Supplementary material for “CO and CO2 hydrogenation to
methanol calculated using the BEEF-vdW functional”
Felix Studt1, Frank Abild-Pedersen1, Joel B. Varley2, Jens K. Nørskov1,2
1
SUNCAT Center for Interface Science and Catalysis, Photon Science, SLAC National
Accelerator Laboratory, Menlo Park, CA 94025
2
Department of Chemical Engineering, Stanford University, Stanford, CA 94305-5025
1
Electronic structure calculations
The first principles calculations have been performed based on Density Functional Theory (DFT).
Calculations were done partly with the ultra-soft pseudopotential1 plane-wave-based code DACAPO2 and
partly with the grid-based projector-augmented wave method (GPAW) code3,4. The plane-wave based
calculations were performed with the RPBE generalized gradient approximation whereas for the grid-based
approach the semi-empirical BEEF-vdW functional5 was used. In both cases we used a 3×3×1 MonkhorstPack6 sampling of the Brillouin zone. For the plane-wave-based code we used 25 Ry as the cut-off energy
for the plane-waves whereas a uniform real-spaced grid with a spacing of 0.18 Å was used for the
representation of the electronic wave functions in GPAW. The stepped surfaces were modeled using the
supercell slab approach with a 9 layer 1x3 (211) unit cell. The distance between slabs is more than 12 Å
and all structures were relaxed such that the average forces were below 0.05 eV/Å. The transition states
shown in Figures S1 – S3 are located using the fixed bond length method. First order saddle points are
identified from vibrational calculations showing one imaginary frequency.
1.1
Adsorbate free energy contributions in RPBE and BEEF-vdW
Table S1 and S2 show all total energies, zero point energies (ZPE), enthalpic temperature corrections, and
entropies (S) of intermediates in the CO and CO2 hydrogenation on Cu(211). All ZPE and S have been
obtained from vibrational calculations for the RPBE optimized structures. This data together with the
RPBE total energies are taken from Reference 8 unless stated otherwise. The calculations shown in Tables
S1 and S2 employing the BEEF-vdW functional were performed using the RPBE optimized Cu lattice
parameter. For the adsorbed intermediates in Table S2 the RPBE optimized adsorption structures were
taken as initial guess for the optimization with BEEF-vdW. However, all intermediates were allowed to
relax fully together with the topmost (111) surface layer on the stepped Cu(211) surfaces. The two
remaining (111) layers were kept fixed in their bulk structure.
The total energies for the BEEF-vdW transition states in Table S2 were obtained by performing a
static calculation on the RPBE optimized transition state structures. To justify this approach we note that all
re-optimized adsorption total energies, using BEEF-vdW, shown in Table S1 changed by less than 0.05 eV
during the optimization. Since the transition states are primarily determined by the strength of the covalent
bond in the final state and covalent bonding is much stronger than the vdW part it is reasonable to assume
errors on the transition state energies of the same magnitude.
Table S1. Total energies, ZPE, enthalpic temperature correction, and S of intermediates and molecules
involved in the hydrogenation of CO and CO2 to methanol on Cu(211) calculated using the RPBE and
BEEF-vdW functional. aCorrected as described in Ref. 9. bCorrected using the fitting procedure described
in Section 1.2. cTaken from Ref. 8.
Species
RPBE
BEEF-vdW
ZPE
S
∫ 𝐶𝑃 d𝑇
(eV)
(eV)
(eV)
(eV/K)
Cu211 slab
-45195.146c
-2041.749
0.000
0.000
0.000000
H*
-45211.285c
-2045.825
0.165
0.007
0.000063
CO*
-45787.246c
-2077.148
0.192
0.085
0.000452
HCO*
-45802.816c
-2080.642
0.440
0.092
0.000661
H2CO*
-45819.234c
-2085.056
0.550
0.104
0.000627
H3CO*
-45836.258c
-2090.085
1.083
0.099
0.000638
HCOO*
-46241.720c
-2101.153
0.624
0.105
0.000751
HCOOH*
-46256.450a,c
-2104.356b
0.900
0.105
0.002515
H2COOH*
-46273.310c
-2108.722
1.231
0.135
0.001288
OH*
-45648.760c
-2065.151
0.359
0.046
0.000318
COOH*
-46240.590c
-2100.003
0.588
0.120
0.000911
H2COO*
-46256.130c
-2103.541
0.861
0.115
0.000911
CO
-591.433c
-34.724
0.130
0.091
0.002092
H2
-32.030c
-7.937b
0.270
0.091
0.001380
CO2
-1029.562a,c
-54.262b
0.320
0.098
0.002263
H2O
-469.751c
-27.163
0.560
0.104
0.001999
CH3OH
-657.207c
-52.120
1.376
0.117
0.002539
Table S2. Total energies, ZPE, enthalpic temperature correction, and S of transition states involved in the
hydrogenation of CO and CO2 to methanol on Cu(211) calculated using the RPBE and BEEF-vdW
functional. aTaken from Ref. 8. bLowest energy transition-state, see Figures S1-S3 for a detailed description
of the geometric differences of the transition state structures
Transition state
RPBE
BEEF-vdW
ZPE
Entropy
∫ 𝐶𝑃 d𝑇
(eV)
(eV)
(eV)
(eV/K)
H* + CO* -> HCO*
-45802.462a
-2080.264
0.086
0.312
0.000606
H*+HCO* -> H2CO*
-45818.385a
-2084.129
0.094
0.602
0.000715
H*+H2CO* -> H3CO*
-45835.040a
-2088.797
0.105
0.889
0.000814
H*+H3CO* -> CH3OH + *
-45851.585b
-2093.350
0.118
1.184
0.000976
H*+CO2 -> HCOO*
-46240.400a
-2099.771
0.108
0.399
0.000985
H*+HCOO* -> HCOOH*
-46256.545b
-2103.835
0.120
0.700
0.000922
H*+HCOOH*-> H2COOH*
-46272.450a
-2107.938
0.131
1.050
0.001019
H2COOH* -> H2CO*+OH*
-46272.849a
-2108.458
0.150
1.141
0.000931
H*+OH* -> H2O + *
-45664.075b
-2068.249
0.062
0.459
0.000524
H*+CO2 -> COOH*
-46239.416a
-2098.792
0.122
0.418
0.000780
Figure S1. Transition state geometries for the splitting of water, a) when the product species are bound to
distinct surface atoms (lowest energy transition-state) and b) when the product species are bound to a single
step-edge surface atom.
Figure S2. Transition state geometries for the dehydrogenation of methanol, a) when the product species
are bound to distinct surface atoms (lowest energy transition-state) and b) when the product species are
bound to a single step-edge surface atom.
Figure S3. Transition state geometries for the dehydrogenation of formic acid, a) when the reaction
involves distinct surface atoms (lowest energy transition-state) and b) when the reaction only involves a
single step-edge surface atom.
1.2
Gas-phase corrections for RPBE and BEEF-vdW
Several previous studies using the RPBE functional have shown that a number of calculated gas-phase
thermochemical reaction energies pertinent to CO and CO2 reduction exhibit large errors when compared to
experiment.9,10 Addressing such errors is critical to properly describe reactions involved in a number of
important industrial processes such as steam methane reforming, 10 the water-gas shift reaction,9 and
methanol synthesis, the topic of this work. One correction approach by Peterson et al. used a statistical
sensitivity analysis to attempt to identify the origins of such errors and apply a systematic correction. 9 This
study identified and quantified how poorly RPBE describes gas-phase molecules with an OCO backbone,
such as CO2, HCOOH, CH3COOH, and HCOOCH3, finding a correction of +0.45 eV necessary to
minimize the error in calculated reaction energies for a set of 21 reactions involving CO and CO2.
Interestingly, RPBE was found to describe CO well in such an analysis, with the largest contribution of the
error due to the OCO species. Here we apply a similar statistical analysis, both repeating the study for
RPBE and extending it to the BEEF-vdW functional. We find that compared to RPBE, BEEF-vdW
systematically introduces more error into each individual species and can be most improved by applying a
systematic simultaneous correction to the OCO containing species and H 2. We find these corrections to be
critical to improve the descrip- tion of reaction energies not just for methanol synthesis, but for all tested
reactions involving the problematic gas-phase species.
As in Reference 9, we use the set of 21 reactions included in Table S3 to perform a statistical
sensitivity analysis of the gas-phase errors in RPBE and BEEF-vdW using the GPAW code.3 Of particular
importance to this study are reactions 5 and 6 involved in methanol synthesis. The analysis consists of
applying a systematic correction to an individual molecular species such as CO, H2, H2O, or those
containing an OCO backbone, and determining the mean absolute error (MAE) with respect to the
experimental values for the reactions in Table S3. For a given molecular species, the optimal correction is
obtained by minimizing the MAE of all reactions in Table S3. This is equivalent to minimizing the MAE of
the “affected reactions”, i.e. the reactions that include the corrected species, because the MAE of the
“bystander reactions” remains constant. Without any corrections, we find a significant MAE for the set of
reactions as calculated with both functionals and shown in Table S4. We find the error for the chosen set of
reactions to be slightly larger for BEEF-vdW than for RPBE.
For RPBE, we reproduce the result of Peterson et al., finding that the error in the reaction set is
largely dominated by the OCO-containing species while the error in CO is essentially negligible. Similarly,
corrections on H2 or H2O can lead to a reduction in the MAE, but the magnitudes of such corrections are
much smaller than the +0.45 eV necessary to correct the energetics of the OCO species (see Table S4).
Therefore, for a correction on a single type of species, the empirical correction of +0.45 eV to those with an
OCO backbone leads to the greatest improvement in the description of the gas-phase species and reaction
energies of Table S3.
Considering BEEF-vdW, we also find that the magnitude of the correction to OCO species is
largest and that the effects of a correction to CO are negligible. As shown in Figure S4, the optimal OCO
correction of +0.59 eV is significantly larger for BEEF-vdW than for RPBE, highlighting that a poor
description of the OCO backbone is still present with BEEF-vdW. In contrast to RPBE, we find that
corrections on H2 or H2O lead to a similar reduction in the MAE as compared to the OCO correction, with a
correction of H2O leading to the greatest improvement in Table S4.
While the analysis for RPBE clearly shows that the majority of observed error resides in the
description of the OCO backbone, the results for BEEF-vdW do not justify a similar correction for only the
OCO species. The BEEF-vdW correction for only H2O does result in an improved MAE comparable to that
in RPBE, but it does not address the error obviously still present in the OCO species. We note that in its
construction, BEEF-vdW is optimized over a much larger set of reaction energies and chemical species
than included in Table S3,6 effectively smearing out the error over a larger number of chemical species not
considered. This may be manifested in the universally larger individual corrections necessary to improve
the MAE for the 21 chosen reactions. To attempt to address the failure of both functionals in describing the
OCO backbone and simultaneously minimize the MAE for the reactions in Table S3, we extended the
analysis to two independent corrections of the OCO species and CO, H 2, or H2O.
We summarize the results for the optimal corrections and resulting MAE in Tables S4 and S5.
From the data it can be seen that a combination of simultaneously correcting the OCO backbone and H 2
leads to the most significant improvement in the energetics of the gas-phase species. In particular we
emphasize that the reaction energetics relevant for methanol synthesis (reactions 5 and 6 in Table S5) are
reduced to nearly 0. For BEEF-vdW, the optimal corrections are +0.09 eV for H2 and +0.33 eV for OCO,
again emphasizing that the majority of error resides in the description of OCO. However, we find that the
simultaneous correction to OCO and H2 is necessary for the most effective improvement to BEEF-vdW.
Figure S4. The effect of adding a correction to OCO-containing species for the reaction set in Table S3, as
calculated with RPBE (a) and BEEF-vdW (b). “Affected reactions” refers to the error in the reactions in
Table S3 containing an OCO backbone, i.e. CO2, HCOOH, CH3COOH, or HCOOCH3. “Bystander
reactions” refers to the reactions that are unaffected by any OCO correction.
(a)
(b)
Ec Correction: 0.45 eV
Ec Correction: 0.59 eV
Table S3. Reactions analyzed for the gas-phase ∆H comparisons (at 25°C and 101325 Pa) as in Ref. 9.
Reaction
Stoichiometry
0
CO2 + H2 -> CO + H2O
1
CO2 + 4 H2 -> CH4 + 2 H2O
2
3 H2 + CO -> CH4 + H2O
3
CO2 + H2 -> HCOOH
4
H2O + CO -> HCOOH
5
CO2 + 3 H2 -> CH3OH + H2O
6
2 H2 + CO -> CH3OH
7
CO2 + 3 H2 -> 1/2 CH3CH2OH + 3/2 H2O
8
2 H2 + CO -> 1/2 CH3CH2OH + 1/2 H2O
9
CO2 + 10/3 H2 -> 1/3 C3H8 + 2 H2O
10
7/3 H2 + CO -> 1/3 C3H8 + H2O
11
CO2 + 7/2 H2 -> 1/2 C2H6 + 2 H2O
12
5/2 H2 + CO 2 -> 1/2 C2H6 + H2O
13
CO2 + 3 H2 -> 1/2 C2H4 + 2 H2O
14
2 H2 + CO -> 1/2 C2H4 + H2O
15
CO2 + 11/4 H2 -> 1/4 CH2=CHCH=CH2 + 2 H2O
16
7/4 H2 + CO -> 1/4 CH2=CHCH=CH2 + H2O
17
CO2 + 2 H2 -> 1/2 CH3COOH + H2O
18
H2 + CO -> 1/2 CH3COOH
19
CO2 + 2 H2 -> 1/2 HCOOCH3 + H2O
20
H2 + CO -> 1/2 HCOOCH3
Table S4. Comparison of the gas-phase errors of the reactions in Table S3 encountered with the BEEFvdW and RPBE exchange-correlation functionals using corrections on one to two independent species.
MAE compares the mean absolute error of the reactions in Table S3. The Ec values next to the molecules
indicate the optimal correction to each species to minimize the MAE, considering the combinations of
pairing OCO with H2, H2O, and CO. A combination of simultaneously correcting the OCO-containing
species and H2 leads to the best description for both RPBE and BEEF-vdW.
BEEF-vdW
RPBE
Species 1
Species 2
Ec1 (eV)
Ec2 (eV)
MAE (eV)
Ec1 (eV)
Ec2 (eV)
MAE (eV)
None
None
0.355
0.248
OCO
None
+0.59
0.143
+0.45
0.058
CO
None
+0.15
0.332
+0.02
0.245
H2
None
+0.17
0.145
+0.12
0.159
H 2O
None
-0.31
0.107
-0.23
0.108
OCO
H2
+0.33
+0.09
0.040
+0.39
+0.02
0.039
OCO
H2 O
+0.23
-0.21
0.079
+0.37
-0.05
0.047
OCO
CO
+0.61
+0.18
0.061
+0.46
+0.05
0.042
Table S5. Reaction enthalpies (in eV) of the reactions listed in Table S3 as calculated with the BEEF-vdW
functional, shown without any corrections (∆Hunc) and corrected (∆Hcor) for H2 and the OCO-containing
species (CO2, HCOOH, CH3COOH, and HCOOCH3). As described in the text, the corrections are +0.33 eV
for OCO and +0.09 eV for H2 species using the BEEF-vdW functional. All errors are reported to the
reference values from NIST (∆Href ).11
Reaction
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1
ΔHref
0.43
-1.71
-2.14
0.15
-0.27
-0.55
-0.98
-0.89
-1.32
-1.30
-1.72
-1.37
-1.80
-0.66
-1.09
-0.65
-1.08
-0.67
-1.10
-0.17
-0.60
ΔHunc
0.87
-1.03
-1.90
0.39
-0.47
0.06
-0.80
-0.29
-1.16
-0.64
-1.51
-0.71
-1.58
-0.07
-0.94
-0.08
-0.94
-0.21
-1.07
0.12
-0.75
error
0.44
0.68
0.24
0.24
-0.20
0.61
0.18
0.60
0.16
0.66
0.21
0.66
0.22
0.59
0.15
0.57
0.14
0.46
0.03
0.29
-0.15
ΔHcor
0.45
-1.72
-2.17
0.30
-0.14
-0.54
-0.98
-0.89
-1.34
-0.27
-1.72
-1.36
-1.80
-0.67
-1.12
-0.66
-1.10
-0.55
-1.00
-0.23
-0.67
error
0.02
-0.01
-0.03
0.15
0.13
0.01
0.00
0.00
-0.02
0.03
0.00
0.01
0.00
-0.01
-0.03
-0.01
-0.02
0.12
0.10
-0.06
-0.07
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2