# Calculation of the Free Energy of Methanediol

Supplementary Material

Calculation of the free energy of Methanediol

The reaction free energy of the formation of gaseous formaldehyde from CO2 and hydrogen

CO2(g) + 2 H2(g) → CH2O(g) + H2O(l)

is 54.7 kJ/mol or 0.57 eV based on experimental thermodynamic data [1], see table S1. In aqueous solution, formaldehyde is hydrated to methanediol

CH2O(aq) + H2O ↔ H2C(OH)2(aq)

with a hydration constant Khyd around 2103 [2, 3].

Experimental Henry’s law constants for aldehydes are apparent constants that include the effect of hydration [4]. For formaldehyde Khyd > 1 and the apparent Henry’s law constant is

KH,app = KH Khyd,

where KH is the Henry’s law constant for non-hydrated formaldehyde

CH2O(g) ↔ CH2O(aq).

The standard reaction free energy of

CH2O(g) + H2O → H2C(OH)2(aq)

can be obtained by combining CH2O solvation and hydration from the apparent Henry’s law constant

G = -kBT ln(KH) - kBT ln(Khyd) = -kBT ln(KH,app)

Here, we use the literature value of KH,app = 3103 M/atm [4], which gives G0 = -0.21 eV. The standard reaction free energy of

CO2(g) + 2 H2(g) → H2C(OH)2(aq)

is then 0.36 eV. Methanediol is deprotonated in alkaline solution

H2C(OH)2(aq) ↔ H2C(OH)O-(aq) + H+ (aq)

with a pKa from 13.05 to 13.55 [5]. The present discussion focus on methanediol reduction in neutral solution, in alkaline solution the initial reduction step of methanediol will be a bit more difficult.

fH0 [kJ / mol] / fG0 [kJ / mol] / S0 [J / mol K] / Cp [J/mol K]
H2(g) / 0.0 / 0.0 / 130.7 / 28.8
H2O(l) / -285.8 / -237.1 / 70.0 / 75.3
CO2(g) / -393.5 / -394.4 / 213.8 / 37.1
CH2O(g) / -108.6 / -102.5 / 218.8 / 35.4

Table S1: Experimental enthalpy and entropy of formation, entropy and heat capacity at 298.15 K from ref. [1] used to calculate the free energy of CH2O.

The Computational Hydrogen Electrode Model

Free energy diagrams are constructed using the Computational Hydrogen Electrode (CHE) model [6, 7]. The method is identical to the method used in previous work on CO2 reduction [7], but given here for completeness. To calculate the reaction free energy of a coupled proton electron transfer step such as CO hydrogenation

CO* + H+ + e- → CHO*,

where the * denotes an adsorbed species, we first calculate the reaction free energy, G0, of the equivalent chemical hydrogenation reaction

CO* + ½ H2(g) → CHO*.

If H2 is at standard pressure, the reaction

H+ + e- ↔ ½ H2(g)

is in equilibrium at the potential of 0 V versus the Reversible Hydrogen Electrode (RHE). Therefore the reaction free energy of the coupled proton electron transfer step is identical to the free energy of the hydrogenation step at 0 V vs. RHE. In general, the reaction free energy of the electrochemical (reduction) step as a function of potential is

G(U) = G0 + e U

where we have neglected corrections to the binding energy due to the electric field and potentials are measured on the RHE scale. We see that at more reducing (negative) potentials the thermodynamic driving force for the reduction reaction is increased.

Computational Details

Density functional theory calculations are performed using the ASE and Dacapo codes [8] and follows previous studies of CO2 reduction on Cu(211) [7, 9, 10]. The ionic cores are described using Vanderbilt ultrasoft pseudopotentials [11]. The Kohn-Sham one electron wavefunctions are expanded using plane waves with kinetic energies below 340.15 eV, while the electron density is expanded in plane waves with kinetic energies below 500 eV. The one-electron states are populated using a Fermi-Dirac distribution with kBT = 0.1 eV and the total energies are extrapolated to kBT = 0 eV. The effect of exchange and correlation is approximated with the RPBE functional [12].

The reaction thermodynamics on Cu(211) is modeled using a 333 slab with 3 close packed planes in an orthorhombic super cell. The atoms in the topmost plane are allowed to relax while the 2 bottom planes are fixed in their bulk positions. Adsorbates are placed on the topside of the slab and the self-consistent dipole correction [13] has been used to decouple the electrostatic dipole correction between the periodically repeated images of the slab. The first Brillouin-zone is sampled using a 441 grid of Monkhorst-Pack k-points [14]. The geometry of adsorbates and the topmost close-packed Cu plane is optimized until all force components are less than 0.05 eV/Å. Adsorbate geometries are shown in figure S1.

The energy of the CO2 molecule has been shifted by +0.45 eV in order to correct for systematic errors in reactions involving the O=C=O backbone as described in Ref. [7]. The effect of hydrogen bonding is included in an approximate way also following Ref. [7], by stabilizing OH* by 0.5 eV [6, 15], hydroxyl functional groups (R-OH*) by 0.25 eV [16], and intermediates containing CO (such as CO* and CHO*) by 0.1 eV [7].

Free energies of adsorbate states at 0 V vs RHE are calculated by adding the vibrational free energy, where all degrees of freedom of the adsorbate have been approximated as harmonic vibrations. Vibrational frequencies are calculated on the Cu(211) surface using finite difference displacements of +/- 0.01 Å.

Figure S1. Adsorbate geometries of possible intermediates in the reduction of CO2, CO and H2C(OH)2.

The free energy of an adsorbate or a molecule is calculated from

G = E + ZPE +  Cp dT - TS

with contributions given in table S2 and S3 for adsorbates and molecules respectively. Vibrational frequencies taken from a previous study [7] or calculated in the present study. Molecular species are treated using statistical mechanics for ideal gas molecules [17]. The partial pressures of molecules are taken to be 101325 Pa (1 atm), except for the liquid phase products H2O and CH3OH. For H2O a partial pressure of 3534 Pa is used corresponding to the vapor pressure of liquid water at room temperature, and 6080 Pa for CH3OH corresponding to an aqueous phase activity of 0.01 [7].

Reaction energies and reaction free energies calculated versus CO2, H2 and H2O are listed in Table S4. For e.g. CO* the reaction energy and reaction free energy is calculated according to the reaction

CO2(g) + H2(g) + * → CO* + H2O(l)

Such that

E = ECO* + EH2O – ECO2 – EH2 – E*

where stabilization due to hydrogen bonding is included in E in Table S4. The reaction free energy is obtained similarly by applying the vibrational corrections listed in Tables S2 and S3.

Adsorbate / ZPE /  Cp dT / -TS / G-Eelct
(eV) / (eV) / (eV) / (eV)
* / 0 / 0 / 0 / 0
CO* / 0.19 / 0.08 / -0.16 / 0.11
OH* / 0.36 / 0.05 / -0.08 / 0.33
CHOH* / 0.78 / 0.09 / -0.18 / 0.69
O* / 0.07 / 0.03 / -0.04 / 0.06
CHO* / 0.45 / 0.09 / -0.18 / 0.35
CH3OH* / 1.42 / 0.12 / -0.27 / 1.27
OCH3* / 1.11 / 0.10 / -0.19 / 1.02
CH2* / 0.61 / 0.05 / -0.09 / 0.57
CH2OH* / 1.07 / 0.10 / -0.22 / 0.95
CH2O* / 0.76 / 0.09 / -0.20 / 0.65
COH* / 0.48 / 0.08 / -0.14 / 0.42
COOH* / 0.62 / 0.10 / -0.19 / 0.54

Table S2. Adsorbate free energy contributions based on the harmonic vibrational free energy at 298.15 K.

Molecule / Fugacity / ZPE /  Cp dT / -TS / G-Eelct
(Pa) / (eV) / (eV) / (eV) / (eV)
CO2 / 101325 / 0.31 / 0.1 / -0.66 / -0.25
CO / 101325 / 0.14 / 0.09 / -0.61 / -0.39
H2 / 101325 / 0.27 / 0.09 / -0.4 / -0.04
CH3OH / 6070 / 1.37 / 0.12 / -0.81 / 0.67
H2O / 3534 / 0.58 / 0.10 / -0.67 / 0.01
CH4 / 101325 / 1.20 / 0.10 / -0.58 / 0.72
CH2O / 101325 / 0.70 / 0.10 / -0.68 / 0.13

Table S3. Free energy contributions for gas phase molecules at 298.15 K.

State / Eelct (eV) / G
(eV)
* + CO2(g) + 8(H+ + e-) / 0 / 0
COOH* + 7(H+ + e-) / -0.39 / 0.42
CO(g) + * + 6(H+ + e-) + H2O / 0.43 / 0.35
CO* + 6(H+ + e-) + H2O / -0.34 / 0.07
CHO* + 5(H+ + e-) + H2O / 0.09 / 0.76
COH* + 5(H+ + e-) + H2O / 0.78 / 1.52
CH2O* + 4(H+ + e-) + H2O / -0.24 / 0.75
H2C(OH)2(aq) + 4(H+ + e-) / - / 0.36
CHOH* + 4(H+ + e-) + H2O / 0.00 / 1.03
CH2OH* + OH* + 4(H+ + e-) / -1.23 / 0.38
OCH3* + 3(H+ + e-) + H2O / -1.28 / 0.10
CH2OH* + 3(H+ + e-) + H2O / -0.63 / 0.69
O* + CH4(g) + 2(H+ + e-) + H2O / -1.60 / -0.43
CH3OH + 2(H+ + e-) + H2O / -1.26 / -0.21
CH2* + H2O + 2(H+ + e-) + H2O / -0.39 / 0.58
CH3OH* + 2(H+ + e-) + H2O / -1.64 / 0.02
OH* + 1(H+ + e-) + H2O + CH4(g) / -3.12 / -1.67
* + 2 H2O + CH4(g) / -2.52 / -1.36

Table S4. Reaction energy and reaction free energy at 0 V versus RHE of adsorbate states calculated vs. CO2, H2 and H2O. The free energy of H2C(OH)2 is taken from experiment. Free energies are calculated at 298.15 K. The reference states for molecules are given in Table S3.

Reduction Pathways for Methanediol

Figure S1 compares the free energy diagrams for the reduction of methanediol through initial O hydrogenation, C hydrogenation, and C-O bond dissociation. As discussed in the main text, initial hydrogenation of the carbon atom could be prohibited as an elementary step because the C atom already has 4 bonds. The limiting potentials for the three pathways are nearly identical, with -0.33 V for O hydrogenation (CH2OH* formation) and -0.31 V for OH* removal (C hydrogenation and C-O bond dissociation). The relative prevalence of O hydrogenation and C-O bond breaking may depend on potential, because the reaction free energies change differently with potential, compare Figure S2(a) to S2(b).

Figure S2. Reduction of methanediol through initial O hydrogenation, C hydrogenation, and C-O bond dissociation at (a) -0.25 V vs. RHE and (b) -0.40 V vs RHE.

Barriers for coupling of CH2* to form C2H4*

Figure S3 shows the result of the climbing image nudged elastic band (CI-NEB) [18] minimum energy path as calculated for the binding of two CH2* adsorbates. This calculation was performed in a 433 supercell with the 4-atom wide dimension parallel to the (211) step to ensure minimal interactions through the periodic boundary conditions, since two adsorbates were included within one supercell. Besides this, the calculation parameters are identical to those detailed above. The transition state found by the CI-NEB was verified to have exactly one imaginary mode, and corrections for the zero-point energy (ZPE) and heat capacity calculated in the harmonic approximation detailed above were included in the reported value of 0.54 eV for the coupling barrier.

Figure S3. CI-NEB result for coupling of 2 CH2* to form C2H4*.

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