Quantum Methods For Adsorption ChE 553 Lecture 5 Quantum Methods For Adsorption

Objective Overview of abinitio quantum mechanical methods to calculate adsorption properties What do all of the words mean How do you apply it to surfaces Weaknesses in current calculations

Outline Review quantum methods, notation Describe how quantum used to make practical calculations for adsorption

Quantum Methods For Adsorption Solve Schroedinger equation (11.39) Calculate Energies of adsorption

Approximation To Solve Shroedinger Equation Hartree Fock (HF) Approximation Treat each electron as though it moves independently of all others (i.e. In the average field of all others) Configuration Interaction (CI) Consider how motion of each electron affects the motion of all of the other electrons

Hartree Fock Approximation = Wavefunction for molecule = Antisymmenizer … One electron wavefunctions

Solution Of HF Equation For Stationary Atoms ( ) ( ) Kinetic Energy Electron-Electron of Electrons Repulsions Electron Core Exchange Energy Attraction Exchange energy: Extra energy term that eliminates electron-electron repulsion when electrons pair up in a bond. E = + ( ) ( ) -

Algebra For Exchange Plug into Schroedinger Equation: (11.55)

Correlation Energy Missing From Hartee Fock Physics: When one atom moves into an area others move out of the way. Leads to a lowering of electron-electron repulsion. Correlation Energy – Lowers the total energy. For Ethane – Total energy = ~50,000 kcal/mole Correlation energy = ~170 kcal/mole.

Approximations Used To Solve Shroedinger Equation Exchange Correlation HF Exact MP2/3 Analytical Expansion DFT CI Expand in infinite series solve for coefficients CC Expand in a series solve for first n coefficients, use analytic expression for higher terms

Common Approximations Method Description HF One electron wavefunctions, no correlation energy as described in section 11.5.2 CI One of a number of methods where the configuration interaction is used to estimate the correlation energy as described in section 11.5.4. In the literature the CI keyword is sometimes erroneously used to denote the CIS method GVB A minimal CI calculation where the sum in equation 11.57 includes 2 configurations per bond. The configurations are to improve bond dissociation energies.

Common Approximations Continued MCSCF, CASSCF A CI calculation where the sum in equation 11.57 includes the all the single excitations of the "active orbitals" and ignores excitations of the "inactive" orbitals. In the limit of large number of configurations, MCSCF gives to the exact result. However, Often people only use a few configurations and still call the calculation MCSCF. CIS A CI calculation where the sum in equation 11.57 includes all the single excitations. This method tends to not be very accurate. CID A CI calculation where the sum in equation 11.57 includes all the double excitations. CISD A CI calculation where the sum in equation 11.57 includes all the single and double excitations.

Common Approximations Continued CISDT A CI calculation where the sum in equation 11.57 includes all the single double and triple excitations. CCSD, QCISD An improved version of a CISD calculation, where the single and double excitations are included exactly, and an approximation is used to estimate the coefficients for the higher order excitations. CCSD(T) QCISD(T) An improved version of a CCSD calculation, where triple excitations are included. MP2, MP3, MP4 A calculation where Moller-Plesset perturbation theory is used to estimate the correlation energy as described in section 11.5.5. The various numbers refer to the level of perturbation theory used in the calculation.

Common Approximations Continued G2(MP2) A combined method where you substitute a MP2 calculation for the MP4 calculation in the G2 calculation G2, G3 A combined method where you compute the energy as a weighted sum of CCSD(T) with different basis sets, an MP4 calculation with a large basis set, plus other corrections. CBS A different combined method, where you use a series of intermediate calculations to extrapolate the CCSD(T) results to infinite basis set size.

Density Functional Methods: Approximate Exchange & Correlation

Correlation Approximation For DFT

Mixed Methods

How Well Does Methods Do?: Bond Energies Approximation Mean Deviation From Experiments Kcal Largest Energy kcal + - B3LYP 3.1 13.6 - 19.7 GGA 18.1 81.0 – 10.1 LDA 24.9 89.3 – 10.4 HF 46.1 8.8 – 173.8 (Tests on Gausian test set with the 6-311+G (2d.P) bases set.

How Well Does Methods Do?: Activation Energies

How To Apply To Surface Reactions Jellium Model Cluster Model Slab Model

Jellium Model Jellium model: Figure 3.33 The electron density outside of a charge compensated jellium surface for rs = 2 and 5, after Halloway and Nørskov, [1991]. (a) Actual electron density, (b) scaled electron density.

Key Predictions Of Jellium Model Adsorbed species are radicals held with rapidly exchanging electrons Very mobile species No preferred orientation of molecules No effects of surface structure on adsorption/reaction

Cluster Model Figure 3.27 A 12, 7, 6 cluster model of a (111) surface of an FCC metal. Slab model: assume planes of atoms with periodic boundary conditions

How Big Of A Cluster Do You Need? TABLE 3.12. The Variation in the Properties of a Series of Cubic Clusters Number of Atoms on a Side Total Number of Atoms (work function 2 8 1.50 3 27 2.12 1.41 4 64 2.42 1.21 5 125 2.59 1.03 512 2.82 0.71 10 1000 2.87 0.58 50 1.26 x 105 2.994 0.12 100 106 3.00 0.06

Typical Cluster Results Table 3.13. The Heat of Adsorption of H2 on a Threefold Hollow on a Series of Nickel Clusters Designed Simulate Ni(111)* Cluster Heat of Adsorption (kcal/mole) Ni4(3,1) -8.3 Ni10(3,7) 24.7 Ni13(12,1) -23.9 Ni17(3,7,7) -22.5 Ni19(12,7) 48.5 Ni20(3,7,7,3) -8.7 Ni22(12,7,3) 27.9 Ni25(12,7,6) 18.7 Ni28(12,7,6,3) 1.9 Ni40(21,13,6) 10.5 Experiment Ni(111) -23 By comparison electronegativity equalization predicts -19

Slab Calculations Figure 3.51 A slab of metal atoms covered by adsorbate.

Weaknesses In Current Calculations Insufficient layers Poor functionals Figure 2.50 A schematic of the surface relaxations when forming a Pt(210) surface as measured by LEED. (Data of Zhang et al. [1991].) Approximation Mean Deviation From Experiments Kcal Largest Energy kcal + - B2LVP 3.1 13.6 - 19.7 GGA 18.1 81.0 – 10.1 LDA 24.9 89.3 – 10.4 LIF 46.1 8.8 – 173.8 (Tests on Gausian test set with the 6-311+G (2d.P) bases set.

Summary QM can be used to calculate heats of adsorption Metals: Usually use approximations Metals: Jellium Model – exchanging electrons Slabs – local effects Clusters – hard to use