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Adsorption on Surfaces Bonding at Surfaces Potential Energy

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Adsorption on Surfaces Bonding at Surfaces Potential Energy Adsorption on surfaces Bonding at surfaces Example: H 2/H/Pd(210) Theoretical description Lennard-Jones picture of dissociative adsorption Nature of surface bonds Adsorption of molecules on surfaces technolo- < gically of tremendous importance 4 Physisorption (E ad ∼ 0.1 eV): Noble gas ad- sorption, molecular adsorption through Van- Theoretical description by ab initio methods 3 der-Waals interaction possible A + B + S 2 Chemisorption (E ad ∼ 1 - 10 eV): Chemical Analysis of electronic structure very useful for bond between surface and adsorbate 1 D AB understanding of chemical trends E E a ad Nature of surface chemical bonds: 0 Potential energy (eV) A+B AB + S Approximate model Hamiltonian can pro- Ead vide qualitative insight: Newns-Anderson, -1 Metallic bonding Effective-Medium-Theory, Embedded-Atom- 0 1 2 Method . Distance from the surface z(Å) Alkali-metal bond (mainly ionic) Adsorption of molecular and atomic hydrogen on Covalent bonds Pd(210) Potential energy curves for molecular and dissociative adsorption Potential energy surfaces (PES) Physisorption PES central quantity to describe gas-surface interaction; Example: catalyst Physisorption mediated by Van der Waals forces 1D representation Multidimensional representation 1.2 without catalyst Adsorbed Activation barrier + - with catalyst e e reactants on the surface 1.0 ~ r’ -R R r R = (0 , 0, Z ) 0.8 - O + 0.6 ~r = ( x, y, z ) 0.4 metal vacuum 0.2 0.0 Potential energy (eV) -0.2 Gas phase -0.4 Reactants barrier Image potential of a hydrogen atom in front of a metal surface: -5 -4 -3 -2 -1 0 1 2 3 4 5 Products A+B Reaction path coordinate (Å) AB e2 1 1 1 1 A. Groß, Surf. Sci. Vol. 500 V = − + − − im ~ ~ ~ ~ ~ ~ 2 "|2R| |2R + ~r + r′| |2R + ~r | |2R + r′| # 1D representation misleading: e2 1 1 2 = − + − . (103) Catalysts provides detour in multi-dimensional configuration space 2 2Z 2( Z + z) ~ with lower activation barriers " |2R + ~r | # van der Waals interaction van der Waals interaction energy Taylor expansion of image force in |~r |/|R~|: Frequency of the atomic oscillator e2 x2 + y2 3e2 z m ω2 2 2 2 2 3 e 2 2 meω 2 Vim = − + z + (x + y ) + z + ... (104) V = k x + y + ⊥ z . (107) 8Z3 2 16 Z4 2 atom 2 2 h i ! 3 with van der Waals interaction ∝ Z− 2 2 Assumption: electronic motion in atom can be modeled by 3D-oscillator: e e ω = ωvib − 3 and ω = ωvib − 3 . (108) k 8meωvib Z ⊥ 4meωvib Z m ω2 V = e vib x2 + y2 + z2 . (105) atom 2 ! van der Waals binding energy: change in zero-point energy of atomic oscillator Atom potential in the presence of the surface: ~ −~e2 E (Z) = (ω + 2 ω − 3ω ) = . (109) 2 2 2 2 vdW vib 3 m ω e x + y 2 ⊥ k 4meωvib Z V = e vib x2 + y2 + z2 − + z2 + . atom 2 8Z3 2 m ω2 ! 2 e 2 2 meω 2 ≈ k x + y + ⊥ z . (106) 2 2 ! van der Waals constant Zaremba-Kohn picture of physisorption Atomic polarizability Taylor expansion of image force of the hydrogen atom corresponds basically to dipole-dipole interaction at distance 2Z e2 α = 2 . (110) However, hydrogen atom in the ground state has no permanent dipole moment ⇒ quantum meω vib treatment neccessary van der Waals binding energy: Quantum derivation of the long-range interaction between a neutral atom and a solid surface (Zaremba, Kohn) ~ωvib α Cv EvdW (Z) = − 3 = − 3 (111) 8Z Z H = Ha + Hs + Vas . (113) Cv = ~ωvib α/ 8 van der Waals constant, related to the atomic polarizability a atom, s solid, Vas interaction term: Fourth-order correction defines dynamical image plane at Z0 s a 3 3 ρˆ (~r )ˆρ (~r ′) s,a s,a s,a V = d ~rd ~r ′ with ρˆ (~r ) = n (~r ) − nˆ (~r ) . (114) Cv 3CvZ0 5 Cv 5 as + |~r − ~r ′| Vim (Z) = − 3 − 4 + O(Z− ) = − 3 + O(Z− ) (112) Z Z (Z − Z0) Z Pertubation treatment of physisorption Pertubation treatment of physisorption II Assumption: negligible overlap of the wave functions ⇒ C H = H + H + V . (115) (2) v 5 a s as E (Z) = − 3 + O(Z− ) (117) (Z − Z0) First-order contribution vanishes; second order: with a s a s |h ψ ψ |V ′ |ψ ψ i| (2) 0 0 as α β 1 ∞ ǫ(iω ) − 1 E = a a s s (E − E ) + ( E − E ) Cv = dω α (iω ) (118) α=0 β=0 0 α 0 β 4π ǫ(iω ) + 1 X6 X6 Z 1 1 0 = − d3~r d3~r d3~x d3~x ′ ′ ~ ~ Z Z Z Z |R + ~x − ~r | |R + ~x ′ − ~r ′| ∞ dω 1 ∞ ǫ(iω ) − 1 × χa(~x, ~x ′, iω )χs(~r, ~r ′, iω ) (116) Z0 = dω α (iω ) z¯(iω ) (119) 2π 4πC v ǫ(iω ) + 1 Z0 Z0 χa,s retarded response functions α atomic polarizability, ǫ dielectric function, z¯ centroid of induced charge density Physisorption potential for He on noble metals van der Waals interaction in DFT Physisorption potential Theoretical description Localized electron hole in current exchange-correlation functionals αz ⇒ veff (z) ∝ e− for z → ∞ instead of veff (z) → 1/z Zaremba and Kohn, PRB 15, 1769 (1977): et al. Interaction potential divided in two parts: Hult : Adiabatic connection formula: Short-range Hartree-Fock term 2 1 1 3 3 e E [n] = d ~rd ~r ′ dλ [hn˜(~r )˜n(~r ′)i − δ(~r − ~r ′)h(~r )i] (122) Longe-range van der Waals interaction xc 2 |~r − ~r | n,λ Z ′ Z0 V (Z) = VHF (Z) + VvdW(Z) (120) ⇒ Second order Noble gas: repulsive interaction proporptio- ∆E (R~) = E − d3~r d3~r d3~x d3~x V (R~ + ~x − ~r ) V (R~ + ~x − ~r ) nal to surface charge density xc xc∞ ′ ′ as as ′ ′ V (Z) ∝ n(Z) (121) Z Z Z Z HF ∞ dω × χa(~x, ~x ′, iω )χs(~r, ~r ′, iω ) (123) Potential as a function of Z 2π Z0 Approximations: Response treated in Random Phase Approximation (RPA) local approximation for screened response Chemisorption Newns-Anderson Model Chemisorption corresponds to the creation of a true chemical bond between adsorbate and Developed by Newns based on a model proposed by Anderson for bulk impurities substrate Describes the interaction of a adatom orbital φ with metal states φ Energetic contributions to chemisorption discussed within DFT: a k Model Hamiltonian (ignoring spin) N Etot = εi + Exc [n] − vxc (~r )n(~r ) d~r − EH + Vnucl nucl + + − H = εa na + εk nk + (Vak ba bk + Vka b ba) (125) i=1 Z k X k k N X X = εi + Exc [n] − veff (~r )n(~r ) d~r + EH + Vel nucl + Vnucl nucl − − i=1 Z X + N ni = bi bi i = a,k . (126) = εi + Exc [n] − veff (~r )n(~r ) d~r + Ees (124) i=1 X Z + ni number operator, bi , bi creation and annihilation operator of the orbital φi, respectively. Adsorbate LDOS in the Newns-Anderson Model Adsorbate level in the Newns-Anderson Model Direct solution of the Schr ¨odinger equation Single-particle Green function H ~c i = εi ~c i (127) 1 Gaa (ε) = (130) ε − εa − Σ( ε) intractable due to the infinite number of metal states Self-energy Σ( ε) = Λ( ε) − i∆( ε): Consider local density of states (LDOS) on the adsorbate level: ∆( ε) = π |V |2 δ(ε − ε ) (131) 1 ak k ρ (ε) = |h i|ai| 2 δ(ε − ε ) = − Im G (ε) (128) k a i π aa X i X G single particle Green function ( δ = 0 +): P ∆( ε′) Λ( ε) = dε ′ , (132) π ε − ε Z ′ |iih i| G(ε) = (129) ε − ε + iδ P denotes principal part integral i i X Parameters in the Newns-Anderson Model Variation of adsorbate levels Ionization energy and affinity level Level variation Adsorbate LDOS: Ionization energy I: energy to remove an A: Gain additional energy due to the inter- electron from a neutral atom and bring it action of the electron with its own image: 1 ∆( ε) e2 ρa(ε) = 2 2 . (133) to infinity π (ε − εa − Λ( ε)) + ∆ (ε) Aeff (z) = A + . (136) Electron affinity A: energy that is gained 4z when an electron is taken from infinity to 6 vacuum level ∆( ε) lifetime broadening 2 the valence level of an atom 4 A = A + e /4z v (z) eff Φ eff Position of affinity level εa(z) + Λ( z) and level width ∆( z) usually enter as parameters in the I and A modified in front of a surface: 2 Newns-Anderson model Let us consider a hydrogen atom in front (eV) 0 F ε of a perfect conductor F ε−ε -2 Newns-Anderson model rather for explanatory purposes than for predictive purposes helpful 2 2 I = I - e /4z e 1 1 2 -4 eff V = − + − (134) im Energy Recently Newns-Anderson model predominantly used to describe charge transfer processes in 2 2Z 2z (Z + z) -6 molecule-surface scattering -8 I: attraction of electron to its own image -10 charge overcompensated by repulsion with -1 0 1 2 3 4 respect to the image of the nucleus Distance from the surface z(Å) ∞ 2 ∂V im e ∆I = dz ′ = − . (135) ∂z ′ 4z z=ZZ Occupied levels increase and affinity levels decrease in front of a surface Adsorbate affinity level variation Effective medium theory Schematic picture Theoretical description Idea: Adsorbate can be considered as being embedded in an inhomogeneous electron gas set up by the substrate 4 Initially empty affinity levels shifts down: Determine average electron density in the vicinity of the adsorbate v (z) e 2 eff 2 ∆(z) εa(z) = ε − (137) ∞ 4z 0 Width of the level increases: n¯i = < nj(~r ) > (139) εF (eV) F 2 ∆(z) αz j=i -2 ∆( z) = ∆ 0 e− (138) X6 ε−ε -4 When εa(z) crosses the Fermi level, the Energy cohesive -6 level becomes filled Energy estimated as the embedding energy of the adsorbate ε (z) -8 a in a homogeneous electron gas, the effective medium: Close to the surface adsorbate is then ne- -10 -1 0 1 2 gatively charged E ≈ Eci (¯ni) (140) Distance from the surface z(Å) Variation of affinity level εa(z) Cohesive energy Ec(¯n) universal function Embedding cohesive energies Qualitative picture of hydrogen adsorption Cohesive energies Discussion Hydrogen embedding energy as a function of the electron density Two contributions to Ec: kinetic and elec- Electron density and potential variation Energetics 5 trostatic energy ) -3 3 4 He Kinetic energy: increases with density due Vacancy Dashed line: Optimum density 3 O to the Pauli principle 2 (n) (eV) 2 c H 1 Electrostatic energy: becomes more attrac- Bulk Surface Chemisorption minimum direct reflecti- 0 tive at higher densities 1 on of the minimum of the Ec(n) curve -1 Competition of kinetic and electrostatic 0 -2 energy leads to a minimum (except for Hydrogen sits off-center in the vacancy -3 Cohesive energy E noble gases) since the electron density is to low in -4 -1 the center of the vacancy 0,00 0,05 0,10 0,15 0,20 -3 n → 0: Ec → A: electron affinity Electron density ( Å ) -2 Embedding energy for H,O and He obtained by LDA-DFT et al.
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