# 快速时时彩: Understanding electrocatalysis From reaction steps.pdf

Understanding electrocatalysis: From reaction steps to first-principles calculations C. S′anchez and E. Leiva Unidad de Matem′atica y F′?sica, Facultad de Ciencias Qu′?micas, INFIQC, Universidad Nacional de C′ordoba, Argentina 1 INFORMATION PROVIDED BY POTENTIAL ENERGY SURFACES In the previous chapter, it was seen that in electrochem- ical kinetics a more or less complicated reaction can be formulated, as a number of events usually denominated elementary steps or simple reaction steps. As we shall see below, knowledge on the potential energy surface (PES) for nuclear motion (see the function V({R I }) defined in equation (12) of The role of adsorption, Volume 2), can be of great help to decide the validity of different proposed mechanisms for a given reaction, on the basis of the relative reaction rates of the different reaction steps. What kind of chemical information can PESs provide? We shall try to answer this question by analyzing a series of schematic representations of simple PESs. Let us start with the simplest example, in Figure 1(a) and (b) we can see a three-dimensional representation of a simple PES and its corresponding contour plot. This PES has been plotted in a space of two generalized coordinates, x and y,thatarein some way related to the nuclear coordinates of the system. The PES gives us the total energy of the atom arrangement that may be called the reactants or products according to some criteria. For example, in the case of an exchange reac- tion such as AB + C → A + BC, the distinction between atomic arrangements in reactants or product states can be given by the interatomic distances. For short AB and long BC distances, the system is termed to be in the reactant state and for short BC and long AB distances, the system is said to be in the product state. For an atom adsorbed on a surface, long and short surface/atom distances would distinguish the reactants (free adsorbate and clean surface) from products (adsorbed atom). The choice of the general- ized coordinates on which to represent the PES is dictated by some chemical intuition about what kind of coordinates are chemically relevant for the problem. Of course the PES can be represented on cartesian atomic coordinate space but usually some clever cuts of the PES along selected mani- folds may provide an easier grasp of the relevant chemical features. For the case schematically depicted in Figure 1(a) and (b), two potential energy wells can be distinguished, one corresponding to the reaction products (marked as P) and one corresponding to the reactants (marked as R). The reaction R → P is exothermic since the product well is deeper than the reactant well. In order for the reaction to proceed, the system must move in configuration space from the R zone to the P zone along some path, in this case any path connecting reactants and products must cross the zones of the configuration space for which the system must increase its potential energy with respect to the bot- tom of any of the wells. This reaction is thus activated since the system must raise its energy in order to overcome the barrier separating both wells. The reactant particles may acquire the energy necessary to overcome this barrier by coupling to the surroundings via thermal fluctuations. The most probable path joining reactants and products will be that with the lowest possible energy, in our case the straight line joining both wells. The potential energy as a function of the position along this path is shown in Figure 2, the x coordinate in the plot is called the reaction coordinate. Edited by Wolf Vielstich, Hubert A. Gasteiger, Arnold Lamm and Harumi Yokokawa. ? 2010 John Wiley this term is the universal part of the density functional, which is the same for every possible system. As we have already stated, the exact form of the universal part is unknown, but there exists some good approximations. We can dissect some more the 5 Understanding electrocatalysis: From reaction steps to first-principles calculations energy functional, separating it into: E[n(r)] = T [n(r)] + integraldisplay v(r)n(r)dr + integraldisplayintegraldisplay n(r 1 )n(r 2 ) |r 2 ? r 1 | dr 2 dr 1 + E xc [n(r)] (7) where the first term represents the kinetic energy of the electrons, the second denotes the interaction between the electrons and the external potential v(r), the third the classical electrostatic energy of the electronic density and the fourth term, called exchange/correlation, contains the electron/electron interaction energy of quantum origin not included into the third term. The third term is called the Hartree energy, this accounts for the most important part of the electron/electron interaction energy but overestimates this repulsive interaction. The exchange energy corresponds to the energy lowering obtained from the application of the Pauli exclusion principle, which does not allow two electrons of the same spin to be in the same point in space, and the correlation energy lowers further the energy by allowing the “motion” of different electrons to be correlated to each other (this is the same kind of energy that gives origin to the dispersion forces described in The role of adsorption, Volume 2). Another component that must be part of the exchange/correlation energy is the correction to the self interaction of the electronic density, which is included into the Hartree energy. The approach proposed by Kohn and Sham [6] in the second seminal paper of the DFT to approximate the functional is that the kinetic energy of the interacting electrons can be calculated to a very good approximation by the energy of a system of non-interacting electrons with the same density. (The existence of such a system usually tempts the beginner to challenge the validity of theorem 1 since it might imply that there exist two systems with the same density but different energy. One must keep in mind that theorem 1 is applied to interacting electron systems, and thus the existence of a non-interacting electron system with the same density and a different energy is no way is contradictory.) For this system of non-interacting electrons, the wavefunction is a product of single particle orbitals and thus the electronic density is the sum over these: n(r) = summationdisplay occ |? i (r)| 2 (8) where the sum goes over the occupied orbitals, ? i ,which are filled in simple aufbau fashion, that is, two electrons per orbital. The orbitals are found by the solution of the so-called Kohn–Sham equations. We have thus concentrated all our ignorance in the exchange/correlation energy: this term includes everything we are not able to figure out easily, so we now need a way to obtain it. The first approach to follow is to obtain it as the integral of a local exchange/correlation energy density, ε xc [n(r)]: E LDS xc [n(r)] = integraldisplay ε xc [n(r)]n(r) dr(9) This is called the local density approximation (LDA) because the exchange/correlation energy density is a func- tion of the local value of the electron density at r.A general non-local functional for the exchange/correlation energy would be of the form E LDS xc [n(r)] = integraldisplayintegraldisplay ε xc (r, r)n(r) dr drd prime (10) The non-locality of this functional comes from the fact that in order to be able to calculate the density in one point one needs information about every other point in space. Another approximation to the exchange/correlation energy that works reasonably well for systems containing molecules is the generalized gradient approximation (GGA), in which E xc has the form: E GGA xc [n(r)] = integraldisplay ε GGA xc [n(r), ?n(r)]n(r) dr(11) Recent examples of ab initio GGAs are Perdew and Wangs’s PW91 [7] and Perdew, Burke and the “GGA made simple” from Perdew, Burke and Ernzerhof, [8] this last being very attractive for its accuracy and the simplicity of its derivation. A semi-empirical density functional that is in common use for the study of molecules and is widely used for the study of adsorption using cluster models is the so-called B3LYP, [9] this functional combines two dif- ferent gradient corrected density functionals together with some exchange energy calculated from the Hartree–Fock approximation. The coefficients used to mix these function- als are determined by least-square fitting to a large number of experimental atomization energies, ionization potentials, proton affinities and atomic total energies. [10] The results obtained from this functional are apparently very good, and as we said it is widely used by the quantum chemistry com- munity. Nevertheless, the functional is semi-empirical since experimental information is used in order to obtain it, and work using this functional cannot be described as ab initio or from first principles. In order to solve Kohn–Sham equations in practical cases, the orbitals are expanded into a suitable basis set as: ? i (r) = summationdisplay j c ij φ j (r) (12) 6 Introduction After replacing this expansion into the Kohn–Sham equations, a generalized eigenvalue problem of the form H KS c i = ε i Sc i (13) is obtained, where the vector c i contains all the coeffi- cients of orbital i and H KS and S are Hamiltonian and overlap matrices. The most common basis sets used are localized orbitals, which are usually the solutions of the isolated atomic problem, gaussian orbitals and plane waves. Since core orbitals are mainly confined near the nuclei, they have almost no participation in normal chemical phenomena. Thus, they are assumed to remain unaffected and only valence electrons are treated explicitly. In the Kohn–Sham Hamiltonian, the interaction of the valence electrons with the nuclei and the core electrons is usually replaced by an operator denominated “pseudopotential”. [11] Another possibility to avoid dealing with core orbitals is to obtain them from a nuclear calculation and keep them frozen in the molecular calculation. This frozen core approximation is used for example in the program ADF together with Slater type basis functions. [12] Many studies of chemisorption processes over metallic clusters have been made using the DFT with pseudopo- tentials and their associated basis sets originally developed from atomic Hartree–Fock calculations such as those from Hay, Wadt and others. [13] Few works have validated this procedure, [14] and only for small molecules containing tran- sition metal atoms. These pseudopotentials are normally used without any particular validation and the authors never make explicit mention of the fact that the pseudopotentials used in their calculations come from Hartree–Fock calcu- lations. Apparently, an appropriate description of valence electrons is in general not very sensitive to the nature of the pseudopotential used to describe the core but from a strict theoretical point of view this mixing of methods may not be justified. In order to treat adsorption and surface reaction prob- lems using the formalism just outlined, some adequate representation of the surface is necessary. With today’s computational capabilities, it is possible to treat explicitly a number of around 100 atoms, clearly this is not enough to treat a metallic surface in all its features, which may contain 10 20 atoms. So we need a method to represent all relevant characteristics of the surface within the 100-atom boundary. Two methods are currently commonly used to solve this problem: the supercell and cluster approaches. In the supercell method, a metallic slab is used to repre- sent the surface, a sufficiently large surface unit cell of this slab is periodically repeated in space as shown in Figure 8. The vacuum space between the periodic images of the slab Figure 8. Schematic representation of the “supercell” method for the study of adsorption on surfaces at fractional small coverage degrees. The unit cell within dashed line is repeated over space to simulate an infinite surface in vacuum. should be large enough so as to rule out any possible inter- actions; in practice, the vacuum region is grown up to the point where the results of interest are independent of its size. The same occurs for the size of the surface unit cell if one wants to study the adsorption of a single entity over the surface at a coverage degree near zero. There might exist some difficulties in estimating these adsorption ener- gies at zero coverage, since at least for some adsorbates, substrate-mediated interactions exist that are long ranged and thus require extremely large surface unit cells in order to be eliminated. The periodicity of the system can be use- ful if one wants to study periodic adsorbate phases, we shall see some examples of this latter. The adsorbate is normally placed on both sides of the slab since the adsorp- tion on a single side creates a net dipole moment in the cell perpendicular to the slab direction, which might need a huge vacuum region in order to avoid interaction between images. Some techniques have been developed that create an artificial dipole within the vacuum region that is self con- sistently varied in order to cancel out the surface dipole. [15] The width of the slab should also be wide enough in order to retain its relevant bulk properties in the middle region, in 7 Understanding electrocatalysis: From reaction steps to first-principles calculations our experience for metallic adsorbates of around five (111) atomic layers are enough for a transition metal substrate, but the number of layers needed for convergence might be more for other adsorbates that polarize the surface more strongly. The basis set to be used for the supercell method or any other periodic system should satisfy the Bloch condition: [16] φ(r + R) = e ik·R φ(r)(14) where R is a Bravais lattice vector and k a vector within the first Brillouin zone in the reciprocal lattice. Plane waves with the periodicity of the Bravais lattice are almost a natural choice for these kind of calculations and have many advantages. For example, the kinetic operator is diagonal and other matrix elements may be easily calculated by means of fast Fourier Transforms (FFT). [17] Some examples of programs using plane waves basis sets for the representation of Kohn–Sham orbitals are fhi98md, [18] VASP, [19] CASTEP [20] and CPMD. [21] Localized orbitals may also be a proper basis set for the study of periodic systems; in this case the basis functions to be used that have the proper periodicity are truly linear combinations of orbitals localized at different cells: φ i I (k,r) = summationdisplay χ i (r ? r I ? R)e ik·(r i +R) (15) where the index i runs over the different localized orbitals, χ, centered on I and r I is the position of center I within the unit cell. The program SIESTA [22] applies some very clever ideas about the construction of the basis set and is a good and fast example of the application of localized basis sets to periodic systems. Contrary to plane waves, there is no systematic way to improve the quality of localized basis sets, a