applications of computational quantum mechanics

“First-principles kinetic Monte Carlo simulations for heterogeneous catalysis versus meanfield rate theory.” to be submitted.. We have undertaken a thorough comparison between the phenom

University of California Santa Barbara

Applications of Computational Quantum Mechanics

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor of Philosophy

in Chemistry

by

Burçin Temel

Committee in charge:

Professor Horia Metiu, Chair

Professor Joan-Emma Shea Professor Steven K Buratto

Professor Bernard Kirtman

September 2006

UMI Number: 3233007

3233007 2006

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by ProQuest Information and Learning Company

The dissertation of Burçin Temel is approved

Applications of Computational Quantum Mechanics

Copyright © 2006

by Burçin Temel

This thesis is dedicated to the two dearest people in my life:

My mom Ayla and my grandma Mahmure

ACKNOWLEDGEMENTS

I would like to thank my advisor Professor Horia Metiu for his guidance and support throughout my graduate study Being his student not only contributed to my scientific knowledge a lot but also to my personal character, work discipline, ethics and professionalism With his enormous knowledge on everything one can imagine in chemistry and physics, Horia taught me how to stand up and defend my opinions, how to question every small detail and not to take anything for granted His perfectionism, eventhough made me suffer time to time, at the end

it allowed me to gain high scientific standards

I can say that I really enjoyed my 5 years in Santa Barbara, I do not only mean weather and beach but also the fun company with great friends A lot of thanks: To Brandon McKenna for his ever ending tries

to make me feel happy, especially during those down times of a Ph.D student, and to Lauren Aubin, Steeve Chretien and Andrij Baumketner for their always exciting and joyful friendship

Also, I would like to express my gratuity to: Greg Mills, with whom

I had great discussions on the projects we worked together Celia Wrathall for helping me with every official document I had to fill in Paul Weakliem for the computational support Jane and Horia Metiu for the great two summers we spent together in Berlin, Germany, and for

their kind invitations and support in Santa Barbara that assured my parents that I was in good hands here

Finally, thanks: to Prof Eric McFarland for allowing a theoretician

to go into his lab and start doing experiments, to Prof Brad Chmelka for valuable conversations, and to Prof Matthias Scheffler for his hospitality at Fritz-Haber Institute (Berlin, Germany)

VITA OF BURđễN TEMEL

August 2006 EDUCATION

Ễ Doctor of Philosophy, Chemistry

University of California Santa Barbara, CA, August 2006

(anticipated)

Thesis title: ỀApplications of computational quantum

mechanicsỂ

Ễ Bachelor of Science, Chemistry

Bilkent University, Ankara, Turkey, June 2001

AWARDS & SCHOLARSHIPS

Ễ Full Scholarship Award

University of California, Santa Barbara, 2001-2006

Ễ Presidential Work-Study Research Grant

University of California, Santa Barbara, 2001-2004

Ễ Full Scholarship Award

with Prof Horia Metiu

Ễ Currently, using both experimental and theoretical

techniques (DFT, solid state calculations) to study the heterogeneous catalysis of methanol to olefins

(Collaboration with Prof Eric McFarland)

Ễ Studied the phenomenological kinetics and the kinetic Monte Carlo approaches for CO oxidation on RuO2

(110) surface Obtained the phase diagrams and found

the most catalytically active states (Collaboration with Prof Matthias Scheffler, Fritz-Haber Institute, Berlin, Germany)

• Used different numerical techniques (iterative and

matrix methods) to solve quantum mechanics problems

by time-dependent and time-independent approaches

• Applied these fast and accurate numerical techniques to calculate elastic and inelastic phase-shifts and

resonances in time-independent scattering theory

• Developed a new basis set method using Chebyshev polynomials to solve non-periodic boundary value

problems fast and accurately

• Studied photoexcited semiconductor-liquid interfaces to investigate the factors that affect the charge separation in the semiconductor Proposed to utilize time-dependent Density Functional Theory to learn more about the

charge dynamics at the interface

Fritz-Haber Institute, Berlin, Germany

Visiting Graduate Researcher, Theory Department, July-Aug.2005 with Prof Matthias Scheffler

• Rate theory for CO oxidation on RuO2 surface

Bilkent University, Ankara, Turkey Undergraduate Researcher, Chemistry Department, 2000-2001

with Prof Ulrike Salzner

• Designed conjugated polymer semiconductors and computed their band gaps with Density Functional Theory

Max-Planck Institute, Frankfurt, Germany

Visiting Undergraduate Researcher, Theoretical Biophysics

Department, July-Sept.2000 with Prof Volkhard Helms

• Studied quantum chemistry of isotopic substitution effects on vibrational IR frequencies of common organic

EXPERIMENTAL

University of California, Santa Barbara

Graduate Researcher, Chemistry and Biochemistry Department,

January 2006-present with Prof Horia Metiu and Prof Eric

McFarland

• Currently, using both experimental (material synthesis,

characterization, heterogeneous catalytic reactions, MS,

GC, IR) and theoretical techniques (DFT, solid state calculations) to study the heterogeneous catalysis of methanol to olefins

Bilkent University, Ankara, Turkey

Undergraduate Researcher, Chemistry Department, June-July 1999

with Prof Omer Dag

• Characterized inorganic metal complexes with UV and

IR spectroscopy techniques

Ministry of Environment, Ankara, Turkey Summer Intern, Golbasi Research Lab., August 1999

• Used analytical methods such as HPLC, GC, MS, Ion

Chromatography, separation, purification to characterize molecules

PUBLICATIONS & PRESENTATIONS

1 Temel, B.; Mills, G.; Metiu, H “Time-independent potential

scattering with variational methods.” (submitted to Journal of Physical

Chemistry A)

2 Temel, B.; Mills, G.; Metiu, H “Inelastic scattering with

Chebyshev polynomials.” (to be submitted)

3 Temel, B.; Reuter, K.; Metiu, H.; Scheffler, M “First-principles

kinetic Monte Carlo simulations for heterogeneous catalysis versus

meanfield rate theory.” (to be submitted)

4 Mills, G.; Wang, B.; Temel, B.; Metiu, H “Chebyshev

polynomials for quantum mechanical problems (bound states).” (to be

submitted)

5 Temel, B.; Tang, W.; Chou, J.; Sushchikh, M.; McFarland, E.; Metiu, H “Methanol conversion to DME on doped zirconia catalyst.” (to be submitted)

6 Temel, B.; Mills, G.; Metiu, H “Inelastic scattering with Chebyshev polynomials.” , poster presentation at American Conference

on Theoretical Chemistry, July 2005

7 Temel, B., “Time-Independent Potential Scattering with Variational Principles”, talk at Fritz-Haber Institute, August 2004

PROFESSIONAL AFFILIATIONS

• American Physical Society student membership

• American Chemical Society student membership TEACHING EXPERIENCE

University of California, Santa Barbara

Chemistry and Biochemistry Department

• Teaching Assistant, Fall 2005

Mathematica as a programming language course

• Teaching Assistant, Spring 2004

Quantum chemistry (graduate level)

• Teaching Assistant, Winter 2004

Mathematica as a programming language course

• Teaching Lab Assistant, Winter 2002

General chemistry

• Discussion Lecturer, Fall 2001 and Spring 2002

Physical chemistry SKILLS

THEORETICAL

• Ab initio programs: Gaussian98, NWChem, CHARMM, VASP

• Programming languages: Fortran77, Fortran 90, C, Mathematica

• Special courses taken: Advance Quantum Chemistry, Molecular Spectroscopy, Density Functional Theory, Non-linear Statistical Mechanics (stochastic phenomena)

EXPERIMENTAL

• Materials synthesis (sol-gel), characterization; X-ray diffraction (XRD), surface area and pore size analysis (BET), heterogeneous catalytic reactions, mass spectrometry (MS), gas chromatography (GC), infrared spectroscopy (IR)

Minimum Error Method (MEM), a least-squares minimization method, provides an efficient and accurate alternative to solve systems

of ordinary differential equations Existing methods usually utilize matrix methods which are computationally costful MEM, which is based on the Chebyshev polynomials as a basis set, uses the recursion relationships and fast Chebyshev transforms which scale as O(N) For large basis set calculations this provides an enormous computational efficiency in the calculations Chebyshev polynomials are also able to represent non-periodic problems very accurately We applied MEM on

accurate than traditionally used Kohn variational principle, and it also provides the wave function in the interaction region

Phenomenological kinetics (PK) is widely used in industry to predict the optimum conditions for a chemical reaction PK neglects the fluctuations, assumes no lateral interactions, and considers an ideal mix

of reactants The rate equations are tested by fitting the rate constants to the results of the experiments Unfortunately, there are numerous examples where a fitted mechanism was later shown to be erroneous

We have undertaken a thorough comparison between the phenomenological equations and the results of kinetic Monte Carlo (KMC) simulations performed on the same system The PK equations are qualitatively consistent with the KMC results but are quantitatively erroneous as a result of interplays between the adsorption and desorption events

The experimental study on methanol coupling with doped metal oxide catalysts demonstrates the doped metal oxides as a new class of catalysts with novel properties Doping a metal oxide may alter its intrinsic properties drastically A catalytically non-active material can

be activated by doping In this study, we showed that pure zirconia (ZrO2) has almost no activity in methanol coupling reaction, whereas when it is doped with aluminum, the doped catalyst produces dimethyl ether (DME), which is valuable as an alternative future energy source

TABLE OF CONTENTS

Approval page……….ii

Dedication……… iv

Vita………vii

Abstract……….xii

I From Scattering of Molecules to Heterogeneous Catalysis………….1

II The Minimum Error Method Applied to Time-Independent Scattering Problems……… 23

A Elastic Scattering with Minimum Error Method……….24

B Inelastic Scattering with Minimum Error Method……….53

III The Kinetics of Heterogeneous Catalysis: A Comparison between Kinetic Monte Carlo Simulations and Phenomenological Kinetics… 88

IV The Methanol Coupling Reaction with a Doped Metal Oxide Catalyst………145

V Conclusions and Future Directions………169

VI Appendix……… 173

Chapter I

From Scattering of Molecules to Heterogeneous Catalysis

In this original thesis dissertation, I will be covering two major topics: Scattering theory and heterogeneous catalysis In the scattering theory part, I present a new numerical technique based on Chebyshev polynomials that is applied onto elastic and inelastic scattering problems In the heterogeneous catalysis part, I present a comparison study on CO oxidation by RuO2 which is done by kinetic Monte Carlo and phenomenological kinetics approaches I also present an experimental study on methanol conversion to dimethyl ether by aluminum doped zirconium oxide, doped oxides as a new class of catalysts The aim of this introduction chapter is to give some background information on the topics that are covered in the following chapters

1 Scattering Theory

Background:

Since 1960s, scattering phenomena has been a useful tool for both chemists and physicists By scattering a beam of particles (e.g electron, neutron, proton or a molecule) from a target or other particles, one can obtain the structural properties of these particles Discovery of the quarks by inelastic electron-nucleon scattering experiments has been one of the most exciting structural investigations of the last century Scattering is a dynamic event: it provides details on the (1) types of forces between the particles hence structures, (2) chemical

dynamics information in the case of reactive scattering of the molecules, and (3) energy transfer between the scattering particle and the target such as excitation, ionization and dissociation All these have great scientific impact in the complex excited media, ranging from stellar atmospheres to laser systems

Figure 1: Schematic drawing of a scattering process

The result of a scattering experiment (see Figure 1) is usually expressed in terms of cross sections Cross sections are defined as: number of events per unit time and per unit scatterer divided by the relative flux of the incident particles.1,2 Therefore, it is equal to the outgoing flux of particles scattered through the spherical surface r dΩ 2for r→ ∞, divided by the incident flux To be able to calculate the cross section we need to consider the probability current density:

( )( ) ( ) ( ) ( )

incoming plane wave

outgoing spherical wave

the analysis of Coulomb potential in this particular case

We find that the outgoing flux is given as

d

f

υσ

υ

Ω

What is a Variational Principle?

For a certain functional J, if we can write δJ =0, then this functional is said to be stationary.3-6 With a stationary functional we can look for the solutions of δJ =0 that depend on a number of variational parameters {c c1, , ,2 c n} Then, the functional J becomes a function of these variational parameters J c c( 1, , ,2 c n) The stationary

property of the functional yields a system of equations 0

i

J c

Hulthén-Kohn Variational Principle:

Hulthén-Kohn variational principle7,8 is used to calculate shifts Let us consider an example of it for an s-wave scattering The radial equation is

Ψ is the trial wave function

Let us take the trial wave function to be

We can express the trial wave function in terms of some variational parameters, which in turn makes the functional Jdepend on these parameters Note that one of the parameters is taken to be the phase-shift since it is an unknown quantity that the trial wave function depends on:

1

N

i i i

The trouble with this method shows itself in the M matrix: Because ij

we are in the continuous spectrum, M may have an eigenvalue that is ij

very close to zero and as a result one of the c ’s might be extremely j

large This would cause a vey large error in the calculated phase-shift This problem is usually referred as the anomalies in the phase-shift or pseudo-resonances of Hulthén-Kohn variational principle.9

Miller’s S-matrix Version of the Kohn Variational Principle:

Miller et al.10 chose the following functional

where ext stands for extremizing the functional They wrote the trial

wave function in terms of the S-matrix with the boundary conditions

Inserting the wave function Ψ% into the Eq.32 gives

, ' 1

0 1

l l l

Here, f r( )=e−αr is taken as the cut-off function

The problem with Miller’s S-matrix method is it does not guarantee a unitary S-matrix; as a result the phase-shift comes out to have a complex part, which is erroneous It also has a numerical instability: When the number of basis functions increases the cut-off function is not able to regularize the wave function properly As a result, phase-shift values come out to be erroneous

Temel and Metiu’s Minimum Error Method:

Our method of finding the phase-shifts is based on least-squares minimization idea We write the error as the following functional:

we keep the unitarity property of the S-matrix We use Chebyshev

polynomials as the basis set and utilize their fast transforms to calculate the derivatives The details of this method is given in Chapter II

2 Phenomenological Kinetics versus Kinetic Monte Carlo Simulations in Heterogeneous Catalysis

Phenomenological kinetics (PK), also so-called mean-field approximation, is based on two assumptions: (1) all adsorbed species are distributed randomly over the catalyst surface and (2) there is no interaction between the adsorbed species How frequently these assumptions are valid is questionable The reaction between two adsorbed molecules, A and B, is proportional to the probability that they are in the right position (e.g nearest neighbors) not with the product of their coverages Using the product assumes a perfect and rapid mixing of the reactants, and the lack of spatial inhomogeneity in the distribution of reactants on the surface In most catalytic reactions this may not be the case The interaction between the adsorbed molecules can lead to segregation into domains of mostly one kind of molecules Even if this does not happen, the time in which the adsorbate distribution becomes uniform may be longer than the time in which the chemical reactions taking place on the surface destroy spatial uniformity Another failure of the PK equations is displayed by the detailed balance A reversible reaction, in which the forward and the

backward rates are proportional to the concentrations, leads to an equilibrium constant that depends on concentrations This assumes that the chemical potential of the participants is linear in the logarithm of the concentration and has no other concentration dependence This is not the case when the reactants interact with the other adsorbates or with each other, in which case we must replace concentrations with activities11 in the chemical potentials and the equilibrium constants This implies that such the PK rate equations are correct only for ideal mixtures.12,13 Finally, the rate equations provided by PK do not take into account fluctuations This could lead to detectable errors when the measurements involve a small number of molecules

Practical catalytic research and much of surface science uses phenomenological rate equations, in which the reaction rate is proportional to the concentration of the reactants One proposes a mechanism, writes the corresponding rate equations and varies the rate constants to fit the data Very often the fitting is rather good and this is taken as a confirmation of the proposed mechanism Due to its usefulness in reactor design, this procedure has reached a high level of sophistication.14 There are however many examples in which a given kinetic mechanism fitted data well, only to fall apart as the data was extended to other temperatures or concentrations, or when it was discovered that reactions not included in the analysis took place in the

which were considered a bimolecular elementary reaction, and whose kinetics was fitted well by a second order equation Two decades of subsequent work established that, without doubt, this is a complicated chain reaction having more than five steps.15 For a practical engineer this may not be a problem: even if the proposed mechanism is erroneous, he can use the equations to design a reactor, as long as they fit the data However, such usage presumes that the equations can be used for conditions for which measurements are not available As many examples show, such extrapolation is dangerous: if the mechanism is incorrect, the extrapolation could be erroneous, even though the equations fitted well a limited set of data Furthermore, once the data are fitted, the resulting rate constants are used to extract activation energies If the fit is good, but accidental, the activation energies extracted from the rate constants are meaningless and are a source of confusion

An alternative to phenomenological kinetics is to perform kinetic Monte Carlo (KMC) simulations The input to KMC is a list of the rate constants of all elementary steps that can take place on the surface: adsorption, desorption, diffusion and chemical reactions A simulation may start, for example, with a clean surface and with the molecules in the gas phase, deposit the molecules on the surface, let them move from site to site and let them react when they are in the

executed with the appropriate rate This is done by using the rates of all processes possible in a given molecular configuration, to construct probabilities for each event that can occur next Then, one event is executed by a Monte Carlo procedure that uses these probabilities After this execution, the time is advanced appropriately, and the probabilities are adjusted to correspond to the new molecular configuration Then, a new event is executed Unlike the ordinary Monte Carlo procedure, KMC provides the evolution of the system out

of equilibrium, in real time (not in number of Monte Carlo moves) Different methods of implementation differ mainly in their efficiency and in the complexity of the events taken into account The KMC method solves exactly the rate equations for the events included in the model

When performed properly, kinetic Monte Carlo simulations do not have any of the shortcomings displayed by phenomenological kinetics Because KMC takes into account the interactions between the molecules and their diffusion on the surface, it builds up the correct spatial distribution of the molecules and two molecules react only when they are in the proper positions Since interactions are accounted for correctly, the thermodynamic activity of the reactants and products is automatically taken into account and there should be no conflict with the detailed balance Since the simulations are stochastic, and each

event takes place with the appropriate probability, fluctuations are also treated correctly

In Chapter III, we give a comparison of phenomenological kinetics and kinetic Monte Carlo simulations applied onto the same system: CO oxidation on RuO2 We compare the optimum reaction conditions, map out the phase diagrams of the CO2 reaction, find the rate determining steps in the CO2 formation mechanism, fit the rate constants to a kinetic Monte Carlo computer experiment, and show multiple steady states obtained by two methods

3 Methanol Conversion to Dimethyl Ether on Aluminum Doped Zirconium Oxide:

In heterogeneous catalysis, the catalysts can be metals, oxides, sulfides, carbides, nitrides, acids, and zeolites etc…A successful catalyst should have high activity and selectivity, long life time, regeneration ability, thermal and mechanical stability against sintering and crushing

For the methanol to dimethyl ether (DME) reaction, a hybrid catalyst (specially treated HZM-5 zeolite added to Cu/ZnO/Al2O3) has been developed at Haldor-Topsoe for this process.16,17 However, the presently existing metal oxides, including zeolites, are limited by coke formation and lower conversions for the synthesis of DME Zeolites

are ubiquitous in the chemical industry and used in reactions with methanol, e.g for the methanol to olefins reaction Earlier, they were selected as the target catalysts because small-pore zeolites gave high selectivities for ethylene and propylene On the other hand, rapid coke formation deactivates the catalyst Coke formation is attributed to the coverage of strong acid sites, where the olefin oligomerization occurs and these species adsorb strongly causing blockage of the pores.18-27

Recent studies on mixed or doped bulk metal oxides have revealed that these materials contain active surface sites, which are also present in zeolites and molecular sieves Furthermore, same catalytic properties are exhibited by both mixed metal oxides and molecular sieves.28-33 Bulk structures of oxides are considered to be composed of positive metal ions (cations) and negative oxygen ions (anions) Cations

at the surface possess Lewis acidity (they accept electrons) and the oxygen ions have Brønsted basicity (proton acceptors) Defects (especially oxygen vacancies) and metal ions inserted into the lattice structure (dopants) are considered to be the active sites in metal oxides

In the Chapter IV, we demonstrate that doping zirconium oxide with aluminum improves its catalytic properties drastically Zirconium oxide, an inactive catalyst for methanol to DME reaction, becomes active when it is doped with aluminum It has a moderate conversion rate, very high selectivity (no other by-products, but only DME), and it

does not have coke formation This is a novel example for a new class

of catalysts: doped metal oxides

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(2) Mott, N F.; Massey, H S The Theory of Atomic Collisons,

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York, 1977

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(5) Moiseiwitsch, B L Variational Principles, 1 ed.; Interscience

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(6) Nesbet, R K Variational Methods in Electron-Atom Scattering Theory; Plenum: New York, 1980

(7) Hulthén, L K Fysiogr Saellsk Lund Foerh 1944, 14, 21

(8) Kohn, W Phys Rev 1948, 74, 1763

(9) Nesbet, R K Physical Review 1968, 175, 134

(10) Zhang, J Z H.; Chu, S I.; Miller, W H Journal of Chemical

Physics 1988, 88, 6233

(11) Metiu, H Physical Chemistry: Thermodynamics, 1 ed.; Garland

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(12) Boyd, R K Chem Revs 1977, 77, 93

(13) Denbigh, K The principles of chemical equilibrium; Cambridge

University Press: Cambridge, 1993

(14) Froment, G F Catalysis Rev 2005, 47, 83

(15) White, J M Other reactions involving halogen, nitrogen and

sulfur compounds In Comprehensive chemical kinetics; Bamford, C

H., Tipper, C F H., Eds.; Elsevier Publishing Company: Amsterdam, 1972; Vol 6; pp 201

(16) Hansen, J B.; Joensen, F H.; Topsoe, H F A Process for preparing acetic acid, methyl acetate, acetic anhydride or mixtures thereof; Patent, U S., Ed.; Haldor Topsoe A/S (DK), 1988

(17) Jorgen, T.-J Process for the preparation of catalysts for use in ether synthesis; Patent, U S., Ed.; Haldor Topsoe A/S (DK), 1985

(18) Bjorgen, M.; Olsbye, U.; Kolboe, S Journal Of Catalysis 2003,

215, 30

(19) Chen, D.; Moljord, K.; Fuglerud, T.; Holmen, A Microporous

And Mesoporous Materials 1999, 29, 191

(20) Dahl, I M.; Kolboe, S Journal Of Catalysis 1996, 161, 304

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Chapter II

The Minimum Error Method Applied to Time-Independent Scattering Problems

Section A

Elastic Scattering with Minimum Error Method

*A related version of this chapter has been published as:

Burcin Temel, Greg Mills, and Horia Metiu, “The minimum-error method for scattering problems in quantum mechanics: two stable and efficient implementations”, J Chem Phys A, 2006 (in press)

Abstract:

We examine here, by using a simple example, two implementation of the minimum error method (MEM), a least-squares minimization for scattering problems in quantum mechanics and show that they provide an efficient, numerically stable alternative to Kohn variational principle MEM defines an error-functional consisting of the sum of the values of (HΨ-EΨ)2 at a set of grid points The wave function Ψ, is forced to satisfy the scattering boundary conditions and

is determined by minimizing the least-square error We study two implementations of this idea In one, we represent the wave function as

a linear combination of Chebyshev polynomials and minimize the error

by varying the coefficients of the expansion and the R-matrix (present

in the asymptotic form of Ψ) This leads to a linear equation for the coefficients and the R-matrix, which we solve by matrix inversion In the other implementation, we use a conjugate-gradient procedure to minimize the error with respect to the values of Ψ at the grid points and the R-matrix The use of the Chebyshev polynomials allows an efficient and accurate calculation of the derivative of the wave function,

by using Fast Chebyshev Transforms We find that, unlike KVP, MEM

is numerically stable when we use the R-matrix asymptotic condition and gives accurate wave functions in the interaction region