Supporting materials Bond-Energy-Integrated Descriptor for Oxygen Electrocatalysis of Transition-Metal Oxides

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1 Supporting materials Bond-Energy-Integrated Descriptor for Oxygen Electrocatalysis of Transition-Metal Oxides Deyao Wu, Cunku Dong,, Hongbing Zhan and Xi-Wen Du, Institute of New Energy Materials, School of Materials Science and Engineering, Tianjin University, Tianjin 30072, China School of Materials Science and Engineering, Fuzhou University, Fuzhou , P. R. China and 1 Computational details. We performed the density functional theory calculations using the Vienna Ab initio Simulation Package (VASP) code, [1] making use of the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional and the projector augmented-wave (PAW) pseudopotential. [2] A plane-wave cutoff energy of 500 ev was used for all geometry optimizations. The adsorbate-surface systems with the vacuum layer of 15 Åwere optimized until the maximum force on each atom is less than 0.05 ev/å. The Brillouin zone was sampled with a Monkhorst-Pack mesh. For β-mno 2, previous work demonstrated that PBE +U method can give a good description of lithium intercalation, band gaps, and magnetic interactions. [3] In this work, we employed U = 6.1 ev and J = 1.0 ev for β-mno 2. [3, 4] For RuO 2, PBE fucntioanl within the framework of GGA [5] was used for the exchange-correlation functional. The adsorption energies of adsorbed *OOH, *OH and *O intermediates were calculated as follows, [6] E OOH = E OOH E( ) (2E H2O 3/2E H2 ) (1-1a) E OH = E OH E( ) (E H2O 1/2E H2 ) E O = E O E( ) (E H2O E H2 ) (1-1b) (1-1c) where E OOH, E OH and E O are the energy of OOH, OH and O intermediates adsorbed on substrate surface (*), respectively. E H2O and E H2 are the energy of free H 2 O and H 2 molecules, respectively. The standard free energy (G) of the adsorbed intermediate on the DFT scale is calculated by [7] G = E DF T + ZP E T S (1-2) where E DF T is the total electronic energy obtained with DFT. ZP E is the zero-point-energy correction and S is the entropy of the system obtained through vibrational-frequency analysis, using the harmonic oscillator approximation. The oxygen evolution/reduction reaction (OER/ORR) was explored by means of the computational hydrogen electrode (CHE) approach. [8] The energetics of gaseous H 2 is used to approximate that of proton and electron in an electrochemical environment, according to the following equilibrium: H 2 = 2(H + + e ) (1-3) The overall reaction scheme of O 2 reduction to OH in alkaline solution is: [9, 10] O 2 + 2H 2 O + 4e 4OH (1-4) 1

2 and the following four-step associative mechanism is dominant in ORR and considered in our calculations, + O 2 + H 2 O + e OOH + OH (1-5a) OOH + e O + OH O + H 2 O + e OH + OH OH + e + OH (1-5b) (1-5c) (1-5d) To avoid the difficulty in treating the triplet state of gaseous O 2, the electronic energy of O 2 was calculated from experimental H 2 O formation energy and the electronic energies of gaseous H 2 O and H 2. We used the experimental value of 2.46 ev of the standard free energy change of the reaction: H 2 O(l) 1/2O 2 (g) + H 2 (g). [11] Thereby, we get the following relation: G H2O 1/2G O2 G H2 = 2.46eV (1-6) The free energy for OH (G OH ) was calculated by G OH = G H2O G H +, where G H + = 1/2G H2 k B T ln10 ph (k B is Boltzmann constant). Hence, the equilibrium potential U 0 for ORR at ph = 13 is determined to be V vs. NHE where the reactant and product are at the same energy level. The potential correlated to free energy change is calculated by, where n is the number of electrons transferred in the reaction. U = G/n (1-7) 2 The definition of bond energy-integrated orbitalwise coordination number CN α. We use moments theorem and linear combination of atomic orbitals (LCAO) model to express bond energy-integrated orbitalwise coordination number on atom site i (CN α ) in terms of the α-electron (α= s, d or sd) hopping integrals. The moments theorem, which is derived by Cyrot- Lackmann (1968), correlates the moments of local density of states to the topology of the local atomic enviroment. Let us consider the n-th moment of the local density of states on atom i, [12] µ (n) i = (E H ii ) n ρ i (E)dE (2-1) whole band where H ii is Extended Hückle Hamiltonian operator, and ρ i (E) the local density of states on atom site i. Consider the local density of states for atom i, we have ρ i (E) = all E k δ(e E k ) < i Ψ k >< Ψ k i > de (2-2) where Ψ k and E k are the eigenstate and its corresponding eigenvalue of the system. By inserting Eq.(2-2) for ρ i (E) into Eq.(2-1), the n-th moment of the the local density of states becomes µ (n) i = (E H ii ) n δ(e E k ) < i Ψ k >< Ψ k i > de (2-3) whole band all E k We expand (E H ii ) n in a Taylor series at E = E k, and find that the integral of (E H ii ) n times δ(e E k ) is just the first term of the series, i.e. (E H ii ) n. By summing over all eigenvalues E k, we get µ (n) i = all E k < i Ψ k > (E H ii ) n < Ψ k i > (2-4) We view the item sandwiched by i > and < i as the i i matrix element of an operator W, W = all E k Ψ k > (E H ii ) n < Ψ k (2-5) 2

3 then the operator is simply defined as as may be easily verified by using It follows that H = W = (H H ii ) n (2-6) all eigenstates Ψ k > E k < Ψ k (2-7) µ (n) i =< i (H H ii ) n i > (2-8) Obviously,zeorth moment µ (0) i =< i i >= 1, and first moment µ (1) i =< i (H H ii ) i >= H ii H ii < i i >= 0, which suggests that these two moments cannot give distinctive information concerning electronic structure of surface atoms. The next moment, µ (2) i, is the lowest moment containing the environment information of an atom (the root mean square width of the DOS), which thus determines the variation in chemical activity of an surface site influenced by its neighboring atoms. To evaluate µ (2) i, we use the completeness of the basis set µ (2) i =< i (H H ii ) 2 i >=< i (H H ii )(H H ii ) i >= i < i H H ii i >< i H H ii i > (2-9) In a nearest-neighbour model, only those i that are neighnours of atom i have nonzero Hamiltonian matrix elements < i H i >. Moreover, < i H H ii i >= 0, and < i H H ii i >= H ii, for i i, thus µ (2) i = H ii H i i (2-10) i i Each term H ii H i i describes an electron starting at site i, hopping out to a neighbouring site i, and hopping back to i. This is described as a path of length two hops from atom i. µ (2) i is thus the sum of all such paths of the length two hops. In transition-metal compounds, one natural choice of the hopping matrix of s and d orbitals between atom site i and its neighboring atom i is the case in which the bond vector of atom site i and i is along the z axis (i.e. l = 0, m = 0, n = 1). Thus, the hopping matrix is given as, [13] Vαν ij s p x p y p y s ssσ 0 0 spσ d xy d yz 0 0 dpπ 0 d xz 0 dpπ 0 0 d x 2 y d z 2 dsσ 0 0 dpσ The above hopping integrals can be calculated according the following expressions, which are abbreviated to ssσ, spσ, dsσ, dpσ, V ssσ = η ssσ h 2 /md 2 V spσ = η spσ h 2 /md 2 (2-11a) (2-11b) V pdσ = η pdσ h 2 r p r 3 d /md4 (2-11c) V pdπ = η pdπ h 2 r p r 3 d /md4 (2-11d) where η is symmetry dependent coefficient; d nearest-neighbor distance, η ssσ = 1.32, η spσ = 1.42, η pdσ = π and η pdπ = 3 5 2π ; r d is the d-orbital radius of transition metal atom and r p the p-orbital radius of neighboring coordinating oxygen atoms in TM oxides. The parameters ssσ, spσ, dsσ, dpσ, and dpπ are fundamental hopping integrals corresponding to s and d orbitals on neigbouring oxygen atoms. In order to consider the hopping matrix of s and d orbitals, we rewrite µ (2) i to µ α 2,i for clarity. From the above equation, the second moment for s and d bands of the atom site i with its neighbouring oxygen atom i can be approximated as 3

4 µ s 2,i = (ssσ 2 + spσ 2 ) (2-12a) µ d 2,i = (dsσ 2 + 2dpπ 2 + dpσ 2 ) (2-12b) µ sd 2,i = (ssσ 2 + spσ 2 + dsσ 2 + 2dpπ 2 + dpσ 2 ) (2-12c) Considering the relationship between bond energy and µ α 2,i [14], we drive a generic definition α of CN i on atom site i, we sum up the second moment of α band (s, d or sd) at atom site i to its all 1st nearest neighbors α-electron hopping integrals, and normalize with the hopping integral of α 1st nearest neighbors in bulk, and thus the definition of CN i is given as µ s 2,i µ s 2,i s i i CN i = i i (2-13a) (V s, (ssσ2 + spσ 2 ) nn ) 2 = µ d 2,i d i i CN i = = (Vnn d, ) 2 µ sd 2,i sd i i CN i = = (Vnn d, ) 2 i i µ d 2,i (dsσ2 + 2dpπ 2 + dpσ 2 ) i i µ sd 2,i (ssσ2 + spσ 2 + dsσ 2 + 2dpπ 2 + dpσ 2 ) (2-13b) (2-13c) where Vnn α, is a sum of the square of the α-electron hopping integrals to 1st nearest-neighbor atom in the optimize bulk crystal. 3 Correlation between CN sd and adsorption energy of adsorbate ( E ads ) According to the bond-order conservation principle in chemisorption, the total bond order χ of a surface metal atom i, which is coordinated with its nearest neighbors and a covalently bonded adsorbate A at an atop site., is conserved and normalized to unity along all surface metal atoms. [15, 16] χ io + χ ia = 1 (3-1) where χ io and χ ia are the bond order of a metal atom i with the surrounding oxygen atoms and with an adsorbate A (*OOH,*OH or *O) in this work, respectively. The bond-order conservation principle has been proven to be effective in scaling the adsorption energies of a series of species, including NH x, CH x, OH x, SH x on transition metal (TM) compounds surfaces [17]. Therefore, we argue that *OOH, *OH and *O intermediates adsorbed on the surface of TM oxides obey bond-order conservation principle Eq.(3-1). According to the Friedel model, [12] µ sd 2,i can be used to estimate bond energy between two neighboring atoms in a solid, so the contribution to the bond energy from site i can be written in the following form, E (i) totalbond = constant µ sd 2,i (3-2) i j That is to say, E (i) totalbond = constant CN sd according to Eq.(2-13c). Because χ io can describe the bond energy between metal atom i and oxygen atom in TM oxides, [18 20] we can get χ io CN sd (3-3) In TM oxides, the adsorbed adsorbates A such as *O can be views a part of TM oxides themself, and thus the two-center interatomic potential of the adsorbate A with the center of metal atom i in the TM oxides can be represented by a ionic potential [21] U(r ij ) = q iq j r ij (3-4) 4

5 where q i and q j are the charge on the neighboring atoms (i and j) in a ionic crystal and r ij is the bond length. And the bonding energy E ads can be defined as, [22, 23] E ads = 1 U(r ij ) (3-5) 2NΩ a where Ω a is the average volume per atom in an ionic crystal, ionic bond order (χ ia ) is proportional to 1/r ia, [24] χ ia 1/r (3-6) where r ia is bond length between site i on the surface of TM oxide and adsorbed atom on the adsorbate. With the aid of Eq.(3-1), Eq.(3-2) and Eq.(3-5), we can get Combining χ io CN sd (Eq.(3-3)), we can draw the conclusion, i j E ads χ ia (1 χ io ) (3-7) E ads CN sd (3-8) 5

6 Figure S1. Optimized geometries of various β-mno2 extended surfaces.the Mn atom marked by sites is the one to be investigated. Ov represents an oxygen vacancy. (001)-ten means that tension is exterted on (001) surface. 6

7 Figure S2. Adsorption energies of *OOH and *O atop on β-mno 2 extended surfaces described by CN. 7

8 Figure S3. Scaling relation between adsorption energies of *OOH, *OH and *O atop on β-mno 2 extended surfaces described by CN s. 8

9 Figure S4. Scaling relation between adsorption energies of *OOH (a), *OH (b) and *O (c) atop on β-mno 2 extended surfaces described by CN d. 9

10 Figure S5. Optimized geometries of various RuO 2 extended surfaces.the Ru atom marked by sites is the one to be investigated. 10

11 Figure S6. Trends in adsorption energy of *OH atop on RuO 2 extended surfaces described by CN sd. 11

12 Figure S7. Adsorption energy of *OH atop on β-mno 2 extended surfaces described by CN d. 12

13 Figure S8. Adsorption energy of *OH atop on RuO 2 extended surfaces described by CN s. 13

14 Figure S9. Schematic illustration of the ORR pathway on the (110) and (001) surfaces of β-mno 2. 14

15 Table S1. Tabulated data for *OOH, *OH and *O atop β-mno 2 extended surface for the corresponding descriptors and average-bond-length(l). Site CN sd CN s CN d cn CN E OOH E OH E O L (001) (001)-vanO-site (001)-vanO-site (001)-van2O (001)-van2O (001)-van2O (001)-vanMn (001)-ten (001)-ten (001)-ten (001)-com (001)-com (001)-com (110) (110)-site (110)-site (111) (001)-GB (002) (101) (101)-vanO (110)-vanO (200) (220) (310)

16 Table S2. Tabulated data for various coordination numbers for RuO 2. Facet CN sd CN s CN d cn CN E OH (001) (001)-vanO (100) (101) (110)-site (110)-site (110)-vanO (111)-site (111)-site (210)

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