 11
-2535.2641
-2534.613
M = 17
-2535.2024
-2534.5337
Table S9. Energies in Hartree for various spin multiplicities (M) for CO+20Pd and for 20Pd from
unrestricted spin (open shell) calculations. The M-value and energies for the ground state are
indicated in bold (and blue color).
5
From the above results, we determine the CO adsorption energy at the br site. As
in Sec.S3, this adsorption energy is obtained from subtracting the energy of the
(separated) 8Pd cluster and the CO from that for the CO+8Pd system. One could
perform the analysis restricting consideration to the singlet state (M = 1). Alternatively,
performing the analysis without this restriction, we choose the lowest energy state for
each of the CO+nPd and the nPd systems (which may have different multiplicity).
From a singlet-state analysis for 8-atom Pd clusters, we obtain an adsorption
energy, EaCO, of -1.98 eV for PBE (versus -2.04 eV from the plane-wave PBE analysis),
and -1.89 eV for PBE0 (versus -1.99 eV from the plane-wave HSE06 analysis). From an
unpaired spin analysis for 8-atom Pd clusters, we obtain an adsorption energy, EaCO, of
-1.55 eV for PBE (versus -1.87 eV from the plane-wave PBE analysis), and -1.31 eV for
PBE0.
From a singlet-state analysis for 20-atom Pd clusters, we obtain an adsorption
energy, EaCO, of -1.87 eV for PBE (versus -2.13 eV from the plane-wave PBE analysis),
and -1.61 eV for PBE0 (versus -1.96 eV from the plane-wave HSE06 analysis). From an
unpaired spin analysis for 20-atom Pd clusters, we obtain an adsorption energy, EaCO,
of -2.00 eV for PBE (versus -2.23 eV from the plane-wave PBE analysis), and -2.08 eV
for PBE0.
S5. Extended 8-site and 9-site models for O2/Pd(100) with reorientation
As indicated in Sec.4.5, for oxygen adsorption models where one of two
adsorption orientations (1 or 2) are possible, the sticking probability has the form
SO2 = P(1) + P(2) – P(12)
where P(j) denotes the probability that orientation j is available and P(12) denotes the
probability that both are available (so this quantity is subtracted to avoid double
counting). For our models, P(1) = P(2) = P8h+8b (P9h+8b) and P(12) = P12h+12b (P13h+16b)
for the 8-site (9-site) model. The configurations associated with P8h+8b (P9h+8b) which
have 8 (9) empty 4fh sites and 8 empty br sites are shown in Figure 8, and the
factorization of these quantities is described in detail in the text in Sec.4.3 and Sec.4.4.
The configurations associated with P12h+12b (P13h+16b) which have 12 empty 4fh sites and
12 empty br sites (13 empty 4fh sites and 16 empty br sites) are shown in Figure S2.
We now describe the appropriate factorization of these latter quantities.
For the 8-site model with reorientation, P12h+12b (see Figure S2a) is determined
from
P12h+12b = P12hQ12b|12h with P12h = (P2h)16/(P1h)20 and Q12b|12h = (Q4b)5(Q3b)4(Q1b)4/(Q2b)12. (S1)
The factorization of P12h uses the feature that the ensemble of 12 4fh sites involves 16
NN pairs which overlap and share many single 4fh sites. The factorization of Q 12b|12h
uses the feature that the ensemble of 12 br sites can be regarded as consisting of 5
6
quartets and 4 triangles which overlap and share many NN pairs of br sites. For the
9-site model with reorientation, P13h+16b (see Figure S2b) is determined from
P13h+16b = P13h Q16b|13h, with P13h = (P2h)16/(P1h)19 and Q16b|13h = (Q1b)4(Q4b)9/(Q2b)12.
(S2)
The factorization of P13h uses the feature that the ensemble of 13 4fh sites involves 16
NN pairs which overlap and share many single 4fh sites. The factorization of Q 16b|13h
uses the feature that the ensemble of 16 br sites can be regarded as consisting of 9
quartets which overlap and share many pairs of NN br sites. As an aside, one can show
that for this 9-site model with reorientation, SO2 decreases quadratically in CO as 2 CO’s
are required to block adsorption.
Figure S2. Configurations for the 8-site and 9-site models for O2 adsorption with reorientation
which allow adsorption in either of the two possible orthogonal orientations.
Finally, we mention that other formulations with more flexible O2 adsorption (e.g.,
by introducing funneling rather than reorientation, or by incorporating both mechanisms)
can also reduce the initial decrease in SO2. See Ref.22.
7