ChE 553 Lecture 5 Quantum Methods For

Adsorption

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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

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Outline• Review quantum methods, notation• Describe how quantum used to make

practical calculations for adsorption

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Quantum Methods For Adsorption

Solve Schroedinger equation

(11.39)

Calculate Energies of adsorption

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H r,R r,R E r,R

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

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Hartree Fock Approximation

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321Hf ∀=′

HF′

∀21

= Wavefunction for molecule

= Antisymmenizer

… One electron wavefunctions

Solution Of HF Equation For Stationary Atoms

Kinetic Energy Electron-Electron

of Electrons Repulsions Electron Core Exchange

EnergyAttraction

Exchange energy: Extra energy term that eliminates electron-electron repulsion when electrons pair up in a bond.

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(E =( )+( )( )) -

Algebra For Exchange

Plug into Schroedinger Equation:

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e ee e1 2

1 2

r r

r r

1 11 1 1 2

21 1 1 2 1 2

1 2 1 12

1 2 1 1 1 2

12

12

V e

e

s s s s

s s s s

r r r r dr dr

r r r r dr dr

* *

* *

(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.

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Approximations Used To Solve Shroedinger Equation

Approximation

Exchange Correlation

HF Exact 0MP2/3 Exact Analytical ExpansionDFT Approximatio

nApproximation

CI Exact Expand in infinite series solve for coefficients

CC Exact Expand in a series solve for first n coefficients, use analytic expression for higher terms

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Common Approximations

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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

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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

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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

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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

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Method Exchange approximations

Slater A simple exchange approximation where the exchange energy is approximated as being 2/3 times the integral of the electron density to the 4/3 power. This is exact for a uniform electron gas.

Becke A modification of the Slater approximation, where corrections are included for changes in the exchange energy due to gradients in the electron density

Perdew-Wang

A modification of the Becke approximation. Perdew and Wang fit the exchange

energy for an electron gas as V F FddxeX e

e

1 2

where Vex is the exchange

energy, e is the electron density, and F1 and F2 are functions that were fit to calculations for electron gases.

Modified Perdew Wang

Modifications of the Perdew-Wang where different functionals are used.

Correlation Approximation For DFT

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VWN A correlation approximation due to Vosko, Wilk, and Nusair, which assumes that the correlation is only a function of the local electron density. The function is calculated assuming that you have a uniform electron gas and the given density. This method is sometimes called the local spin density method.

Becke A modified version of the VWN approximation, where an extra correction is added to account for variations in the correlation energy due to the gradients in the electron density. The Becke gradient approximation is optimized for metals.

LYP A different modification of the VWN approximation, where an extra correction is added to account for variations in the correlation energy due to the gradients in the electron density. The LYP gradient approximation is optimized for molecules

Perdew Wang

A modification of the Becke approximation. Perdew and Wang fit the exchange

energy for an electron gas as V F Fddxcor e

e

3 4

where Vcor is the exchange

energy, e is the electron density, and F3 and F4 are functions that were fit to calculations for electron gases.

Modified Perdew Wang

There are a number of modifications of Perdew-Wang in the literature. Generally, people use different functions to tailor F3 and F4 for a specific application. I find it difficult to use the methods, because one never knows whether they are going to work. However, they are common in the literature.

Mixed Methods

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X-alpha An approximation that uses the Slater approximation for the exchange integral and ignores the correlation. It is common in X-alpha to also change the coefficent in the Slater approximation from 2/3 to 0.7

LDA, LDSA

A method which used the Slater approximation for the exchange and the VWN approximation for the correlation

B3LYP A hybrid method where the exchange energy is approximated as a weighted average of the Hartree-Fock exchange energy and the Slater-Becke exchange energy, while the correlation is calculated via the LYP approximation.

AM1,

MINDO, CINDO, Huckel

These are several semiempirical methods. These methods are similar, in spirit, to the DFT methods, but one also approximates the coulomb repulsions with empirical functions

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.

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How Well Does Methods Do?: Activation Energies

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Table 11.6 The energy of the transition state of the reaction H + H2 H2 + H calculated by a number of methods. All of the calculations used a 6-311G++(3df, 3pd) basis set. Results of Johnson et al Chem. Phys. Lett. 221 100 (1994).

DFT Methods Exchange Approximation Correlation Approximation Transition State Engery,

kcal/mole Slater VWN -2.81 Slater LYP -3.45 Slater Perdew-Wang -3.58 Becke VWN +3.65 Becke LYP +2.86 Becke Perdew-Wang +2.84

Perdew-Wang VWN +2.75 Perdew-Wang LYP +1.98 Perdew-Wang Perdew-Wang +1.95

Non-DFT Methods

MP2 +13.21 CCSD (T) +9.91

G-2 +9.8 Experiment +9.7

How To Apply To Surface Reactions

• Jellium Model• Cluster Model• Slab Model

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Jellium Model

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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

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Cluster Model

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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?

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Number of Atoms on a Side

Total Number of Atoms

(work function

2 8 1.50 1.503 27 2.12 1.414 64 2.42 1.215 125 2.59 1.038 512 2.82 0.7110 1000 2.87 0.5850 1.26 x 105 2.994 0.12

100 106 3.00 0.06

eV,3 eV,3TABLE 3.12. The Variation in the Properties of a Series of Cubic Clusters

Typical Cluster Results

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Cluster

Heat of Adsorptio

n (kcal/mol

e) Cluster

Heat of Adsorption (kcal/mole

) 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) -8.7Ni22(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

Table 3.13. The Heat of Adsorption of H2 on a Threefold Hollow on a Series of Nickel Clusters Designed Simulate Ni(111)*

By comparison electronegativity equalization predicts -19

Slab Calculations

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Figure 3.51 A slab of metal atoms covered by adsorbate.

Weaknesses In Current Calculations

• Insufficient layers

• Poor functionals

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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.

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].)

SummaryQM can be used to calculate heats of adsorption

– Usually use approximations

Metals:Jellium Model – exchanging electronsSlabs – local effectsClusters – hard to use

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