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

2. 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

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

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

5. 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

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

7. 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 = + ( ) ( ) -

8. Algebra For Exchange
Plug into Schroedinger Equation:
(11.55)

9. 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.

10. 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

11. Common Approximations
Method
Description HF
One electron wavefunctions, no correlation energy as described in section CI
One of a number of methods where the configuration interaction is used to estimate the correlation energy as described in section 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 includes 2 configurations per bond. The configurations are to improve bond dissociation energies.

12. Common Approximations Continued
MCSCF, CASSCF
A CI calculation where the sum in equation 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 includes all the single excitations. This method tends to not be very accurate.
CID
A CI calculation where the sum in equation includes all the double excitations.
CISD
A CI calculation where the sum in equation includes all the single and double excitations.

13. Common Approximations Continued
CISDT
A CI calculation where the sum in equation 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 The various numbers refer to the level of perturbation theory used in the calculation.

14. 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.

15. Density Functional Methods: Approximate Exchange & Correlation

16. Correlation Approximation For DFT

17. Mixed Methods

18. How Well Does Methods Do?: Bond Energies
Approximation
Mean Deviation From Experiments Kcal
Largest Energy kcal
B3LYP
3.1
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 G (2d.P) bases set.

19. How Well Does Methods Do?: Activation Energies

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

21. Jellium Model Jellium model:
Figure 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.

22. 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

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

24. How Big Of A Cluster Do You Need?
TABLE 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

25. Typical Cluster Results
Table 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

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

27. Weaknesses In Current Calculations
Insufficient layers
Poor functionals
Figure 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
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 G (2d.P) bases set.

28. 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