WIR SCHAFFEN WISSEN – HEUTE FÜR MORGEN

Dr. Dennis Palagin :: Paul Scherrer Institut / ETH Zürich

ETH Zurich, 14.10.2020

Introduction to first-principles modelling of catalysis at surfaces: Lecture 1

Who are we? What do we do?

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Theory division: TheoCat

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https://www.psi.ch/lsk/theocat-project

Theoretical Catalysis

1) Intro

2) Computational catalysis and first-principles modelling

3) Structure optimization

4) “ab initio” thermodynamics

5) “Descriptors” and catalysts screening

Why catalysis?

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Catalysis is a multibillion-dollar industry with a profound influence on everyday life

What is a catalyst?

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Catalysts lower the activation barrier of a reaction without being consumed

Early Nobel prizes in catalysis

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Wilhelm Ostwald, 1909

“investigations into fundamental

principles governing chemical equilibria

and rates of reaction”

Paul Sabatier , 1912“for his method of hydrogenating

organic compounds in the presence of

finely disintegrated metals”

Fritz Haber, 1918

“for the synthesis of ammonia

from its elements”

Cyril Hinschelwood, 1956

“for researches into mechanism

of chemical reactions”

Irving Langmuir, 1932

“for discoveries and investigations

in surface chemistry”

(almost) 21st century Nobel prizes

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John Pople, 1998

“for his development of

computational methods in

quantum chemistry”

“for the development of multiscale models

for complex chemical systems”

Walter Kohn, 1998

“for his development of the

density-functional theory”

Theoretical Catalysis

1) Intro

2) Computational catalysis and first-principles modelling

3) Structure optimization

4) “ab initio” thermodynamics

5) “Descriptors” and catalysts screening

Computational modeling can assist in understanding catalysis

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Computational catalyst design

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Toward realistic time scales simulation

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Problem: too many parameters!

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Schrödinger Equation

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Calculate the total energy

𝐸tot = 𝜓 𝐻 𝜓

in order to find stable states (minimum energy configurations).

𝜓 – the system’s total eigenfunction

𝐻 – total Hamiltonian

In general solvable ⇒ Born Oppenheimer Approximation

𝜓 = 𝜓electrons + 𝜓nuclei

Nuclei mostly treated as classical point particles RI

⇒ only 𝜓e (and thus 𝐻𝑒) to consider.

Electronic Hamiltonian

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𝐻𝑒 = 𝑇𝑒 + 𝑉𝑒−𝑒 + 𝑉𝑒−𝐼

𝑇𝑒 = σ𝑖𝑁𝑒 𝑝𝑒

2

2𝑚kinetic energy of electrons

𝑉𝑒−𝑒 = σ𝑖𝑁𝑒σ𝑖′

𝑁𝑒 1

𝑟𝑖−𝑟

𝑖′ electron-electron interaction

𝑉𝑒−𝐼 = σ𝑖𝑁𝑒σ𝑗

𝑁𝐼 𝑞𝑗

𝑟𝑖−𝑅

𝑗

electron-nuclei interaction

Everything known but too complicated to solve.

Simplify!

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The electron density contains all the information and is more

tractable than the wave function

Density Functional Theory

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⇒ density that minimizes 𝐸{𝑅𝐼}[𝜌] is the ground state electron density.

For a fixed set of nuclear coordinates RI

𝐸e(𝑅𝐼) = 𝜓𝑒 𝐻𝑒(𝑅𝐼) 𝜓𝑒

Idea:

Express total electronic energy as a functional of the electronic density 𝜌(RI).

𝐸 𝑅𝐼 𝜌 = 𝑇 𝜌 + න𝑑3𝑟𝜌 𝐫 𝑣 𝑅𝐼

𝐼 𝐫 +1

2ඵ𝑑3𝑟𝑑3𝑟′

𝜌(𝐫)𝜌(𝐫′)

𝐫 − 𝐫′+ 𝐸XC [𝜌]

kinetic energy

of system with

non-interacting

electrons

potential energy of

electron-nuclei

interaction

inter-nuclei

repulsion

Exchange-

correlation:

Non-classical

interactions

So what is now EXC?

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“ab initio”

DFT

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Density Functional Theory (DFT)

is widely applied to model catalysis at surfaces

The choice of the exchange-correlation (XC) functional

is critical for accurate predictions!

Practitioner level

GGA (metals)

Hybrids (molecules, insulators)

Major problems

Self-interaction error

van der Waals interaction

Inaccuracies in binding

energies and activation

barriers of ~0.2–0.4 eV!

J. P. Perdew and K. Schmidt,

AIP Conf. Proc. 557, 1 (2001)

DFT for catalysis?

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Theoretical Catalysis

1) Intro

2) Computational catalysis and first-principles modelling

3) Structure optimization

4) “ab initio” thermodynamics

5) “Descriptors” and catalysts screening

Potential Energy Surfaces

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Optimization problem

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PES is a function of 3N–6 coordinates

Very smooth for clusters

Optimization methods

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Next local minimum

Exploring configurational space

Local optimization Global optimization

• Steepest descent

• Conjugate gradient

• Broyden-Fletcher-Goldfarb-Shanno

• Simulated annealing

• Basin-hopping

• Genetic algorithms

Global minimum

Local optimization

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Rα, i+1 = Rα, i + ϒiFα({Rα, i})

Strictly downhill to local minimum

• Conjugated gradient:

fraction of the previous search direction to the atomic forces

• Broyden-Fletcher-Goldfarb-Shanno:

adding Hessian information

• Steepest descent:

Global optimization

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• Simulated annealing:

acceptance criterion

• Genetic algorithms:

“survival of the fittest”

Ẽ{Rm} = min E{R}

Transform PES intoset of staircases

P(∆E)=exp(−∆E/kBT)

• Basin hopping:

Transition states: Nudged Elastic Band

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• Intermediate configurations are bonded together with springs, so

that they are always constrained to remain between the

configurations that precedes and follow them. Then the

configurations act as an elastic chain.

• Of the true force that applies to atoms, only the component normal

to the reaction path is used. Along the reaction path it is the force

due to the springs that applies to atoms.

Theoretical Catalysis

1) Intro

2) Computational catalysis and first-principles modelling

3) Structure optimization

4) “ab initio” thermodynamics

5) “Descriptors” and catalysts screening

Real vs. model materials: materials and conditions

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ab initio thermodynamics

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From statistical thermodynamicswe know:

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𝐺 𝑇, 𝑝 = 𝐸 electr. +

−𝑅𝑇ln8π2

𝜎

2π𝑘B𝑇

ℎ2

32

∙ 𝐼A𝐼B𝐼C12

+

𝑖

𝑁Aℎ𝜈𝑖4π

+ 𝑅𝑇ln 1 − exp−ℎ𝜈𝑖𝑘B𝑇

−𝑅𝑇ln2π𝑘B𝑇𝑚

32

ℎ3∙𝑘B𝑇

𝑝

electronic

vibrational

translational

rotational

Phase diagram for Pt-O system

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Pt PtO2

Pt + O2 ⇌ PtO2

Pt-O system on ceria oxide support

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(Pt)ads

(PtO)ads

(PtO2)ads

ab initio thermodynamics

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Theoretical Catalysis

1) Intro

2) Computational catalysis and first-principles modelling

3) Structure optimization

4) “ab initio” thermodynamics

5) “Descriptors” and catalysts screening

Activity descriptors

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Activity descriptors

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Activity descriptors

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Activity descriptors

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Activity descriptors

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Microkinetic modelling

Page 41A. Bruix et al., Nature Catalysis 2, 659 (2019)

structure/energy → TS barriers → reaction rates → probabilities of system evolution

Summary: Power of theory

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References:

DFT:

“The ABC of DFT” by Kieron Burke:

http://chem.ps.uci.edu/˜kieron/dft/book/

ab initio thermodynamics:

“Ab Initio Atomistic Thermodynamics for Surfaces: A Primer” by Karsten Reuter:

https://th.fhi-berlin.mpg.de/th/publications/EN-AVT-142-02.pdf

Volcano plots:

“Towards the computational design of solid catalysts” by Jens Nørskov:

https://www.nature.com/articles/nchem.121

Theoretical catalysis:

“First-principles-based multiscale modelling of heterogeneous catalysis” by

Karsten Reuter:

https://www.nature.com/articles/s41929-019-0298-3

www.psi.ch/lsk/theocat-project

Interested in a

PhD project?

Contact me!

dennis.palagin@psi.ch