Computational Systems Chemistry with Rigorous Uncertainty Quantiﬁcation

DISS ETH NO. 24842

A thesis submitted to attain the degree of

DOCTOR OF SCIENCES of ETH ZÜRICH (Dr. sc. ETH Zürich)

presented by Jonny Proppe

Master of Science (M.Sc.) Chemistry University of Hamburg

born on August 18, 1987 citizen of Germany

accepted on the recommendation of

Prof. Dr. Markus Reiher, examiner Prof. Dr. Sereina Riniker, co-examiner

ETH Zürich Zürich, Switzerland 2018 Jonny Proppe: Computational Systems Chemistry with Rigorous Uncertainty Quantification Dissertation ETH Zürich No. 24842, 2018. To Louisa.

Contents

Abstract vii

Zusammenfassung ix

Acknowledgments xi

Abbreviations xiii 1 Complexity and Uncertainty: Challenges for Computational Chemistry 1

2 Computational Chemistry from a Network Perspective 5 2.1 Exploration of Chemical Reaction Networks ...... 6 2.2 Determination of Rate Constants from First Principles ...... 12 2.3 Kinetic Modeling of Reactive Chemical Systems ...... 14

3 Computational Chemistry from a Statistical Perspective 25 3.1 Statistical Calibration of Parametric Property Models ...... 25 3.2 Uncertainty Classification ...... 27 3.3 Prediction Uncertainty from Resampling Methods ...... 31 4 Mechanism Deduction from Noisy Chemical Reaction Net- works 37 4.1 Kinetic Modeling of Complex Chemical Reaction Networks ...... 38 4.2 Overview of the KiNetX Meta-Algorithm ...... 40 4.3 Automatic Network Generation ...... 41 4.4 Hierarchical Network Reduction ...... 42 4.5 Propagation of Free-Energy Uncertainty ...... 48 4.6 Analysis of Kinetic Sensitivity to Free-Energy Uncertainty ...... 49 4.7 Integration of Stiff Ordinary Differential Equations ...... 51 4.8 Exemplary Workflow of the KiNetX Meta-Algorithm ...... 52 4.9 Accuracy and Efficiency of KiNetX-Based Kinetic Modeling ...... 55 4.10 KiNetX as a Guide for Reaction Space Exploration ...... 58 4.11 Conclusions ...... 60 5 Reliable Estimation of Prediction Uncertainty for Physico- chemical Property Models 61 5.1 Isomer Shift Calibration in Theoretical 57Fe Mössbauer Spectroscopy . . 62 5.2 Reference Set of Molecular Iron Compounds ...... 63 5.3 Effect of Experimental Uncertainty on Model Parameters ...... 64 5.4 Model Selection Based on Occam’s Razor ...... 68 5.5 Assessment of Data Inconsistency Based on Jackknife-after-Bootstrapping 69

v 5.6 How Reliable are Density Functional Rankings Based on a Speciﬁc Data Set? ...... 76 5.7 Eﬀect of Exact Exchange on Model Prediction Uncertainty ...... 80 5.8 Conclusions ...... 83 6 Case Study: Thermochemical, Kinetic, and Spectroscopic Modeling of Iron Porphyrin Carbene Chemistry 87 6.1 Thermochemical Analysis of Iron Porphyrin Carbene Reactivity . . . . . 87 6.2 Density Functional Assessment Based on Kinetic Modeling ...... 105 6.3 Mössbauer Spectroscopy for the Discrimination of Spin–Charge States . . 111 7 A Computational Perspective on the Study of Complex Chem- ical Systems 115 Appendix A Kinetic Modeling of Complex Reactions: Techni- cal Details 119 A.1 Computational Singular Perturbation ...... 119 A.2 Construction of Random Covariance Matrices ...... 122 A.3 Encoding Chemical Logic into Network Graphs ...... 123 Appendix B 57Fe Mössbauer Isomer Shift Prediction: Technical Details 125 B.1 Statistical Calibration Analysis ...... 125 B.2 Quantum Chemical Calculations ...... 128

References 131

Publications 153

Curriculum Vitae 155

vi Abstract

The success of in silico design approaches for molecules and materials that attempt to solve major technological issues of our society depends crucially on knowing the uncertainty of property predictions. Calibration is an essential model-building approach in this respect as it renders the inference of uncertainty-equipped predictions based on computer simulations possible. However, there exist various pitfalls that may affect the transferability of a property model to new data. By resorting to Bayesian inference and resampling methods (bootstrapping and cross-validation), we discuss issues such as the proper selection of reference data and property models, the identification and elimination of systematic errors, and the rigorous quantification of prediction uncertainty. We apply this statistical calibration approach to the prediction of 57Fe Mössbauer isomer shifts from electron densities obtained with density functional theory. Our findings reveal that the specific selection of reference iron complexes can have a significant effect on the ranking of density functionals with respect to model transferability. Further- more, we show that bootstrapping can be harnessed to determine the sensitivity of such model rankings to changes in the reference data set, which is inevitable to guide future computational studies. Such a statistically rigorous approach to calibration is almost unknown to chemistry. Our study is one of the very few addressing this issue and its results can be applied by all chemists to arbitrary property models with our open- source software reBoot. In this thesis, we define a new standard for the calibration of computational results due to the rigor, transparency, and generality of our statistical approach, which is completely automatable. Black-box uncertainty quantification can also be applied to macroscopic systems by propagating the uncertainties inferred for single- molecule properties, which will ultimately allow modeling in chemistry to accelerate the discovery of important drugs, organic materials for solar cells, electrolytes for flow bat- teries, etc. A rather fundamental application area of this systems-focussed uncertainty quantification approach is the understanding of complex chemical reaction mechanisms, which is therefore another focus of this thesis. For an approach that accounts for all elementary processes within a reactive mixture, it is essential to know all relevant intermediates and transition states, to determine relative (free) energies, to quantify their uncertainties, and to model the systems kinetics based on uncertainty propagation. The advantage of a holistic in silico approach to chemistry is that the origin of all data can be rigorously controlled, which allows for reliable uncertainty quantification and propagation. In this thesis, we present the first automated exploration of parts of chemical

vii reaction space based on quantum mechanical descriptors at the example of synthetic nitrogen fixation. Moreover, an extension to the exploration strategy considering uncertainty propagation through all stages of in silico modeling is presented in detail at the example of the formose reaction. It is generally hard to model the kinetics of such complex reactive systems as they usually constitute processes spanning multiple time scales. Here, we present a simple and efficient strategy based on computational singular perturbation, which allows us to model the kinetics of complex chemical systems at arbitrary time scales. To study arbitrary reaction networks of dilute chemical systems (low-pressure gas or low-concentration solution phase), we implemented a generalized scheme of our kinetic modeling approach referred to as KiNetX. Main features of the completely automated KiNetX meta-algorithm are hierarchical network reduction, uncertainty propagation, and global sensitivity analysis, the latter of which detects critical (uncertainty-amplifying) regions of a network such that more complex electronic structure models are only employed if necessary. We also developed an automatic generator of abstract reaction networks encoding chemical logic, named AutoNetGen, which is coupled to KiNetX and allows us to examine a multitude of different chemical scenarios in short time. In a final case study, we apply the insights gained from computational systems chemistry with rigorous uncertainty quantification to model the thermochemistry, kinetics, and spectroscopic properties of iron porphyrin compounds, which constitute a crucial type of active centers in metalloenzyme research.

viii Zusammenfassung

Der Erfolg von in-silico-Designansätzen für Moleküle und Materialien, welche zur Lö- sung wichtiger technologischer Probleme unserer Gesellschaft entwickelt werden, hängt signifikant von unserem Wissen über die Unsicherheit von Eigenschaftsvorhersagen ab. Die Kalibrierung von Simulationsergebnissen stellt in dieser Hinsicht einen essentiellen Modellbildungsschritt dar, auf dessen Grundlage unsicherheitsbehaftete Vorhersagen ermöglicht werden. Diese potentiell überaus effektive Vorgehensweise ist jedoch mit einer Reihe von Schwierigkeiten verbunden, ohne deren Berücksichtigung die Über- tragbarkeit eines Eigenschaftsmodells auf neue Daten beeinträchtigt werden kann. In dieser Dissertation diskutieren wir solche Schwierigkeiten – die geeignete Wahl von Re- ferenzdaten und Eigenschaftsmodellen; die Identifizierung und Eliminierung systema- tischer Fehler; die rigorose Quantifizierung der Unsicherheit einer Vorhersage – im Lichte Bayes’scher Inferenz und auf Wiederholungsprobennahmen basierender Metho- den (Bootstrapping, Kreuzvalidierung). Wir wenden unseren statistischen Ansatz auf die Kalibrierung berechneter Elektronendichten an, um ab-initio-Vorhersagen über 57Fe- Mößbauer-Isomerieverschiebungen treffen zu können. Unsere Ergebnisse deuten darauf hin, dass die spezifische Wahl der Referenzeisenkomplexe einen signifikanten Effekt auf das Ranking der verwendeten Modelle (hier: Dichtefunktionale) bezüglich ihrer Übertragbarkeit ausübt. Des Weiteren zeigen wir, dass Bootstrapping genutzt werden kann, um die Sensitivität solcher Modellrankings auf Änderungen im Referenzdatensatz zu quantifizieren, was unerlässlich für richtungsweisende Entscheidungen in künftigen computergestützten Studien ist. Ein derart statistisch rigoroser Kalibrierungsansatz ist in der chemischen Forschung beinahe unbekannt. Unsere Studie ist eine der weni- gen, die dieses Thema adressieren, und die darin vorgestellten Methoden können von allen Chemikern mithilfe unserer Open-Source-Software reBoot routinemäßig auf beliebige Eigenschaftsmodelle angewendet werden. Damit definieren wir einen neuen Stan- dard für die Kalibrierung von quantenchemischen Simulationsergebnissen aufgrund der Rigorosität, Transparenz und Allgemeingültigkeit unseres statistischen Ansatzes, der vollständig automatisierbar ist. Solch eine Blackbox-Unsicherheitsquantifizierung kann auch auf makroskopische Systeme angewendet werden, indem die für Einzelmoleküleigen- schaften inferierten Unsicherheiten propagiert werden. Die Realisierung dieses system- fokussierten Ansatzes kann die Beschleunigung von wichtigen Erkenntissen in Bereichen wie z.B. Wirkstoff-, Solarzellen- und Elektrolytforschung vorantreiben. Ein eher funda- mentaler Anwendungsbereich dieses Ansatzes sind komplexe chemische Reaktionsmecha-

ix nismen, die einen weiteren Fokus dieser Dissertation darstellen. Um allen Elemen- tarprozessen innerhalb einer reaktiven Mischung Rechnung zu tragen, ist die Kenntnis aller relevanten Intermediate und Übergangszustände, die Bestimmung relativer (freier) und unsicherheitsbehafteter Energien sowie die kinetische Modellierung des Systems basierend auf Fehlerfortpflanzung von entscheidender Bedeutung. Der Vorteil eines solchen holistischen in-silico-Ansatzes für die Chemie liegt in der Möglichkeit, den Ur- sprung aller Daten genau kontrollieren zu können, was unerlässlich für eine zuverläs- sige Unsicherheitsanalyse ist. In dieser Dissertation präsentieren wir die erste automa- tische Exploration chemischer Reaktionsnetzwerke basierend auf quantenmechanischen Deskriptoren am Beispiel der synthetischen Stickstofffixierung. Darüber hinaus stellen wir eine Erweiterung dieser Explorationsstrategie am Beispiel der Formosereaktion vor, welche die Fehlerfortpflanzung durch alle in-silico-Modellierungsstufen berücksichtigt. Im Allgemeinen ist es schwierig, die Kinetik solch komplexer, reaktiver Systeme zu studieren, da die zugrundeliegenden Prozesse für gewöhnlich diverse Zeitskalen um- fassen. Hier präsentieren wir eine einfache und effiziente Strategie basierend auf der Computational-Singular-Perturbation-Methode, die es uns erlaubt, die Kinetik komplexer chemischer Systeme für beliebige Zeitskalen zu modellieren. Um zudem beliebige Reaktionsnetzwerke verdünnter chemischer Systeme (Niedrigdruckgasphase und Niedrigkonzentrationslösungsphase) untersuchen zu können, haben wir ein verallgemei- nertes Schema unseres kinetischen Ansatzes, genannt KiNetX, implementiert. Haupt- funktionen unseres vollständig automatisierten KiNetX-Metaalgorithmus sind eine Hierarchie von Netzwerkreduktionsmethoden, Fehlerfortpflanzung sowie eine globale Sensitivitätsanalyse, auf deren Grundlage kritische (unsicherheitsverstärkende) Regio- nen eines Netzwerkes identifiziert werden können, sodass komplexere Elektronenstruk- turmodelle nur dann zum Einsatz kommen werden, wenn diese auch benötigt werden. Zudem haben wir einen automatischen Generator abstrakter Reaktionsnetzwerke na- mens AutoNetGen entwickelt, der chemische Logik in Form spezifischer Graphenstruk- turen enkodiert und KiNetX zuführt, sodass in kurzer Zeit eine Vielzahl von verschiede- nen chemischen Szenarien untersucht werden kann. In einer abschließenden Fallstudie wenden wir die gewonnen Erkenntnisse auf die thermochemische, kinetische und spek- troskopische Modellierung von Eisenporphyrinverbindungen an, die einen hohen Stellen- wert als aktive Zentren in der Metalloenzymforschung einnehmen.

x Acknowledgments

Many people have accompanied me during my doctoral studies at ETH Zürich, which has been an amazingly exciting journey for me. First of all, I would like to thank my PhD supervisor Prof. Dr. Markus Reiher for giving me the opportunity to work on thrilling projects that I could explore independently. His unconventional ideas and his enthusiasm for science have encouraged me to explore new directions for computational chemistry, which is a fascinating endeavor that requires both rigor and creativity. I am very grateful for his generosity and support during my time as a PhD student. I appreciate that Prof. Dr. Sereina Riniker has proof-read this dissertation and accepted to be the co-examiner of its defense. At this point, I would also like to thank Prof. Dr. Carmen Herrmann, who supervised my Master’s thesis at the University of Hamburg and supported me in getting prepared for the next step of my academic career. Now that I oﬃcially ﬁnished my academic studies, there is no more time to waste to thank my biology teacher from high school, Dr. Karin Stäb, for encouraging me to study the sciences when I was indecisive about my professional future. I am grateful to the whole Reiher research group for an inspiring working atmosphere and relaxing feierabend beers: Dr. Stefan Knecht, Dr. Leon Freitag, Dr. Arseny Kovyrshin, Dr. Christopher Stein, Adrian Mühlbach (Gucci, Gucci), Andrea Muolo, Jan- Grimo Sobez, and Alain Vaucher. In particular, I would like to thank Dr. Thomas Wey- muth, Tamara Husch, and Gregor Simm for the enjoyable teamwork that resulted in some exciting papers. Many thanks also go to the former members of the group: PD Dr. Hans-Peter Lüthi, Dr. Arndt Finkelmann, Dr. Moritz Haag, Dr. Maike Bergeler, Dr. Erik Hedegård, Dr. Florian Krausbeck, Dr. Yingjin Ma, Dr. Halua Pinto de Magal- haes, and our former secretary Romy Isenegger. I wish our new secretary Sarah Katsche- witz and our new PhD student Christoph Brunken a good time in the group. Stefan Gugler, who is currently a semester student in our group and working with me on an exciting project, proof-read this dissertation. I acknowledge his support and wish him good luck for his Master’s project in the group of Prof. Dr. Heather J. Kulik at the Massachusetts Institute of Technology. Moreover, there is a number of other (former) members at ETH’s Laboratory of Physical Chemistry that I would like to send my thanks: Prof. Dr. Philippe Hünen- berger and Dr. Erich Meister for enjoyable and instructive teaching collaborations; Clau-

xi dia Hilty, Veronika Sieger, and Christine Siegrist for helping me getting started at ETH and with several other administrative issues; and Dr. Noah Bieler, Dr. Pavel Oborsky, Dr. Alice Lonardi, Dr. Pascal Merz, Dr. Francesco Ravotti, Jagna Witek, Joël Gubler, David Hahn, Eno Paenurk, and Annick Renevey for a pleasant social atmosphere (and, again, feierabend beers). What should not be forgotten is the institution that provided sustenance while writing this thesis: The St. Laurentius brewery of Bülach (only a 10-minute train ride away from Zürich). Visit the amazing craft beer brewers of St. Laurentius in their taproom should you be around Zürich next time. It is a cozy place where you can meet local people while having some really tasty beers. Thank you and good luck for the future! Clearly, it is not easy to withstand such an intensive and sometimes exhaustive period without good friends and family. I thank my mother Sabine Proppe for giving me the space that I needed during early life to encounter unconventional ideas and plenty of creative activities. Unfortunately, my father Jörg Proppe cannot share this big moment with me; he would have known how to celebrate it in the best possible way. Furthermore, I thank my grandmother Karin Schulz for sparking my interest in writing and literature. During my time in Zürich, I luckily got to know Jagna, Arndt, Christopher, Moritz, and Noah, on whose support I could count at all times and with whom I enjoyed several cultural trips through Europe. Many thanks also go to my friends from home: Fabian Schröder, Dawid Wieczorek, Jana Moer, Dr. Jos Tasche, Michael Prenzlow, and Felix Köster. They steadily supported me in achieving my goals, which I will not forget. In particular, I thank my good friend Dr. Verena Kraehmer, who has been and is a great source of intellectual inspiration and an invaluable advisor to me. I hope that she will not forget that the unconventional way is the only way to make a diﬀerence. Eventually, I thank my beloved wife Dr. Louisa Proppe for her trust in me and her tireless support, be it emotional or intellectual. She has been my closest friend during the last ﬁve years, and I am both happy and lucky having her by my side. I am very much looking forward to experiencing the next exciting chapters of life with her.

xii Abbreviations

ACED absorber contact electron density

CED contact electron density

CRN chemical reaction network

CSP computational singular perturbation

DFT density functional theory

EDA ethyl diazoacetate

HTVS high-throughput virtual screening

IPC iron porphyrin carbene

JPDF joint probability density function

MAP maximum posterior

MPU model prediction uncertainty

MPV mean predictive variance

MSE mean squared error

ODE ordinary diﬀerential equation

PDF probability density function

PES potential energy surface

PUI parameter uncertainty inﬂation

RMSD root-mean-square deviation

RMSE root-mean-square error

TST transition state theory

xiii xiv 1 Complexity and Uncertainty: Challenges for Computational Chemistry

Due to the continuous advancements in computer hardware development and algorith- mic research, the detailed computational study of complex (including macroscopic) reactive systems based on first-principles physics has recently started to rise.* Examples are the nanoreactor1 by Pande and Martínez, Aspuru-Guzik’s heuristics-aided quantum chemistry,2 and the global reaction route mapping3 scheme by Morokuma, Ohno, and Maeda. All these approaches make different assumptions on the microscopic system processes studied such that, from a feasibility point of view, none is generally applicable. To illustrate this point, we consider two examples. On the one hand, the dynamics on a rugged energy landscape will demand advanced sampling methods from molecular dynamics simulations rather than a standard quantum chemical approach that considers only a few selected stationary points on that surface.4,5 On the other hand, for processes on a well-structured potential energy surface (PES) with nonshallow minima, explicit dynamics may suffer from sampling problems and is often replaced by kinetic models that eventually allow one to access long time and length scales beyond the reach of explicit dynamical approaches.6 Quantum chemical models are well suited for describing energy changes due to changes in the electronic structure of reacting molecules if these electronic effects govern the overall energetics of the process. Usually, structures considered relevant as stable intermediates or transition states are optimized and their energies are compared to identify

*This chapter is reproduced in part from J. Proppe, T. Husch, G. N. Simm, M. Reiher, Faraday Discuss. 2016, 195, 497, with permission from the Royal Society of Chemistry.

1 Chapter 1 Complexity and Uncertainty

the relevant reaction pathways. Clearly, this approach is limited, especially if carried out manually, to a rather small number of structures only. For predictive work on systems for which little or no experimental information is known, the exploration of potentially important structures becomes an immense task. Several approaches exist to overcome this issue. For instance, in reactive molecular dynamics simulations,1,7–15 the nuclear equations of motion are solved to explore and sample configuration space. By contrast, heuristics-guided exploration approaches are based on rules derived from chemical concepts.2,16–22 By applying predefined (possibly alchemical) transformation rules to create new chemical species, explorations in configuration space are greatly accelerated. Recently, we proposed a fully automated heuristics-guided exploration pro- tocol20 in which the heuristic rules rest on reactivity descriptors derived from quantum mechanics making the approach applicable to all classes of molecules.

It is important to understand that to theoretically grasp the kinetics of complex reactive systems, we must be prepared to investigate an enormous number of possible intermediates (on different PES’s) not generated by simple conformational changes but by the sheer number of chemically different species. Moreover, to take full account of all processes in the system, an astronomic number of elementary reactions needs to be considered. At the same time, the underlying processes usually span multiple time and length scales, which renders any explicit dynamics description effectively unfeasible unless very harsh reaction conditions are enforced1 to accelerate the discovery of rare events. From an electronic structure point of view, it would be preferable to transform the multidimensional and continuous phase space that provides a full system description into a lower-dimensional and discrete reaction network representation. This kind of phase space discretization is based on the concept of metastability23 and would allow us to partition a complex system composed of, e.g., one mole of species into a set of much smaller subsystems (ideally, single molecules) that can be studied independently of each other.

For truly complex chemical reaction networks (CRNs), we need to establish a holistic protocol based on quantum chemical calculations that would span the whole range of steps from molecular and electronic structure optimization to detailed kinetic modeling. Clearly, various choices and approximations need to be made and hence the protocol to be established will not be unique. Still, we demand the development of such a protocol to be subjected to constraints that will make it universally applicable. Besides, we are faced with the fact that quantum chemical raw data need to be augmented by nuclear motion and temperature corrections before they can be subjected as rate constants to kinetic modeling (state-of-the-art reaction rate theories and applications thereof have been presented and discussed at the 2016 Faraday Discussions on Reaction Rate Theory24–27). Hence, if we must be prepared to make certain assumptions and

2 approximations, we will expect from our protocol that the violation of an approximation can be identified within the protocol and overcome by approaches beyond the realm of the protocol’s standard methods. This way, we may be able to identify possible breaches that point to more sophisticated theoretical approaches to be applied. If the number of such breaches is small, then the general basis of the protocol, in our case quantum chemical models, will remain valid. And, in fact, in view of the successes of quantum chemical reaction mechanism elucidation, we have good reason to believe that this is feasible. Obviously, such an endeavor will only be possible if we do have error and uncertainty measures for all quantum chemical raw data at hand that allow us to assess the accuracy and precision of individual simulation results. As essentially every computational model employed for property calculations is based on a set of approximations (including state-of-the-art electronic structure models), we need to study their biasing effects on property predictions. This type of systematic error, referred to as model inadequacy,28 can generally not be deduced, which makes it necessary to include (unbiased!) reference data against which the model approximations can be assessed. Without the identification and elimination of systematic errors based on rigorous statistical methods,29,30 we cannot infer statistically valid prediction uncertainties for microscopic properties derived from quantum chemical calculations, which is crucial to arrive at reliable uncertainty measures for their macroscopic counterparts. This line of argument illustrates the importance of uncertainty analysis for the rigorous study of complex reactive systems, which constitutes novel challenges for the field of computational chemistry. One rare application of uncertainty analysis for reactive systems derived from first principles is a study by Vlachos and co-workers where the uncertainty of chemical surface processes was inferred from a Bayesian estimation of species energy uncertainties.31 To summarize, the ingredients of a general protocol for the generation and analysis of CRNs are: (1) the automated exploration of possibly relevant intermediates and transition states, (2) the determination of rate constants from free energies equipped with reliable uncertainty estimates, and (3) the kinetic modeling of the reaction network emerging including uncertainty propagation. While protocols by other groups attempt to achieve a similar goal,32–34 none of them combines all essential ingredients for the holistic approach to systems chemistry proposed by us. In this work, we discuss all components of our general protocol at (i) a catalytic Chatt-like nitrogen fixation cycle,20 (ii) the formose reaction,35 (iii) the formation of a cyclopropane derivative catalyzed by an iron porphyrin complex,36 and (iv) abstract CRNs encoding chemical logic.37 Furthermore, we introduce a rigorous statistical calibration approach38 that allows for reliable uncertainty estimation of properties for each vertex/edge in a CRN, which will subsequently be propagated35 through the entire net-

3 Chapter 1 Complexity and Uncertainty

work. It is our goal to establish protocol-inherent validation measures that keep track of the validity of the assumptions made and that may point to advanced theoretical approaches to deliver more reliable data if needed. Moreover, our analysis is intended to be a general feasibility analysis of this protocol that will, as we shall show, point to interesting future developments.

4 2 Computational Chemistry from a Network Perspective

For the investigation of CRNs, the efficient and accurate determination of all relevant intermediates and elementary reactions of the underlying chemical system is manda- tory.*,†,‡ The complexity of such a network may grow rapidly, in particular if reactive species are involved that might cause myriads of side reactions. Without automation, a complete investigation of complex reaction mechanisms is tedious and possibly unfeasible. Here, we conceptualize a computational protocol that constructs reaction networks based on quantum mechanical reactivity descriptors in a fully automated manner. Further- more, to gain a quantitative understanding of the dynamics underlying such complex chemical systems, it is, in general, necessary to accurately determine the corresponding free-energy surface and to solve the resulting continuous-time reaction rate equations for a continuous state space. For a complex reaction network, it is computationally hard to fulfill these two requirements. However, it is possible to approximately address these challenges in a physically consistent way. On the one hand, it may be sufficient to consider approximate free energies if a reliable uncertainty measure can be provided. On the other hand, a highly resolved time evolution may not be necessary to determine quantitative fluxes in a reaction network if specific time scales are in demand. We

*This chapter is reproduced in part with permission from M. Bergeler, G. N. Simm, J. Proppe, M. Reiher, J. Chem. Theory Comput. 2015, 11, 5712. Copyright 2015 American Chemical Society. †This chapter is reproduced in part from J. Proppe, T. Husch, G. N. Simm, M. Reiher, Faraday Discuss. 2016, 195, 497, with permission from the Royal Society of Chemistry. ‡This chapter is reproduced in part from J. Proppe, M. Reiher, 2018, arXiv:1803.09346.

5 Chapter 2 Computational Chemistry from a Network Perspective

present discrete-time kinetic simulations in discrete state space taking free-energy uncertainties into account. The method builds upon thermochemical data obtained from electronic structure calculations in a condensed-phase model. Our kinetic approach supports the analysis of general reaction networks spanning multiple time scales, which is here demonstrated at the example of the formose reaction.

2.1 Exploration of Chemical Reaction Networks

Complex reaction mechanisms are found in transition metal catalysis,39 polymerizations,33 cell metabolism,40 flames and environmental processes,41 and are the objective of systems chemistry.42 Knowing all chemical compounds and elementary reactions of a specific chemical process is essential for its understanding in atomistic detail. Even though many chemical reactions result in the selective generation of a main product,43 generally, multiple reaction paths compete with each other leading to a variety of side products. In such cases, a reactive species (such as a radical, a valence-unsaturated species, a charged particle, a strong acid or base) can be involved or the energy de- posited into the system may be high (e.g., due to a high reaction temperature). For a detailed analysis of a chemical system, relevant intermediates and transition states are to be identified according to their relative energies. Manual explorations of complex reaction mechanisms employing well-established electronic structure models are slow, tedious, and error-prone. They are limited to the search for expected dominant reaction paths (e.g., a catalytic cycle or an enzymatic cascade). It is therefore desirable to develop a fully automated protocol for an efficient and accurate exploration of configuration spaces involving both intermediates and transition states. Existing approaches comprise, for example, global reaction route mapping,3 and reactive molecular dynamics.1,9,13 Starting from a given structure, the global reaction route mapping procedures evolves along the corresponding PES by exploiting local curvature information. Since the dimension of a PES scales with the number of atomic nuclei, the global reaction route mapping methods, though highly systematic, are not suitable for the exploration of large reactive systems. Contrary to such stationary approaches, in reactive molecular dynamics simulations, the nuclear equations of motion are solved to explore and sample configuration spaces. The capability of reactive ab initio molecular dynamics for studying complex chemical reactions was shown at the example of the prebiotic Urey–Miller experiment.1,13 To overcome the high computational demands of first-principles calculations in ab initio molecular dynamics, reactive force fields44 can be employed, which accelerate calculations by some orders of magnitude.9 They are, however, not generally available for any type of system. A complementary strategy is to exploit the conceptual knowledge of chemistry from quantum mechanics to explore reaction mechanisms.45 By applying predefined transfor-

6 Exploration of Chemical Reaction Networks 2.1 mation rules to create new chemical species based on reactivity concepts, searches in the chemical configuration space are accelerated without resorting to expert systems as applied for synthesis planning.46–54 In this context, the efficiency of heuristics-guided quantum chemistry was explored recently.2,16–22 The idea behind heuristic guidance in quantum chemistry is to propose a large number of hypothetical molecular structures, which are subsequently optimized by electronic structure methods. Although this approach cannot guarantee to establish a complete CRN, heuristic methods allow for a highly efficient and directed search based on empiricism and chemical concepts. Crucial for the construction of such heuristic rules is the choice of molecular descriptors. For the study of chemical reactions, graph-based descriptors dominate the field,2,16,17,19,22,49,50,52,55–57 which are based on the concept of the chemical bond. Zim- merman developed a set of rules based on the connectivity of atoms to generate molecular structures and to determine elementary reactions.16,17 Quantum chemical structure optimizations and a growing string method16,58,59 for transition state search were applied to study several textbook reactions in organic chemistry. Aspuru-Guzik and co-workers2,19 developed a methodology for testing hypotheses in prebiotic chemistry. Rules based on formal bond orders and heuristic functions inspired by Hammond’s postulate to estimate activation barriers were applied to model prebiotic scenarios and to determine their uncertainty. Recently, a new algorithm for the discovery of elementary reaction steps was published60 that uses freezing string61 and Berny optimization62–64 methods to explore new reaction pathways of organic single-molecule systems. While graph-based descriptors perform well for many organic molecules, they may fail for transition metal complexes, where the chemical bond is not always well defined.65 Complementary to the approaches described above, we aim at a less context-driven method to be applied to an example of transition metal catalysis. Clearly, such an approach must be based on information directly extracted from the electronic wave function so that no additional ad hoc assumptions on a particular class of molecules are required. In the first step of our heuristics-guided approach, we identify reactive sites in the chemical system. When two reactive sites are brought into close proximity, a chemical bond between the respective atoms is likely to be formed (possibly after slight activation through structural distortion). In addition, we define reactive species which can attack target species at their reactive sites. This concept is illustrated in Fig. 2.1. A simple example for the first-principles identification of reactive sites is the localization of Lewis base centers in a molecule as attractors for a Lewis acid. Lone pairs are an example for such Lewis base centers and can be detected by inspection of an electron localization measure such as the electron localization function by Becke and Edgecombe66 or the Laplacian of the electron density as a measure of charge concen-

7 Chapter 2 Computational Chemistry from a Network Perspective

Reactive Species

Reactive Site

Target Species

Figure 2.1: A reactive site of a reactive species approaching the reactive sites of a target species. The color of a reactive site represents the value of some chemical reactivity descriptor. This ﬁgure is reproduced with permission from M. Bergeler, G. N. Simm, J. Proppe, M. Reiher, J. Chem. Theory Comput. 2015, 11, 5712. Copyright 2015 American Chemical Society.

tration.67 Other quantum chemical reactivity indices can also be employed,45 such as Fukui functions,68 partial atomic charges,69,70 or atomic polarizabilities.71,72 With these descriptors, reactive sites can be discriminated, i.e., not every reactive site may be a candidate for every reactive species (indicated by the coloring in Fig. 2.1). For example, an electron-poor site is more likely to react with a nucleophile rather than with an elec- trophile. Moreover, reactive species consisting of more than one atom may have distinct reactive sites. Naturally, the spatial orientation of a reactive species toward a reactive site is important. In the second step, reactive species are added to a target species resulting in a set of candidate structures for reactive complexes. Such compound structures should resemble reactive complexes of high energy (introduced by sufficiently tight structural positioning of the reactants, optionally activated by additional elongation of bonds in the vicinity of reactive sites), which are then (third step) optimized employing electronic structure methods. By means of standard structure optimization techniques73 we search for potential reaction products from the high-energy reactive complexes. Several structure optimizations of distinct candidates may result in the same minimum-energy structure. Such duplicate structures must be identified and discarded to ensure the uniqueness of intermediates in the network (fourth step). It should be noted that each intermediate of a CRN can be considered a reactive species to every other intermediate of that network. Through a structural comparison based on a metric (such as the root-mean-square deviation, RMSD, of atomic positions), pairs of structures which can be interconverted by an elementary reaction, i.e., a single transition state, are identified (fifth step). If no such pair can be found for a certain structure, the local configuration space in the vicinity of that structure needs to be explored further to ensure that no intermediate will be overlooked. In the sixth step, the automatically identified elementary reactions

8 Exploration of Chemical Reaction Networks 2.1 are validated by transition state searches73 and subsequent intrinsic reaction path calculations.74 In the seventh and last step of our heuristics-guided approach, a CRN comprising all determined intermediates and transition states is automatically generated. The visualization of results as network graphs in which vertices and edges represent molecular structures and elementary reactions, respectively, supports understanding a chemical process in atomistic detail. The readability of a network graph can be enhanced if vertices and edges are supplemented by attributes such as colors or shapes chosen with respect to their relative energy or to other physical properties (Fig. 2.2).

+ H H N

N N Mo N N

+ H N H

N N Mo N N

Figure 2.2: Visualization of a CRN. Each vertex represents a chemical species, here, a singly or doubly protonated molybdenum complex ligated with dinitrogen. The color of each vertex represents the equilibrium energy of the corresponding species and ranges from dark blue (very low energy) to dark red (very high energy). Isomeriza- tion reactions (interconverting chemical species that are located on the same PES) are indicated by a solid edge, whereas protonation reactions (leading to products that are located on a different PES) are indicated by a dashed edge. Here, isomerization reactions involve a shift of a proton from one to another nonhydrogen atom. Low- energy and high-energy transition states of isomerization reactions are indicated by dark-gray and light-gray edges, respectively. This ﬁgure is reproduced with permission from M. Bergeler, G. N. Simm, J. Proppe, M. Reiher, J. Chem. Theory Comput. 2015, 11, 5712. Copyright 2015 American Chemical Society.

Even though our heuristics-guided approach aims at restricting the number of possible minimum-energy structures, the number of generated intermediates may still be exhaus- tively large as the following example illustrates. For a protonation reaction (Fig. 2.2), we may assume that the number of diﬀerent protonated intermediates can be determined from the unprotonated target species by identifying all reactive sites which a proton, the reactive species, can attack. This number is given by a sum of binomial coeﬃcients,

9 Chapter 2 Computational Chemistry from a Network Perspective

n ( ) ∑RS n N = RS = 2nRS − 1, (2.1) p p=1

where nRS is the number of reactive sites and p is the number of protons added to the target species. Even for such a simple example, the number of possible intermediates increases exponentially. For example, for a target species with ten reactive sites, N = 1023 intermediates will be generated. Obviously, the transfer of several protons to a single target species is not very likely from a physical point of view as charge will increase so that the acidity of the protonated species might not allow for further protonation. In the presence of a reductant, however, these species can become accessible in reduced form. We applied this protonation–reduction scheme20 for studying Chatt- like catalytic nitrogen fixation under ambient conditions at the example of Schrock’s molybdenum complex (Fig. 2.2).75–77 This catalyst, like all others developed for this purpose,78–80 is plagued by a very low turnover number. Of all chemical species generated by the application of heuristic rules, some will be kinetically inaccessible under certain physical conditions. By defining reaction conditions (in general, a temperature T ) and a characteristic time scale of the reaction under consideration, one can identify those species that are not important for the evolution of the reactive system under these conditions. Even if these intermediates are thermodynamically favored, they may not be populated on the characteristic time scale at temperature T . By removing these species from the network, one can largely reduce its complexity, which in turn simplifies subsequent analyses (such as kinetic modeling, cf. Chapter 4). For the following discussion, we introduce the notation that a CRN is to be understood as a connected graph built from a set of intermediates (vertices) and a set of elementary reactions (edges). A path shall denote a directed sequence of alternating vertices and edges, both of which occur only once. A subnetwork is a connected subgraph of a CRN uniquely representing a single PES defined by the number and type of atomic nuclei, the number of electrons, and by the electronic spin state. Subnetworks can be related to each other according to the heuristic rules which describe addition or removal of reactive species (defined by their nuclear framework, i.e., by their nuclear attraction potential and charge) and electrons. Substrates are species that represent the reactants of a complex chemical reaction. The initial population of all other target species is zero. For the exclusion of nonsubstrate vertices from the CRN, we propose a generic energy cutoff rule: If each path from a substrate vertex to a nonsubstrate vertex comprises at least one sequence of consecutive vertices with an increase in energy larger than a cutoff

EC, then we remove the nonsubstrate vertex from the network.

10 Exploration of Chemical Reaction Networks 2.1

3 2 1

≜ EC

Energy 7 0 5

6 Cutoﬀ 4 Rule

Initial Set Accessible Set

Figure 2.3: Illustration of the process of removing intermediates (shown as vertices) from a CRN by applying the energy cutoff EC. The vertex representing the substrate is blue-colored, vertices to be removed are red-colored. This ﬁgure is reproduced with permission from M. Bergeler, G. N. Simm, J. Proppe, M. Reiher, J. Chem. Theory Comput. 2015, 11, 5712. Copyright 2015 American Chemical Society.

The application of this energy cutoﬀ rule is illustrated in Fig. 2.3. Starting from substrate 0, intermediate 1 can only be reached via a transition state higher than EC, and therefore, it can be removed from the network. Since intermediates 2 and 3 can only be reached via intermediate 1, they can also be omitted. Despite being similar in energy to substrate 0, intermediate 5 can be discarded, since it can only be formed by a transition state higher than EC. Even though the transition state between intermediates

6 and 7 is below EC, the population of intermediate 7 is negligible, since, starting from substrate 0, it can only be formed through intermediate 4. Note that this energy cutoff rule is conservative as we compare energy differences of stable intermediates, which are a lower bound for activation energies of reactions from a low-energy intermediate to one that is higher in energy. Therefore, intermediates can be removed prior to the calculation of transition state structures, which significantly saves computational resources. Once transition states are calculated, this rule can be reapplied to further reduce the complexity of the CRN in order to arrive at a minimal network of all relevant reaction steps.

The introduced kinetic cutoﬀ EC depends on temperature T and on the characteristic time scale of the reaction. For instance, assuming a reactive system following Eyring’s quasi-equilibrium argument,81 one can determine the average time for a unimolecular reaction to occur. For this purpose, the Eyring equation (2.2) is employed to obtain the unimolecular decay rate constant, k. We understand the half life ln(2)/k as the time after which a molecule has reacted with a probability of 50%. For an activation free energy of 25 kcal mol−1 and a temperature of T = 298 K, the average time for a unimolecular reaction to occur equals three days. This time may well be considered an upper limit

11 Chapter 2 Computational Chemistry from a Network Perspective

for a practical chemical reaction. If one can aﬀord longer reaction times, the energy cutoﬀ needs to be increased. Similarly, if one is interested in a range of temperatures,

∆T = Tmax − Tmin, the energy cutoﬀ has to be adapted to the maximum temperature,

Tmax. Otherwise, intermediates would be removed from the CRN which are accessible at

Tmax. In a conservative exploration, a reasonable choice for the maximum temperature may be the decomposition temperature of an important compound class studied. Special attention needs to be paid to the energy differences between intermediates of different subnetworks, since our protocol divides the PES of the chemical system into various subsystem PES’s. For instance, if two intermediate structures differ by one reactive species, say a proton, the energy for supplying that reactive species by a strong acid has to be taken into account. Otherwise, different subnetworks of a CRN cannot be compared as the total energies to be compared depend on the number of (elementary) particles. In Chapter 4, we present more sophisticated algorithms based on kinetic modeling to further reduce the number of vertices and edges in a CRN. Such network reduction algorithms can guide and accelerate the computationally expensive first-principles exploration of chemical reaction space as described above.

2.2 Determination of Rate Constants from First Principles

A CRN of all relevant intermediates and transition states of a chemical process sets the frame to study population trajectories through the network. In solution chemistry, typically trajectories of molar concentrations are studied, which depend on several conditions such as the initial feed of reactants and temperature. While the correlation of these conditions with the product distribution can be determined quite straightfor- wardly by a suitable experimental setup, it remains a challenge to analyze why a certain product distribution was found. To resolve this issue, studying the kinetics of a chemical process is inevitable. Only then, intermediates relevant for the product distribution but not contained in it can be identiﬁed. This time-resolved picture would allow us to develop strategies to support the formation of a desired product or to suppress the formation of unintended side products. As experimental kinetics can only examine a limited number of chemical species, thorough theoretical kinetic models corresponding to complex CRNs spanning several time scales are desired. For the construction of a general-purpose kinetic model based on mass action, rate constants are the essential elements to be determined. Conventional transition state theory (TST) provides a simple approach to calculate rate constants for isothermal reactions. In conventional TST, it is assumed that a reaction coordinate along a Born–Oppenheimer PES is intersected orthogonally by a hyperplane in such a way that once crossed by a trajectory starting from a reactant

12 Determination of Rate Constants from First Principles 2.2 state, that trajectory ends in the corresponding product state.82,83 This crossing point is approximated by the first-order saddle point of a reaction coordinate. Given a canonical ensemble of microstates, for which the number of molecules Nm, the temperature T , and the volume V are constant, the thermal rate constant k(T ) of a reaction from a reactant to a product crossing the corresponding transition state depends on the Helmholtz free- energy difference between reactant and transition state, ∆A‡,∗, through an exponential function,81 { } ‡,∗ kBT ∆A k(T ) = exp − , (2.2) h RT where kB is the Boltzmann constant, R the ideal gas constant, and h the Planck constant. Eq. (2.2) is referred to as Eyring equation. Here, we refer to a standard state 23 of Nm ≈ 6.022 × 10 and V = 1 L (indicated by a superscript asterisk to the free energy). It is a known problem that conventional TST cannot (a) ensure recrossing-free trajectories through the approximated dividing hyperplane (overestimation of rate constants) and (b) account for quantum effects such as tunneling (underestimation of rate constants). Both phenomena can be accounted for in conventional TST by introduc- ing a fudge factor, κ, to the right-hand side of Eq. (2.2). Extended approaches such as variational TST84 and quantum TST85 provide ways to circumvent these problems, but require much more information on the PES than its low-order stationary points. However, it was shown that conventional TST works surprisingly well even for large molecules such as enzymes.86,87 To ensure the subsequent kinetic analysis of the CRN to be reliable, we require our set of rate constants to be equipped with an uncertainty measure. Only this way, we can assess the validity of the output of a kinetic simulation, e.g., a product ratio or a proposed reaction mechanism. The only quantity we need to determine for the construction of a kinetic model based on TST is the Helmholtz free energy, A, for all intermediates and transition states contained in a given CRN. In our case, A is determined by the canonical partition function Q = Q(Nm,V,T ) through A(T,Q) = −kBT lnQ, where all energy states of the system of Nm molecules enter Q. The Helmholtz free energy is composed of the electronic energy at 0 K and the zero-point vibrational energy as well as thermal energetic and entropic contributions due to translational, rotational, vibrational, and electronic motion (see Table 6.1 for a comprehensive list of thermochemical contributions to free energies). For the inference of a comprehensive uncertainty measure, it may be necessary to determine all systematic errors in these quantities introduced by the computational models employed for their calculation. Recently, Simm and Reiher88 showed how to estimate systematic errors in electronic energy differences to assess the reliability of reaction mechanisms. The procedure is based on sampling from a probability density function (PDF) corresponding to a pa-

13 Chapter 2 Computational Chemistry from a Network Perspective

rameter in the LC∗-PBE0 density functional, from which sets of electronic energy differences are calculated. These sets can be transformed into sets of rate constants through Eq. (2.2), each of which yields a unique set of concentration profiles (concentration–time plots) for the species of the CRN. This procedure was also applied to the formose reaction studied here (see Section 2.3.2 and Ref. 35 for more details). While an uncertainty measure for electronic energies may suffice if electronic effects dominate the overall energetics of the underlying systems process, a more general protocol would consider systematic errors in all contributions to the Helmholtz free energy. Furthermore, the effect of the specific model employed for the calculation of rate constants (here, based on conventional TST) on the output of a kinetic simulation remains to be studied.

2.3 Kinetic Modeling of Reactive Chemical Systems

A detailed understanding of reactive chemical systems on arbitrary time scales would support the optimization of chemical processes through directed manipulations of pro- moting and interfering factors during the course of a reaction. For this purpose, we need to uncover the kinetic principles of a general (complex and noisy) CRN from a ﬁrst-principles perspective. Our objective is to model the kinetics of complex chemical processes on a given time scale with rigorous uncertainty control. Such a procedure will ultimately allow us to automatically deduce product distributions, reaction mechanisms, and other network properties from noisy quantum chemical data, and to assess their reliability.

For the construction of an elementary kinetic model, activation free energies from first-principles calculations are required. From the rate constants calculated by, e.g., Eq. (2.2), differential equations describing the time propagation of population densities (here, molar concentrations) of all chemical species can be constructed. By integrating these differential equations, the underlying systems process can be modeled. Since differential equations describing chemical processes are generally coupled, analytical integration becomes rapidly impossible. Explicit numerical integration is the standard method of choice for solving a system of differential equations. However, if the underlying systems process spans multiple time scales differing by several orders of magnitude — systems featuring such a spread in time scales are said to be stiff —, explicit numerical integration becomes highly inefficient.89 For this purpose, a variety of approaches were designed concerning both the implicit numerical intergration (as discussed in Chapter 4) of a system of differential equations and the simplification of kinetic models (as discussed in Section A.1).90

14 Kinetic Modeling of Reactive Chemical Systems 2.3

2.3.1 Network Structure and Properties

We describe the structure of a CRN by a graph of N vertices and 2L unidirectional edges. As we assume every chemical transformation to be reversible, the graph is strictly bidirectional, which explains the even number of edges. Either of both edges corresponding to a reaction pair is assigned an arbitrary but unique direction (forward or backward). We deﬁne the N-dimensional column vector of time-dependent species concentrations,

( )⊤ y(t) ≡ y = y1, ··· , yN , (2.3) which keeps track of the population density of each vertex at a given time. Here, yn refers to the concentration of the n-th chemical species. The 2L-dimensional column vector of rate constants, ( ) ( ) k+ ⊤ k = = k+, ··· , k+, k−, ··· , k− , (2.4) k− 1 L 1 L contains scaling factors that determine the transition rate for each unidirectional edge. We deﬁne the (N × L)-dimensional stoichiometry matrix of forward reactions,   S+ ··· S+  11 1L  +  . . .  S =  . .. .  , (2.5) + ··· + SN1 SNL

+ ∈ { } ∈ { } where the element Snl (n 1, ..., N and l 1, ..., L ) describes the number of molecules of the n-th species that is consumed in the l-th forward reaction. Analogously to S+, we deﬁne the (N × L)-dimensional stoichiometry matrix of backward reactions,   S− ··· S−  11 1L  −  . . .  S =  . .. .  . (2.6) − ··· − SN1 SNL

The combined (N × L)-dimensional stoichiometry matrix (of reaction pairs) reads

S = S− − S+ . (2.7)

From the quantities introduced above and assuming mass action kinetics, we can express the L-dimensional column vector of forward reaction rates as

( )⊤ + + ··· + f = f1 , , fL , (2.8)

15 Chapter 2 Computational Chemistry from a Network Perspective

with components N ∏ + + + Snl fl = kl yn . (2.9) n=1

∏ + N Snl By deﬁnition, the product n=1 yn will only be zero if a species involved in the l-th + forward reaction (indicated by a positive value of Snl) has a zero concentration, since + + Snl for all species not involved (Snl = 0), yn = 1 (regardless of the unit of measurement). Analogously, the L-dimensional column vector of backward reaction rates is deﬁned as

( )⊤ − − ··· − f = f1 , , fL , (2.10)

with components ∏N − − − Snl fl = kl yn . (2.11) n=1 From the stoichiometry matrix, S, and the L-dimensional column vector of (paired) reaction rates, f = f + − f − , (2.12)

we can express the time-dependent change in the species concentrations as ( ) ⊤ d g = g1, ··· , g ≡ y = S · f , (2.13) N dt which is an N-dimensional column vector, and allows us to construct the (N × N)- dimensional Jacobian matrix, J, with elements

∂ Jij = gi . (2.14) ∂yj

To keep track of the concentration that has been transported by every edge from + − t = 0 to t = tmax, one integrates the corresponding reaction rates f and f over that time interval, ∫ t=tmax F+ = f +(t) dt , (2.15) t=0 ∫ t=tmax F− = f −(t) dt . (2.16) t=0 Here, F+ and F− refer to fluxes of forward reactions and backward reactions, respectively. By analogy with reaction fluxes, we can define the N-dimensional column vector of species fluxes as ∫ t=tmax ( ) G = S+ · f −(t) + S− · f +(t) dt = S+ · F− + S− · F+ . (2.17) t=0

16 Kinetic Modeling of Reactive Chemical Systems 2.3

Figure 2.4: Possible mechanism of the ﬁrst steps of the formose reaction (a) and the corresponding network representation (b). This ﬁgure is reproduced from J. Proppe, T. Husch, G. N. Simm, M. Reiher, Faraday Discuss. 2016, 195, 497, with permission from the Royal Society of Chemistry.

Note that S+ is multiplied with F− since in a backward (−) reaction, the reactants or left-hand-side species (+) are formed. Hence, we only consider incoming fluxes, but no outgoing fluxes, since the outgoing flux of a specific vertex equals the incoming flux of all adjacent vertices. An equivalent argument holds for the multiplication of S− with F+.

2.3.2 Results and Discussion

Formose reaction is the collective term for a plethora of possible autocatalytic oligomerization reactions of formaldehyde in aqueous solution.91,92 The reaction aﬀords a highly complex (racemic) mixture of linear and branched monosaccharides (tetroses to octoses), polyols, and several degradation products. Hence, the formose reaction is an example of a large and highly entangled reaction network. The key challenge of this network is the presence of coupled reactions spanning multiple time scales. In Section 2.1, we have shown how such a network can be explored in general. Since the exploration of the formose reaction is beyond the scope of this section, only a subnetwork of the formose reaction is investigated here. The structure coordinates were adapted from Ref. 93 (see the Supporting Information of Ref. 35). The heuristics-guided exploration of the whole formose network has been published recently.21 This subnetwork, which already features many conceptual challenges of the entire formose reaction, is shown in Fig. 2.4. It represents a possible mechanism for the ﬁrst steps of the formose reaction as described by Kua et al.93 and comprises six chemical

17 Chapter 2 Computational Chemistry from a Network Perspective

‡,∗ −1 Table 2.1: Activation free energies ∆A (in kJ mol , with uncertainty estimates σ∆A‡,∗ ) and rate constants − − − k (in s 1 and L mol 1 s 1 for unimolecular and bimolecular reactions, respectively) for the reactions in the formose model network (Fig. 2.4).

‡,∗ Reactant(s) Product(s) ∆A σ∆A‡,∗ k R1 1 2 95.4 4.8 6.7 × 10−3 R2 2 1 124.9 13.2 8.1 × 10−10 R3 1 + 1 3 215.4 14.2 1.2 × 10−25 R4 3 1 + 1 311.1 23.0 1.9 × 10−42 R5 3 4 157.3 11.6 1.7 × 10−15 R6 4 3 130.8 10.2 7.5 × 10−11 R7 3 5 100.3 3.2 9.2 × 10−4 R8 5 3 119.2 12.3 8.0 × 10−9 R9 1 + 4 6 112.5 13.4 1.2 × 10−7 R10 6 1 + 4 185.4 23.1 2.0 × 10−20

species and five reversible reactions (ten elementary reactions Ri). We obtained all free energies in single-point calculations as described in the Supporting Information of Ref. 35. In water, formaldehyde (1) is in equilibrium with its hydrated form, methane- diol (2). 1 dimerizes to glycolaldehyde (3), which is a reaction with a high activation free energy (Table 2.1). The exact mechanism of the dimerization has not been unraveled yet.94–97 From experimental studies it is, however, well known that the dimerization proceeds very slowly. 3 can react with water to 1,1,2-ethanetriol (5). Another possible reaction of 3 is the enolization to 1,2-ethenediol (4). The addition of 1 to 4 yields glyceraldehyde (6). This bimolecular reaction introduces a significant entanglement in the model network. The model network does not capture the autocatalytic nature of the formose reaction, in which 3 can be regenerated from intermediates produced in subsequent reactions. Table 2.1 presents (standard-state Helmholtz) activation free energies (in solution), ‡,∗ ∆A , and the resulting rate constants, k, together with uncertainty estimates, σ∆A‡,∗ , for the reactions in the model network. It can be seen that ∆A‡,∗ is high (above 100 kJ mol−1) for most reactions. In addition, most reactions have estimated errors of above 10 kJ mol−1, which reflects the large uncertainty of the respective reaction rates. In Section 4 of Ref. 35, we showed that the LC*-PBE0 functional provides reliable error estimates above 4.2 kJ mol−1. The estimated error for reaction R7 is below that value, and therefore, most likely too small. For the kinetic simulation based on computational singular perturbation (CSP),98,99 we selected an absolute temperature of 298.15 K, a 1 M solution of 1 in water as initial feed, and a time gap criterion of ϵ = 10−3 (for technical details of the kinetic modeling approach employed here, see Section A.1 and the Supporting Information of Ref. 35).

18 Kinetic Modeling of Reactive Chemical Systems 2.3

Figure 2.5: Fast (bottom left) and slow (top right) subnetworks of the CRN shown in Fig. 2.4. This ﬁgure is reproduced from J. Proppe, T. Husch, G. N. Simm, M. Reiher, Faraday Discuss. 2016, 195, 497, with permission from the Royal Society of Chemistry.

For every set of activation free energies, it was found that all reaction pairs except (R3, R4) — the dimerization of 1 to 3 and the reverse reaction — contribute to the fast processes. Therefore, only reaction pair (R3, R4) constitutes the slow subnetwork (Fig. 2.5). The concentration profiles obtained from kinetic simulations of the formose model CRN are shown in Fig. 2.6. The red curves correspond to the profiles obtained from the activation free energies listed in Table 2.1. The black curves correspond to the profiles obtained from the activation free energies calculated from the ensemble of density functionals generated by our error estimation scheme (cf. Section 4 of Ref. 35). The simulated time scale of the global process exceeds the age of the universe in each case. This finding should not be interpreted in absolute terms, but it indicates that the uncatalyzed thermal formose reaction is very unlikely to occur if one starts from 1 alone, provided that the activation free energies and their estimated uncertainties are reliable. It should be noted that 3 is autocatalytically regenerated in the formose reaction, which is not considered in our model network. This way, reaction R3 can be circumvented, which would most likely lead to an acceleration of the overall process. The concentration profiles show clearly how sensitive rate constants may be to varia- tions in activation free energies. For instance, the variation in time of the concentration profiles of 2 spans almost 23 orders of magnitude (8.7 × 1022 at an arbitrarily chosen −1 concentration of y2 = 0.01 mol L ). Since only reactions R3 and R4 contribute to the time resolution of the chemical process, uncertainties in the corresponding activation free energies need to be responsible for this significant variation. In Table 2.2, properties of the fastest and slowest concentration trajectories (Fig. 2.6, species 2, left-most and right-most curves) are compared. For reaction R3, the activation free energy spans a range of about 60 kJ mol−1, and for reaction R4, this range is about 100 kJ mol−1, leading to a deviation in rate constants of about 10 and 17 orders of magnitude, respec-

19 Chapter 2 Computational Chemistry from a Network Perspective

Figure 2.6: Concentration profiles with respect to time for chemical species 1−6 according to the CRN shown in Fig. 2.4. The profiles resulting from the activation free energies listed in Table 2.1 are shown in red. The other profiles (black) result from activation free energies calculated from the ensemble of density functionals generated by our error estimation scheme. 35 Note that the time scale of the equilibration process is extremely large, which originates from neglecting relevant intermediates and elementary reactions in our model network. For

readability reasons, all plots start after the first global time step τ1,slow (cf. Section A.1), which depends on the respective set ofi activation free energies, and therefore, the onset of the profiles is different. This figure is reproduced from J. Proppe, T. Husch, G. N. Simm, M. Reiher, Faraday Discuss. 2016, 195, 497, with permission from the Royal Society of Chemistry.

tively. Taking into account the concentrations of 1 and 3 (the constituents of reactions −1 R3 and R4) at our arbitrarily chosen concentration of y2 = 0.01 mol L , the rates of both reactions can be calculated. For both the fastest and slowest concentration trajectories, reaction R3 is much faster than reaction R4. Therefore, we assume only reaction R3 to be relevant for the kinetic simulation. The reaction time can be roughly estimated by the inverse of the current reaction rate. In our case, the reaction time of the slowest concentration trajectory is higher than that of the fastest trajectory by a factor of 1.4 × 1023, which is quite close to the factor of 8.7 × 1022 determined from the concentration data of 2. Obviously, an error of this magnitude with respect to the activation free energy is far too large to quantify concentration profiles in terms of absolute time. Moreover, it should be noted that the error introduced by choosing conventional TST to calculate rate constants is not considered here. Even though the uncertainty in the activation free energies strongly affects absolute time, it does not affect the qualitative trend of the concentrations profiles (Fig. 2.7). This

20 Kinetic Modeling of Reactive Chemical Systems 2.3

‡ ∗ − − − − Table 2.2: Activation free energies ∆A , (in kJ mol 1), rate constants k (in s 1 and L mol 1 s 1 for unimolec- −1 −1 ular and bimolecular reactions, respectively), concentrations y1 and y3 (in mol L ) at y2 = 0.01 mol L , and − − reaction rates r (in mol L 1 s 1) for reactions R3 and R4 (Fig. 2.4) of the fastest and slowest concentration trajectories (Fig. 2.6, species 2, left-most and right-most curves).

‡,∗ ‡,∗ ∆AR3 ∆AR4 kR3 kR4 fastest 183.4 259.3 4.6 × 10−20 2.4 × 10−33 slowest 243.8 357.2 1.2 × 10−30 1.6 × 10−50 2 y1 y3 rR3 = kR3(y1) rR4 = kR4y3 fastest 1.4 × 10−4 2.9 × 10−2 9.3 × 10−28 6.9 × 10−35 slowest 7.4 × 10−11 1.5 × 10−7 6.5 × 10−51 2.5 × 10−57

finding can be explained by the distinct separation of the magnitude of the activation free energies. For instance, the activation free energy for reaction R3 in the slowest case is even lower than that for reaction R4 in the fastest case (Table 2.2). Furthermore, the activation free energies and their uncertainties listed in Table 2.1 show that all reaction barriers are well separated from each other, which does not allow for an alternative reaction mechanism. Clearly, for small activation energy differences, such as found in enantioselective/asymmetric organocatalysis,100,101 large uncertainties would also lead to qualitatively different results. Qualitative validity of our kinetic simulations is also underlined by the fact that in all cases, 5 is the main product at chemical equilibrium. The population dominance of 5 over 3 was also found experimentally by Kua et al.102 However, their calculated Gibbs ‡,∗ − ‡,∗ activation free energies for the corresponding reaction pair (R7, R8) (∆G3→5 ∆G5→3 = 2.5 kJ mol−1)93 are very similar to each other. Their Gibbs activation free energies can be directly compared to our Helmholtz activation free energies, because volume changes are neglected. Our activation free energies for the reaction pair (R7, R8) differ ‡,∗ − ‡,∗ − −1 significantly from each other on average (∆A3→5 ∆A5→3 = 18.9 kJ mol ). A reason for the observed difference is the choice of computational models for the calculations (e.g., different density functionals and solvation models). It might seem surprising that 5 is the main product in our simulation even though 6 is a thermodynamic sink. However, one should keep in mind that the concentration profile of 6 is temporally significantly populated. To understand this finding, we need to discriminate between the fast and slow processes. Considering the fast subnetwork (Fig. 2.5), we understand that there are two uncon- nected channels to form 6, i.e., (1, 2) and (3, 4, 5). This picture is equivalent to a reaction of type A + B ⇌ C, where the initial concentration difference between A and B is conserved over the course of the reaction. It follows that

∆ ≡ (y1 + y2) − (y3 + y4 + y5) (2.18)

21 Chapter 2 Computational Chemistry from a Network Perspective

Figure 2.7: Concentration profiles with respect to reaction progress for chemical species 1−6 according to the CRN in Fig. 2.4. The profiles resulting from the activation free energies listed in Table 2.1 are shown in red. The other profiles (black) result from activation free energies calculated from the ensemble of density functionals generated by our error estimation scheme. 35 This figure is reproduced from J. Proppe, T. Husch, G. N. Simm, M. Reiher, Faraday Discuss. 2016, 195, 497, with permission from the Royal Society of Chemistry.

is the conserved quantity in our case. If one of the two channels is unpopulated, 6 cannot be formed. This case holds in the beginning (dominant population of 1) and in the end (dominant population of 5) of the reaction process. The slow subnetwork (Fig. 2.5) now connects these two channels. Since channel (1, 2) is dominantly populated in the beginning of the reaction process, flux occurs towards channel (3, 4, 5) and, hence, towards 6. The concentration of 6 increases while the magnitude of ∆ decreases. At a certain point in time, approximately when the concentration of channel (3, 4, 5) starts to becoming dominant over that of channel (1, 2), the magnitude of ∆ increases again so that the concentration of 6 decreases. Recall that here, we are studying a small segment of a complex CRN, where 6 can isomerize to more stable intermediates or reacts with 1 to higher sugars. Therefore, the reflux of 6 is most likely an artifact resulting from the particular choice of the network. Another feature of the conservation of ∆ at short time scales is that the kinetic model can be reduced to a single differential equation (see the Supporting Information of Ref. 35). This differential equation can be integrated by any conventional explicit numerical integration algorithm. Here, we chose the standard fourth-order Runge–Kutta algorithm. We compared the result to that of our CSP-type method, where we employ

22 Kinetic Modeling of Reactive Chemical Systems 2.3 an explicit Euler algorithm according to Eq. (A.6), which is the simplest ansatz for numerical integration and known to be unstable due to the lack of an inherent time step selection. However, our CSP-type method provides the time step for the explicit Euler algorithm by continuously analyzing the Jacobian (Section A.1). We emphasize that both approaches to model the kinetics of the network (CSP/Euler vs. Runge–Kutta) yield identical results.

23 Chapter 2 Computational Chemistry from a Network Perspective

24 3 Computational Chemistry from a Statistical Perspective

One of the major challenges in computational science is to determine the uncertainty of a property prediction based on data from computer simulations.*,† As highly accurate first-principles calculations are in general unfeasible for most physical systems, one usually relies on a number of approximations. The effect of such approximations on properties derived from them can generally not be deduced, leading to predictions that are sensitive to systematic errors such as inconsistent reference data, parametric model assumptions, or inadequate computational models. Therefore, it is challenging to quantify the uncertainty of a computational result, which, however, is necessary to assess the suitability of a computational model. In this chapter, we discuss problems and solutions for the performance assessment of computational models. For this purpose, we eluci- date the different sources of uncertainty, the elimination of systematic errors, and the combination of individual uncertainty components to the uncertainty of a prediction.

3.1 Statistical Calibration of Parametric Property Models

Predicting observables from a combination of scientiﬁc (knowledge-based) and statistical (data-based) information is at the heart of any parametric property model applied in computational science.28 In chemical physics, property models are applied whenever it

*This chapter is reproduced in part with permission from J. Proppe, M. Reiher, J. Chem. Theory Comput. 2017, 13, 3297. Copyright 2017 American Chemical Society. †This chapter is reproduced in part with permission from G. N. Simm, J. Proppe, M. Reiher, Chimia 2017, 71, 202. Copyright©Swiss Chemical Society.

25 Chapter 3 Computational Chemistry from a Statistical Perspective

is unfeasible or too demanding to calculate a target observable for a desired range of chemical systems with benchmark methods. To resolve this issue, the target observable is represented by a property model, which is a parametric representation of the former. The statistical variables of a property model (parameters) represent its unknown part, and their optimization (calibration of the model) requires both a reference data set and an objective, e.g., minimization of the mean squared error (MSE). The reference data set comprises pairs of values for (a) the target observable (obtained from measurements or benchmark calculations) and (b) the corresponding input variable. The input variable can represent the target observable itself or another physically motivated variable. It is also possible that the input variable is a vectorial quantity representing, e.g., nuclear coordinates.

The purpose of calibration is the estimation of parameter values that maximize the transferability of a property model to measurements or benchmark results of its target observable not included in the reference data set. In the field-specific literature, there are numerous applications of property models, e.g., for the calibration of force fields,103,104 coarse-grained models for biomolecular simulation,105 exchange–correlation density functionals,88,106 dispersion-corrected potentials for density functional theory (DFT),107 semi-empirical electronic structure models,108 vibrational frequencies,109–111 kinetic models,31,112,113 ionization potentials,114 thermochemical properties,115 properties of semi-conductors and insulators,116 linear free-energy relationships,117 or melting point models,118 to name only a few.

Concomitant with the maximization of transferability is the assessment of model prediction uncertainty (MPU), i.e., the expected random deviation of a prediction from a measurement or benchmark result. MPU can be estimated analytically (or at least iteratively) if certain parametric assumptions are made on the population distribution underlying the reference data set. For a continuous variable, the most likely of all other parametric population distributions is the normal distribution, which is parameterized by mean and variance.119 In that case, Bayesian inference is an eﬃcient way to estimate MPU, which we will address in Chapter 5. Generally, the more input–target pairs are included in the reference data set, the more reliable it does represent the underlying population distribution, and the less ambiguous is the selection of an adequate property model. Consequently, MPU estimation becomes increasingly reliable for an increasing number of data points given a reasonable property model and a speciﬁc input domain (the interval of the input variable studied). However, in the usual case of a limited number of data points, MPU estimation is error-prone and requires a thorough analysis of parametric population assumptions. Moreover, the property model under consideration may be inadequate such that systematic deviations of predictions from measurements or benchmark results are observed.30,116,120,121 Another source of systematic errors are

26 Uncertainty Classification 3.2 inconsistent data (e.g., outliers).30,122,123 In Section 3.2, the diﬀerent sources of uncertainty we are faced with in the calibration of parametric property models are discussed in greater detail. Subsequently, in Section 3.3, we recapitulate nonparametric bootstrapping introduced by Efron,124 a statistical inference method that meets the challenge of unknown population distributions by sampling from available data. This approach produces an empirical population distribution, where the reference data set itself represents the underlying population. By drawing samples from the empirical population distribution, uncertainty in the parameters can be inferred, which is an essential part in MPU estimation, and hence, for the assessment of transferability.

3.2 Uncertainty Classification

One usually distinguishes between three major sources of uncertainty given the reference data at hand are reliable: parameter uncertainty, numerical uncertainty, and model inadequacy.30 Except for stochastic and chaotic models, we expect that numerical uncertainty can be systematically controlled such that its contribution to the overall uncertainty is negligible.125–127

3.2.1 Parameter Uncertainty

For the prediction of properties of chemical systems not included in the training of a computational model, one needs to estimate the uncertainty of its parameters in addition to their “best” values. Otherwise, one would neglect a (potentially essential) component in determining the prediction uncertainty of a computational model. Pa- rameter uncertainty is a result of noise and systematic errors in both the reference data and the computational model under consideration (cf. Section 3.2.2), in particular if the number of reference data is small. Only for a large number of data and a given domain of application (e.g., a speciﬁc region of chemical space), parameter uncertainty becomes negligible. Parameter uncertainty can be estimated in many ways, e.g., through Bayesian in- ference119 (cf. Section B.1.1) or through sampling methods (see below). Due to their conceptual simplicity, we focus on the latter family of methods to demonstrate the eﬀect of parameter uncertainty on MPU. To discuss the general concepts of sampling methods, we focus on the common case of a single target observable, y, linked to a scalar input variable, x, or to a vectorial input variable, x. In the following, we choose the more ⊤ general notation x = (x0, x1, ...) . Given a reference data set, D, comprising N data points (D ≡ {(xn, yn)} with n ∈ {1, ..., N}) we would like to learn predictions of the

27 Chapter 3 Computational Chemistry from a Statistical Perspective

target observable by calibration of the underlying property model, f(x, w) ≈ y, where ⊤ w = (w0, w1, ...) is the vector of parameters.

The MSE and the coeﬃcient of determination, R2, are determined with respect to a reference data set, D, and represent common performance measures of a property model, ∑N ( ) 1 2 MSE ≡ MSED = y − f(x , w) (3.1) ,w N n n n=1 and 2 2 minw(MSED,w) R ≡ RD = 1 − N ∑ , (3.2) N − 2 n=1(yn y¯)

where y¯ is the arithmetic mean of the target values, {yn}, and w is the vector of param-

eters. Minimizing the MSE with respect to the parameters, ∂MSED,w/∂wm = 0 ∀m,

is equivalent to the method of least squares and yields minw(MSED,w). In the following, we refer to the corresponding parameter vector as wD, and the shorthand notation

MSED ≡ minw(MSED,w) implies the least-squares objective. It may be important to distinguish the coeﬃcient of determination, R2, from the squared coeﬃcient of correlation, r2, the latter being independent of the parametric model. Only in some cases (such as linear least-squares regression with a single input variable), both quantities are equivalent.

2 MSED and R are established measures of the model performance conditioned on D. However, in terms of transferability, we would like to know the model performance independently of a speciﬁc data set. In the following thought experiment, we assume that we can generate an asymptotically large number, B, of new (training) samples, D∗ ≡ {(x∗ , y∗ )} with n ∈ {1, ..., N} and b ∈ {1, ..., B}, where the asterisk means b nb nb “drawn from the population distribution underlying D”. If the data points are randomly drawn from a smooth population distribution, we can safely assume that all samples have no data in common (the importance of independent samples for MPU estimation D∗ will be discussed in Section 3.3.1). For every training sample, b , we learn the least- ∗ ∗ squares parameters, wb , of the corresponding property model, f(x, wb ), and evaluate its deviation from the target values of the reference sample, D,

∑N ( ) 1 ∗ 2 MSED,w∗ = yn − f(xn, w ) ≥ MSED . (3.3) b N b n=1

{ } D { ∗ } In Eq. (3.3), the target values yn are elements of , whereas the predictions f(xn, wb ) D∗ D E have been learned from b . To estimate the -independent model performance, [MSE], D∗ we average over all training samples, b ,

28 Uncertainty Classification 3.2

1 ∑B E[MSE] = MSED,w∗ ≥ MSED , (3.4) B b b=1 ∗ ∀ where the equality only holds in the artiﬁcially ideal case of f(xn, wb ) = f(xn, wD) n, b. Compared to the MSED, the E[MSE] additionally incorporates uncertainty in the parameters. Note that under the assumption of normally distributed data with respect to a parametric model, E[MSE] can be calculated analytically, i.e., sampling is not required in that case. Here, however, we examine the implications of such a critical assumption, which is why we explicitly refrain from considering a parametric population distribution underlying D.

3.2.2 Model Inadequacy

An inadequate computational model is not able to reproduce reference data within their uncertainty range,30 i.e., the model under- or overestimates the uncertainty of the reference data. Underestimating prediction uncertainty is a result of overfitting, where the computational model is too flexible (features too many parameters) such that it does not only fit the explainable part of the reference data (the underlying physics), but also its unexplainable part (noise). By contrast, underfitting is caused by models which are too rigid (possess too few parameters) to fit the explainable part of the reference data, leading to overestimation of prediction uncertainty. Moreover, from a results perspective, model inadequacy can be divided into a systematic and a quasi-random part,116 which is illustrated in Fig. 3.1. While both types of model inadequacy can be reduced by an internal correction of the computational model itself, an external correction of the results produced with a computational model can only eliminate the systematic part of model inadequacy, but associated with considerably lower effort. The simplest external corrections are linear functions, which are applied in the prediction of, e.g., vibrational frequencies109–111 or Mössbauer isomer shifts.38,128–132 A drawback of the external correction approach is the loss of transferability to other observables since the correction is applied to an expectation value of a specific observable instead of its underlying wave function, which is the unique common physical ground of all observables. Another efficient but complementary approach to the correction of models and results is parameter uncertainty inflation (PUI).121 PUI ensures that the covariance matrices of the model parameters are transferable to any model comprising the same parameters. Increasing parameter uncertainty is straightforward as it only requires modification of the unknown part (parameter distributions) of an otherwise known model. However, the resulting predictions may not reflect the correct dependence on the input variable(s),

29 Chapter 3 Computational Chemistry from a Statistical Perspective benchmark result

result of computational model

Figure 3.1: Illustration of systematic and quasi-random model inadequacy for synthetic data. For an adequate computational model, the data would scatter around the dashed line of origin. Here, however, the results of the computational model reveal a nonconstant deviation from the benchmark results (obtained from measurements or highly accurate calculations), which is an indication of systematic model inadequacy. An a posteriori correction of the computational model can be realized by ﬁtting a linear calibration function to the data (solid line). The scatter of data around the calibration line appears to be random, but the residuals are on average significantly larger than the uncertainty in the benchmark results (indicated by error bars). We refer to this effect as quasi-random model inadequacy, which implies that the uncertainty of the prediction (represented by the yellow prediction band) does not match with the uncertainty of the benchmark result. Here, the error bars are obviously narrower than the prediction band. This ﬁgure is reproduced with permission from G. N. Simm, J. Proppe, M. Reiher, Chimia 2017, 71, 202. Copyright©Swiss Chemical Society.

which is determined by the sensitivity coeﬃcients of the model (the partial derivatives of a model prediction at a certain point in input space with respect to the model parameters at their expected values). Pernot referred to this issue as the PUI fallacy 121 and illustrated it with three examples: (i) linear scaling of harmonic vibrational frequencies,111 (ii) calibration of the mBEEF density functional against heats of formation,106,133 and (iii) inference of Lennard–Jones parameters for predicting temperature-dependent vis- cosities based on a Chapman–Enskog model.103 In these cases, PUI resulted in correct average prediction uncertainties, but uncertainties of individual predictions were systematically under- or overestimated. Furthermore, PUI does not resolve the issue of model inadequacy per se. For instance, in multiscale modeling, where the target observable is built on a hierarchy of other observables with shorter time and/or length scales, all uncertainties inferred at low levels (short time/length scales) will propagate to the ﬁnal prediction uncertainty (cf. Section 2.3). Consequently, increasing parameter uncertainty at low levels can lead to a prediction uncertainty so large that no sensible conclusions can be drawn about the underlying problem.

30 Prediction Uncertainty from Resampling Methods 3.3

3.3 Prediction Uncertainty from Resampling Methods

Since statistically rigorous calibration of physicochemical property models is an important objective of this dissertation, we provide a concise review of the relevant concepts and notation needed for this purpose. Here, we focus on resampling methods. A Bayesian perspective on this topic is provided in Section B.1.1.

3.3.1 Nonparametric Bootstrapping

The bootstrapping class of sampling methods by Efron124 has been continuously developed.134,135 In nonparametric bootstrapping, the reference sample, D, itself acts as population and, hence, new samples are drawn from D. Consequently, given N input– −1 target pairs in D, these pairs are drawn with equal probability, p(xn, yn) = N ∀n. This procedure of sampling from available data is referred to as resampling. The term nonparametric refers to the exclusion of a parametric population distribution. We drop this term in the following and bootstrapping implies, unless otherwise mentioned, its (original) nonparametric variant. D∗ D To generate a bootstrap sample, b , N elements are drawn from with replacement. This procedure is repeated B times, say B = 1,000. There exist two variants of bootstrapping if the reference data set is composed of input–target pairs.134,135 The ﬁrst variant is independent of the property model under consideration and referred to as pair resampling. In that case, the input–target pairs themselves are drawn with replacement, D∗ ≡ { ∗} ∈ { } i.e., the bootstrap samples are constructed as b (xn, yn)b with b 1, ..., B . The second variant requires a precalibrated property model conditioned on the reference data set, D. The resulting residuals, {en = yn − f(xn, wD)}, are then subject to resampling, which is referred to as residual resampling. Since independent and identically distributed residuals are assumed in that case, they have the same probability to occur anywhere along the input domain for which data is available. Consequently, the input values,

{xn}, are ﬁxed in residual resampling, whereas the residuals, {en}, are randomly drawn with replacement and added to the precalibrated property model, {f(xn, wD)}. Hence, ∗ ∗ the corresponding bootstrap samples are constructed as D ≡ {(x , f(x , wD) + e )}. b n n nb Here, we will exclusively apply pair resampling since calibration prior to bootstrapping imposes further parametric assumptions. Bootstrapping allows us to sample mean and (co)variance of the parameters contained in a calibration model (and of arbitrary other statistics) without relying on parametric population assumptions. Mean and covariance of w read

1 ∑B w¯ ∗ = w∗ (3.5) B b b=1

31 Chapter 3 Computational Chemistry from a Statistical Perspective

and ∑B 2 1 ∗ ∗ ∗ ∗ ⊤ σ ∗ = (w − w¯ )(w − w¯ ) , (3.6) w B − 1 b b b=1 respectively. The model performance can be estimated from bootstrapping according to Eq. (3.4), where the square root of E[MSE] represents an estimate of the MPU. Note that this bootstrapped variant of the E[MSE] is diﬀerent from that derived in the thought experiment introduced in Section 3.2.1. Here, the reference data set, D, and the training data D∗ sets, b , have many data points in common. This overlap promotes underestimation of MPU, but there exist straightforward corrections to the bootstrapped E[MSE], for instance, the .632 estimator,136,137

1 ∑N E[MSE] = 0.368 MSED + 0.632 E[MSE]− , (3.7) .632 N n n=1

where E[MSE]−n refers to the E[MSE] with respect to the complete reference data set minus the n-th data point, ∑ ( ) E 1 − ∗ 2 [MSE]−n = yn f(xn, wb ) . (3.8) |B−n| b∈B−n

Here, B−n and |B−n| represent the set and number of bootstrap samples not comprising the n-th input–target pair, respectively. The constant 0.632 ≈ 1 − e−1 relates to the probability of a data point to be included in a bootstrap sample.137 In overﬁtting situations, the .632 estimator can be biased. In such a case, the constant 0.632 requires correction, resulting in the improved .632+ estimator.137,138 Another resampling method that is most popular with respect to MPU estimation is referred to as cross-validation.137 In k-fold cross-validation, one splits the reference data set into k subsets of most similar size. The model is trained on k − 1 subsets and validated with respect to the remaining one. This procedure can be performed in k distinct ways. Leave-one-out cross-validation is a frequently applied variant of k- fold cross-validation where k = N. The corresponding measure of model performance,

E[MSE]LOO, reads

∑N ( ) 1 2 E[MSE] = y − f(x , wD ) , (3.9) LOO N n n −n n=1

where wD−n refers to the least-squares estimate of w with the n-th data point removed from the reference data set. Since cross-validation usually deals with a limited number of 2 ≤ k ≤ N training samples, it is, in general, computationally more eﬃcient than

32 Prediction Uncertainty from Resampling Methods 3.3 bootstrapping. One of the advantages of bootstrapping over cross-validation is, however, the direct assessment of variability in the regression parameters. Furthermore, cross- validation may suffer from a poor bias–variance tradeoff.137 This issue relates to the observations that (a) for large values of k (in particular for k = N), the variance of performance measures is overestimated, whereas (b) for small values of k (in particular for k = 2), the expected value of performance measures is biased. Both effects may result in significant misestimates of E[MSE]. In bootstrapping, this issue is resolved by the .632 and .632+ estimators. We would like to highlight that the sampled parameter distributions resemble empirical variants of posterior distributions employed in Bayesian inference.139,140 The latter approach allows for a direct estimation of MPU, which bypasses the need for corrections. For the inference of arbitrary posterior distributions, a Bayesian approach is generally much more involved than bootstrapping. However, if Gaussian posterior distributions are enforced (by choosing Gaussian prior distributions and a Gaussian likelihood function), the MPU can be estimated efficiently through Bayesian inference,119 which we will discuss in the next subsection. In Chapter 5, we will compare the .632 bootstrap estimate of MPU to that obtained from Bayesian inference, assuming normality of the population distribution in the latter case. Bootstrapping is an appealing alternative to statistical methods that resort to parametric assumptions on population distributions. On the one hand, modern general- purpose computers can generate and analyze thousands of bootstrap samples in a few seconds given a light fitting problem, i.e., one where calibration is not the limiting step such as in linear least-squares regression. On the other hand, it is an objective approach as it allows for inferring statistical quantities solely from available information (the reference data set). Clearly, if the available data is biased in a way that it badly represents the true underlying population distribution, application of bootstrapping or any other statistical method is not sensible. Major sources of data bias are small sample sizes134 and gross outliers. For the latter exist established detection methods.137 Strictly speaking, even bootstrapping builds upon a population assumption, i.e., the reference data set itself being the population. To express it in Chernick’s words, bootstrapping does not mean “getting something for nothing”, but “getting the most from the little that is available”.134

3.3.2 Prediction Uncertainty of Linear Regression Models

Linear regression refers to a class of calibration procedures, where a target observable, y, is estimated to be a linear combination, f(x, w), of M input variables, {xm}

(m ∈ {1, ..., M}), and M + 1 parameters, {wm} (m ∈ {0, 1, ..., M}, where the zero-index refers to the intercept, which is quasi-multiplied by x0 = 1),

33 Chapter 3 Computational Chemistry from a Statistical Perspective

∑M ⊤ f(xn, w) = wm(xm)n = xn w . (3.10) m=0

Given a least-squares objective, the parameter vector wD reads

( )− ⊤ 1 ⊤ wD = X X X y , (3.11)

119 ⊤ where the so-called design matrix X = (x1, ..., xN ) contains all instances of the ⊤ input vector x contained in D, and y = (y1, ..., yN ) is the vector of target values.

Even though linear least-squares regression is a well-established approach with eligi- bility for many applications, it implies certain critical assumptions:134 (a) independent

and identically distributed residuals, {en = yn − f(xn, wD)}, (b) ﬁnite variance of residuals, and (c) variance-free input variables. Only if these assumptions are valid, the least-squares approach yields the best linear unbiased estimate of regression parameters. For instance, violation of assumption (c) is a ubiquitous phenomenon in the calibration of property models. It is a central topic of this thesis and will be discussed in detail in Chapter 5. Furthermore, if assumption (a) is violated because the variance of the residuals is correlated with one or more input variables, one should instead minimize the weighted MSE, WMSE,141 for calibration,

∑N ( ) 1 −2 2 WMSE ≡ WMSED = u y − f(x , w ) , (3.12) ,w,U N n n n U n=1

where U is a diagonal weight matrix (assumption of independent residuals),   u2 ··· 0  1   . . .  U =  . .. .  . (3.13) ··· 2 0 uN

2 For property models, the elements un usually represent experimental variances. The parameter vector obtained from weighted least-squares regression, wD,U, reads ( ) −1 ⊤ −1 ⊤ −1 wD,U = X U X X U y . (3.14)

This expression is a generalization of the special case covered by Eq. (3.11) where all

elements un are equal, which is why we drop the subscript U in the following (it will be evident from the context to which deﬁnition of U we are referring). For the analysis

34 Prediction Uncertainty from Resampling Methods 3.3 of MPU in the special case of weight equality, it is suﬃcient to consider the average (root-mean-square) experimental uncertainty, v u u ∑N t 1 ⟨u⟩ ≡ u2 . (3.15) N n n=1

In bootstrapped linear least-squares regression, each bootstrap sample yields a parameter vector ( ) −1 ∗ ∗ ⊤ ∗ −1 ∗ ∗ ⊤ ∗ −1 ∗ wb = (Xb ) (Ub ) Xb (Xb ) (Ub ) yb , (3.16) where the asterisk indicates that X, y, and U are aﬀected by the resampling procedure. Properly speaking, when explicitly considering uncertainty on the target observable, pair resampling is replaced by triple resampling where bootstrap samples are constructed as D∗ ≡ { ∗} b (xn, yn, un)b . Locally resolved MPU for a prediction at the input value x0, u(x0), can be estimated from the MSE increased by a factor of N/(N − M − 1) and the 2 116 covariance matrix of the model parameters, σw∗ , introduced in Eq. (3.6), √ N ⊤ 2 u(x ) = MSED ∗ + x σ ∗ x . (3.17) 0 N − M − 1 ,w¯ 0 w 0

2 Note that u (x0) estimates the prediction variance of the target observable at the input 2 value x0 with respect to the average experimental variance, ⟨u⟩ , which would converge to the increased MSE if all systematic errors have been removed (cf. Section B.1.2). 2 For a new series of measurements at input value x0 with experimental variance u0, it 2 − ⟨ ⟩2 is necessary to add the diﬀerence u0 u to the increased MSE in Eq. (3.17). The relation between the E[MSE].632 and the locally resolved MPU is approximately given by 1 ∑N E[MSE] ≈ u2(x ) , (3.18) .632 N n n=1 where the approximation sign mainly arises from the assumption of normality of the parameter distributions as indicated by the second term on the right-hand side of Eq. (3.17). The entire right-hand side of Eq. (3.17) can be replaced by its sampled analog (bootstrapping of prediction intervals135), but for the sake of user-friendliness, we decided to specify a limited number of distinct characteristic values (two for a prediction, w¯ ∗, 2 and three for the corresponding uncertainty, σw∗ ) to facilitate comparisons of diﬀerent computational models (e.g., density functionals) with respect to their performance.

Bayesian linear regression is an alternative to bootstrapped linear least-squares regression. If there is good reason to assume that the parameter distributions are Gaussian, Bayesian linear regression can be much more eﬃcient than sampling-based regression.

35 Chapter 3 Computational Chemistry from a Statistical Perspective

Bayesian linear regression provides analytical posterior distributions of parameters,119

the maxima of which represent the best-ﬁt parameter vector, wMAP (MAP, maximum posterior). The procedure is outlined in Section B.1.1 and yields a measure of model performance referred to as mean predictive variance (MPV),

1 ∑N MPV = s2(x ) , (3.19) N n n=1

2 2 where s (xn) is the analog of u (xn) obtained from bootstrapping. Alternatively, the MPV can be obtained by summing over a dense grid of input values (with a number of grid points ≫ N),116 which yields a smoother result in the sense that it is less dependent on the particular choice of reference input values. In that case, it is important to specify the bounds of the input interval and the spacing between or distribution of grid points.

3.3.3 Jackknife-after-Bootstrapping — Data Diagnostics

So far, the discussion of statistical inference was built on the implicit assumption of data sets representing their underlying population distributions well. Verifying the validity of this assumption is a tedious task, but a diagnostic referred to as jackknife-after- bootstrapping provides a good approximation to the problem.134,142 In the first step of that diagnostic (given a data set with N input–target pairs), the sampled ensemble of { ∗} { ∗} ∈ { } parameters, wb , is decomposed into N different sets, wb −n (n 1, ..., N ), the ∗ n-th of them containing only those parameters wb learned from bootstrap samples in which the n-th input–target pair is not included. For instance, one obtains a bootstrap ∗ estimate of the mean of a parameter with the n-th data point removed, w¯ −n. If this jackknifed mean deviates significantly from that inferred from the complete reference data set, w¯ ∗, we have an indication that the n-th data point biases the calibration. An appealing feature of the jackknife-after-bootstrapping method is its efficiency. Instead of running N extra bootstrap simulations, one only performs a single bootstrap simulation on the complete reference data set. The reason is that some bootstrap samples do not contain certain data points. This way, the decomposition of parameters into N subsets can be performed simultaneously to the bootstrap simulation on the complete reference data set. Since the probability of a data point to be excluded from a bootstrap sample is roughly 0.368 ≈ e−1,137 we recommend to increase the default number of bootstrap samples by a factor of approximately 3 to preserve the intended calibration accuracy.

36 4 Mechanism Deduction from Noisy Chemical Reaction Networks

We introduce KiNetX, a fully automated meta-algorithm for the kinetic simulation and analysis of general (complex and noisy) CRNs with rigorous uncertainty control.*,† It is designed to cope with model-inherent errors in quantum chemical calculations on elementary reaction steps. We developed and implemented KiNetX to possess three features. First, KiNetX identifies and eliminates all kinetically irrelevant species and elementary reactions to simplify subsequent kinetic analyses as well as to guide, and thereby, accelerate the exploration of chemical reaction space. Second, KiNetX prop- agates the correlated uncertainty in the network parameters (activation free energies) obtained from ensembles of quantum chemical models to infer the uncertainty of product distributions and entire reaction mechanisms. Third, KiNetX estimates the sensitivity of species concentrations toward changes in individual activation free energies, which allows us to systematically select the most efficient quantum chemical model for each elementary reaction given a predefined accuracy. For a rigorous analysis of the KiNetX algorithm, we developed a random generator of artificial reaction networks, AutoNetGen, which encodes chemical logic into their underlying graph structure. AutoNetGen allows us to consider a vast number of distinct chemical scenarios which is necessary to investigate the reliability and efficiency of KiNetX in a statistical context.

*This chapter is reproduced from J. Proppe, M. Reiher, 2018, arXiv:1803.09346. †In this chapter, we adopt the notation introduced in Chapter 2.

37 Chapter 4 Mechanism Deduction from Noisy Chemical Reaction Networks

4.1 Kinetic Modeling of Complex Chemical Reaction Networks

In this chapter, we introduce the generalization of the kinetic modeling workflow discussed in Section 2.3 to arbitrary CRNs, which we call KiNetX. Our discussion starts from a CRN for which the initial species concentrations and all activation free energies, including measures for their correlated uncertainty, are supposed to be known. A reaction network endowed with free energy information is the graph representation of a kinetic model, i.e., a system of ordinary differential equations (ODEs) consisting of variables (species concentrations) and parameters (rate constants derived from relative free energies). We require a comprehensive analysis of complex reaction networks to be based on the following steps: (i) translation of CRNs to kinetic models (systems of ODEs), (ii) numerical integration of (possibly stiff) kinetic models, (iii) simplification of kinetic models (mechanism reduction), (iv) propagation of uncertainty in free-energy differences to uncertainty in time-dependent species concentrations, and (v) sensitivity analysis of species concentrations toward changes in relative free energies.

There exist several kinetic modeling software packages that comprise one or more of these steps. For instance, the Kinetic PreProcessor (KPP) translates a predefined reaction network to a system of ODEs in different programming languages (e.g., For- tran90, C, and Matlab).143,144 Hence, KPP can be coupled to programs that carry out numerical integration and the actual kinetic analysis. Since numerical integration is not restricted to chemical kinetics but is relevant in basically all areas of science, there exists a plethora of computer programs for solving any type of differential equation. Numerical integration of ODE systems is a particularly well-studied field. We refer to Chapter 9.1 in Ref. 90 for a concise overview of ODE solvers employed in the field of chemical kinetics.

One of the most widely applied software packages for comprehensive kinetic modeling is CHEMKIN.145–147 Depending on the version (e.g., CHEMKIN,148 CHEMKIN- PRO,149 or Reaction Workbench150), some or even all of the ﬁve steps mentioned above are contained. As the development of CHEMKIN was and is particularly driven by gas phase chemistry, it comprises application-speciﬁc features that take care of, e.g., transport processes or changes in pressure and/or temperature during the course of a reaction. An open-source alternative in gas phase chemistry to CHEMKIN with comparable feature scope is Cantera.151 A particularly appealing feature of Cantera is its interface to several programming languages (Python, Matlab, C++, and Fortran90). Another open-source software package developed for comprehensive kinetic modeling is COPASI (COmplex PAthway SImulator).152 The objective of COPASI is the kinetic study of biochemical networks, but it can also be applied to homogeneous, isothermal, multicomponent chemical reaction systems. COPASI is a stand-alone program suite

38 Kinetic Modeling of Complex Chemical Reaction Networks 4.1 that allows for several kinetic model representations and contains many analysis features. Due to its focus on biochemical systems, where particle numbers are likely to be very small, COPASI contains an implementation for stochastic kinetic simulations153 in addition to its deterministic counterpart (numerical integration of ODEs).

The master equation, which is the fundamental equation for stochastic chemical kinetics, is also crucial in cases where the time scales of reactive events and collisional relaxation compete with each other, such that a nonequilibrium description of state transitions becomes necessary. MESMER (Master Equation Solver for Multi-Energy well Reactions) has been developed for this purpose by Glowacki and colleagues154,155 based on their results for both gas phase156 and solution phase157,158 chemistry. Like other comprehensive kinetic modeling software packages, MESMER automatically de- rives the kinetic model from a given reaction network and solves the corresponding master equation. In terms of mechanism reduction, MESMER is able to transform the set of elementary rate constants to a reduced set of phenomenological rate constants, which can potentially be observed in experiments.

In this chapter, we introduce KiNetX, a meta-algorithm that accomplishes the five tasks mentioned in a fully automated fashion. The purpose of KiNetX is the kinetic analysis of first-principles reaction networks that consist of elementary reaction steps for which activation free energies were obtained from a series of calculations based on quantum chemical models. The innovative key feature of KiNetX is its appreciation of recent developments in the uncertainty quantification of properties, in particular reaction energies, derived from quantum chemical models.35,88,106,121,133,159–162 Instead of making educated guesses, consulting rules, or fitting to experimental data, it has become possible to infer the correlated uncertainty of free-energy predictions from ensembles of quantum chemical models, which allows for a direct evaluation of the underlying covariance matrix. This core functionality of KiNetX significantly increases the reliability of mechanism reduction, uncertainty propagation, and sensitivity analysis.

A major challenge for reaction space exploration is the factorial increase of possible reaction channels for an increasing number of species. Two strengths of our KiNetX meta-algorithm are that it (a) can steer the exploration of chemical reaction space in order to accelerate this exploration and (b) identifies the most critical reactions in a chemical network that might require a reevaluation of their activation free energies based on more sophisticated quantum chemical models. To offer a large pool of examples and to set up performance statistics for KiNetX, we developed an automated network generator, AutoNetGen. AutoNetGen creates artificial reaction networks based on chemical logic with which we can study an arbitrary number of possible chemical scenarios.

39 Chapter 4 Mechanism Deduction from Noisy Chemical Reaction Networks

4.2 Overview of the KiNetX Meta-Algorithm

Our meta-algorithm KiNetX written in Matlab163 operates on CRNs consisting of N vertices (including initial species concentrations) and 2L unidirectional edges (including rate constants derived from activation free energies). All reaction networks are provided by AutoNetGen, which is also written in Matlab (Section 4.3). The workﬂow of KiNetX can be split into three major tasks:

1. Identify and eliminate all kinetically irrelevant vertices and edges of the original CRN by applying a hierarchy of network reduction algorithms resulting in a sparse reaction network (Section 4.4): (a) Maximum-ﬂux analysis (Section 4.4.1) (b) Detailed ﬂux analysis (Section 4.4.2)

2. Examine the precision of network properties (e.g., product distributions, reaction mechanisms) through propagation of free-energy uncertainty (Section 4.5). 3. Determine the eﬀect of changes in activation free energies on the uncertainty of time-dependent species concentrations through a global sensitivity analysis (Sec- tion 4.6).

Furthermore, for all algorithms but the maximum-ﬂux analysis, the integration of a system of (generally stiﬀ) ODEs is required several times. We interface to the ode15s module of Matlab for the integration of ODE systems.164 The minimal input requirements for KiNetX are:

• A CRN with N vertices and L bidirectional edges (here, provided by AutoNetGen).

• A set of N initial concentrations, y0 (here, provided by AutoNetGen). • A set of N arrays specifying properties for each of the corresponding PES’s, e.g., atomic numbers, number of atoms per atomic number, total electric charge, or electronic spin multiplicity (here, provided by AutoNetGen). • A set of 2L Helmholtz activation free energies,

( )⊤ ≡ ‡,∗,+ ‡,∗,+ ‡,∗,− ‡,∗,− A0 ∆A1 , ..., ∆AL , ∆A1 , ..., ∆AL (4.1)

(here, provided by AutoNetGen). The double-dagger symbol, ‡, refers to the Helm- holtz free-energy difference between an intermediate state and a transition state, whereas the asterisk symbol, ∗, indicates that the Helmholtz free-energy difference is given with respect to a predefined standard state. To keep the notation ‡,∗,+/− ≡ +/− uncluttered, we define ∆Al Al for the remaining chapter.

• A covariance matrix, ΣA, of dimension 2L × 2L representing the correlated uncertainty of activation free energies (here, provided by AutoNetGen), respectively. We assume that the uncertainty of species concentrations is negligible.

40 Automatic Network Generation 4.3

• A maximum reaction time, tmax, representing a practical time scale or a time scale of interest. • A thermostat temperature, T , for the calculation of rate constants.

At present, we require the input to be provided in SI units. Optional input parameters (with default values in SI units if not dimensionless) are:

• The number of samples, B, to be drawn from ΣA (default: B = 100).

• The number of time points, U, between t = 0 and t = tmax at which species concentrations will be evaluated based on cubic spline smoothing (default: U = 1,000).

• A ﬂux threshold, Gmin, above which a chemical species will be considered kinetically relevant (default: 1.0 × 10−6 mol L−1).

• A concentration threshold, ymin, reﬂecting a detection limit of measuring species concentrations (i.e., species which do not reach this threshold from below during a kinetic simulation will be considered nondetectable; default: 1.0 × 10−3 mol L−1).

Currently, KiNetX is limited to the simulation and analysis of homogeneous, isothermal, reactive chemical systems in dilute solution, which constitute a major category of chemical systems in synthetic chemistry.

4.3 Automatic Network Generation

Before the algorithms of KiNetX are detailed, we introduce AutoNetGen, which produces sample networks for a systematic and rigorous analysis of KiNetX. AutoNetGen generates a CRN in a layer-by-layer fashion. A new layer is formed by combinatorially finding all possible reactions between all species of the previous layers. As AutoNetGen deals with abstract (unspecified) chemical species, we cannot resort to descriptors identifying reactive sites20 of molecules. Instead, we implemented random classifiers that either allow or forbid the formation of an edge between a pair of vertices. In the current version of AutoNetGen, only closed systems (no particle influx or outflux) are constructed. This limitation is introduced here for the sake of clarity and not for technical reasons; the numerical solver implemented in KiNetX can also integrate kinetic models of open systems. The first layer of a CRN represents the reactants, which must be specified on input. Further input parameters are the number of layers, X, to be generated (yielding X + 1 layers in total), and lower and upper bounds to both the activation free energy, Amin and Amax (with respect to the higher-energy intermediate state of a reversible reaction), and the corresponding maximum standard deviation, σA,max. These three boundary values serve to randomly construct the mean vector A0 and the covariance matrix ΣA (see Section A.2 for details).

41 Chapter 4 Mechanism Deduction from Noisy Chemical Reaction Networks

For the construction of a chemically sensible reaction network, we created a set of rules encoding chemical logic into the underlying graph structure. First, uni- and bimolecular reactions of the following types are considered in AutoNetGen:

• A ⇌ P • A + A ⇌ P • A + B ⇌ P • A + A ⇌ P + Q • A + B ⇌ P + Q

Here, two diﬀerent characters imply two diﬀerent species, and two identical characters imply two identical species. Second, for all reactions, the sum of atom numbers per chemical element is forced to be identical for the reactant (left-hand side) and product (right-hand side) states. For simplicity, we work with hypothetical masses of species (integers) in AutoNetGen and require that the sum of species masses is identical for the reactant and product states. See Section A.3 for details of the AutoNetGen algorithm.

4.4 Hierarchical Network Reduction

For hierarchical network reduction, a series of increasingly sophisticated algorithms (in terms of both rigor and required computing resources), each of which eliminates a set of kinetically irrelevant vertices and edges, is implemented. As a result, each higher-level reduction algorithm may become more efficient through the elimination of vertices and edges by a lower-level reduction algorithm. At the same time, the maximum error introduced by eliminating network constituents is a user-defined threshold, which allows for controlling the accuracy loss caused by the network reduction algorithms employed. Here, we introduce a hierarchy of two reduction algorithms, namely the Maximum- Flux Analysis and the Detailed Flux Analysis, which represent our interpretation of established kinetic modeling concepts.90 In the results section, we will analyze the computational efficiency gained by the application of the two reduction algorithms and the errors they introduce in the calculation of time-dependent species concentrations.

4.4.1 Maximum-Flux Analysis

The appealing characteristic of the maximum-ﬂux analysis is that it solely operates on the graph structure and the parameters of a CRN, i.e., it does not involve the computationally expensive integration of ODE systems. In the ﬁrst step, a set of pseudo-

maximum rate constants, kmax, is generated by subtracting three standard deviations

(square-root diagonal elements of ΣA) from the mean value of each activation free energy. In the current version of KiNetX, all activation free energies will be transformed to

42 Hierarchical Network Reduction 4.4 rate constants based on the Eyring equation (2.2), which requires a predefined constant temperature, T , ( +/− ) A − 3σ +/− +/− kBT 0,l A k = exp − l , (4.2) l,max h RT where h, kB, and R are Planck’s constant, Boltzmann’s constant, and the gas constant, respectively. The definition of kmax ensures that every vector k is smaller than kmax regarding all of its elements in more than 99.9% of the cases. Hence, we may consider kmax a good approximation to the set of rate constants producing the highest flux for all of the L reversible elementary reactions [Eqs. (2.15) and (2.16)]. Because of our assumption of dilute chemical systems, collisions involving three or more reactive species can be considered negligible from a statistical point of view. For this reason, each element of the forward and backward reaction rate vectors, f + and f −, is of the form +/− +/− fl = kl y +/− y +/− , (4.3) nl,1 nl,2 where the vertex index n is determined by the edge index l, the direction, + or −, and the position in the rate equation, 1 or 2. In case of a unimolecular reaction, we +/− simply define that the second population density has an index nl,2 = 0 (denoting a max hypothetical null-species), which implies a value of y0 = y0 = 1 independent of the unit of measurement. Here, we consider closed systems where the total number of atoms is a conserved quantity. In this respect, we define the time-invariant atomic population density, Y ,

∑N ∑I Y = ai,nmiyn , (4.4) n=1 i=1 where ai,n is the number of atoms of the i-th chemical element in the n-th chemical species with atomic mass mi. Y is a conserved quantity in the case of a closed system. For simplicity, AutoNetGen assigns hypothetical species masses (integers), i.e., Eq. (4.4) reduces to ∑N Y = Mnyn , (4.5) n=1 where Mn is the mass of the n-th species. With this deﬁnition of Y , we can determine max the maximum population density that can be observed for the n-th state, yn = Y/Mn. Consequently, the maximum rate that can be observed for a chemical reaction reads ( ) +/− 1 +/− max max fl max = kl y +/− y +/− , (4.6) x nl,1 nl,2 where x = 1 for a unimolecular reaction or a homo-bimolecular reaction (type 2A → P),

43 Chapter 4 Mechanism Deduction from Noisy Chemical Reaction Networks

and x = 4 for a hetero-bimolecular reaction (type A + B → P). In the case of a homo- 2 max bimolecular reaction, the product of yn will be maximized if yn = yn . The factor 1/4 for a hetero-bimolecular reaction arises from the fact that the product of two population densities will be maximized if the individual maximum population densities are divided max × max by 2, i.e., (ynym)max = (yn /2) (ym /2).

+/− The product of fl and tmax has the dimension of a population density. The maximum population density passing through the l-th unidirectional edge after tmax reads ( ) ( ) +/− +/− Fl max = fl maxtmax . (4.7)

Likewise, the maximum population density passing through the n-th vertex after tmax reads { ( ) } max +/− −/+ Gn = max Snl Fl max . (4.8) ( ) ( ) + − − + Note that we consider the products Snl Fl max and Snl Fl max as indicated by the max permuted superscripts of Snl and Fl. If Gn < Gmin, the n-th vertex will be removed, except for the case where the n-th vertex represents a reactant. All bidirectional edges that were connected to at least one of the removed vertices will be removed, too.

After this procedure, the number of vertices and edges should have decreased( ) signif- +/− icantly. However, the decrease may be suboptimal, because multiplying fl max by +/− tmax is based on an artificial scenario. It implies that fl is stuck in its maximum value from t = 0 to t = tmax, meaning that the maximum rate of its consumed species is considered without them being consumed effectively. For a single irreversible reaction A → P, this scenario can be interpreted in terms of an artificial open system: Whenever a certain quantity of A is consumed, A will be immediately refilled by that quantity. max max Hence, both Fl and Gn will be highly overestimated in general. Moreover, note that the definition of Y in Eqs. (4.4) and (4.5) is rather conservative. Imagine two reactants located on different PES’s that are not directly connected to each other, i.e., they themselves or their derivatives (e.g., isomers, dissociation products, or oligomers) only react with each other, but they are never interconverted. In that case, the concentra- max tions of many species may even hypothetically never reach yn . A prominent example are catalytic reactions, where the concentration of the catalyst is usually much smaller than the concentration of the substrate, and hence, the maximum hypothetical catalyst max concentration will be smaller than our definition of ycat by several orders of magnitude. A graph-theoretical analysis would allow to define a set of Y -values — instead of a single value — to appreciate the existence of interacting but not directly connected subnetworks. However, while it may occur that kinetically irrelevant vertices and edges are kept, kinetically relevant vertices and edges will not be removed.

44 Hierarchical Network Reduction 4.4

4.4.2 Detailed Flux Analysis

Having removed a (possibly large) number of vertices and edges, the efficiency of integrating the underlying system of ODEs will significantly increase. This increase may even improve the scaling of the integration with the size of the network as the stiffness of the ODEs may decrease due to the possible removal of vertices and edges that are kinetically irrelevant for the time scale considered. Before a detailed flux analysis can be performed, the ODEs of the reduced reaction network, where kmax is chosen as the set of rate constants, are integrated from t = 0 to t = tmax. U time points will be saved, with t1 = 0 and tU = tmax. For u ∈ {2, ..., U}, we define ( ) ζ(tu−1/2) ≡ ζ(tu−1) + ζ(tu) /2 (4.9) for any time-dependent property ζ. We can now approximate the expressions for the edge fluxes,

∑U [ ]( ) ( ) +/− +/− +/− ≈ − − ≡ Fl fl (tu−1/2) tu tu 1 Fl approx , (4.10) u=2 and for the vertex ﬂuxes, ( ) ∑L ( ) ( ) ≈ + − − + Gn Snl Fl approx + Snl Fl approx . (4.11) l=1

Vertices and edges will be removed in the same fashion as during a maximum-ﬂux analysis. Since Gmax is an upper bound to G, we may now identify and eliminate further kinetically irrelevant vertices and edges.

4.4.3 Assessment of Sparse Networks

To verify the accuracy of the two network reduction algorithms based on flux analysis, we may integrate the system of ODEs for the original network (before any flux analysis), for the intermediate network (after maximum-flux analysis and before detailed flux analysis), and for the resulting sparse network (after detailed flux analysis). For a given time point tu, we can assess the deviation, δpq(tu), in the concentration vectors yp(tu) and yq(tu) — where p, q ∈ {original, intermediate, sparse} — through our metric v u u ∑N ( ) t1 2 δ (t ) = M 2 y (t ) − y (t ) . (4.12) pq u 2 n n,p u n,q u n=1

45 Chapter 4 Mechanism Deduction from Noisy Chemical Reaction Networks

Both the factor 1/2 and the masses Mn ensure that δpq(tu) is bounded by the values

zero and Y . δpq(tu) = 0 represents full identity, whereas δpq(tu) = Y reﬂects maximum disparity and can only be reached if all species populated in the p-th network are

unpopulated in the q-th network. The deviation in the concentration vectors, ∆pq, can also be determined over the entire course of the reaction and is a time-weighted

average over all δpq(tu) values, v u u ∑U t 1 [ ]( ) ∆ = δ2 (t − ) t − t − . (4.13) pq t pq u 1/2 u u 1 max u=2

By analogy with δpq(tu), ∆pq is bounded by the values zero and Y .

There are two technical issues associated with the comparison of concentration vectors obtained through kinetic simulations of different reaction networks that are subsets of the same original network. First, the number of vertices, N, usually differs for every network. Second, the number of time points, U, and/or their values usually differ for every network, which is a result of the implicit scheme applied in the numerical integration of stiff ODE systems. The first issue can be solved by considering only those vertices that remain in the sparse reaction network. For the original and intermediate networks, we will therefore observe a slightly reduced total population density, Y − ε,

where ε should be at most on the order of Gmin.

The second issue can be solved by interpolating the discrete data points of each concentration–time plot with cubic splines. Only this way, we can introduce a uniform

number of time points, U, with identical values tu for every concentration profile considered. Note that spline smoothing is only reasonable if data noise is negligible, a property which one can assess easily in the case of one control variable (here, time). To evaluate the reliability of the cubic spline interpolations, we compared them against results from Gaussian process regression.165 Under suitable assumptions, Gaussian process regression yields an optimal trade-off between fitting and smoothness (cubic splines only ensure the latter property). Hence, it can be employed to predict concentrations from previously unconsidered values in the time domain (here, the domain between t = 0 and

t = tmax). Furthermore, as Gaussian process regression is a strictly Bayesian method, one may obtain reliable uncertainty estimates for each prediction. However, as this regression method scales cubically with the number of data points, it is not suitable for repetitive applications (usually, hundreds of regressions would be necessary). In all cases studied, we found that both the mean deviation between the two interpolation methods (cubic spline smoothing versus Gaussian process regression) and the MPV obtained from Gaussian process regression are negligibly small compared to the mean

46 Hierarchical Network Reduction 4.4 variance of the concentrations themselves (by a factor of <10−6). Hence, data noise is indeed negligible and cubic splines work well for interpolating concentration profiles. At the end of the two flux analyses, one can already obtain a first hint of the underlying reaction mechanism. For this purpose, one would repeat the integration for k0 derived from the original set of activation free energies, A0, and plot all concentration profiles of non-steady-state species, i.e., those which exceed ymin at any time during the kinetic simulation. However, since we generally lack knowledge about the exact values of activation free energies,35 we do not know how much their correlated uncertainty — more precisely, their joint PDF (JPDF) — affects the initially determined reaction mechanism. This information could only be derived analytically for the vanish- ing relative number of the simplest of all reaction networks. Consequently, numerical propagation of the correlated uncertainty in the activation free energies based on sampling will be necessary to assess the reliability of the initial reaction mechanism.

4.4.4 Synergies Between Network Reduction and Exploration

A major advantage of network reduction algorithms is their ability to guide explorations of the vast chemical reaction space. New species and elementary reactions may be discovered by combining the reactive sites of all chemical species already present in a CRN.20 Obviously, this combinatorial problem will rapidly force every exploration algorithm to stop after a few layers, the number of which may be smaller than what would be needed for a truly comprehensive understanding of a complex chemical reaction mechanism. This unsystematic search (in terms of kinetics) will lead to exploring regions of reaction space that are kinetically irrelevant under given external reaction conditions. In this respect, reduction algorithms can significantly increase the exploration breadth and depth and, hence, be considered a way to guide and accelerate the exploration of chemical reaction subspaces. In turn, chemical explorations guided by network reduction can increase the efficiency of a detailed flux analysis and other methods based on kinetic simulations as fewer vertices and edges may be submitted to KiNetX, which affect the computational time required for the numerical integration of ODE systems. However, this increase in efficiency may only be observed if the vertices and edges provided to KiNetX will not be identified as kinetically irrelevant by a maximum-flux analysis. The coupling of kinetic simulations with mechanism generation is well-known in the reaction engineering community. It has been introduced by Broadbelt, Green, and coworkers,166 and recently revisited by Green and coworkers167 for their Reaction Mechanism Generator,32 an exploration software originally designed for the study of gas phase reactions,168 which has recently been extended to understand the kinetics of solution phase chemistry.169 Here, we follow a similar strategy but with a focus on

47 Chapter 4 Mechanism Deduction from Noisy Chemical Reaction Networks

the uncertainty of free-energy diﬀerences derived from ensembles of quantum chemical models and their propagation in kinetic simulations, which is a key feature of KiNetX.

4.5 Propagation of Free-Energy Uncertainty

The objective of uncertainty propagation90 is to assess the accuracy (or bias) and precision (or variance) of concentration profiles obtained from kinetic simulations. To obtain reliable results, it may be necessary to infer the correlation of the underlying model parameters (here, activation free energies), which is usually difficult to estimate. One way to estimate parameter correlation is the backward propagation of uncertainty observed in measured concentration profiles,170 which requires knowledge of the complete underlying mechanism, i.e., a sparse reaction network containing all kinetically relevant vertices and edges. This strategy is clearly appealing for verifying mechanism complete- ness, but it does not serve our purpose of understanding chemical reactivity from a first-principles perspective. It has recently been shown that the neglect of parameter correlation in kinetic models derived from electronic structure calculations is error-prone.31 As the parameters of a quantum chemical model are (unknown) functions of chemical space, the activation free energies of reactions comprising similar species will not change independently of each other when the value of an electronic structure parameter is changed. Fortunately, recent statistical developments in quantum chemistry35,88,106,121,133,159–162 enable the modeling of correlated uncertainty in free-energy differences. Still, inferring the correlation between activation free energies is computationally hard as it requires sampling from quantum chemical models35,88 and, hence, repeated first-principles calculations for all species of the chemical system studied (including transition states, and possibly, other non-minimum-energy structures). Even if machine learning models are employed, a comprehensive training set based on a vast number of quantum chemical calculations would be necessary.171 We have already demonstrated the steps required for the propagation of correlated uncertainty in activation free energies for a model network of the formose reaction.35 Here, we will generalize the procedure for the study of arbitrary chemical systems. In the first step of our uncertainty propagation algorithm, B samples are drawn from

the JPDF of activation free energies with mean vector E[A] ≡ A0 and covariance matrix

V[A] ≡ ΣA. Each sample is labeled Ab with b ∈ {1, ..., B}. Note that this setup neglects third- and higher-order moments of the JPDF of activation free energies, which may be a weak assumption for actual reaction networks derived from first principles. To avoid these limitations, one can always sample directly from the underlying quantum chemical model,35,88 which requires repeated first-principles calculations and is, therefore, rather inefficient compared to sampling from a covariance matrix. Another possibility not yet

48 Analysis of Kinetic Sensitivity to Free-Energy Uncertainty 4.6 explored by us is the application of matrix algebra to construct special matrices that simplify expressions for higher-order moments of JPDFs.172 Subsequently, the ODEs of the sparse reaction network are integrated for all B samples of activation free energies. Concentration profiles are interpolated by cubic spline smoothing for each of the N chemical species and for every sample. The number of time values, U, and the time domain (the set of tu-values) are chosen to be identical for all interpolations. For each of the U values of tu and for each of the N chemical species, the median value, y0.5,n(tu), and both an upper and lower prediction uncertainty value, y(1+c)/2,n and y(1−c)/2,n, respectively, are determined for a predefined confidence level c with 0 < c < 1. KiNetX is now able to assess whether the product distribution, the reaction mechanism, or another network property derived from A0 is reliable. For this purpose, various measures may be taken. For the product distribution, one could determine a mean deviation with respect to the A0 sample based on the δ0b(tmax) metric, where the subscripts 0 and b refer to the solutions derived from A0 and Ab, respectively. For the reaction mechanism, one could determine a mean deviation based on the ∆ metric. √ ∑ √ ∑ 0b −1 B 2 −1 B 2 For instance, if B b=1 δ0b(tmax) is very small, but B b=1 ∆0b is rather large, then there is more than one possible route to the same metastable sink of the reaction network.

4.6 Analysis of Kinetic Sensitivity to Free-Energy Uncertainty

In the following, we assume that the concentration profiles of the sparse reaction network reveal pronounced uncertainties such that it is difficult to correctly assign the main product(s) or to suggest a specific reaction mechanism. To find out to which activation free energies the concentration profiles are most sensitive, a sensitivity analysis is required.90 For a general sensitivity analysis, the activation free energies are perturbed to study how such perturbations affect the time-dependent species concentrations. For a local sensitivity analysis, the model parameters are perturbed one by one from their nominal values (here, the elements of A0). While a local sensitivity analysis is straightforward to conduct and computationally feasible (usually, 2L kinetic simulations are required), it has a significant disadvantage in that it does take into account the correlation between the model parameters (we already discussed this issue in the context of uncertainty propagation). Consequently, one may overlook important correlation effects on the uncertainty of concentration profiles. For a global sensitivity analysis, the correlation between the model parameters is taken into account, but the process requires significantly more computational resources than a local sensitivity analysis. Furthermore, the design of a global sensitivity analysis

49 Chapter 4 Mechanism Deduction from Noisy Chemical Reaction Networks

is not as unambiguous as for a local sensitivity analysis, which explains the existence of several approaches. While a local sensitivity analysis requires 2L kinetic simulations, global sensitivity analysis methods may require 2CL (Morris method, where C is an integer usually much smaller than B), B (polynomial chaos), or 4BL (Sobol method) kinetic simulations.90 The Morris method173 is among the simplest of global sensitivity analyses as it is not designed to induce a mapping between input and target quantities. Instead, its purpose is to categorize the input quantities (here, activation free energies) as either “important” or “unimportant” depending on how strongly changes in them affect the target quantities (here, time-dependent species concentrations). We are particularly interested in this categorization since we aim for a descriptor that informs us about the quality of activation free energies obtained from a basic quantum chemical model. If we find that the uncertainty of some species concentrations is too large to derive sensible conclusions about specific system properties, the results of a Morris analysis will support us in identifying the most critical activation free energies that require a reevaluation based on more sophisticated quantum chemical models. The original Morris method does not take into account the JPDF of the input quantities. Here, we introduce an extended variant of the Morris method that explicitly considers this information as it is the central element of our KiNetX meta-algorithm.

In our implementation of the Morris method, one starts from the nominal sample A0

and changes its ﬁrst element to the ﬁrst element of A1. For this new vector of activation

free energies, A01 , a numerical integration of the corresponding ODE system from t = 0

to t = tmax will be carried out. Subsequently, the second element of A01 is replaced

by the second element of A1 and a numerical integration based on the new vector A02

will be carried out. This procedure is repeated 2L times until we arrive at A02L = A1, for which we have already carried out the numerical integration in the context of uncertainty propagation. The entire procedure is repeated, now by an element-wise change

from A1 to A2, and generally by an element-wise change from Ab to Ab+1 until b = C−1 is reached. In the end, we will have generated solutions for another C(2L − 1) kinetic

models in addition to the B + 1 solutions obtained for the free-energy sets {A0, ..., AB}. For our Morris analysis, we are interested in the C(2L − 1) new solutions and the ﬁrst C +1 of the B +1 solutions obtained previously, amounting to a total number of 2CL+1 solutions.

For each of these solutions, KiNetX determines the δpq(tmax) and ∆pq metrics, where p and q represent two adjacent A-vectors as constructed by our extended Morris algorithm. Hence, 2CL values will be obtained for each metric, C thereof for every activation { +/−} { +/−} free energy. We deﬁne the sensitivity coeﬃcients zl,δ and zl,∆ as

50 Integration of Stiff Ordinary Differential Equations 4.7

v u u C∑−1 +/− ≡ t 1 zl,δ δ +/− +/− (tmax) (4.14) C bl ,bl−1 b=0 and v u u C∑−1 +/− ≡ t 1 zl,∆ ∆ +/− +/− , (4.15) C bl ,bl−1 b=0

+ − respectively. The subscripts bl and bl refer to the samples Abl and AbL+l , respectively, ≡ and we define Ab0 Ab. Note that we do not divide the δpq(tmax) and ∆pq metrics by the respective changes in activation free energy in Eqs. (4.14) and (4.15), which is implied in the usual definition of sensitivity coefficients. Instead, we define the dimension of each sensitivity coefficient to be a concentration divided by the correlated standard deviation of the associated activation free energy. This way, we obtain a direct measure of the average effect a free-energy perturbation has on the solution of the corresponding kinetic model. We justify this unsual approach by the fact that all perturbations applied originate from actual samples of the underlying JPDF of activation free energies. +/− Finally, we can set up a ranking of sensitivity coefficients. The larger zl,δ and +/− +/− zl,∆ , the larger the effect of changes in Al on the concentration profiles. With this ranking at hand, one can systematically improve on both the accuracy and precision of +/− the concentration profiles to reliably suggest product distributions (based on zl,δ ) or +/− specific reaction mechanisms (based on zl,∆ ). For an actual CRN derived from first principles, one would recalculate the critical activation free energies with a high-accuracy quantum chemical model.

4.7 Integration of Stiff Ordinary Differential Equations

For a system of stiff ODEs (the general case), an implicit numerical integration algorithm is necessary, which approximates the solution vector (here, of species concentrations) at a new time point tu+1 by iteratively solving a nonlinear equation involving the current 90 state at tu and the future state at tu+1. Implicit numerical integration requires information on the Jacobian matrix to determine the local behavior of the system of ODEs, which is essential in terms of accuracy, numerical stability, and efficiency. For nonlinear kinetic models, as studied by us, the Jacobian matrix is a (N ×N)-dimensional function of time (N being the number of species/vertices present in the underlying network). The Jacobian can be approximated by finite differences, but we found that the implementation of an analytical expression significantly improves the accuracy per solution vector. Based on this finding, we decided to always provide an analytical Jacobian to the ode15s numerical integration module164 of Matlab. Other optional arguments passed by us to

51 Chapter 4 Mechanism Deduction from Noisy Chemical Reaction Networks

the ode15s module are relative and absolute error tolerances, and the condition that all solutions must be strictly nonnegative. Compared to implicit numerical integration, its explicit counterpart is a noniterative method, and hence, efficient for the evaluation of the next time step. However, explicit numerical integration only yields stable solutions if the time step chosen reflects the fastest process in the kinetic model. Therefore, explicit numerical integration becomes rapidly unfeasible for stiff systems where some processes are slower than the fastest one by several orders of magnitude, and, hence, too many time steps would be necessary to model the slower system processes. A stable alternative to implicit numerical integration is the coupling of time scale separation with an explicit numerical integration algorithm. CSP98,99 is an iteratively improvable approach for the separation of slow and fast system processes, which has also been explored by us.35 However, the effort required for repeated time scale separation during a kinetic simulation may compen- sate for the savings gained by avoiding implicit numerical integration.90 Especially for systems of ODEs, it has been shown that CSP combined with explicit numerical integration does not outperform implicit numerical integration in terms of computational time requirements.174

4.8 Exemplary Workflow of the KiNetX Meta-Algorithm

In the following, we demonstrate the KiNetX workﬂow at a speciﬁc four-layer reaction network generated with AutoNetGen (Fig. 4.1). We refer to this reaction network as CRN-1. We started from two reactants with masses m =( 1 and )m = 2 (arbitrary units), ⊤ −1 respectively, and an initial concentration vector y0 = 1.0, 1.0 mol L , leading to the conserved quantity Y = 3.0 mol L−1 according to Eq. (4.5). Furthermore, we chose −1 −1 −1 −1 Amin = 25 kJ mol , Amin = 150 kJ mol , and σA,max = 4.18 kJ mol ≈ 1 kcal mol . The latter quantity serves as control parameter in the random construction of the co-

variance matrix of activation free energies, ΣA, by AutoNetGen. For the calculation of Eyring rate constants [Eq. (2.2)], a temperature of T = 298.15 K was deﬁned. For all

kinetic simulations, we set tmax = 3,600 s. The original CRN-1 consisted of 372 vertices and 272 (bidirectional) edges. After maximum-flux analyis, we obtained the intermediate CRN-1 with 248 vertices and 179 edges, which was further reduced to 23 vertices and 18 edges by detailed flux analysis. While the computational time required for numerical integration is negligible in this case (3.5 s for the original CRN-1 on a 2.4 GHz central processing unit), we will later show that computing time increases superlinearly with the initial number of vertices. −3 −1 For a chosen flux threshold of Gmin = 1.0 × 10 mol L , we find a total deviation of −5 −1 ∆original, sparse = 3.6 × 10 mol L between the concentration profiles of the original and the sparse CRN-1, i.e., the accuracy loss is below our predefined tolerance threshold.

52 Exemplary Workflow of the KiNetX Meta-Algorithm 4.8

16 9 10 17

23 8 4 5 14

1 3 15 11 2 13

12 18 19

6 7

22 20 21

Figure 4.1: Graphical representation of the sparse CRN-1. Enumerated vertices refer to chemical species. Green squares (edges) represent reversible elementary reactions. They can also be interpreted as transition states or, more generally, as reaction paths. Two opposing sides of every green square are connected with one or two lines each (the remaining sides are not connected with any line), which in turn are connected with one vertex each. The two sides per square connected with vertices this way represent the reactant and product states of that reaction. Examples of unimolecular, homo-bimolecular, and hetero-bimolecular reactions are (1) → (4), (4) + (4) → (9), and (1) + (2) → (3), respectively. Red lines denote (unidirectional) elementary reactions to which the time-dependent species concentrations are highly sensitive.

For uncertainty propagation, we drew B = 1, 000 samples from ΣA. In Fig. 4.2, we show the resulting concentration profiles for a selection of dominant chemical species −3 −1 (those which exceed ymin = 1.0 × 10 mol L between t = 0 and t = tmax). While the concentration profiles reveal similar shapes for a given chemical species, the absolute profile spread is quite pronounced for the potential main products (species 6, 7, 8, 13, and 15) and for one of the reactants (species 2). In Fig. 4.3, we present median values (dashed lines) and 90% confidence intervals (green bands) for the concentration profiles of some dominant chemical species. The spread in the final atomic concentrations (species concentrations multiplied by their

relative mass compared to the hypothetical atomic mass matom = 1) of the potential main products varies from 0.5 mol L−1 (species 6) to 1.0 mol L−1 (species 13) as mea-

53 Chapter 4 Mechanism Deduction from Noisy Chemical Reaction Networks

sured by the 90% conﬁdence interval, which is clearly too large to assign a reliable

product distribution after tmax = 3,600 s given a ﬁxed total atomic concentration of Y = 3.0 mol L−1.

Figure 4.2: Sampled concentration proﬁles (B = 1,000) of some dominant chemical species in the sparse CRN-1.

Figure 4.3: Average (median) concentration profiles and associated 90% confidence bands obtained from B = 1,000 samples before (dashed lines and green bands) and after (solid lines and blue bands) global sensitivity analysis of some dominant chemical species in the sparse CRN-1. Improvements in accuracy and precision are indicated by shifted averages and tighter confidence bands, respectively.

54 Accuracy and Efficiency of KiNetX-Based Kinetic Modeling 4.9

To study the effect of individual changes in activation free energies on the uncertainty of the concentration profiles, we performed a Morris-type sensitivity analysis with C = 20. We ranked the 2L = 36 sensitivity coefficients and imitated a refinement of a critical activation free energy (corresponding to large sensitivity coefficients) by randomly drawing one value from its conditional PDF and defined this values as exact (uncertainty-free) reference activation free energy. We “refined” all activation free energies corresponding to the top 90% ranking of sensitivity coefficients, which concerns only 11 out of 36 unidirectional elementary reactions (Fig. 4.1). Fig. 4.3 illustrates the accuracy and precision gain obtained through our global sensitivity analysis. Now, we can clearly assign species 8, 13, and 15 as the dominant products after t = tmax. This finding suggests that global sensitivity analyses can be harnessed to systematically improve on activation free energies derived from first principles without employing high-accuracy quantum chemical calculations for all constituents of a reaction network. Having a method at hand that systematically selects the most efficient prediction model for a given accuracy is a long-sought goal in computational chemistry.

4.9 Accuracy and Efficiency of KiNetX-Based Kinetic Modeling

As the dimension of reactive chemical systems grows with the advancement of computer hardware and intelligent exploration algorithms, so do the computer time requirements for kinetic modeling. While kinetic simulations can be carried out independently of each other for the different samples of activation free energies (and, hence, trivially parallelized), the speed-up in parallelizing a single kinetic simulation has a less favorable, sublinear scaling with respect to the number of processors. Therefore, kinetic modeling may become a significant cost factor in addition to the first-principles-based exploration of chemical reaction space. Furthermore, as trivial parallelization is also limited in practical terms (number of available processors, amount of available storage), it is important to take into account the number of kinetic simulations required for uncertainty propagation and sensitivity analysis in the design of a kinetic modeling study. In Table 4.1, we show that for intermediate-sized reaction networks (about L = 100 edges), already thousands of kinetic simulations are required for a number of C = 20 Morris samples. While we expect the uncertainty of the species concentrations to converge rather rapidly with an increasing number of activation free-energy samples, B, the possible number of edges in a sparse reaction network is arbitrarily large, which indicates that global sensitivity analysis dominates uncertainty propagation in terms of computer time requirements. Our experience shows that this trend is already pronounced for reaction networks of intermediate size, and motivates the study of network reduction in terms of accuracy and efficiency.

55 Chapter 4 Mechanism Deduction from Noisy Chemical Reaction Networks

Table 4.1: Number of kinetic simulations required in a kinetic network analysis depending on the number of bidirectional edges, L, and the number of activation free-energy sets, B. For a complete kinetic analysis, C(2L − 1) + B + 1 kinetic simulations are to be carried out. Here, we chose a number of C = 20 Morris samples.

L = 20 L = 100 L = 500 L = 2,500 B = 10 791 3,991 19,991 99,991 B = 100 881 4,081 20,081 100,081 B = 1,000 1,781 4,981 20,981 100,981

For a sample of 1,000 relatively small two-reactant networks (Table 4.2) consisting of four layers, we found that the numbers of vertices and edges were reduced on average (median) by a factor of larger than 5 after flux analysis. Interestingly, the dimension (numbers of vertices and edges) of the original reaction networks correlates perfectly with the dimension of the intermediate networks (Table 4.3), whereas the correlation of both to the dimension of the sparse networks is significantly lower (about 60%). As the first simplification step is based on a graph-theoretical analysis, it appears that reaction networks based on chemical logic (at least those generated by AutoNetGen, but possibly also actual CRNs) comprise a collective structural property that highly correlates with the extent of dimensionality reduction during a maximum-flux analysis.

Table 4.2: Statistical measures (median, 95% quantile, and maximum) of several quantities (number of vertices, N, number of edges, L, computational time required for a single kinetic simulation, tCPU, and concentration met- − rics, ∆) obtained from a kinetic analysis of 1,000 artiﬁcial four-layer, two-reactant networks (Y = 2 mol L 1) generated with AutoNetGen. The subscripts 1, 2, and 3 denote original, intermediate, and sparse networks, −1 −1 −1 respectively. We chose Amin = 25 kJ mol , Amax = 150 kJ mol , σA,max = 0 kJ mol , T = 298.15 K, and tmax = 300 s.

median 95% quantile maximum N1 36 262 2,476 N2 26 183 1,774 N3 6 22 96 L1 27 200 1,881 L2 18 139 1,344 L3 3 15 72 CPU t1 / s <0.1 0.8 147.9 CPU t2 / s <0.1 0.4 77.1 CPU t3 / s <0.1 0.1 11.6 −1 −16 −9 −5 ∆1,2 / (mol L ) 3.8 × 10 8.7 × 10 2.7 × 10 −1 −9 −7 −4 ∆1,3 / (mol L ) 4.4 × 10 5.0 × 10 5.3 × 10

At the same time, the accuracy of the concentration proﬁles was lowered by a maxi- −4 −1 mum of ∆1,3 = 5.3×10 mol L (Table 4.2), which is larger than our chosen tolerance −6 −1 (ﬂux threshold Gmin = 1.0×10 mol L ). We postulate two main reasons for exceeding

56 Accuracy and Efficiency of KiNetX-Based Kinetic Modeling 4.9 the flux threshold. First, as edges with minimal fluxes are removed during flux analysis, these fluxes will be redistributed over the remaining network, which may induce an accumulation of error. Second, relative and absolute tolerances specified for numerical integration may be too large such that numerical uncertainty dominates ∆1,3. The first issue addresses an inherent consequence of flux analysis and cannot be resolved with the network reduction strategy chosen. The second issue could be resolved by tightening of the relative and absolute tolerances employed in the numerical integration scheme. By −9 −9 −1 default, we chose yrel = 1.0 × 10 and yabs = 1.0 × 10 mol L , respectively. When −12 −1 choosing yabs = 1.0 × 10 mol L instead, we observe a decrease of the maximum −4 −1 −5 −1 value of ∆1,3 from 5.3 × 10 mol L to 3.0 × 10 mol L , and a further decrease to −6 −1 −12 6.8 × 10 mol L when additionally setting yrel = 1.0 × 10 . It appears that this accuracy issue can be resolved to a good degree by tightening the tolerance parameters specified for numerical integration. Note that the accuracy loss as measured by ∆1,2 and

∆1,3 does neither correlate with the network dimension nor with the computational time required for numerical integration (Table 4.3). Hence, we expect the performance of our hierarchical network reduction scheme to be independent of the kinetic model, which is a crucial requirement for studying time-dependent properties of arbitrary reaction networks.

Table 4.3: Correlation matrix measuring the covariance of different quantities (number of vertices, N, number of edges, L, computational time required for a single kinetic simulation, tCPU, and concentration metrics, ∆). For further details, see Table 4.2.

CPU CPU CPU N1 N2 N3 L1 L2 L3 t1 t2 t3 ∆1,2 ∆1,3 N1 1.00 1.00 0.60 1.00 1.00 0.59 0.72 0.70 0.04 −0.01 0.03 N2 1.00 1.00 0.61 1.00 1.00 0.61 0.73 0.71 0.04 −0.01 0.03 N3 0.60 0.61 1.00 0.60 0.61 0.99 0.42 0.43 0.06 −0.02 0.01 L1 1.00 1.00 0.60 1.00 1.00 0.60 0.72 0.70 0.04 −0.01 0.03 L2 1.00 1.00 0.61 1.00 1.00 0.61 0.73 0.71 0.04 −0.01 0.03 L3 0.59 0.61 0.99 0.60 0.61 1.00 0.42 0.43 0.05 −0.02 0.01 CPU t1 0.72 0.73 0.42 0.72 0.73 0.42 1.00 0.99 0.32 0.00 0.00 CPU t2 0.70 0.71 0.43 0.70 0.71 0.43 0.99 1.00 0.24 0.00 0.00 CPU t3 0.04 0.04 0.06 0.04 0.04 0.05 0.32 0.24 1.00 0.00 0.00 ∆1,2 −0.01 −0.01 −0.02 −0.01 −0.01 −0.02 0.00 0.00 0.00 1.00 0.04 ∆1,3 0.03 0.03 0.01 0.03 0.03 0.01 0.00 0.00 0.00 0.04 1.00

Furthermore, we ﬁnd that the average computational time required for the kinetic simulation of a four-layer reaction network (Table 4.2) is insigniﬁcant (less than one decisecond), whereas the computational time required to simulate the kinetics of the largest original network deviates by a factor of about 500 from the time required to solve the corresponding sparse network (147.9 s versus 0.3 s). While it is not surprising that larger reaction networks tend to be associated with more exhaustive calculations, it is

57 Chapter 4 Mechanism Deduction from Noisy Chemical Reaction Networks

important from an efficiency point of view to study the scaling of computational time as a function of the network dimension. For this purpose, we additionally generated 100 five- layer networks and plotted the number of vertices of all 1,100 original networks against the corresponding computational times required for numerical integration (Fig. 4.4). There is a clear superlinear trend in the data, suggesting that the relative efficiency of kinetic modeling is a decreasing function of the network dimension. Bootstrapping regression models38,175 results in an estimated mean computational time of 90 20 minutes for 10,000 vertices (and approximately 7,500 edges), which would render any sensitivity analysis unfeasible. Clearly, network reduction may lead to considerably smaller networks, but even a single kinetic simulation that needs to be carried out prior to a detailed flux analysis may significantly slow down the entire workflow of KiNetX. Our findings underpin once more the importance of network reduction and further motivate its application during instead of after network exploration. In the next section, we will discuss the advantages of such an on-line reduction scheme carried out in parallel to the exploration of chemical reaction space.

400

300

200 computational / s time

100

0 0 500 1000 1500 2000 2500 number of vertices

Figure 4.4: Computational time required for numerical integration as a function of the number of vertices. Com- putational times were recorded for 1,000 four-layer and 100 five-layer reaction networks (red dots) generated with AutoNetGen. The quadratic fit function (black line) and the 90% confidence band (enclosed by dashed lines) were estimated through bootstrapping of regression models (1,000 samples).

4.10 KiNetX as a Guide for Reaction Space Exploration

In this section, we will draft how KiNetX could be coupled with network exploration programs such as our Chemoton21 software in an on-line fashion. This way, kinetic

58 KiNetX as a Guide for Reaction Space Exploration 4.10 network reduction could be applied throughout the entire exploration procedure, e.g., subsequent to the formation of a new layer. While this alternating exploration–reduction scheme may increase the kinetic modeling efficiency (cf. Section 4.9), we focus on the implications for network exploration. For demonstration purposes, we consider two successive layers (third and fourth) of 1,000 artificial reaction networks generated with AutoNetGen. We chose Amin = −1 −1 −1 25 kJ mol , Amax = 150 kJ mol , σA,max = 0 kJ mol , T = 298.15 K, tmax = 300 s, −6 −1 and Gmin = 1.0 × 10 mol L . The average (median) three-layer network consisted of 43 vertices and 33 edges, whereas the average four-layer network already consisted of 758 vertices and 574 edges. After our two-step flux analysis of the three-layer network, the numbers of vertices and edges decreased on average by 33 and 26, respectively. Restarting the generation of the fourth layer from these sparse three-layer networks, we arrived at an average of 79 vertices and 50 edges, which is equivalent to an average reduction of network elements of about 90% compared to the ensemble of four-layer networks that was constructed without on-line coupling to a mechanism reduction algorithm. The largest four-layer network consisted of 98,188 vertices and 72,766 edges, and of 501 vertices and 371 edges at the three-layer stage, which were reduced to 27 vertices and 17 edges after flux analysis. A repeated generation of the fourth layer resulted in a significantly sparser network consisting of 786 vertices and 454 edges, corresponding to a reduction of network elements of >99%. Assuming that all elementary reactions are represented by three points on a PES (two minima, one first-order saddle point), the generation of the sparse fourth-layer network would require 2,068 free-energy calculations (taking into account the vertices and edges eliminated after flux analysis), whereas the generation of the original fourth-layer network would require another 168,886 free- energy calculations. Note that a network reduction algorithm would have already been reasonable prior to the generation of the third layer, which could have led to a further saving of about 1,000 free-energy calculations. It is difficult to determine a fixed value that represents the possible relative saving of free-energy calculations by an on-line network reduction. For instance, the degree of reduction depends on the height of reaction barriers. If the activation free energies are too large for the temperature and time scale chosen, one will certainly observe larger savings. In distinct cases, one could even merely apply an energy threshold instead of carrying out a kinetic analysis.20 Conversely, we expect the relative savings to decrease when decreasing the barrier heights. For an upper activation free-energy bound of 100 kJ mol−1 instead of 150 kJ mol−1, e.g., we find that the average reduction of network elements decreases from 90% to about 75% (in both cases, the same ensemble of reaction networks has been considered). Nevertheless, our results suggest that

59 Chapter 4 Mechanism Deduction from Noisy Chemical Reaction Networks

network reduction algorithms could signiﬁcantly increase the possible search depth and breadth in explorations of chemical reaction space to arrive at a truly comprehensive understanding of chemical reactivity.

4.11 Conclusions

We have demonstrated the strong capability of advanced kinetic modeling techniques for the deduction of product distributions and reaction mechanisms from input data equipped with uncertainty measures. Our approach is designed, but not limited to be fed with raw data from quantum chemical calculations as we aim to develop a flexible kinetic modeling framework rooted in the first principles of quantum mechanics. For this purpose, we developed the meta-algorithm KiNetX, which enables to carry out kinetic analyses of complex CRNs in a fully automated manner, including hierarchical network reduction, uncertainty propagation, and global sensitivity analysis. We demonstrated the entire workflow of KiNetX at a noisy reaction network generated with our program AutoNetGen, which constructs artificial reaction networks and encodes chemical logic into their underlying graph structure. Our results show that KiNetX can systematically identify regions in a network that require more accurate free-energy data, without the need to carry out high-accuracy quantum chemical calculations for all species considered. To study the accuracy and efficiency of KiNetX, we had to explore a multitude of reaction networks to cover a wide chemical spectrum, which is very time-consuming regarding the number of quantum chemical calculations required for this purpose. With the development of AutoNetGen, we were able to examine a large number of distinct chemical scenarios in short time. Most importantly, we found that the accuracy loss induced by network reduction is controllable and does not correlate with the dimension of the network, which suggests that KiNetX can serve as a general-purpose meta-algorithm for the kinetic analysis of CRNs. Furthermore, we showed that the computational resources required for the numerical integration of ODE systems scale superlinearly (presumably exponentially) with the number of vertices, which underlines the importance of network reduction algorithms. Therefore, we investigated the benefits of employing network reduction algorithms to guide and accelerate network exploration on the fly. We showed that, if network reduction algorithms are employed during network exploration, the number of quantum chemical calculations required can be significantly reduced, by 75–90% in the cases studied. In future work, we will couple KiNetX to our automated network exploration program Chemoton.21

60 5 Reliable Estimation of Prediction Uncertainty for Physicochemical Property Models

We apply bootstrapping to assess a linear property model linking the 57Fe Mössbauer isomer shift to the contact electron density (CED) at the iron nucleus for a diverse set of 44 molecular iron compounds.*,† The CED is calculated with twelve density functionals across Jacob’s ladder (PWLDA, BP86, BLYP, PW91, PBE, M06-L, TPSS, B3LYP, B3PW91, PBE0, M06, TPSSh). We provide systematic-error diagnostics and reliable, locally resolved uncertainties for isomer shift predictions. Pure and hybrid density functionals yield average prediction uncertainties of 0.06–0.08 mm s−1 and 0.04–0.05 mm s−1, respectively, the latter being close to the average experimental uncertainty of 0.02 mm s−1. Furthermore, we show that both model parameters and prediction uncertainty depend signiﬁcantly on the composition and number of reference data points. Accordingly, we suggest that rankings of density functionals based on performance measures (e.g., the squared coeﬃcient of correlation, r2, or the root-mean-square error, RMSE) should not be inferred from a single data set. Our statistical calibration approach is of general applicability and not restricted to 57Fe Mössbauer isomer shift prediction. We provide the statistically meaningful reference data set MIS39 and a new calibration of the isomer shift based on the PBE0 functional.

*This chapter is reproduced with permission from J. Proppe, M. Reiher, J. Chem. Theory Comput. 2017, 13, 3297. Copyright 2017 American Chemical Society. †In this chapter, we adopt the notation introduced in Chapter 3.

61 Chapter 5 Reliable Estimation of Prediction Uncertainty

5.1 Isomer Shift Calibration in Theoretical 57Fe Mössbauer Spec- troscopy

Calibration has been frequently applied to predict the isomer shift observed in 57Fe Mössbauer spectroscopy.128–132,176–197 The corresponding theory198 postulates a linear

relationship between the measurable isomer shift, δexp, and the diﬀerence in the CED, 57 ρabsorber − ρsource, of the Fe isotope embedded in two diﬀerent chemical environments (referred to as absorber and source), ( ) ∆r δ = g(r) [ρ − ρ ] . (5.1) exp r absorber source

Here, g(r) is a function of the average charge radius, r, of an iron nucleus, and ∆r is the diﬀerence between the charge radii of the excited state and the ground state of an ⊤ iron nucleus. In the corresponding property model, δ(ρabsorber, w) with w = (w0, w1) ,

all quantities of the right-hand side of Eq. (5.1) except for ρabsorber are hidden in the

regression parameters w0 (intercept) and w1 (slope),

δ(ρabsorber, w) = w0 + w1ρabsorber ≈ δexp . (5.2)

The absorber CED (ACED), ρabsorber, is determined on the basis of an electronic structure model, which in turn is typically based on Kohn–Sham DFT. Since every speciﬁcation employed in an electronic structure calculation (e.g., density functional,

basis set, integration grid, convergence criteria) may affect the value of ρabsorber, the property model needs to be calibrated every time a specification is changed. As noted in Section 3.3.2, one of the key assumptions in applying linear least-squares regression is variance-free input values. While electronic structure calculations yield virtually variance-free results (neglecting numerical errors and convergence threshold effects), the expectation value of an observable remains unpredictable due to model-inherent systematic errors, which are collectively referred to as model inadequacy (cf. Section 3.2.2). Model inadequacy causes the average uncertainty of the reference isomer shifts

to be nonreproducible, no matter which values we choose for w0 and w1. Consequently, one should have serious doubt on the validity of the free-of-variance assumption. Another frequent source of systematic errors are inconsistent data. On the one hand, the reference isomer shifts employed in this study have been recorded at diﬀerent temperatures (4–100 K), which can lead to signed deviations of about −0.02 mm s−1 (second- order Doppler shift).198 Bochevarov, Friesner, and Lippard proposed to consider only those isomer shifts recorded at liquid helium temperature (4.2 K).131 This situation is clearly desirable, but it would have limited our reference data set in terms of chemical di-

62 Reference Set of Molecular Iron Compounds 5.2 versity. On the other hand, it is difficult to ensure that the molecular structure representations employed are sufficiently accurate for reliable ACED calculations. Even though crystal structures guide the search for the correct minimum on the Born–Oppenheimer surface, there is no guarantee that structure optimization yields reliable results. Fur- thermore, not only the iron-containing compound itself may be important for isomer shift calibration, but also the closer environment of the solid sample such as adjacent iron complexes, counterions, or solvent molecules. The effect of the molecular structure representation on the ACED remains an issue to be studied, which is beyond the scope of this work. In essentially all previous calibration studies of the isomer shift,128–132,176–197 the squared coefficient of correlation, r2, served as a measure to assess the performance of an electronic structure model (note that in linear least-squares regression with a single input variable, as applied here, r2 is identical to the coefficient of determination, R2). However, there is no guarantee that R2 (= r2) allows for a reliable comparison of two electronic structure models with respect to their transferability, because this performance measure does not take into account uncertainty in the model parameters. The incompleteness of R2 (or the RMSE) as model performance measure, the possible existence of inconsistent reference data, and the unpredictable variability in the input variable has motivated us to reexamine isomer shift calibration in the light of a statistically rigorous analysis. We assess the reliability of different performance measures and study the transferability of 12 density functionals with respect to isomer shift predictions for 44 iron compounds of considerable chemical diversity (Table 5.1). Details on the computational protocol employed for both statistical calibration analysis and quantum chemical calculations are provided in Section B.2. Our statistical calibration program reBoot developed for the analysis presented in this chapter is available on our webpage.175 In combination with the data provided in the Supporting Information of Ref. 38, reBoot allows one to reproduce all results of this chapter. Furthermore, reBoot can be harnessed to apply the statistical calibration methods presented here to arbitrary polynomial property models that are linear with respect to their parameters. Note that the statistical methods implemented in reBoot are not limited to this family of models. For instance, implementation of nonpolynomial models or models being nonlinear in their parameters would be straightforward.

5.2 Reference Set of Molecular Iron Compounds

In previous calibration studies of the isomer shift,128–132,176–197 a variety of iron complexes was considered. Here, we chose a diverse set of molecular iron compounds (Ta- ble 5.1, and Figs. S6–S16 in Ref. 38) representing wide ranges of formal oxidation states, spin multiplicities, total charges, and ligand environments (type, number, and spatial

63 Chapter 5 Reliable Estimation of Prediction Uncertainty

orientation). We blended parts of previous reference sets by Noodleman,129 Neese,130 Friesner and Lippard,131 and our group132 with iron(I) complexes.80,199–205 We explicitly excluded linear and T-shaped iron compounds with a formal oxidation state of +1,206–210 since these species are known to reveal pronounced spin–orbit coupling not considered in our computational approach. While the effect of strong spin–orbit coupling on iron CEDs remains a subject to be studied in more detail, scalar-relativistic effects have been found to induce only a constant shift of nonrelativistic CEDs for iron-containing molecules, which is why isomer shift calibration is frequently based on nonrelativistic calculations.198 In our previous study on 57Fe isomer shift calibration, we found that scalar-relativistic calculations lead to a slightly higher correlation between the target observable and the input variable compared to nonrelativistic calculations.132 However, in preparation of Ref. 38, we detected a specification error for iodine in the EMSL basis set database211 for the def2-TZVP basis set and the Molcas computer program that turned out to be the source of the slightly better performance. Correction of this error reveals equivalent input–target correlation for scalar-relativistic and nonrelativistic results.

5.3 Effect of Experimental Uncertainty on Model Parameters

In the following, we will report the square root values (RMSE, R632, RMPV, RLOO) of

the performance measures introduced (MSED, E[MSE].632, MPV, E[MSE]LOO) for the sake of better comparability with experimental uncertainty. When applying linear least-squares regression to all reference isomer shifts (N = 44, Fig. 5.1, left), we find an RMSE of 0.07 mm s−1 for all hybrid density functionals (B3LYP, B3PW91, PBE0, M06, TPSSh). For the pure density functionals (PWLDA, BP86, BLYP, PW91, PBE, M06-L, TPSS), the RMSE ranges from 0.08 mm s−1 (M06-L, TPSS) to 0.10 mm s−1 (BLYP); see Table 5.2. Even though the RMSE is a lower bound to the MPU inferred from this specific set of reference isomer shifts, it is already significantly larger than the average experimental uncertainty of ⟨u⟩ = 0.02 mm s−1 found for molecular iron compounds.222 However, it is possible that ⟨u⟩ is larger than 0.02 mm s−1 for the compounds studied here, since several isomer shift measurements were reported without uncertainty (Table 5.1). Se- lecting only those measurements for which uncertainty has been reported (N = 30), we also find an average experimental uncertainty of ⟨u⟩ = 0.02 mm s−1. The RMSE still ranges from 0.06 mm s−1 (B3LYP, B3PW91, PBE0, M06) to 0.10 mm s−1 (BLYP). This discrepancy between the average experimental uncertainty and the RMSE indicates that explicit consideration of experimental uncertainty (through weighted least-squares regression) may have a minor effect on the model parameters. This hypothesis can be examined by iteratively reweighted linear least-squares regression (see Section B.1.2

64 Effect of Experimental Uncertainty on Model Parameters 5.3 , - ∗ o 132 c KEFFEG FUJQOQ YEZSAZ VIYHER − PTSQFE10 N09-14 PTHPFE10 BUYKUB10 VIYHOB DEDWUE QAJNUL CACZIP10 MATVOT DIBXAN10 N09-24 CELVEU NIHQIF N09-23 N09-22 N09-6 RUDHAB , dianion of 2,6-bis(2- 4 213 215 203 201 204 216 130 216 218 201 218 219 220 199 221 130 223 224 130 130 130 205 -diketiminate; cyclam, 1,4,8,11- β ref. coords. code -methylbenzothiohydroxamato) anion; N , hydrotris-1-(pyrazolyl)borate; MAC exp 3 δ 80 , bulky 212 214 203 201 204 216 217 216 218 201 218 200 220 199 221 222 223 222 222 225 205 Me ’); HBpz -DKI S , β S (K) ref. S; 80 80 80 80 80 80 80 77 80 77 4.2 4.5 4.2 4.2 4.5 4.2 4.2 4.2 4.2 4.2 4.2 4.2 3 exp H 6 T -C 2 b ) 1 − ) ) ) ) ) ) ) ) − − − − − − − − -butylpyridine; tmc, 1,4,8,11-tetramethyl-1,4,8,11-tetraazacyclotetradecane; (mm s 0.76(3) 0.63(1) 0.62(3) 0.62(1) 0.52(3) 0.50(1) 0.47(3) 0.43(1) 0.34(2) 0.28(6) 0.57(2) 0.44(2) 0.33(2) 0.28(1) 0.97( 0.88( 0.66( 0.79( 0.68( 0.67( 0.67( 0.38( tert exp δ NC; ArS, 2,6-Me 3 py, 4- H 6 -7,8-dihydro-octaethylporphyrin; OEP, dianion of octaethylporphyrin; opda, Bu t -C 2 trans -iron at room temperature. α ’ dianion; +2+2 4 +1 6 4 +2+2+1 4 +2 4 5 +3 4 +1+3 6 4 +3 6 +2 5 +3 6 6 +1+1 4 +2+3 8 4 5 +1 4 +2+1 6 +2 5 +1 6 6 S , S ox. state CN ; ArNC, 2,6-Me − + 1 5 5 4 5 3 4 3 1 4 1 6 1 2 4 2 5 6 4 1 2 2 2 S The R13-Y code refers to the structure #Y as found in Table 1 of our previous isomer shift calibration study. CO 2 3 130 − ) 2 –N; Dipp, 2,6-diisopropylphenyl; DTSQ, bis(1,2-dithiosquarato- 2 -butylthio)-methyl)borate; por, porphyrin; PyO, 6-methyl-2-pyridonate; PyS, 6-methyl-2-thiolate; pyS tert 2+ 2 –CH 2 N–Dipp) Me + 2 2 ) Bu 3 t ) ) 3 TACN) Me -DKI 3 Pr i 3 ) β 2 (OAc) 4 ) , phenyltris(( 2 ) 4 ) (Me 2 4 − -DKI 2 )(PEt )(PhCCPh) 3 Bu − 2 ) t β 2 2 dae) a 2 Cl − 2 2 )N Bu Bu − 2 2 4 )(pyS t t 2 4 3 )(pyS Pr i 3 3 py) -benzene) -Fe(cyclam)(N 6 Bu O(HBpz O(OAc) t η 2 2 Fe(NCS) Fe(opda) Fe( Fe(PhTt Fe( Fe(por)(O Fe(PhTt Fe(PMe)(PhBP Fe Fe(HCCPh)(HC(C Fe(SiP trans Reference set of molecular iron compounds compiled for this study (to be continued). : e e d e e d d e e e d 5.1 Abbreviations in alphabetical order: ac, CH All experimental isomer shiftsThe reported six-letter in codes this refer study to refer the to Cambridge Structural bulk Database identiﬁers. The N09-X codeThis refers compound to refers the to structureThe an #X in molecular inconsistent ascending structure input–target order was pair as truncated according found prior to in to the the Supporting structure jackknife-after-bootstrapping optimization. method. no.1 compound 2 3 4 5 67 Fe(DTSQ) 89 Fe(SPh) 10 Fe(OEC) 1112 Fe(OEP) 13 14 1516 Fe 17 Fe(SMe 18 FeCl(MBTHx) 1920 Fe(PH 21 Fe(NO)(pyS 22 LiFe(trop TACN, 1,4,7-trimethyl-1,4,7-triazacyclononane; OEC, dianion of e c b a d 3 Information of the studyFor by two Römelt, compounds, Ye, no and code Neese. is available (–). tetraazacyclotetradecane; dae, N–CH tetraanion ofMe 1,4,8,11-tetraaza-13,13-diethyl-2,2,5,5,7,7,10,10-octamethyl-3,6,9,12,14-pentaoxocyclotetradecane; MBTHx,phenylenediamine; PhTt bis( mercaptophenylthiomethyl)pyridine; “S2”, 1,2-benzenedithiolato- trop, 5H-dibenzo[a,d]cyclo-hepten-5-yl. Table

65 Chapter 5 Reliable Estimation of Prediction Uncertainty otnaino Table of Continuation 4Fe(OEC)C Fe(CO) 44 43 Fe(PyS)(ac)(CN)(CO) 42 Fe(PyS)(ac)(CO) 41 Fe(PyS)(ac)(CO) 40 Fe(SC 39 Fe(NO)(pyS 38 Fe 37 Fe(PyO)(ArS)(CO) 36 Fe(PyS)I(CO) 35 Fe(PyO)I(CO) 34 Fe( 33 Fe(PPh 32 Fe 31 Fe(PPh 30 29 28 NaFe(trop 27 26 Fe(SEt) 25 24 compound 23 no. e d e e e FeCl( Fe(NO) Fe(OEC)Cl Fe(SiP Fe(OEP)CO 2 = η (PyS) O(tmc)(NCCH 4 -butadiene)(CO) η 5 3 i 4 H Pr 5 2 3 3 5.1 -MAC 4 − (S( )(“S2”) ) 4 2 )CO 2 N-CO)I(CO) (ac) 2 a . (“S2”) 6 dae) p H 4 -Me)Ph) ) 5 ∗ + 2 2 2 ) (CO) PPh PPh − 2 2 2 2 PPh (ArNC) 3 2 3 3 ) 4 PPh 2+ 2 − 3 2 3 2 AN)1 (ArNC) 3 2 S 1 3 1 1 1 1 1 1 1 3 3 3 2 6 1 3 5 1 1 6 2 1 + x tt CN state ox. 46 +4 26 6 6 +2 +2 6 +2 +2 6 6 +2 5 +2 6 +4 5 +4 6 +4 4 +1 5 +3 +2 26 +2 45 5 +4 +4 6 6 +2 +2 4 +3 5 +1 0 0 7 5 δ exp − − − − 0.12( 0.19( 0.25( 0.27( 0.21( 0.17(1) 0.00(2) 0.00(2) 0.04(2) 0.10(2) 0.10(2) 0.12(1) 0.17(1) 0.20(1) 0.01(2) 0.06(1) 0.06(2) 0.18(2) m s (mm 0.11( 0.09(1) 0.02(2) 0.04(2) − − − − − − ) ) ) ) ) − ) 1 ) b T exp 100 100 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 77 78 77 77 77 80 77 55 77 77 77 77 K ref. (K) 228 238 237 235 235 235 235 222 235 235 235 234 232 230 231 230 229 228 205 202 227 226 δ exp e.cod.code coords. ref. 228 130 237 132 236 132 235 130 236 132 132 234 233 230 130 230 229 228 205 202 227 226 SUMXED N09-20 JESGUJ R13-Q FUMZET R13-O YAKZAN N09-21 FUMZAP R13-K R13-L PUCFID − SOCWAI N09-18 SOCVUB SONMUE SUMWUS RUDGUU UTOXOR CANDAW10 YEQPOA c

66 Effect of Experimental Uncertainty on Model Parameters 5.3

Figure 5.1: Results of bootstrapped (B = 105) linear least-squares regressions in the presence (left) and in the absence (right) of the inconsistent data points #1, #2, #7, #13, #28 (Table 5.1). The width of the 95% conﬁdence band (turquoise area, obtained from the 0.025 and 0.975 quantiles of the bootstrapped JPDF of model parameters) decreases by a factor of about 2 when removing the inconsistent data points. The solid lines represent ∗ mean values of regression parameters over all bootstrap samples, w¯ , and are visually indistinguishable from the least-squares solutions to the regression problem, wD. The dashed line (right) is a replication of the solid line shown in the left frame (identical values for w0 and w1). The corresponding ACED values were obtained from B3LYP/def2-TZVP calculations. This ﬁgure is reproduced with permission from J. Proppe, M. Reiher, J. Chem. Theory Comput. 2017, 13, 3297. Copyright 2017 American Chemical Society. for more details).116,141 Table 5.3 summarizes the results for B3LYP, which was re- peatedly found to perform superior to other density functionals in isomer shift calibration.128,130,131,177,182,192

While the slope, w1, is almost invariant to the approaches applied, the intercept, w0, changes by 57.41 mm s−1 when switching from ordinary (unweighted) least-squares regression to weighted least-squares regression, whereas ordinary least-squares regression and iteratively reweighted least-squares regression yield almost identical results (deviation of 3.65 mm s−1). This result suggests that experimental uncertainty does not need to be taken into account since model inadequacy and/or data inconsistency appear to play a dominant role. Furthermore, the bootstrapped standard deviation of the intercept, which is larger than 100 mm s−1 in each case, indicates that the diﬀerences between all calibration procedures are insigniﬁcant. Qualitatively analogous results were obtained for the remaining density functionals. As a consequence, we assume that calibration of our property model, δ(ρabsorber, w), is not perturbed by the inclusion of reference isomer shifts for which experimental uncertainty was not reported. Unless otherwise mentioned, the application of bootstrapping will imply ordinary least-squares regression, which is equivalent to assigning the average experimental uncertainty of ⟨u⟩ = 0.02 mm s−1 to all reference isomer shifts.

67 Chapter 5 Reliable Estimation of Prediction Uncertainty

Table 5.2: Performance measures (RMSE, R632, RMPV, RLOO) of density functionals regarding isomer shift prediction. The reference data set is the original one (N = 44). RMSE, R632, and RLOO were calculated on the basis of ordinary least-squares regression. The RMPV was obtained from Bayesian linear regression based on the evidence approximation. For the calculation of the .632 estimator, R632, B = 105 bootstrap samples were − generated. All performance measures are reported in mm s 1. The corresponding ACED values were obtained with the def2-TZVP basis set for ligand atoms.

functional RMSE R632 RMPV RLOO PWLDA 0.09 0.10 0.09 0.10 BP86 0.09 0.09 0.09 0.09 BLYP 0.10 0.10 0.10 0.10 PW91 0.09 0.09 0.09 0.09 PBE 0.09 0.09 0.09 0.09 M06-L 0.08 0.08 0.08 0.08 TPSS 0.08 0.09 0.09 0.09 B3LYP 0.07 0.07 0.07 0.07 B3PW91 0.07 0.07 0.07 0.07 PBE0 0.07 0.07 0.07 0.07 M06 0.07 0.07 0.07 0.07 TPSSh 0.07 0.08 0.08 0.08

Table 5.3: Parameters of the linear isomer shift model, δ(ρabsorber, w), optimized with different calibration procedures based on linear least-squares regression. Only those reference isomer shifts with reported experimental uncertainty (N = 30) were employed. The corresponding ACED values were obtained from B3LYP/def2-TZVP calculations. Standard deviations in parentheses were obtained from bootstrapping (B = 104).

−1 −1 3 calibration procedure w0 (mm s ) w1 (mm s bohr ) ordinary (unweighted) 4454.80(15303) −0.38(1) weighted 4397.39(14260) −0.37(1) iteratively reweighted 4451.15(14963) −0.38(1)

5.4 Model Selection Based on Occam’s Razor

In this and in the next subsection, we will explore the possibilities to unravel the eﬀects of model inadequacy and data inconsistency on the model residuals. First of all, we examine whether systematic model inadequacy is present. Systematic model inadequacy would result in residuals which show a trend with respect to the underlying property model instead of random scatter. While Gaussian process regression165 is a reliable approach to infer the model complexity with the highest transferability, we will apply a simple alternative based on Occam’s razor. In addition to the linear model, which is based on a physical theory,198 we choose quadratic, cubic, and quartic models,

68 Assessment of Data Inconsistency Based on Jackknife-after-Bootstrapping 5.5

∑M m δM (ρabsorber, w) = wmρabsorber , (5.3) m=0 with M = 2, 3, 4. In bootstrapping procedures, the model parameters are calibrated with respect to a bootstrap sample and then validated at a reference sample. Therefore, if the model is too rigid or too ﬂexible (i.e., when it features too few or too many parameters, respectively), we will observe decreased transferability compared to a balanced model through an increase of the MPU. Table 5.4 summarizes the results for B3LYP.

Table 5.4: Estimated MPU (R632) for four property models of increasing polynomial degree, M, on the basis of B = 104 bootstrap samples (N = 44), respectively. The corresponding ACED results were obtained from B3LYP/def2-TZVP calculations. The linear and quadratic models reveal the lowest MPU.

property model M R632 (mm s−1) linear 1 0.07 quadratic 2 0.07 cubic 3 0.08 quartic 4 0.10

The linear and quadratic models reveal the lowest MPU as measured by the .632 estimator. Applying Occam’s razor, we choose the simpler of both models, which is also the only one built on physical grounds.198 By inspection of Fig. 5.1, the residuals of the linear model appear randomly distributed. Qualitatively identical results were obtained for the remaining density functionals. Consequently, we may assume that the discrepancy between RMSE and ⟨u⟩ is rooted in inconsistent data and/or quasi-random model inadequacy. The latter eﬀect would be a consequence of nonconstant systematic errors introduced by the electronic structure model under consideration, which cause an increase in data noise.116

5.5 Assessment of Data Inconsistency Based on Jackknife-after- Bootstrapping

In Fig. 5.1 (left), uncertainty in the regression parameters is represented by the 95% confidence band (turquoise area) obtained from bootstrapping (B = 105). The corresponding ACED values were obtained from B3LYP calculations. The black line represents the mean of regression parameters over all bootstrap samples, w¯ ∗, as defined in Eq. (3.5). R632, RMPV, and RLOO equal the RMSE (0.07 mm s−1); see Table 5.2. Note that the experimental resolution is limited to 0.01 mm s−1. If instead it would be artificially increased to 0.001 mm s−1, which can be approximated by adding a trailing zero to the reference isomer shifts, we obtain RMSE = 0.066 mm s−1, R632 = 0.070 mm s−1,

69 Chapter 5 Reliable Estimation of Prediction Uncertainty

RMPV = 0.069 mm s−1, and RLOO = 0.070 mm s−1. For details on a statistically valid increase of the experimental resolution, see Section B.1.4. Hence, the low experimental resolution masks the effect of parameter uncertainty on the MPU as measured by R632, RMPV, and RLOO, at least for this specific composition and number of isomer shifts. Moreover, the RLOO appears to be an efficient alternative to the R632 for isomer shift calibration. Qualitatively similar results were also obtained for the remaining density functionals (Table 5.2).

1 w0 N = 44 w1 N = 44

0 1 w0 N = 39 w1 N = 39 standardized probability density probability standardized density probability standardized 0 0.26 0.28 0.30 0.32 0.34 0.36 −0.40 −0.36 −0.32 intercept (mm s−1) slope (mm s−1 bohr3)

Figure 5.2: Bootstrapped parameter distributions (B = 105, histograms with 50 bars each) and Gaussian posterior parameter distributions (solid curves) obtained in the presence (top) and in the absence (bottom) of the inconsistent data points #1, #2, #7, #13, #28 (Table 5.1 and Fig. 5.1, left). After their removal, the standard deviation of

both intercept, w0, and slope, w1, introduced in Eq. (5.2) decreases by a factor of about 2, and the corresponding mean values are obviously shifted. Furthermore, the similarity between the two types of parameter distributions

increases. Here, we employed an isomer shift model with a centered ACED, i.e., δ(ρ) = w0 + w1(ρ − ρ¯), where we dropped the subscript 'absorber'. Here, ρ¯refers to the mean ACED of the reference data set considered. The corresponding statistics of the model parameters are summarized in Table 5.5. All ACED values were obtained from B3LYP/def2-TZVP calculations. This ﬁgure is reproduced with permission from J. Proppe, M. Reiher, J. Chem. Theory Comput. 2017, 13, 3297. Copyright 2017 American Chemical Society.

In Fig. 5.2 (top left and top right), histograms of the bootstrapped parameter distributions (B = 105) are shown. The solid curves are Gaussian posterior distributions

70 Assessment of Data Inconsistency Based on Jackknife-after-Bootstrapping 5.5 obtained from Bayesian linear regression. We find systematic deviation of the posterior slope distribution from its bootstrapped variant, in particular with respect to higher- order moments (skewness), which may explain the slight difference between R632 and RMPV when artificially increasing the experimental resolution. The skewness of the bootstrapped slope distributions may arise from data points with an above-average effect on the objective (here, sum of least squares) such as those at the boundaries of the input domain or apparent outliers. By contrast, the agreement between the posterior intercept distribution and its bootstrapped variant appears to be sufficiently high. Q–Q plots of bootstrapped and Gaussian parameter distributions (Fig. S1 in Ref. 38) support these findings.

Table 5.5: Statistics of model parameters corresponding to the distributions shown in Fig. 5.2. w¯m and σwm (m ∈ {0, 1}) correspond to mean and standard deviation of the inferred parameter distributions, respectively. 2 r{w0,w1} refers to the correlation between w0 and w1 as measured by the square root of r . For better comparabil- − ity, the results have not been rounded to the experimental resolution (0.01 mm s 1).

inference method N w¯0 w¯1 σw0 σw1 rw0,w1 bootstrapping 44 0.3295 −0.3674 0.0101 0.0157 −0.4720 39 0.2888 −0.3805 0.0056 0.0082 −0.0776 Bayesian 44 0.3288 −0.3658 0.0102 0.0137 0.0000 39 0.2886 −0.3801 0.0057 0.0085 0.0000

To examine the reliability of the sampled parameter distributions and the MPU associated with them, we apply the jackknife-after-bootstrapping method. An overview in Fig. 5.3 (left) of the RMPV versus the normalized unsigned deviation in the intercept,

∆w0,n, ∗ ∗ |w¯ − − w¯ | ∆w ≡ ∑ 0, n 0 , (5.4) 0,n N | ∗ − ∗| n=1 w¯0,−n w¯0 optimized with respect to both D and D−n, reveals no clear correlation between the quantities (here, D−n refers to the reference data set with the n-th data point removed, and the RMPV is given with respect to D−n). The same holds true for the correlation between the RMPV and the normalized unsigned deviation in the slope (Fig. 5.3, center),

∗ ∗ |w¯ − − w¯ | ∆w ≡ ∑ 1, n 1 . (5.5) 1,n N | ∗ − ∗| n=1 w¯1,−n w¯1

As the parameters are correlated in the property model, we also calculated the normalized RMSD of the property model, ∆δn, calibrated with respect to both D and

D−n,

71 Chapter 5 Reliable Estimation of Prediction Uncertainty

√ ( ) ∑ 2 N ∗ − ∗ i=1 δ(ρi,absorber, w¯ −n) δ(ρi,absorber, w¯ ) √ ∆δn ≡ ( ) . (5.6) ∑ ∑ 2 N N ∗ − ∗ n=1 i=1 δ(ρi,absorber, w¯ −n) δ(ρi,absorber, w¯ )

A plot of the RMPV versus ∆δn (Fig. 5.3, right) reveals a more distinct correlation. At smaller deviations, the RMPV is almost constant, whereas it decreases overlinearly at larger deviations. Hence, if the removal of a data point significantly changes the functional form of the property model, it also has a significant effect on the MPU.

0.072

) 0.070 − 1 0.068

0.066 1 2 0.064

RMPV (mm s 0.062 13 28 0.060 10−4 10−3 10−2 10−1 10−4 10−3 10−2 10−1 10−3 10−2 10−1

Figure 5.3: RMPV of N jackknife data sets, D−n, versus three measures of the difference between various calibra- 4 tions of δ(ρabsorber, w). The reference calibration (B = 10 ) is performed with respect to the complete reference D ∗ D ∗ data set, , yielding w¯ , which is compared to N calibrations based on the jackknife data sets, −n, yielding w¯−n. ∗ ∗ Left: ∆w0,n, normalized absolute deviation of w¯0,−n from w¯0 (intercept). Center: ∆w1,n, normalized absolute ∗ ∗ ∗ ∗ deviation of w¯1,−n from w¯1 (slope). Right: ∆δn, normalized RMSD of δ(ρabsorber, w¯−n) from δ(ρabsorber, w¯ ). The data points highlighted (#1, #2, #13, #28; Table 5.1 and Fig. 5.1, left) are potentially inconsistent as their removal leads to signiﬁcant changes in the MPU as measured by the RMPV. For better legibility, the results of the RMPV − have not been rounded to the experimental resolution (0.01 mm s 1). The corresponding ACED values were obtained from B3LYP/def2-TZVP calculations. This ﬁgure is reproduced with permission from J. Proppe, M. Reiher, J. Chem. Theory Comput. 2017, 13, 3297. Copyright 2017 American Chemical Society.

The plots of Fig. 5.3 reveal four data points (#1, #2, #13, #28; Table 5.1 and Fig. 5.1,

left) that lead to a distinct decrease of the MPU and high values of ∆δn, respectively. The question arises whether these data points are inconsistent in the sense that if they are present, the underlying population distribution may not be well-represented by the data set under consideration. Since we are studying several density functionals, we can examine whether we find the same potentially inconsistent data points in the remaining cases. We find that three of the four potentially inconsistent data points (#1, #2, #28) identified with B3LYP are also identified as most likely inconsistent with all

other density functionals (we considered the four highest values of ∆δn for each density functional). This finding indicates that these data points are not affected by inadequate ACED calculations as they have been identified to be inconsistent irrespective of the density functional employed. Rather, systematic measurement errors or deficiencies in

72 Assessment of Data Inconsistency Based on Jackknife-after-Bootstrapping 5.5 the molecular structure representation may be responsible for their inconsistent status. Moreover, the fourth potentially inconsistent data point identiﬁed with B3LYP (#13) has also been identiﬁed with three other hybrid density functionals (B3PW91, PBE0,

M06) and is ranked fifth with respect to ∆δn for the remaining hybrid density functional (TPSSh). In this case, we interpret data point #13 as affected by inadequate ACED calculations as it was only identified inconsistent by a particular category of density functionals. Likewise, data point #7 (Table 5.1 and Fig. 5.1, left) is one of the first four data points identified potentially inconsistent by all pure density functionals (PWLDA, BP86, BLYP, PW91, PBE, M06-L, TPSS) and one hybrid density functional (TPSSh), and is ranked fifth with respect to ∆δn for all remaining hybrid density functionals but M06. We do not find obvious similarities in the corresponding compounds regarding the different categories in Table 5.1, but in all five cases, iron is coordinated to nitrogen (Fig. 5.4). We also compared the calculated versus ideal expectation value of the ⟨S2⟩ operator for all open-shell complexes (Table S5 in Ref. 38), but find no anomalies for the inconsistent data points. Likewise, we cannot confirm that the RMSD of atomic positions is particularly high for those iron complexes corresponding to inconsistent input–target pairs (Table S1 in Ref. 38).

#2 #1

#28

#13

Figure 5.4: Molecular structures (tube models) corresponding to the inconsistent input−target pairs #1, #2, #7, #13, #28 (Table 5.1 and Fig. 5.1, left). All complexes exhibit at least one iron−nitrogen bond. Color code: magenta, iron; light gray, carbon; white, hydrogen; green, chlorine; blue, nitrogen; red, oxygen; yellow, sulfur; dark gray, chemical bond. This ﬁgure is reproduced with permission from J. Proppe, M. Reiher, J. Chem. Theory Comput. 2017, 13, 3297. Copyright 2017 American Chemical Society.

To obtain reliable estimates of the MPU, we decided to remove all inconsistent data points (#1, #2, #7, #13, #28; Table 5.1 and Fig. 5.1, left). Repeating bootstrapped (B = 105) linear least-squares regression (Fig. 5.1, right), the updated property model (solid line) clearly deviates from that one calibrated with respect to the reference data set including the inconsistent input–target pairs (dashed line). Fig. 5.2 (bottom left and bottom right) reveals that mean and standard deviation of the bootstrapped pa-

73 Chapter 5 Reliable Estimation of Prediction Uncertainty

rameter distributions (histograms) change significantly, the latter decreases by a factor of 2 (Table 5.5). The corresponding Gaussian posterior distributions obtained from Bayesian linear regression (solid curves) are now both quite similar to their sampled counterparts. This finding is also supported by Q–Q plots of bootstrapped versus Gaus- sian parameter distributions (Fig. S1 in Ref. 38). Even though the 95% confidence band (Fig. 5.1, right) is narrower than before, we now find a difference between the RMSE (0.03 mm s−1) and R632 as well as RMPV and RLOO (0.04 mm s−1, respectively). This finding is an artifact resulting from the low experimental resolution of 0.01 mm s−1. Increasing the experimental resolution artificially from 0.01 mm s−1 to 0.001 mm s−1, we obtain RMSE = 0.035 mm s−1, R632 = 0.036 mm s−1, RMPV = 0.036 mm s−1, and RLOO = 0.036 mm s−1. In this case, we see that the difference between RMSE and R632 decreases from 0.004 mm s−1 to 0.001 mm s−1, which we would expect for a narrowing of the confidence band. Furthermore, due to the higher similarity of the bootstrapped and Gaussian posterior distributions after removal of the inconsistent data points, we also find that the difference between R632 and RMPV vanishes (before: 0.001 mm s−1) for the increased experimental resolution. This result indicates that the normal-population assumption appears to be reasonable for isomer shift calibration after removal of inconsistent data points. Again, the RLOO appears to be an efficient alternative to the R632 for isomer shift calibration. Note that the updated RMSE (0.03 mm s−1 instead of 0.07 mm s−1) is now much closer to the average experimental uncertainty (⟨u⟩ = 0.02 mm s−1). For the remaining density functionals, the RMSE ranges from 0.03 mm s−1 (B3PW91, PBE0) to 0.08 mm s−1 (BLYP), while R632, RMPV, and RLOO range from 0.04 mm s−1 (B3PW91, PBE0, TPSSh) to 0.08 mm s−1 (BLYP); see Table 5.6. Noteworthy, while inclusion of the inconsistent data points in the reference data set (N = 44) leads to overestimation of the MPU as measured by the R632 (86% of the data points lie within the 68% prediction band), their exclusion (N = 39) results in a significant improvement of MPU estimation (72% of the data points lie within the 68% prediction band); see Fig. 5.5. Note that the prediction bands are equivalent to u(x) defined in Eq. (3.17). Because of the decrease of the RMSE, we examined once again the importance of explicitly considering experimental uncertainty. The results are summarized in Table 5.7

and lead us again to the conclusion that calibration of δ(ρabsorber, w) is not perturbed by the inclusion of reference isomer shifts for which experimental uncertainty has not been reported. Furthermore, we applied Occam’s razor to our consistent reference data set (N = 39). In all cases (M = 1, 2, 3, 4), we obtained an R632 of 0.04 mm s−1, which conﬁrms again the validity of the linear (physical) model.

74 Assessment of Data Inconsistency Based on Jackknife-after-Bootstrapping 5.5

Combining all these findings, we are confident that the updated, consistent reference data set represents the underlying population distribution sufficiently well if hybrid density functionals generate the input data. Hence, our reference set of N = 39, already pruned by statistically critical outliers, provides a well-defined starting ground for further parameterization studies or for systematic extensions of the reference data. We may call this special set of data the MIS39 data set (for 39 reference data points of Mössbauer isomer shifts).

Table 5.6: Performance measures (RMSE, R632, RMPV, RLOO) of density functionals regarding isomer shift prediction. The reference data set is the original one with the inconsistent data points (#1, #2, #7, #13, #28; Table 5.1 and Fig. 5.1, left) removed (N = 39). RMSE, R632, and RLOO were calculated on the basis of ordinary least-squares regression. The RMPV was obtained from Bayesian linear regression based on the evidence approximation. For the calculation of the .632 estimator, R632, B = 105 bootstrap samples were generated. All − performance measures are reported in mm s 1. The corresponding ACED values were obtained with the def2- TZVP basis set for ligand atoms.

functional RMSE R632 RMPV RLOO PWLDA 0.07 0.07 0.07 0.07 BP86 0.07 0.07 0.07 0.07 BLYP 0.08 0.08 0.08 0.08 PW91 0.07 0.07 0.07 0.07 PBE 0.07 0.07 0.07 0.07 M06-L 0.06 0.06 0.06 0.06 TPSS 0.06 0.06 0.06 0.06 B3LYP 0.03 0.04 0.04 0.04 B3PW91 0.03 0.04 0.04 0.04 PBE0 0.03 0.04 0.04 0.04 M06 0.05 0.05 0.05 0.05 TPSSh 0.04 0.04 0.04 0.04

Table 5.7: Parameters of the linear isomer shift model, δ(ρabsorber, w), optimized with different calibration procedures based on linear least-squares regression. Only those reference isomer shifts with reported experimental uncertainty and consistent status (N = 28) were employed. The corresponding ACED values were obtained from B3LYP/def2-TZVP calculations. Standard deviations in parentheses were obtained from bootstrapping (B = 104).

−1 −1 3 calibration procedure w0 (mm s ) w1 (mm s bohr ) ordinary (unweighted) 4510.05(11754) −0.38(1) weighted 4434.25(12639) −0.38(1) iteratively reweighted 4495.55(10932) −0.38(1)

75 Chapter 5 Reliable Estimation of Prediction Uncertainty

Figure 5.5: Results of bootstrapped (B = 105) linear least-squares regressions in the presence (top left) and in the absence (top right) of the inconsistent data points #1, #2, #7, #13, #28 (Table 5.1 and Fig. 5.1, left). The cor- ∗ responding diagrams of residuals, δexp − δ(ρabsorber, w¯ ), are shown at the bottom. The 68% prediction band (green area) comprises 86% (left) and 72% (right) of the data points. The solid lines are equivalent to those shown in Fig. 5.1. The corresponding ACED values were obtained from B3LYP/def2-TZVP calculations. This ﬁgure is reproduced with permission from J. Proppe, M. Reiher, J. Chem. Theory Comput. 2017, 13, 3297. Copyright 2017 American Chemical Society.

5.6 How Reliable are Density Functional Rankings Based on a Specific Data Set?

In the previous subsection, we found for the particular composition and number of data points that the effect of parameter uncertainty on the MPU is very small (<0.01 mm s−1 for all density functionals). Hence, given this specific reference data set, we are confident that the RMSE (and with it the squared coefficient of correlation, r2) is a good approximation to performance measures such as R632, RMPV, and RLOO, and therefore, suited to set up a ranking of density functionals. In the following, we only distinguish between first and other places in a ranking as only those density functionals will be

76 How Reliable are Density Functional Rankings Based on a Specific Data Set? 5.6 considered for actual applications that reveal highest transferability. Regarding our reference set of isomer shifts (N = 39), B3LYP, B3PW91, and PBE0 are placed first in all rankings studied (RMSE, R632, RMPV, RLOO), with TPSSh being placed first in the rankings based on R632, RMPV, and RLOO (Table 5.6). This finding is consistent with those of other calibration studies of the isomer shift.128,130,131,177,182,192 However, in practice, it is relevant to know which density functional yields the most accurate predictions independently of the reference data employed for calibration of the property model. So far, we considered rankings of density functionals conditioned on a specific reference data set, but what we aim at is an unconditional ranking of density functionals. Otherwise, we cannot assess the transferability of a property model trained on a specific density functional to data not involved in its calibration. Clearly, the dependency of any statistical measure on a specific choice of reference data cannot entirely be removed, but bootstrapping can yield a good approximation to the problem given the data under consideration is representative of the underlying population distribution, which we have assessed in detail by the jackknife-after-bootstrapping method. To study the reliability of density functional rankings with respect to the composition and number of data points, we have drawn samples of different size (N = 5, 10, 20, 39) with replacement (B = 2.5×103 each) from the empirical population distribution of the consistent reference data set (N = 39). For every bootstrap sample, the isomer shifts have been perturbed randomly according to their experimental uncertainty (for details, see Section B.1.3) to allow for a statistically justifiable variation between the synthetic data sets. In Fig. 5.6, we show the percentage of first places that a density functional has reached for 2,500 different data sets and four different data set sizes. The rankings are based on the RMSE. For all different data sets considered, B3LYP, B3PW91, and PBE0 are still most frequently placed first, but the respective percentage varies significantly. For N = 5 data points, PBE0 reveals the highest number of first places, but only in about 60% of cases. Even all pure density functionals are placed first in at least 10% of cases, although the different performance measures are clearly in favor of hybrid density functionals regarding the reference data set (Table 5.6). Hence, calibration studies based on 5 data points are highly susceptible to random conclusions about the transferability of density functionals. With an increasing number of data points, the percentage of first places continuously decreases for all pure density functionals and for M06, while TPSSh reveals relatively constant results for all numbers of data points considered (between 30% and 40% first places). By contrast, the percentage of first places for B3LYP, B3PW91, and PBE0 increases continuously, with PBE0 being ranked first in > 95% of cases for 39 data points. Qualitatively identical and quantitatively similar results were obtained for density functional rankings based on the RMPV (Fig. S2 in Ref. 38). This finding

77 Chapter 5 Reliable Estimation of Prediction Uncertainty

N = 5 N = 10 TPSSh TPSSh M06 M06 PBE0 PBE0 B3PW91 B3PW91 B3LYP B3LYP TPSS TPSS M06-L M06-L PBE PBE PW91 PW91 BLYP BLYP BP86 BP86 PWLDA PWLDA 0 20 40 60 80 100 0 20 40 60 80 100 percentage of first places percentage of first places

N = 20 N = 39 TPSSh TPSSh M06 M06 PBE0 PBE0 B3PW91 B3PW91 B3LYP B3LYP TPSS TPSS M06-L M06-L PBE PBE PW91 PW91 BLYP BLYP BP86 BP86 PWLDA PWLDA 0 20 40 60 80 100 0 20 40 60 80 100 percentage of first places percentage of first places

Figure 5.6: Percentage of first places a density functional reached for B = 2,500 synthetic data sets of different size (N = 5, 10, 20, 39), which were generated by drawing from the empirical population distribution of the consistent reference data set (N = 39) with replacement. The rankings were determined on the basis of the RMSE, − which was rounded to the experimental resolution of 0.01 mm s 1. Hence, more than one density functional can be placed first for a given synthetic data set, which is why the bars sum up to >100%. For every bootstrap sample, the isomer shifts have been perturbed randomly according to their experimental uncertainty (for details, see Section B.1.3). The corresponding ACED values were obtained with the def2-TZVP basis set for ligand atoms. This figure is reproduced with permission from J. Proppe, M. Reiher, J. Chem. Theory Comput. 2017, 13, 3297. Copyright 2017 American Chemical Society.

indicates that for the given experimental resolution, parameter uncertainty plays no signiﬁcant role, even for 5 data points where the standard deviation in the parameters is about 4 times larger compared to 39 data points (Fig. S5 in Ref. 38). Consequently, both RMSE and r2 can be considered stable performance measures for density functional rankings applied in isomer shift predictions of molecular iron compounds. Note that we did not employ the R632 as performance measure as it is very costly to sample an error for every bootstrap sample, which is equivalent to bootstrapping bootstrap samples (double bootstrapping135). However, as shown above, the RMPV can be expected a good approximation to the R632 (as well as RLOO). The fact that more than one density functional is placed ﬁrst on average is clearly an indicator of the low experimental resolution. When increasing the experimental res-

78 How Reliable are Density Functional Rankings Based on a Specific Data Set? 5.6

olution from 0.01 mm s−1 to 0.001 mm s−1 in a statistically sound way (for details, see Section B.1.4), the results based on the RMSE change significantly (Fig. 5.7). The percentage of first places clearly decreased for 5 data points (all density functionals are placed first in less than 40% of cases), and a clear preference for a density functional (more than 95% first places) can be expected only for a number of data points significantly larger than that employed in this study (N = 39). Similar results were obtained for density functional rankings based on the RMPV (Fig. S3 in Ref. 38).

Figure 5.7: Percentage of first places a density functional reached for B = 2.5 × 103 synthetic data sets of different size (N = 5, 10, 20, 39), which were generated by drawing from the empirical population distribution of the consistent reference data set (N = 39) with replacement. The rankings were determined on the basis of the − RMSE, which was rounded to an artificially increased experimental resolution of 0.001 mm s 1 (for details, see Section B.1.4). Hence, more than one density functional can be placed first for a given synthetic data set, which is why the bars sum up to >100%. For every bootstrap sample, the reference isomer shifts have been perturbed randomly according to their experimental uncertainty (for details, see Section B.1.3). The corresponding ACED values were obtained with the def2-TZVP basis set for ligand atoms. This figure is reproduced with permission from J. Proppe, M. Reiher, J. Chem. Theory Comput. 2017, 13, 3297. Copyright 2017 American Chemical Society.

Reducing the data set dependency of the MPU by the application of bootstrapping and increasing the experimental resolution in a statistically sound way, we ﬁnd a clear preference for PBE0 as the density functional with the highest transferability. This preference even remains (a) when resubstituting the 5 inconsistent input–target pairs

79 Chapter 5 Reliable Estimation of Prediction Uncertainty

(#1, #2, #7, #13, #28; Table 5.1 and Fig. 5.1, left) into our consistent reference data set (N = 44), and (b) when selecting only those reference isomer shifts for which experimental uncertainties were reported (without resubstitution of inconsistent data points, N = 28); see Fig. S4 in Ref. 38. Therefore, we suggest that PBE0 is the density functional of choice for applications (with the CP(PPP) basis for Fe and def2-TZVP for all other elements; see Section B.2), which leads to the following linear isomer shift model obtained from bootstrapped (B = 106) linear least-squares regression,

∗ −1 −1 δ(ρ0, w¯ ) = 2.888(1) × 10 mm s (5.7) PBE0 ( ) −1 −1 3 − ρ0 − ρ¯ × 3.619(1) × 10 mm s bohr , 1 N∑=39 ρ¯ = ρ = 11819.0531906 bohr−3 , (5.8) N n n=1

the corresponding covariance matrix, ( ) × −5 × −6 3 2 2.959(4) 10 5.994(48) 10 bohr 2 −2 σw∗ = − − mm s , (5.9) PBE0 5.994(48) × 10 6 bohr3 6.641(9) × 10 5 bohr6

and the corresponding increased MSE,

N −3 2 −2 MSED,w¯ ∗ = 1.185(1) × 10 mm s . (5.10) N − M − 1 PBE0

We omitted the subscript “absorber” for the ACED, ρ. For reasons of reproducibility, ∗ we specify four signiﬁcant ﬁgures for the characteristic values contained in w¯ PBE0 and σ ∗ , and for the increased MSE. For the mean ACED, ρ¯, we employed the raw- wPBE0 data precision provided in Tables S2 and S3 of Ref. 38. The standard deviations in parentheses were obtained from bootstrapping (B = 103) the 106 parameter estimates.

Given a single new ACED, ρ0, the corresponding prediction uncertainty, u(ρ0) with ⊤ ρ0 = (1, ρ0 − ρ¯) , would, according to Eq. (3.17), read √ N ∗ ⊤ 2 u(ρ0) = MSED,w¯ + ρ0 σw∗ ρ0 . (5.11) N − M − 1 PBE0 PBE0

5.7 Effect of Exact Exchange on Model Prediction Uncertainty

In this section, we briefly discuss our observation why hybrid density functionals yielded significantly lower MPU estimates compared to pure density functionals. A systematic (even linear) behavior of relative energies of states of different spin multiplicity on the admixture of exact exchange (i.e., Hartree–Fock-type exchange) in an energy density functional has already been observed more than 15 years ago.239–241 Since then, the

80 Effect of Exact Exchange on Model Prediction Uncertainty 5.7

exact exchange admixture, measured by the linear parameter c3 following the notation in Ref. 239, has been well recognized as one of the most crucial parameters determining the accuracy of (hybrid) density functionals (see Refs. 242–245 for recent systematic investigations). The linear dependence of such relative energies in many (but not all) cases is remarkable and results from a linear dependence of the absolute electronic energies on c3, in which the effect of a self-consistently optimized (and hence, changing) electron density plays a negligible role. Here, we calculated the ACED of all 44 reference compounds (Table 5.1) for varying amounts of exact exchange from c3 = 0.00 to c3 = 0.50 in steps of 0.05 in the B3LYP density functional (for which c3 = 0.20 was originally set). Interestingly, also the contact 2 density features a linear dependence on c3 (r > 0.9995 in all compounds), although with different slopes. In other words, the first derivative of the ACED with respect to c3 is constant for a given compound, but different for all complexes (Fig. 5.8). This complex-specificity is also confirmed by bootstrapping (B = 103) the uncertainty of the first derivative of the ACED (see the error bars in Fig. 5.8, which represent three standard deviations).

5 10 15 20 25 30 35 40 2

1 ) − 3

0 − mean (bohr 3

dACED / d c / dACED −1

−2 5 10 15 20 25 30 35 40 compound ID

Figure 5.8: Change of the B3LYP/def2-SVP ACED with respect to the exact exchange admixture, dACED / dc3, for all 44 compounds studied (Table 5.1 and Figs. S6−S16 38). The constant ﬁrst derivative indicates the linear behavior of the ACED with respect to the exact exchange parameter c3. For better comparability, we subtracted the mean derivative obtained for all compounds. This ﬁgure is reproduced with permission from J. Proppe, M. Reiher, J. Chem. Theory Comput. 2017, 13, 3297. Copyright 2017 American Chemical Society.

81 Chapter 5 Reliable Estimation of Prediction Uncertainty

Consequently, a change in the exact exchange admixture does not lead to a unique

ACED shift for all compounds, which is why the MPU is a nonlinear function of c3. Plotting the R632 against the exact exchange admixture reveals that this performance

measure is minimized for c3 = 0.20–0.25 (Fig. 5.9, where N = 39), which indicates

why B3LYP, B3PW91 (both featuring c3 = 0.20), and PBE0 (c3 = 0.25) are the most transferable density functionals with respect to isomer shift calibration.

BLYP

0.08 PW91 BP86 PWLDA PBE TPSS 0.07 M06-L ) − 1

M06

0.06 R632 (mm s

TPSSh 0.05

B3PW91 PBE0 0.0 0.1 0.2 0.3 0.4 0.5

exact exchange c3

Figure 5.9: R632 versus exact exchange admixture parameter c3 for isomer shift models obtained from B3LYP

(black dots) with varying values of c3, and for the other density functionals investigated in this study (without further modiﬁcation: red dots). The dashed curve was obtained from cubic spline interpolation. For clarity, we − did not round the results to the experimental resolution of 0.01 mm s 1. We employed our consistent reference data set (N = 39) for calculating the R632 (def2-SVP basis set). This ﬁgure is reproduced with permission from J. Proppe, M. Reiher, J. Chem. Theory Comput. 2017, 13, 3297. Copyright 2017 American Chemical Society.

The observation of linear dependence of the ACED is not straightforward to explain. It is clear that only basis functions with angular momentum quantum number l = 0 (s-type functions) can contribute to the nonrelativistic contact density in an atomic system (note the short-range behavior ∝ rl of the radial function in a one-electron atom246). This short-range behavior is (also in the spherically averaged case of an atom in a molecule247) determined by the lowest-order contributions of the potential to a Taylor series expansion in terms of the radial distance in the short-range quantum mechanical diﬀerential equations for a single electron. The lowest-order constributions are the centrifugal and the nuclear point charge potentials, but not the electron–electron

82 Conclusions 5.8 interaction potentials, which provide a constant contribution at the nuclear position.248 Hence, for a spherically symmetric atom, this is also true for the Hartree–Fock exchange interaction, which contributes a constant term to the potential at the nuclear origin248 and would therefore not, in this case, aﬀect the radial function, and hence, the electron density at the nucleus. For a (nonspherical) molecular system, additional contributions to the contact density need to be considered. Following the partitioning scheme of the ACED by Neese,128 we ﬁnd for compound #43 (iron pentacarbonyl) that the iron s-functions in the CP(PPP) basis set contribute dominantly to the ACED (> 99.9%). Hence, their contribution to the ACED alone already reveals the linear trend of the ACED with respect to exact exchange admixture. However, the three s-functions in the CP(PPP) basis set with the largest exponent are constant with respect to a change of c3, whereas the smallest

11 s-functions change linearly with c3, even though their slope is sometimes positive (for larger exponents) and sometimes negative (for smaller exponents), which makes a detailed analysis diﬃcult. Interestingly, considering not only the charge at one speciﬁc point, i.e., the ACED, but the integral of the electron density over some space, i.e., a partial charge of the iron atom in iron pentacarbonyl, also reveals a linear behavior with c3 although with negative slope (see also Ref. 242). Hence, Mulliken and Löwdin partial charges decrease with increasing c3, while the ACED increases.

5.8 Conclusions

To reliably estimate the prediction uncertainty of a property model, it is important to identify and correct for systematic errors due to, e.g., inconsistent measurements, parametric population assumptions, or inadequate computational models. Here, we studied this issue at the example of 57Fe Mössbauer isomer shift predictions based on a linear model. Twelve density functionals across Jacob’s ladder were considered for the calculation of the ACED for 44 chemically diverse molecular iron compounds (formal oxidation states: 0, +1, +2, +3, +4), whereas the corresponding target data refer to measured isomer shifts. We explicitly considered uncertainty in the model parameters, which may be a crucial ingredient for the estimation of MPU. For this purpose, we employed both bootstrapping124,134,135,137 and Bayesian linear regression based on the evidence approximation.119 First of all, we found that the RMSE, which measures the standard deviation of the model residuals, is signiﬁcantly larger than the average experimental uncertainty (0.07–0.10 mm s−1 versus 0.02 mm s−1). This discrepancy cannot be explained by the simplicity of the linear model as more complex property models (quadratic, cubic, quartic) were found to yield either equal or larger MPU estimates when applying bootstrapping. However, with the jackknife-after-bootstrapping approach, which probes the

83 Chapter 5 Reliable Estimation of Prediction Uncertainty

sensitivity in the MPU with respect to small changes in the reference data set, we could identify 5 inconsistent data points. The reliability of the jacknife-after-bootstrapping approach can be assessed by determining the fraction of data points lying in a certain prediction interval. When including the inconsistent data points, 86% of the reference isomer shifts lie in the 68% prediction band (representing u(x) as defined in Eq. (3.17)), whereas excluding them leads to only 72% of the reference isomer shifts contained in the 68% prediction band, which reveals a significant improvement of the statistical reliability. Furthermore, the high agreement of bootstrapped and Gaussian posterior parameter distributions (the latter were obtained from Bayesian linear regression) suggests that the normal-population assumption is reasonable for isomer shift calibration given the reference data set is carefully selected. The new (consistent) reference data set still leads to overestimation of the average experimental uncertainty (RMSE of 0.03–0.08 mm s−1), which can be assigned to nonconstant systematic errors of the density functionals leading to random shifts of the (unknown) true input values. This random model inadequacy suggests that weighted least-squares regression, where experimental uncertainties are explicitly introduced as a weight, may bias the calibration of the property model. However, on the basis of bootstrapped regression we did not find a significant difference between the ordinary and the weighted setup due to pronounced parameter uncertainty, whereas Grandjean and Long196 recommend the application of weighted least-squares regression for the calibration of isomer shift models. Comparison of the RMSE with more reliable performance measures that consider uncertainty in the parameters (such as R632, RMPV, or RLOO) reveals nearly equal results in most of the cases. This finding can be explained by the low experimental resolution of 0.01 mm s−1 compared to the width of the input domain (here, 1.08 mm s−1), which almost completely masks parameter uncertainty. Therefore, simple performance measures such as the RMSE and the squared coefficient of correlation, r2, are expected to be reliable for the construction of density functional rankings, where relative rather than absolute MPU estimates are required. The same holds for the RMSD and the mean unsigned error as defined and discussed in a calibration study on physical properties of crystals by Pernot et al.116 Next, we examined the sensitivity of density functional rankings with respect to the composition and number of data points. For this purpose, we generated 10,000 synthetic data sets (based on bootstrapping) with 5, 10, 20, and 39 data points. For 5 data points, false conclusions about the performance of a density functional are likely. For instance, the “best” density functional, PBE0, is placed first in only about 60% of cases. In the case of 39 data points, B3LYP and B3PW91 are placed first in 85– 90% of cases, whereas PBE0 is placed first in more than 95% of cases. Still, there

84 Conclusions 5.8 remains a serious chance of favoring a subprime density functional for usual data set sizes (N < 40)128–132,176–191,193–196 employed in previous calibration studies of the isomer shift. RMSE and RMPV produced equivalent density functional rankings. However, due to the low experimental resolution, important effects stemming from parameter uncertainty may be hidden. We examined this hypothesis by artificially increasing the experimental resolution from 0.01 mm s−1 to 0.001 mm s−1. While RMSE and RMPV still yield similar rankings, the percentage of first places clearly decreases. For instance, the “best” density functional, PBE0, is placed first in less than 40% of cases for 5 data points. Even for 39 data points, the identification of PBE0 as “best” density functional is only successful in 70–75% of cases. Finally, we discussed our observation that hybrid density functionals yield significantly lower MPU estimates compared to pure density functionals. When varying the exact exchange admixture of the B3LYP density functional (for which the linear exact 239 exchange parameter c3 was originally set to c3 = 0.20), we found a linear dependence 2 of the ACED on c3 for all 44 iron compounds studied (r > 0.9995). However, the first derivative of the ACED with respect to c3 is complex-specific, which is why the MPU is a nonlinear function of c3. It is minimized for c3 = 0.20–0.25, matching with our observation that B3LYP, B3PW91 (c3 = 0.20), and PBE0 (c3 = 0.25) yielded isomer shift models with the lowest prediction uncertainty. When selecting a density functional for actual applications, one is interested in a reliable uncertainty estimation of an isomer shift prediction for a given ACED. For this purpose, the prediction bands employed in this study represent locally resolved MPU based on the .632 estimator. The validity of the MPU estimate is highest at the mean ACED for a given density functional and decreases continuously from there, which is why extrapolations outside the input domain studied are not recommended. This limitation motivated us to cover a wide range of possible ACED values such that chemically diverse iron complexes can be investigated on the basis of our calibration analysis. Another possibility to approach accurate isomer shift predictions is to calculate the isomer shift directly from first principles. Filatov derived the corresponding computational scheme249 and applied it to examine the isomer shift of molecular iron compounds.250 Postcalibration of the resulting isomer shift pairs (measured versus calculated) would allow for an assessment of the more adequate input variable (isomer shift versus ACED). We discussed several issues in this chapter ranging from error diagnostics, MPU estimation, and the role of experimental resolution to the reliability of conclusions drawn on the basis of a specific data set. Bootstrapping clearly shows that performance assessment of density functionals is error-prone for a small number of data points, and for univariate linear regression models, one should consider at least about 40 data points.

85 Chapter 5 Reliable Estimation of Prediction Uncertainty

While the main message of previous calibration studies of the isomer shift remains (hybrid density functionals perform superior to pure density functionals128,130,131,177,182,192), we now provide a solid statistical framework to examine the certainty of such findings. This framework allowed us to suggest a new, statistically well-justified property model for the isomer shift, given in Eqs. (5.7) to (5.11), based on PBE0 calculations. More- over, we introduced a new reference data set for such parameterizations, MIS39, which may be employed and extended for future work. In particular, this calibration study presents the first statistically rigorous analysis for theoretical Mössbauer spectroscopy providing the practitioner with reliable, locally resolved uncertainties for isomer shift predictions. Moreover, our calibration analysis is of general applicability and not restricted to property models applied in Mössbauer spectroscopy. We provide the code of our statistical calibration software reBoot on our webpage175 so that the methodology presented here can be applied in other parameterization studies of physicochemical property models.

86 6 Case Study: Thermochemical, Kinetic, and Spectroscopic Modeling of Iron Porphyrin Carbene Chemistry

Metallocarbenes constitute an appealing chemical compound class as they facilitate crucial reactions such as N–H insertion, C–H insertion, and O–H insertion.* Iron carbenes are nontoxic and low-cost alternatives to second-row and third-row metallocarbenes, and can mediate reactions such as cyclopropanation and carbonyl olefination. Here, we study an iron porphyrin carbene (IPC) complex with an Fe–C(carbene)–N(porphyrin) bridging configuration. Our theoretical analysis suggests that, although the formation of an inert bridged complex is thermodynamically favored over its more reactive end-on isomer, rapid equilibration between the two isomers ensures efficient cyclopropanation.

6.1 Thermochemical Analysis of Iron Porphyrin Carbene Reactiv- ity

For our theoretical analysis of the IPC complex with a proximal N-methylimidazole ligand, we introduce different energy measures and contributions that are defined in Ta- ble 6.1. The definitions are based on the standard model of gas phase thermochemistry, which assumes a complete decoupling of all degrees of freedom (electronic, vibrational,

*This chapter is reproduced in part with permission from T. Hayashi, M. Tinzl, T. Mori, U. Krengel, J. Proppe, J. Soetbeer, D. Klose, G. Jeschke, M. Reiher, D. Hilvert, Nat. Catal. 2018, 1, 578. Copyright 2018 Springer Nature.

87 Chapter 6 Case Study: Iron Porphyrin Carbenes

rotational, translational), harmonic vibrations, a rigid rotator, and the particle-in-a-box model for translational motion. Within this model, rotational and translational partition functions can be calculated from the equilibrium coordinates of the atomic nuclei alone, whereas the vibrational partition function requires a quantum chemical frequency analysis. In the following sections, we carefully discuss all energy measures and contributions deﬁned in Table 6.1 to provide a reliable thermochemical description of the stability and reactivity of the IPC complex. Although we are considering a condensed-phase system — the thermochemistry of which is usually hard to model —, our discussion will reveal that the standard model of gas phase thermochemistry can be harnessed to derive sensible corrections to the electronic energy of condensed-phase systems. The thermochemical

energy that we propose to use as an approximate enthalpy, Happrox (Table 6.1), neglects translational and global rotational motion.

Table 6.1: Deﬁnitions of thermochemical energy contributions. R and T represent the gas constant and temperature, respectively. We assume that the operating conditions are a pressure of p = 1 atm and a temperature of T = 298.15 K (unless otherwise mentioned). For an ideal gas, the volume V is given by nRT/p, where n is the amount of substance. All energies are deﬁned with respect to an amount of substance of n = 1 mol, which results − − in a standard state concentration of y = n/V = 4.09 × 10 2 mol L 1. For all electronic, vibrational, rotational, and translational contributions, we assume electronic ground state degeneracy, the harmonic approximation, a

rigid rotor (with symmetry number σ = 2 for H2O and N2, and σ = 1 for all other molecules), and a particle in a 3-dimensional box, respectively.

Energy Deﬁnition Eel Electronic energy (T = 0 K) EZPE Zero-point vibrational energy (T = 0 K) E0 ≡ Eel + EZPE Total energy at T = 0 K Evib Vibrational energy excluding EZPE Erot Rotational energy Etrans Tranlational energy Etherm ≡ Evib + Erot + Etrans Thermal correction to energy U ≡ E0 + Etherm Internal energy Uapprox ≡ U − Erot − Etrans Approximate internal energy neglecting rotational and translational contributions H ≡ U + pV/n ≈ U + RT Enthalpy Happrox ≡ H − Erot − Etrans Approximate enthalpy neglecting rotational and translation contributions Sel Electronic entropy Svib Vibrational entropy Srot Rotational entropy Strans Translational entropy S ≡ Sel + Svib + Srot + Strans Total entropy G ≡ H − TS Gibbs free energy

88 Thermochemical Analysis of Iron Porphyrin Carbene Reactivity 6.1

6.1.1 Computational Details

All electronic structure calculations were carried out with the Gaussian 09 suite of pro- grams251 (see Tables 6.2 and 6.3 for an overview of all species and elementary reactions considered). We performed PBE252/def2-TZVP253,254 and PBE0255/def2-TZVP structure optimizations and also PBE0/def2-TZVP and B3LYP⋆ 239/def2-TZVP single-point calculations on the PBE/def2-TZVP optimized structures. To keep the notation uncluttered, we denote these approximations PBE//PBE, PBE0//PBE0, B3LYP⋆//PBE, and PBE0//PBE, respectively. Here, the ﬁrst label refers to the density functional employed for a single-point electronic energy calculation on the optimized structure obtained with the density functional denoted by the second label. For structure optimizations of open-shell singlet states, we performed a single-point calculation of the triplet state on the optimized triplet structure and subsequently optimized it in an un- restricted Kohn–Sham framework with an equal number of α- and β-electrons such that

S = MS = 0 (chkbasis and guess=mix keywords in Gaussian 09), where S and MS are the spin quantum number and the spin projection quantum number, respectively. The PBE//PBE Hessian of the electronic energy with respect to the nuclear coordinates was calculated to analyze thermochemical energies of the species and reactions studied. To obtain guess structures for a transition state search, we performed relaxed scans (PBE/def2-SVP253,254) of internal coordinates (in steps of 0.1 Å or 0.004 Å for bond lengths and 10° for torsional angles) expected to be important for the transition studied. The maximum-energy structure of the relaxed scan was subjected to a transition state optimization (PBE/def2-TZVP) based on the quadratic synchronous transit method256 (opt=qst3 keyword in Gaussian 09), which also requires the two corresponding minimum-energy structures. We found that the minimum-energy structures for the reactions with EDA and styrene are so diﬀerent from the transition state guess that we considered those structures from the relaxed scan adjacent to its maximum- energy structure. Subsequently, reactants and products were validated by following the intrinsic reaction coordinate74 (PBE/def2-TZVP) from the optimized transition state (irc keyword in Gaussian 09), with a full structure optimization of the resulting reactants and products.

6.1.2 Structure and Stability of the Bridged IPC Complex

We truncated the protein crystal structure of the IPC complex (bridged conﬁguration) to the minimal (generic) model shown in Fig. 6.1. In our theoretical analysis, we considered the low-spin state, the intermediate-spin state, and the high-spin state of the formal iron oxidation states +II (neutral species, 1) and +III (cationic species, 1+), respectively, for molecular structure optimizations. Relative energies and structural parameters are summarized in Tables 6.4–6.6.

89 Chapter 6 Case Study: Iron Porphyrin Carbenes

Table 6.2: Notation for chemical species and transition states.

Species Description 1 IPC complex, bridged conﬁguration 1+ Cation of 1 2 IPC complex, end-on conﬁguration 2+ Cation of 2 3 Iron porphyrin complex, carbene ligand replaced by H2O 3+ Cation of 3 4 Ethyl diazoacetate (EDA) 5 Diethyl maleate 6 Styrene 7 Ethyl 2-phenylcyclopropanecarboxylate (+) + Σz Adduct of 1 or 1 with z ≡ {4, 6} TS1,2 Transition state of the reaction 1 ⇌ 2 + + ⇌ + TS1,2 Transition state of the reaction 1 2 TS14,35 Transition state of the reaction 1 + 4 + H2O ⇌ 3 + 5 + N2, here, referring to the elementary step 1 + 4 ⇌ Σ4 + + ⇌ + TS14,35 Transition state of the reaction 1 + 4 + H2O 3 + 5 + N2, + ⇌ + here, referring to the elementary step 1 + 4 Σ4 TS16,37 Transition state of the reaction 1 + 6 + H2O ⇌ 3 + 7, here, referring to the elementary step 1 + 6 ⇌ Σ6 + + ⇌ + TS16,37 Transition state of the reaction 1 + 6 + H2O 3 + 7, + ⇌ + here, referring to the elementary step 1 + 6 Σ6 TS24,35 Transition state of the reaction 2 + 4 + H2O ⇌ 3 + 5 + N2 TS26,37 Transition state of the reaction 2 + 6 + H2O ⇌ 3 + 7

The PBE//PBE results suggest that the singlet state of 1 is slightly more stable than the triplet state (8.5–13.0 kJ mol−1) and significantly more stable than the quintet state (63.5–71.2 kJ mol−1). By contrast, the PBE0//PBE results suggest that the singlet state of 1 is significantly less stable than both the triplet state (40.5–44.9 kJ mol−1) and the quintet state (17.8–25.4 kJ mol−1). For 1+, the findings are similar. The PBE//PBE results suggest that the doublet state is more stable than the quartet and sextet states (22.9–28.5 kJ mol−1 and 93.2–103.8 kJ mol−1, respectively), whereas the PBE0//PBE results suggest the quartet state to be more stable than the doublet state (18.0–23.5 kJ mol−1), or balanced energetics between the doublet state and the sextet state, i.e., −0.2–10.4 kJ mol−1 with respect to the doublet state. Reiher, Salomon, and Hess demonstrated more than 15 years ago that spin state energy splittings depend linearly on the exact-exchange admixture so that pure density functionals favor low-spin states, whereas hybrid density functionals favor high-spin states.239–241 They defined the B3LYP⋆ density functional239 with a reduced exact–

90 Thermochemical Analysis of Iron Porphyrin Carbene Reactivity 6.1

Table 6.3: Chemical transformation steps of chemical species and transition states. See Table 6.2 for deﬁnitions of the boldface numbers and characters.

Symbol Transformation step A1 1 → 2 A2 1 → TS1,2 A3 2 → TS1,2 + + B1 1 → 2 + → + B2 1 TS1,2 + → + B3 2 TS1,2 C1 1 + 4 → Σ4 C2 1 + 4 + H2O → 3 + 5 + N2 C3 1 + 4 → TS14,35 C4 Σ4 → TS14,35 + → + D1 1 + 4 Σ4 + + D2 1 + 4 + H2O → 3 + 5 + N2 + → + D3 1 + 4 TS14,35 + → + D4 Σ4 TS14,35 E1 1 + 6 → Σ6 E2 1 + 6 + H2O → 3 + 7 E3 1 + 6 → TS16,37 E4 Σ6 → TS16,37 + → + F1 1 + 6 Σ6 + + F2 1 + 6 + H2O → 3 + 7 + → + F3 1 + 6 TS16,37 + → + F4 Σ6 TS16,37 G1 2 + 4 + H2O → 3 + 5 + N2 G2 2 + 4 → TS24,35 G3 3 + 5 + N2 → TS24,35 + + H 2 + 4 + H2O → 3 + 5 + N2 J1 2 + 6 + H2O → 3 + 7 J2 2 + 6 → TS26,37 J3 3 + 7 → TS26,37 + + K 2 + 6 + H2O → 3 + 7 exchange admixture of 15% (instead of 20% in B3LYP) to yield more reliable spin state energy splittings, especially for iron compounds (note that PBE0 contains a 25% exact-exchange admixture). The B3LYP⋆//PBE results (Tables 6.4 and 6.5) suggest the triplet state of 1 to be slightly more stable than the singlet state by 14.3–18.8 kJ mol−1, whereas the singlet state is — analogously to PBE//PBE — clearly favored over the quintet state (44.4– 52.1 kJ mol−1). Similarly, we ﬁnd balanced energetics between the doublet and quartet

91 Chapter 6 Case Study: Iron Porphyrin Carbenes

Figure 6.1: Truncated X-ray diffraction structure of the IPC complex including atom labels. Color code: light gray, carbon; white, hydrogen; magenta, iron; blue, nitrogen; red, oxygen; dark gray, chemical bond. This ﬁgure is reproduced with permission from T. Hayashi, M. Tinzl, T. Mori, U. Krengel, J. Proppe, J. Soetbeer, D. Klose, G. Jeschke, M. Reiher, D. Hilvert, Nat. Catal. 2018, 1, 578. Copyright 2018 Springer Nature.

states of 1+ (the quartet state is 2.4–8.0 kJ mol−1 higher in energy) and a distinct destabilization of the sextet state over the doublet state (45.4–56.0 kJ mol−1). Combined with the PBE//PBE results, we expect the low-spin and intermediate-spin states of 1 and 1+ to be potentially relevant from a thermochemical perspective. In comparison to the crystal structure (Table 6.6), the only reasonable optimized structure obtained from both PBE and PBE0 is that of the low-spin state. For PBE, the Fe–C1 and Fe–N1 bond lengths (for a definition of atom labels see Fig. 6.1) are overestimated by all optimized structures (0.05–0.26 Å and 0.08–0.49 Å, respectively), whereas the C1–N1 bond length is estimated fairly accurately by most of the optimized structures (max. 0.07 Å deviation from crystal structure). In cases where bond distances in the optimized structure did not match the values observed in the crystal structure very well, the lower bounds of bond length deviations correspond to the low-spin states of 1 and 1+. For instance, the Fe–N1 bond length and Fe–C1–N1 bond angle are only reproduced with sufficient accuracy by the low-spin states of both 1 and 1+ (+0.08 Å and +0.4–0.9°, respectively), whereas all higher spin states systematically feature larger values for both internal coordinates (+0.41–0.49 Å and +9.6–19.0°, respectively). At the same time, the optimized low-spin states underestimate the Fe–N5 bond length more strongly (−0.21–0.23 Å) than the higher spin states (−0.11–0.14 Å). However, this deviation, although significant, is still small compared to that observed for the Fe– N1 bond length and might originate from the truncation of the protein backbone to the N-methylimidazole ligand. For all optimized structures, we observe that the N1 atom moves slightly out of the

92 Thermochemical Analysis of Iron Porphyrin Carbene Reactivity 6.1 porphyrin plane, which is not observed for the crystal structure. We also observe that the torsional angles Fe–C1–C2–O and N3–Fe–N5–C3 deviate significantly between the crystal and optimized structures. The latter torsional angle describes the rotational orientation of the N-methylimidazole ligand. To study the effect of the imidazole orientation on the out-of-plane porphyrin nitrogen atom N1, we carried out a scan of the N3–Fe–N5–C3 torsional angle (36 steps of 10°, PBE//PBE) while relaxing all other degrees of freedom. We find that the N1–N2–N3–N4 torsional angle (representing the out-of-plane character) does not change significantly (10.2°–11.5° in 1 and 10.8°–12.0° in 1+ versus 0.2° in the crystal structure). Furthermore, we studied the correlation of the Fe–C1–C2–O and N3–Fe–N5–C3 torsional angles of both 1 and 1+ by performing three constraint structure optimizations (PBE//PBE), where we fixed (a) the first, (b) the second, and (c) both torsional angles while relaxing all other degrees of freedom. We find that the electronic energy difference between the fully optimized structure, Eel, and the third constrainedly optimized structure, Eel,c, can be approximated by the sum of the electronic energy differences between the fully optimized structure and each of the first( two contrainedly) ( optimized) structures, Eel,a and Eel,b, respectively, i.e., Eel − Eel,c ≈ Eel − Eel,a + Eel − Eel,b . The error is smaller than 0.2 kJ mol−1, which we consider negligible. We conclude that the rotational orientation of the imidazole ligand has a negligible effect on the reactivity of the carbene carbon atom C1. For PBE0, we find equivalent deviations from the crystal structure compared to PBE (Table 6.6). We conclude that the low-spin states of both 1 and 1+ are favored in terms of thermochemistry and molecular structure. Consequently, we will consider the singlet state of 1 and the doublet state of 1+ in the following analysis.

6.1.3 Stability of the End-On IPC Complex

For the construction of the low-spin states of the neutral, 2, and the cationic, 2+, IPC complexes in the end-on configuration, we performed relaxed scans of the C1–N1 bond in both low-spin structures of 1 and 1+, and optimized the minimum-energy structures. Starting from the optimized low-spin states of 2 and 2+, we performed structure optimizations of the intermediate-spin and high-spin states, respectively. Note that we cannot compare structural parameters in this case as the X-ray diffraction experiment only exhibits a bridged complex. For 2 (Table 6.7), our PBE//PBE results suggest that the singlet state is significantly more stable than the triplet (44.6–48.6 kJ mol−1) and quintet (125.4–136.0 kJ mol−1) states compared to 1 (8.5–13.0 kJ mol−1 and 63.5–71.2 kJ mol−1, respectively). We find a similar but expectedly less pronounced preference for the singlet state of 2 for B3LYP⋆//PBE (20.0–24.0 kJ mol−1 and 75.5–86.2 kJ mol−1 compared to the triplet and

93 Chapter 6 Case Study: Iron Porphyrin Carbenes

Table 6.4: Energies of the intermediate- (triplet) and high-spin (quintet) states of 1 relative to its low-spin state (singlet [the open- and closed-shell states converged to the same solution]) at T = 298.15 K and p = 1 atm. The

electronic energy, Eel, has been calculated on the basis of three different approximations (PBE//PBE, PBE0//PBE, and B3LYP⋆//PBE). To keep the notation uncluttered, we mention each approximation only once as a subscript label. All nonthermal and thermal corrections to the electronic energy have been calculated on the basis of

the PBE//PBE approximation. Here, ∆H = ∆U and ∆Happrox = ∆Uapprox. All energy differences are reported − in kJ mol 1.

Total energy Correction Triplet Quintet

∆Eel,PBE//PBE 9.9 68.4 ∆EZPE −1.5 −4.9 ∆E0 8.5 63.5 ∆Evib 0.6 1.4 ∆Erot 0.0 0.0 ∆Etrans 0.0 0.0 ∆Etherm 0.6 1.4 ∆H 9.1 64.9 ∆Happrox 9.1 64.9 T ∆Sel 2.7 4.0 T ∆Svib 1.1 2.3 T ∆Srot 0.0 0.0 T ∆Strans 0.0 0.0 T ∆S 3.8 6.3 ∆G 13.0 71.2

∆Eel,PBE0//PBE −43.5 −20.5 ∆E0 −44.9 −25.4 ∆H −44.3 −24.0 ∆Happrox −44.3 −24.0 ∆G −40.5 −17.8

∆Eel,B3LYP⋆//PBE −17.3 49.3 ∆E0 −18.8 44.4 ∆H −18.2 45.8 ∆Happrox −18.2 45.8 ∆G −14.3 52.1

quintet states, respectively). In the case of PBE0//PBE, the triplet state is eﬀectively as stable as the singlet state (−2.8–1.4 kJ mol−1 stability gain/loss), whereas the quintet state is less stable than the singlet state by 30.1–41.4 kJ mol−1. As we have discussed, the high exact-exchange admixture of 25% in PBE0 biases spin state energy splitting in favor of higher spin states. Hence, our results indicate that the singlet state of 2 is clearly favored over its quartet and sextet states. For 2+ (Table 6.8), the preference of the low-spin state is even more pronounced.

94 Thermochemical Analysis of Iron Porphyrin Carbene Reactivity 6.1

Table 6.5: Energies of the intermediate- (quartet) and high-spin (sextet) states of 1+ relative to its low-spin state

(doublet) at T = 298.15 K and p = 1 atm. The electronic energy, Eel, has been calculated on the basis of three different approximations (PBE//PBE, PBE0//PBE, and B3LYP⋆//PBE). To keep the notation uncluttered, we mention each approximation only once as a subscript label. All nonthermal and thermal corrections to the electronic energy have been calculated on the basis of the PBE//PBE approximation. Here, ∆H = ∆U and ∆Happrox = ∆Uapprox. − All energy differences are reported in kJ mol 1.

Total energy Correction Quartet Sextet

∆Eel,PBE//PBE 25.4 101.1 ∆EZPE −2.4 −7.8 ∆E0 22.9 93.2 ∆Evib 0.9 2.2 ∆Erot 0.0 0.0 ∆Etrans 0.0 0.0 ∆Etherm 0.9 2.2 ∆H 23.8 95.4 ∆Happrox 23.8 95.4 T ∆Sel 1.7 2.7 T ∆Svib 2.9 5.5 T ∆Srot 0.0 0.1 T ∆Strans 0.0 0.0 T ∆S 4.7 8.4 ∆G 28.5 103.8

∆Eel,PBE0//PBE −21.1 7.7 ∆E0 −23.5 −0.2 ∆H −22.7 2.0 ∆Happrox −22.7 2.0 ∆G −18.0 10.4

∆Eel,B3LYP⋆//PBE 4.9 53.2 ∆E0 2.4 45.4 ∆H 3.3 47.6 ∆Happrox 3.3 47.6 ∆G 8.0 56.0

Here, the results of all density functionals suggest that the doublet state is much more stable than the quartet (>51.8 kJ mol−1) and sextet (>74.4 kJ mol−1) states.

6.1.4 Interconversion Between the Bridged and End-on Complexes

Next, we studied the interconversion between the bridged and end-on conﬁgurations of the IPC complex (for a summary, see Fig. 6.2). We determined the transition states + + + between 1 and 2 (TS1,2), and 1 and 2 (TS1,2). Subsequently, we performed an

95 Chapter 6 Case Study: Iron Porphyrin Carbenes

Table 6.6: Selected internal coordinates of the low-spin (LS), intermediate-spin (IS), and high-spin (HS) states of 1 and 1+, respectively. Deﬁnitions of the atom labels are provided in Fig. 6.1.

Internal coord. X-ray 1-LS 1-IS 1-HS 1+-LS 1+-IS 1+-HS PBE Fe–C1 / Å 1.89 1.94 2.01 2.01 1.95 2.00 2.15 C1–N1 / Å 1.43 1.50 1.43 1.43 1.46 1.43 1.42 Fe–N1 / Å 2.05 2.13 2.52 2.54 2.13 2.50 2.46 Fe–N5 / Å 2.26 2.03 2.15 2.14 2.05 2.12 2.13 Fe–C1–N1 74.9° 75.3° 92.5° 93.9° 75.8° 92.1° 84.5° Fe–C1–C2–O −131.5° −86.3° −110.1° −118.8° −107.2° −115.9° −99.3° N1–N2–N3–N4 2.7° 10.4° 7.5° 5.1° 11.0° 6.8° 9.5° N3–Fe–N5–C3 167.9° −123.5° −91.5° −173.5° −97.9° −170.9° −175.0° PBE0 Fe–C1 / Å 1.89 1.97 1.99 2.16 1.92 1.98 2.13 C1–N1 / Å 1.43 1.49 1.43 1.42 1.46 1.42 1.43 Fe–N1 / Å 2.05 2.17 2.54 2.45 2.09 2.49 2.40 Fe–N5 / Å 2.26 2.02 2.16 2.21 2.05 2.12 2.13 Fe–C1–N1 74.9° 76.7° 94.8° 83.6° 74.9° 92.7° 82.2° Fe–C1–C2–O −131.5° −82.2° −117.1° −91.9° −105.1° −118.9° −98.6° N1–N2–N3–N4 2.7° 10.4° 5.7° 8.4° 11.4° 6.0° 8.6° N3–Fe–N5–C3 167.9° −119.5° 84.4° −176.6° −95.7° −170.5° −175.6°

intrinsic reaction coordinate scan to confirm the correspondence between both configu- rations. Relative energies of both reactions are listed in Tables 6.10 and 6.11. For PBE//PBE, we find that 2 is slightly more stable than 1 (9.7–19.0 kJ mol−1), −1 and that TS1,2 is only 26.4–31.2 kJ mol higher in energy than 1. A striking feature of the thermochemical data is that the difference in the vibrational entropy is quite −1 pronounced between 2 and TS1,2 (TSvib = 8.5 kJ mol ), even though their molecular

structures are distinctly closer to each other compared to 1 and TS1,2. As the low- frequency modes dominate the vibrational entropy, the harmonic oscillator description may fail if these modes correspond to shallow potentials. Contrary to the bridged–end-on interconversion of the neutral IPC complex, our PBE//PBE results suggest that 1+ is significantly more stable than 2+ (46.1– 65.0 kJ mol−1), and that the activation energy for the isomerization of 1+ to 2+ is distinctly higher (72.5–81.0 kJ mol−1). Again, we find a pronounced difference in the −1 + + vibrational entropy (TSvib = 13.5 kJ mol ) between 2 and TS1,2. Noteworthy, we studied the effect of a polarizable environment on the molecular structure, electronic energy, and charge density distribution of the cationic complexes. For this purpose, we repeated the structure optimizations of 1+ and 2+ embedded in a polar continuum solvent (water). The bond lengths listed in Table 6.6, the electronic energy difference between the bridged IPC complex and its end-on isomer, and the partial charges did not change by more than 0.1 Å, 1 kcal mol−1, and 0.02 elementary charges, respectively.

96 Thermochemical Analysis of Iron Porphyrin Carbene Reactivity 6.1

Table 6.7: Energies of the intermediate- (triplet) and high-spin (quintet) states of 2 relative to its low-spin state (singlet [the open- and closed-shell states converged to the same solution]) at T = 298.15 K and p = 1 atm. The electronic energy, Eel, has been calculated on the basis of three different approximations (PBE//PBE, PBE0//PBE, and B3LYP⋆//PBE). To keep the notation uncluttered, we mention each approximation only once as a subscript label. All nonthermal and thermal corrections to the electronic energy have been calculated on the basis of the PBE//PBE approximation. Here, ∆H = ∆U and ∆Happrox = ∆Uapprox. All energy differences are reported − in kJ mol 1.

Total energy Correction Triplet Quintet

∆Eel,PBE//PBE 46.4 134.0 ∆EZPE −1.9 −8.6 ∆E0 44.6 125.4 ∆Evib 0.4 2.2 ∆Erot 0.0 0.0 ∆Etrans 0.0 0.0 ∆Etherm 0.4 2.2 ∆H 45.0 127.6 ∆Happrox 45.0 127.6 T ∆Sel 2.7 4.0 T ∆Svib 0.9 4.3 T ∆Srot 0.0 0.1 T ∆Strans 0.0 0.0 T ∆S 3.6 8.4 ∆G 48.6 136.0

∆Eel,PBE0//PBE −0.7 38.7 ∆E0 −2.6 30.1 ∆H −2.2 32.3 ∆Happrox −2.2 32.3 ∆G 1.4 40.7

∆Eel,B3LYP⋆//PBE 21.8 84.1 ∆E0 20.0 75.5 ∆H 20.4 77.8 ∆Happrox 20.4 77.8 ∆G 24.0 86.2

In the majority of cases, the relative electronic energy is the dominant term of all thermodynamic energy differences considered in this work (see, e.g., Tables 6.10 and 6.11). Other energy contributions are only comparable in magnitude if the electronic energy difference is already quite small (<20 kJ mol−1). Hence, the standard model of gas phase thermochemistry works sufficiently well, even electronic energy differences would suffice if a suitable error margin for density functionals is taken into account. In the case of Fe(II), the average difference between total-energy measures for a given density

97 Chapter 6 Case Study: Iron Porphyrin Carbenes

Table 6.8: Energies of the intermediate- (quartet) and high-spin (sextet) states of 2+ relative to its low-spin state

ergy have been calculated on the basis of the PBE//PBE approximation. Here, ∆H = ∆U and ∆Happrox = ∆Uapprox. − All energy differences are reported in kJ mol 1.

Total energy Correction Quartet Sextet

∆Eel,PBE//PBE 71.0 174.2 ∆EZPE −3.3 −9.6 ∆E0 67.7 164.5 ∆Evib −0.3 1.9 ∆Erot 0.0 0.0 ∆Etrans 0.0 0.0 ∆Etherm −0.3 1.9 ∆H 67.4 166.5 ∆Happrox 67.4 166.5 T ∆Sel 1.7 2.7 T ∆Svib −5.7 −0.5 T ∆Srot 0.0 0.1 T ∆Strans 0.0 0.0 T ∆S −4.0 2.4 ∆G 63.4 168.9

∆Eel,PBE0//PBE 59.3 84.0 ∆E0 56.0 74.4 ∆H 55.7 76.3 ∆Happrox 55.8 76.3 ∆G 51.8 78.7

∆Eel,B3LYP⋆//PBE 66.3 128.7 ∆E0 63.1 119.1 ∆H 62.8 121.0 ∆Happrox 62.8 121.0 ∆G 58.8 123.4

functional is of similar magnitude compared to the energy difference when changing the density functional (about 10–20 kJ mol−1). While the magnitude of the former quantity is preserved in the case of Fe(III), PBE0//PBE leads to an energy difference increase of 57.2 kJ mol−1 between 1+ and 2+ compared to PBE//PBE. Such a pronounced deviation of reaction energy differences indicates that the PBE and PBE0 PES’s are not compatible in the case of Fe(III). Here, this incompatibility even leads to the absurd + + + result that the PBE0//PBE transition state between 1 and 2 , TS1,2, is lower in energy than 2+. Noteworthy, we observe high spin contamination of about 53% in the

98 Thermochemical Analysis of Iron Porphyrin Carbene Reactivity 6.1

OH2 −256.2 III (−139.5) Fe + product

+ EDA, H2O, − N2 59.6 (+ styrene, H2O) 160.6 (59.6) (128.5)

O O O OH 2 O III + EDA III III Fe Fe Fe 46.7 − H2O, N2 59.6 74.0 0.0

555.6 483.8

O O O OH2 II + EDA II O II Fe Fe Fe 3.0 − H2O, N2 −12.1 26.4 0.0 7.6 200.9 (21.0) (133.4) + EDA, H2O, − N2 (+ styrene, H2O)

OH2 −299.9 II (−183.2) Fe + product

Figure 6.2: Proposed mechanism for the reaction of the Fe(II) and Fe(III) IPC complexes with EDA and styrene, −1 respectively. All energy differences are reported in kJ mol and represent approximate enthalpies, ∆Happrox, where all energy contributions have been obtained from the PBE//PBE approximation. Energy differences in blue [Fe(III)] and red [Fe(II)] are given with respect to the bridged complexes. The energy difference between the Fe(II) and Fe(III) species (black font) refers to the energy required to oxidize the isolated Fe(II) IPC complex (note that − the oxidation energy for an isolated reductant such as dithionous acid is 801.6 kJ mol 1 on our energy scale). This ﬁgure is reproduced with permission from T. Hayashi, M. Tinzl, T. Mori, U. Krengel, J. Proppe, J. Soetbeer, D. Klose, G. Jeschke, M. Reiher, D. Hilvert, Nat. Catal. 2018, 1, 578. Copyright 2018 Springer Nature.

Kohn–Sham determinant of 2+ for PBE0//PBE (Table 6.9). Even though spin annihilation as implemented in Gaussian 09 eliminates spin contamination almost entirely (which does not imply correct density matrices, and hence, reliable electronic energy expectation values), a fraction of about 5% spin contamination remains in the case of + 2PBE0//PBE.

6.1.5 Formation of the IPC Complex

While we ﬁnd the low-spin state of 3+ (PBE-optimized) to be lower in energy than its high-spin state, EPR measurements clearly suggest a high-spin state. However, looking closer at the calculated results, we ﬁnd that this structure is a pathological case for spin

99 Chapter 6 Case Study: Iron Porphyrin Carbenes

⟨ 2⟩ + + + Table 6.9: Calculated S expectation values for 1 , 2 , and TS1,2 before and after spin annihilation. Kohn−Sham determinants were obtained from the PBE//PBE and PBE0//PBE approximations. Without spin contamination, one would obtain ⟨S2⟩ = 0.75 for a doublet state.

2 2 PBE//PBE ⟨S ⟩before ⟨S ⟩after 1+ 0.77 0.75 2+ 0.84 0.75 + TS1,2 0.79 0.75 2 2 PBE0//PBE ⟨S ⟩before ⟨S ⟩after 1+ 0.78 0.75 2+ 1.15 0.79 + TS1,2 0.79 0.75

Table 6.10: Relative energies of reaction A (Table 6.3) at T = 298.15 K and p = 1 atm. The electronic energy, Eel, has been calculated on the basis of two different approximations (PBE//PBE and PBE0//PBE). To keep the notation uncluttered, we mention each approximation only once as a subscript label. All nonthermal and thermal corrections to the electronic energy have been calculated on the basis of the PBE//PBE approximation. Here, −1 ∆H = ∆U and ∆Happrox = ∆Uapprox. All energy differences are reported in kJ mol .

Total energy Correction A1 A2 A3

∆Eel,PBE//PBE −9.7 31.2 40.8 ∆EZPE −4.6 −4.1 0.5 ∆E0 −14.2 27.1 41.3 ∆Evib 2.1 −0.7 −2.8 ∆Erot 0.0 0.0 0.0 ∆Etrans 0.0 0.0 0.0 ∆Etherm 2.1 −0.7 −2.8 ∆H −12.1 26.4 38.5 ∆Happrox −12.1 26.4 38.5 T ∆Sel 0.0 0.0 0.0 T ∆Svib 6.9 −1.6 −8.5 T ∆Srot 0.0 0.1 0.0 T ∆Strans 0.0 0.0 0.0 T ∆S 6.9 −1.5 −8.4 ∆G −19.0 27.9 46.9

∆Eel,PBE0//PBE 10.5 42.0 31.6 ∆E0 5.9 38.0 32.1 ∆H 8.0 37.3 29.2 ∆Happrox 8.0 37.3 29.2 ∆G 1.1 38.8 37.7

state splittings, which are well known to depend strongly on the admixture of exact exchange.239–241 In this case, we ﬁnd 78.9 kJ mol−1 for the splitting (electronic energy)

100 Thermochemical Analysis of Iron Porphyrin Carbene Reactivity 6.1

Table 6.11: Relative energies of reaction B (Table 6.3) at T = 298.15 K and p = 1 atm. The electronic energy, Eel, has been calculated on the basis of two different approximations (PBE//PBE and PBE0//PBE). To keep the notation uncluttered, we mention each approximation only once as a subscript label. All nonthermal and thermal corrections to the electronic energy have been calculated on the basis of the PBE//PBE approximation. Here, −1 ∆H = ∆U and ∆Happrox = ∆Uapprox. All energy differences are reported in kJ mol .

Total energy Correction B1 B2 B3

∆Eel,PBE//PBE 65.0 81.0 16.0 ∆EZPE −8.3 −7.1 1.2 ∆E0 56.7 73.8 17.1 ∆Evib 2.9 0.1 −2.8 ∆Erot 0.0 0.0 0.0 ∆Etrans 0.0 0.0 0.0 ∆Etherm 2.9 0.1 −2.8 ∆H 59.6 74.0 14.4 ∆Happrox 59.6 74.0 14.4 T ∆Sel 0.0 0.0 0.0 T ∆Svib 13.5 1.5 −12.1 T ∆Srot 0.0 0.0 0.0 T ∆Strans 0.0 0.0 0.0 T ∆S 13.5 1.5 −12.0 ∆G 46.1 72.5 26.4

∆Eel,PBE0//PBE 122.2 113.4 −8.8 ∆E0 113.9 106.3 −7.6 ∆H 116.8 106.4 −10.4 ∆Happrox 116.8 106.4 −10.4 ∆G 103.3 104.9 1.6

with PBE (0% exact exchange), 30.0 kJ mol−1 with B3LYP⋆ (15% exact exchange), and −10.9 kJ mol−1 with PBE0 (25% exact exchange). Including thermal corrections further −1 −1 ⋆ lowers the gap by 7.0 kJ mol (∆Happrox = 23.0 kJ mol for B3LYP //PBE). Gas phase Gibbs free-energy differences even amount to 5.7 kJ mol−1 with B3LYP⋆//PBE, which is another clear sign that the calculations point to spin states that are very similar in energy. Hence, we may rely on the finding from EPR spectroscopy that the high-spin state of 3+ is the one that is (slightly) lower in energy. Note that the entropy change in the gas phase Gibbs free-energy difference may be taken as an estimate for the condensed-phase entropy change of a spin crossover as discussed by Brehm et al.257 For the reaction of low-spin 3+ with EDA to the end-on IPC complex 2+, we find −1 an activation barrier (∆Happrox) of 12.9 kJ mol , which equals the energy difference + + between H2O + 2 and EDA + 3 . The reaction can be considered irreversible due to (i) loss of dinitrogen and (ii) rapid conversion of 2+ to the thermodynamically more

101 Chapter 6 Case Study: Iron Porphyrin Carbenes

stable bridged IPC complex 1+ (59.6 kJ mol−1 lower in energy than 2+) through a considerably small unimolecular activation barrier of 14.4 kJ mol−1.

6.1.6 Reactions of the IPC Complex with EDA and Styrene

To study the reactivity of the bridged and end-on Fe(II) and Fe(III) complexes, we examined their reactions with EDA and styrene. For a summary of the corresponding reaction mechanisms, see Fig. 6.2. In all cases, we ﬁnd that the bridged complexes 1 + and 1 form an adduct with both EDA and styrene (reactions C1, D1, E1, and F1 in Tables 6.12–6.15, respectively), before the products are formed through a barrier-free reaction path. These adducts are characterized by a covalent bond between the carbene carbon atom of the IPC complex and one of the porphyrin nitrogen atoms, and by a covalent bond between the IPC carbene carbon atom and the carbene carbon atom of EDA or one of the two oleﬁn carbon atoms of styrene. The adducts are less stable than the reactants (52.2–196.2 kJ mol−1 for PBE//PBE and 21.0–144.4 kJ mol−1 for PBE0//PBE), but an equilibrium with the reactant state (favoring the latter) is not expected due to the spontaneous formation of the products. For both PBE//PBE and PBE0//PBE, we observe that the Gibbs free-energy dif-

ference, ∆G, is significantly larger than the approximate enthalpy difference, ∆Happrox, regarding the adduct formation (46.5–71.6 kJ mol−1). This distinct increase can be attributed mostly to the translational and rotational entropy differences, which arise from the different numbers of molecules in the reactant and adduct states (two versus one) through the transformation of three translational and three rotational degrees of freedom to six vibrational degrees of freedom. However, in a spatially confined system such as the protein-embedded IPC complex, this effect can be expected to be much weaker and, therefore, will probably not blur the system-inherent electronic energetics at 0 K. This artificial energy-increasing effect has already been discussed in a study by Reiher and Hess in 2002.258 Combined with the expected bias of vibrational entropy (dominated by low-frequency modes) and the fact that the electronic entropy difference

is zero for a given PES, we conclude that the approximate enthalpy diﬀerence, ∆Happrox, derived from the standard model of gas phase thermochemistry is suitable for the description of the protein-embedded IPC complex. Note that in the standard model of gas phase thermochemistry, the translational entropy is the only thermochemical energy contribution that is an eﬀective function of the volume. Hence, for all chemical species in solution, one would need to correct the translational entropy according to −1 Strans,solution = Strans + RT ln(y/ysolution), where RT ln(y/ysolution) = −7.93 kJ mol −1 for a solution-phase concentration of ysolution = 1 mol L (Table 6.1). Due to the neglect of entropic contributions in our thermochemical analysis, we do not consider this correction here.

102 Thermochemical Analysis of Iron Porphyrin Carbene Reactivity 6.1

Independent of the reactant (EDA or styrene) and the formal oxidation state of iron, we find significant activation energies for the formation of the adducts (∆Happrox = −1 −1 128.5–200.9 kJ mol for PBE//PBE and ∆Happrox = 120.8–200.5 kJ mol for PBE0//PBE). For the end-on complexes 2 and 2+ (Tables 6.16 and 6.17), we do not find the formation of an adduct with either EDA or styrene, but rather the immediate forma- −1 tion of the products via a low-energy transition path (∆Happrox = 18.8–19.7 kJ mol −1 for 2 + EDA and ∆Happrox = 33.2–40.5 kJ mol for 2 + styrene) or even through a barrier-free reaction in the case of 2+ + EDA and 2+ + styrene, respectively. Similar results were found for the reaction of EDA with an Fe(II) IPC complex having a prox- − 259 imal SCH3 ligand instead of N-methylimidazole. For both the bridged and end-on complexes, we find that the reaction with EDA is highly exothermic (∆Happrox = 256.2– −1 −1 315.8 kJ mol for PBE//PBE and ∆Happrox = 303.2–420.0 kJ mol for PBE0//PBE). Our results also suggest that the reaction with styrene is less (but still highly) exothermic −1 compared to EDA (∆Happrox = 139.5–199.1 kJ mol for PBE//PBE and ∆Happrox = 163.1–279.9 kJ mol−1 for PBE0//PBE).

We conclude that the bridged complexes 1 and 1+ are inert towards both EDA and styrene compared to the end-on complexes 2 and 2+. Given that the initial reaction mixture consists of 3+ (catalytic amounts) plus reactant without reducing agent, the formation of 2(+) occurs immediately based on our PBE//PBE data. Rate constants were obtained from approximate enthalpy differences and conventional TST, cf. Eq. (2.2). The pre-equilibration between 1+ and 2+ leads to a very small and temporarily constant concentration of 2+ (according to our PBE//PBE results, the relative molar concentration of 2+ compared to 1+ would be 3.7 × 10−11). Therefore, the rate of the reaction of 2+ plus reactant would be of pseudo-first order. According to conventional TST, the corresponding effective bimolecular rate constant (the product of the bimolecular rate + −3 −1 constant and the constant concentration of 2 ) amounts to kbi,eff = 9.0 × 10 s for an initial concentration of 3+ of 10−5 mol L−1 (actual experimental conditions).

By contrast, the value of the unimolecular rate constant corresponding to the isomer- + + −1 −1 ization of 1 to 2 amounts to kuni = 6.7 × 10 s . In this case, product formation would be the rate-determining step of the overall reaction. Clearly, one should assess this kinetic analysis with caution since the barrier-free product formation would require consideration of nonequilibrium rate theories. There exist simple empirical expressions for rate constants of barrier-free reactions, which work well for small-molecule solution chemistry.260 Assuming water as solvent, the resulting rate constant represents an eﬀec- tive activation barrier of 2.6 kJ mol−1 in conventional TST, which lowers the predicted product formation rate only by a small factor of less than 3. However, the actual IPC complex is embedded in a protein environment, which functions as a complex diﬀusion barrier for the reactants. Therefore, it is not sensible to derive conclusions about the

103 Chapter 6 Case Study: Iron Porphyrin Carbenes

absolute value of kbi,eff based on our computational data. Still, the value of kbi,eff we calculated can be considered an upper bound to the observable rate constant, and we may safely assume that diffusion of EDA and styrene to the IPC complex determines the rate of the overall reaction. This hypothesis is also supported by the pseudo-first order consumption rate of 3+ (through reaction with EDA) that was found in experiments (see Figure 4B of the Supporting Information of Ref. 36).

Table 6.12: Relative energies of reaction C (Table 6.3) at T = 298.15 K and p = 1 atm. The electronic energy, Eel, has been calculated on the basis of two different approximations (PBE//PBE and PBE0//PBE). To keep the notation uncluttered, we mention each approximation only once as a subscript label. All nonthermal and thermal corrections to the electronic energy have been calculated on the basis of the PBE//PBE approximation. All en- − ergy differences are reported in kJ mol 1.

Total energy Correction C1 C2 C3 C4

∆Eel,PBE//PBE 115.5 −305.6 194.0 78.6 ∆EZPE 13.8 4.5 −0.4 −14.2 ∆E0 129.3 −301.0 193.6 64.4 ∆Evib 5.2 1.2 9.7 4.5 ∆Erot −3.7 −1.2 −3.7 0.0 ∆Etrans −3.7 0.0 −3.7 0.0 ∆Etherm −2.2 −0.1 2.3 4.5 ∆U 127.0 −301.1 195.9 68.9 ∆Uapprox 134.5 −299.9 203.4 68.9 ∆H 124.6 −301.1 193.5 68.9 ∆Happrox 132.0 −299.9 200.9 68.9 T ∆Sel 0.0 0.0 0.0 0.0 T ∆Svib 11.6 1.2 25.0 13.4 T ∆Srot −33.9 1.9 −33.9 0.0 T ∆Strans −49.3 2.7 −49.3 0.0 T ∆S −71.6 5.7 −58.2 13.4 ∆G 196.2 −306.8 251.7 55.5

∆Eel,PBE0//PBE 29.6 −372.8 193.6 164.0 ∆E0 43.4 −368.3 193.2 149.8 ∆U 41.2 −368.3 195.5 154.3 ∆Uapprox 48.6 −367.1 203.0 154.3 ∆H 38.7 −368.3 193.0 154.3 ∆Happrox 46.1 −367.1 200.5 154.3 ∆G 110.3 −374.1 251.3 141.0

104 Density Functional Assessment Based on Kinetic Modeling 6.2

Table 6.13: Relative energies of reaction D (Table 6.3) at T = 298.15 K and p = 1 atm. The electronic energy, Eel, has been calculated on the basis of two different approximations (PBE//PBE and PBE0//PBE). To keep the notation uncluttered, we mention each approximation only once as a subscript label. All nonthermal and thermal corrections to the electronic energy have been calculated on the basis of the PBE//PBE approximation. All en- − ergy differences are reported in kJ mol 1.

Total energy Correction D1 D2 D3 D4

∆Eel,PBE//PBE 111.1 −260.6 154.3 43.2 ∆EZPE 0.8 3.0 −2.8 −3.6 ∆E0 111.9 −257.5 151.5 39.6 ∆Evib 12.2 1.4 11.6 −0.6 ∆Erot −3.7 −1.2 −3.7 0.0 ∆Etrans −3.7 0.0 −3.7 0.0 ∆Etherm 4.7 0.1 4.1 −0.6 ∆U 116.7 −257.4 155.7 39.0 ∆Uapprox 124.1 −256.2 163.1 39.0 ∆H 114.2 −257.4 153.2 39.0 ∆Happrox 121.6 −256.2 160.6 39.0 T ∆Sel 0.0 0.0 0.0 0.0 T ∆Svib 36.7 2.3 32.8 −3.9 T ∆Srot −33.9 1.9 −33.9 0.0 T ∆Strans −49.3 2.7 −49.3 0.0 T ∆S −46.5 6.8 −50.4 −3.9 ∆G 160.7 −264.2 203.6 42.9

∆Eel,PBE0//PBE 37.1 −307.6 187.3 150.2 ∆E0 38.0 −304.6 184.5 146.6 ∆U 42.7 −304.4 188.7 146.0 ∆Uapprox 50.1 −303.2 196.1 146.0 ∆H 40.2 −304.4 186.2 146.0 ∆Happrox 47.7 −303.2 193.6 146.0 ∆G 86.7 −311.2 236.6 149.8

6.2 Density Functional Assessment Based on Kinetic Modeling

To gain further insight into the reliability of density functional-based thermochemistry, we studied the kinetics of the mechanism proposed for the reaction of the Fe(III) IPC complex with styrene (Fig. 6.3). There is EPR-spectroscopic evidence that, in the absence of a reducing agent, a low-spin Fe(III) compound is dominantly present in the reaction mixture.36 Based on our former thermochemical analysis, we can safely assume that the bridged Fe(III) complex is the dominantly populated species at the outset of the reaction given that the aqua iron porphyrin complex was completely consumed through addition of equivalent amounts of EDA prior to the actual reaction.

105 Chapter 6 Case Study: Iron Porphyrin Carbenes

Table 6.14: Relative energies of reaction E (Table 6.3) at T = 298.15 K and p = 1 atm. The electronic energy, Eel, has been calculated on the basis of two different approximations (PBE//PBE and PBE0//PBE). To keep the notation uncluttered, we mention each approximation only once as a subscript label. All nonthermal and thermal corrections to the electronic energy have been calculated on the basis of the PBE//PBE approximation. All en- − ergy differences are reported in kJ mol 1.

Total energy Correction E1 E2 E3 E4

∆Eel,PBE//PBE 109.7 −203.7 126.2 16.5 ∆EZPE 7.6 18.1 2.7 −4.9 ∆E0 117.3 −185.6 128.9 11.6 ∆Evib 9.1 4.8 7.0 −2.1 ∆Erot −3.7 −3.7 −3.7 0.0 ∆Etrans −3.7 −3.7 −3.7 0.0 ∆Etherm 1.7 −2.6 −0.5 −2.1 ∆U 119.0 −188.2 128.5 9.5 ∆Uapprox 126.5 −180.7 135.9 9.5 ∆H 116.5 −190.7 126.0 9.5 ∆Happrox 124.0 −183.2 133.4 9.5 T ∆Sel 0.0 0.0 0.0 0.0 T ∆Svib 22.7 6.7 17.5 −5.2 T ∆Srot −33.4 −8.8 −33.4 0.0 T ∆Strans −49.0 −41.4 −49.0 0.0 T ∆S −59.7 −43.6 −64.9 −5.2 ∆G 176.2 −147.1 190.9 14.6

∆Eel,PBE0//PBE 77.9 −247.6 113.6 35.6 ∆E0 85.5 −229.4 116.3 30.7 ∆U 87.2 −232.0 115.8 28.6 ∆Uapprox 94.7 −224.6 123.3 28.6 ∆H 84.8 −234.5 113.4 28.6 ∆Happrox 92.2 −227.1 120.8 28.6 ∆G 144.4 −190.9 178.2 33.8

We consider an initial concentration of 1.0 mol L−1 for both 1+ and styrene. Ex- perimental results indicate that, in the absence of a reducing agent, about 60% of styrene are consumed after 300 s given initial concentrations of 3+, EDA, and styrene of 10 µmol L−1, 20 mmol L−1, and 10 mmol L−1, respectively.36 Even if we assume an immediate conversion of 3+ to 2+ and further to 1+, the initial styrene concentration still exceeds that of 1+ by a factor of 1,000. Due to the presence of 3+ and EDA in the experimental setup, a continuous supply of 1+ can be ensured, which decreases the relative excess of styrene over 1+ during the course of the reaction. In our computational scenario, the initial concentrations of 1+ and styrene are much larger (by factors of 100,000 and 100, respectively), leading to equivalent amounts of the two reactants

106 Density Functional Assessment Based on Kinetic Modeling 6.2

Table 6.15: Relative energies of reaction F (Table 6.3) at T = 298.15 K and p = 1 atm. The electronic energy, Eel, has been calculated on the basis of two different approximations (PBE//PBE and PBE0//PBE). To keep the notation uncluttered, we mention each approximation only once as a subscript label. All nonthermal and thermal corrections to the electronic energy have been calculated on the basis of the PBE//PBE approximation. All en- − ergy differences are reported in kJ mol 1.

Total energy Correction F1 F2 F3 F4

∆Eel,PBE//PBE 52.2 −158.7 119.5 67.3 ∆EZPE 8.6 16.6 2.0 −6.6 ∆E0 60.8 −142.1 121.5 60.7 ∆Evib 7.7 5.0 9.5 1.8 ∆Erot −3.7 −3.7 −3.7 0.0 ∆Etrans −3.7 −3.7 −3.7 0.0 ∆Etherm 0.3 −2.4 2.1 1.8 ∆U 61.1 −144.5 123.6 62.5 ∆Uapprox 68.5 −137.0 131.0 62.5 ∆H 58.6 −146.9 121.1 62.5 ∆Happrox 66.1 −139.5 128.5 62.5 T ∆Sel 0.0 0.0 0.0 0.0 T ∆Svib 19.2 7.8 26.4 7.2 T ∆Srot −33.4 −8.8 −33.3 0.1 T ∆Strans −49.0 −41.4 −49.0 0.0 T ∆S −63.2 −42.5 −56.0 7.2 ∆G 121.8 −104.4 177.1 55.3

∆Eel,PBE0//PBE 21.0 −182.3 168.1 147.1 ∆E0 29.6 −165.7 170.1 140.5 ∆U 29.9 −168.1 172.2 142.3 ∆Uapprox 37.3 −160.7 179.6 142.3 ∆H 27.4 −170.6 169.7 142.3 ∆Happrox 34.8 −163.1 177.1 142.3 ∆G 90.6 −128.1 225.7 135.1 compared to the initial 1,000-fold excess of styrene in the experimental setup. Hence, we expect that the calculated conversion of styrene after 300 s is comparable to or even larger than its measured counterpart.

Rate constants of elementary reactions featuring a barrier, kact, were calculated with the Eyring equation (2.2). For barrier-free elementary reactions, which are diﬀusion- controlled, we calculated the rate constant as260

8k TN k = B A , (6.1) diﬀ 3η where NA is the Avogadro constant and η is the solvent viscosity at temperature T . For

107 Chapter 6 Case Study: Iron Porphyrin Carbenes

Table 6.16: Relative energies of reactions G and H (Table 6.3) at T = 298.15 K and p = 1 atm. The electronic en-

ergy, Eel, has been calculated on the basis of two different approximations (PBE//PBE and PBE0//PBE). To keep the notation uncluttered, we mention each approximation only once as a subscript label. All nonthermal and thermal corrections to the electronic energy have been calculated on the basis of the PBE//PBE approximation. All − energy differences are reported in kJ mol 1.

Total energy Correction G1 G2 G3 H

∆Eel,PBE//PBE −295.9 8.7 247.9 −325.6 ∆EZPE 9.1 4.8 5.5 11.3 ∆E0 −286.8 13.6 253.4 −314.2 ∆Evib −0.9 8.7 16.6 −1.5 ∆Erot −1.2 −3.7 −6.2 −1.2 ∆Etrans 0.0 −3.7 −7.4 0.0 ∆Etherm −2.2 1.2 3.0 −2.8 ∆U −289.0 14.8 256.4 −317.0 ∆Uapprox −287.8 22.2 270.0 −315.8 ∆H −289.0 12.3 251.4 −317.0 ∆Happrox −287.8 19.7 265.1 −315.8 T ∆Sel 0.0 0.0 0.0 0.0 T ∆Svib −5.7 22.7 40.5 −11.3 T ∆Srot 1.9 −34.0 −48.9 1.9 T ∆Strans 2.7 −49.3 −95.0 2.7 T ∆S −1.2 −60.6 −103.4 −6.7 ∆G −287.8 72.9 354.8 −310.3

∆Eel,PBE0//PBE −383.3 7.7 335.1 −429.8 ∆E0 −374.2 12.6 340.6 −418.5 ∆U −376.4 13.8 343.6 −421.2 ∆Uapprox −375.1 21.2 357.2 −420.0 ∆H −376.4 11.3 338.6 −421.2 ∆Happrox −375.1 18.8 352.2 −420.0 ∆G −375.2 71.9 442.0 −414.5

water and T = 298.15 K, we ﬁnd η = 0.89 mPa s.261 The rate constant of the reverse ′ reaction, kdiﬀ, was determined by the thermochemical equilibrium condition,

k k′ = diﬀ , (6.2) diﬀ K where K is the equilibrium constant, ( ) ∆Happrox K = exp − . (6.3) RT

Here, ∆Happrox is the enthalpy diﬀerence between the two intermediate states considered.

108 Density Functional Assessment Based on Kinetic Modeling 6.2

Table 6.17: Relative energies of reactions J and K (Table 6.3) at T = 298.15 K and p = 1 atm. The electronic energy,

Eel, has been calculated on the basis of two different approximations (PBE//PBE and PBE0//PBE). To keep the notation uncluttered, we mention each approximation only once as a subscript label. All nonthermal and thermal corrections to the electronic energy have been calculated on the basis of the PBE//PBE approximation. All − energy differences are reported in kJ mol 1.

Total energy Correction J1 J2 J3 K

∆Eel,PBE//PBE −194.0 21.3 158.6 −223.7 ∆EZPE 22.7 5.7 −7.3 25.0 ∆E0 −171.3 27.0 151.4 −198.8 ∆Evib 2.7 8.7 13.0 2.1 ∆Erot −3.7 −3.7 −3.7 −3.7 ∆Etrans −3.7 −3.7 −3.7 −3.7 ∆Etherm −4.7 1.2 5.5 −5.3 ∆U −176.1 28.2 156.9 −204.1 ∆Uapprox −168.6 35.6 164.3 −196.6 ∆H −178.5 25.7 154.4 −206.5 ∆Happrox −171.1 33.2 161.8 −199.1 T ∆Sel 0.0 0.0 0.0 0.0 T ∆Svib −0.2 27.6 40.0 −5.8 T ∆Srot −8.8 −33.3 −37.5 −8.8 T ∆Strans −41.4 −49.0 −50.6 −41.4 T ∆S −50.5 −54.7 −48.1 −56.0 ∆G −128.1 80.4 202.5 −150.5

∆Eel,PBE0//PBE −258.0 28.6 230.7 −304.6 ∆E0 −235.3 34.3 223.4 −279.6 ∆U −240.0 35.5 229.0 −284.9 ∆Uapprox −232.6 43.0 236.4 −277.5 ∆H −242.5 33.0 226.5 −287.4 ∆Happrox −235.1 40.5 233.9 −279.9 ∆G −192.0 87.7 274.6 −231.4

We compared the PBE//PBE results for the model mechanism (Fig. 6.3) with those of the PBE0//PBE(0) approximation, where the latter refers to PBE0 single-point energy calculations on a PBE0 PES in the case of intermediate states and a PBE PES in the case of transition states (Fig. 6.3 and Table 6.18). With this choice, we can also resolve the + absurd situation observed for PBE0//PBE where the approximate enthalpy of TS1,2 is lower than that of 2+. While all approximate enthalpies relative to 1+ become larger in magnitude for PBE0//PBE(0), the reaction mechanism is qualitatively preserved, i.e., the ranking of approximate enthalpy diﬀerences remains the same. The results of our ﬂux analysis (cf. Section 4.4) suggest that the reaction of 1+ with styrene is kinetically irrelevant (irrespective of the density functional approximation),

109 Chapter 6 Case Study: Iron Porphyrin Carbenes

which is why we do not explicitly take into account the adduct formation of 1+ with styrene in our kinetic model. The resulting concentration profiles for the reduced mechanism comprising two reversible reactions are shown in Fig. 6.4. For PBE//PBE, we find full conversion of styrene after about 20 s, whereas the overall reaction does not even start within the first 300 s when choosing the PBE0//PBE(0) approximate enthalpy differences. While the Fe(III) IPC complex is continuously regenerated in the experiment through addition of EDA to the aqua iron porphyrin complex 1+,36 here, the entire amount of the IPC complex required for full conversion of styrene is present from the outset of the reaction. Therefore, we rather expect a faster consumption of styrene in the computational setup leading to a conversion larger than 60%.

In the following, we consider approximate activation enthalpies obtained with PBE// PBE and PBE0//PBE(0) as bounds of an uncertainty margin for each elementary (unidirectional) reaction. In Fig. 6.4, we show concentration profiles for 98 further sets of approximate reaction enthalpies, which were drawn from uniform PDFs bounded by the PBE//PBE and PBE0//PBE(0) values. We find that the reaction profiles are in- + sensitive to the large activation barrier of the products back to 2 , styrene, and H2O (199.1 kJ mol−1 and 292.8 kJ mol−1, respectively). Furthermore, as the reverse reaction is barrier-free for all sampled enthalpies, it does also not contribute to the variance of the concentration profiles. Only for the two remaining elementary reactions corresponding to the interconversion between 1+ and 2+, we find a dependence of the concentration profiles on the corresponding approximate activation enthalpies. In Table 6.18, we summarize these quantities for both density functional approximations and for one random sample that revealed complete styrene conversion after 300 s. The maximum deviation in the approximate activation enthalpies between PBE//PBE and the random sample is less than 10 kJ mol−1, whereas it is larger than 40 kJ mol−1 between the latter and PBE0//PBE(0). Given that (i) the results for PBE0//PBE(0) do not reflect the experimental findings at all and (ii) all random samples corresponding to styrene conversions >60% after 300 s reveal activation energies that are closer to PBE//PBE, we conclude that the latter is the more reasonable density functional approximation.

We emphasize that reaction mechanism elucidation may require highly accurate quantum chemical data, and that only an uncertainty-based kinetic analysis can provide a reliable assessment of this issue. We suggest that kinetic modeling supported by uncertainty quantiﬁcation should be employed as an additional instrument to assess thermochemical data, and that our program KiNetX allows to perform such analyses routinely. Clearly, reliable uncertainty quantiﬁcation for chemical kinetics requires reaction mechanisms that contain the entirety of kinetically relevant information. For instance, the set of sampled energies that constitutes the best description of styrene consumption in our computational scenario (experimental data required) may not represent the correct ac-

110 Mössbauer Spectroscopy for the Discrimination of Spin–Charge States 6.3

‡,∗ −1 Table 6.18: Approximate activation enthalpies, ∆Happrox, reported in kJ mol obtained from PBE//PBE, PBE//PBE(0), and a random sample drawn from uniform PDFs bounded by the PBE//PBE and PBE0//PBE(0) values.

source 1+ → 2+ 2+ → 1+ PBE//PBE 74.0 14.4 PBE0//PBE(0) 124.2 8.5 random sample 82.9 10.9 tivation barriers. Since our molecular structure model neglects the protein environment of the active iron porphyrin site, the mechanism discussed (Fig. 6.3) may be incomplete. A signiﬁcant number of protein conformations might be necessary in order to model the kinetics of the reaction of styrene with the Fe(III) IPC complex correctly. This issue illustrates once more the importance of the interplay between reaction space exploration and uncertainty-based kinetic modeling.

OH2 −139.5 III (−177.1) Fe + product

59.6 + styrene, H2O 128.5 (115.7) (197.6)

O O O O III III Fe Fe 59.6 74.0 0.0 (115.7) (124.2)

Figure 6.3: Hypothetical mechanism for the reaction of the Fe(III) IPC complex with styrene considered in our kinetic modeling study. The reaction energies, ∆Happrox, were obtained from PBE//PBE and PBE0//PBE(0) (values − for the latter approximation are provided in parentheses) and are reported in kJ mol 1. This ﬁgure is reproduced with permission from T. Hayashi, M. Tinzl, T. Mori, U. Krengel, J. Proppe, J. Soetbeer, D. Klose, G. Jeschke, M. Reiher, D. Hilvert, Nat. Catal. 2018, 1, 578. Copyright 2018 Springer Nature.

6.3 Mössbauer Spectroscopy for the Discrimination of Spin–Charge States

To verify the reaction mechanisms suggested in this chapter, the experimentalist would be faced with the challenge to distinguish between the several possible spin–charge states of the IPC complex over the course of the reaction. Freeze-quench Mössbauer spectroscopy262 is a promising method in this respect as it is designed for the study of

111 Chapter 6 Case Study: Iron Porphyrin Carbenes

Figure 6.4: Calculated concentration profiles of the dominant chemical species presented in Fig. 6.3. The green and red profiles correspond to PBE//PBE and PBE0//PBE(0) results, respectively. The other profiles were sampled from uniform PDFs of approximate activation enthalpies bounded by the PBE//PBE and PBE0//PBE(0) values. The blue profiles highlight a random sample.

reactive intermediates. Furthermore, because of its sensitivity to all iron species in a sample, it also has advantages over other spectroscopic methods for this type of analysis. Here, we apply our PBE0/def2-TZVP-based isomer shift model [Eqs. (5.7)–(5.11)] to six spin–charge states of both the bridged and the end-on IPC complex (Fig. 6.5). Since our statistical calibration approach (cf. Chapters 3 and 5) allows to reliably estimate prediction uncertainties, we can assess the extent to which two signals can be distinguished. Our results suggest that the bridged and end-on complexes should be easily distin- guishable if their low-spin or intermediate-spin states are populated (irrespective of their formal oxidation states). Furthermore, the possible coexistence of the Fe(II) and Fe(III) end-on complexes should be especially well-detectable if their low-spin states are populated. However, we assume the absence of a signal for the low-spin Fe(III) end-on complex due to the signiﬁcantly larger stability of the bridged complex (59.6 kJ mol−1) and the small activation energy required for converting the end-on complex into its bridged isomer (14.4 kJ mol−1). The predicted isomer shifts of the bridged complexes occupy a much closer range of possible values compared to the end-on complexes. Given

112 Mössbauer Spectroscopy for the Discrimination of Spin–Charge States 6.3

O O O O O O O O II III II III Fe Fe Fe Fe ISLS HS ISLS HS ISLS HS ISLS HS

0.5

0.4 ) − 1 0.3

0.2

0.1 isomer shift (mm s

0.0

−0.1

IPC complex in different spin─charge states

Figure 6.5: 57Fe Mössbauer isomer shifts and prediction uncertainties (one and two standard deviations) from our PBE0/def2-TZVP-based calibration model [Eqs. (5.7)−(5.11)] for all structurally optimized (PBE/def2-TZVP) IPC complexes. The computational setup for isomer shift calculations was adopted from Section B.2. that the bridged complex is present in sufficient quantities, our findings indicate that a distinction between its low-spin and intermediate-spin states (for both Fe(II) and Fe(III)) is effectively impossible with Mössbauer spectroscopy, whereas the high-spin states appear to be discriminable from the other spin states with a probability larger than one standard deviation. Here, EPR-spectroscopic measurements could provide further information, which indeed suggest a low-spin Fe(III) IPC complex in the absence of a reducing agent. A similar theoretical Mössbauer analysis without uncertainty quantification has been applied to other IPC complexes.263 The authors also suggest that isomer shift calculations can support the discrimination of spin–charge states. While experimental isomer shifts are not available at present for the systems studied in this chapter, our analysis proposes experiments to measure these data.

113 Chapter 6 Case Study: Iron Porphyrin Carbenes

114 7 A Computational Perspective on the Study of Complex Chemical Systems

This dissertation revealed that the combination of traditional concepts and methods from computational chemistry with the elements of network theory, reaction engineering, and uncertainty quantification allows us to study complex reactive systems from first principles. We demonstrated that uncertainty and sensitivity analyses of dilute and spatially homogeneous chemical systems characterized by free energies of intermediates and transition states can be conducted efficiently. Furthermore, we showed how to infer reliable uncertainty estimates of property predictions for every vertex in a reaction network, which is a prerequisite to infer the precision of macroscopic property predictions through uncertainty propagation. We also showed how uncertainty-based kinetic modeling can be harnessed as a guide for the systematic exploration of reaction networks. However, to render our holistic approach to reaction chemistry truly universal, we need to overcome challenges at every step of our general protocol, which we will elaborate on in the following. The first challenge concerns the systematic discovery of target compounds. A target compound might be a chemical compound with certain properties important for, e.g., medical treatments or optoelectronic applications. It is necessary to search the chemical space for promising compounds, from which a reverse network exploration could be started to identify possible synthetic pathways and available reactants. However, systematic exploration of the essentially infinite chemical space264 is an arduous task as we lack universal similarity measures265 that would define axes along which to carry out purposeful searches. A promising recent approach by Aspuru-Guzik and co-workers

115 Chapter 7 Computational Perspective on Complex Chemical Systems

harnessing the concept of an autoencoder266 tackles the issues of search eﬃciency and ambiguous similarity concepts. An encoder (a deep neural network) transforms discrete molecular representations (here, SMILES strings) into a continuous molecular representation referred to as latent space, which is a mathematically abstract object. A decoder (another deep neural network) transforms the latent space back into discrete representations. Given a property as a function of the latent space, one can perform gradient-based optimizations in the latent space to discover new molecules.

The second challenge addresses the applicability of our holistic approach to other types of chemical systems than those studied in this thesis, e.g., spatially heterogeneous systems, nonequilibrium systems, or stochastically behaving systems. Examples of spatially heterogeneous systems can be found in the areas of heterogeneous catalysis31 or reacting-flow chemistry,267 which require an explicit consideration of spatial coordinates to account for transport processes. The description of transport processes involves the integration of a system of partial differential equations, which is significantly less efficient compared to ODE integration and opens new possibilities for approaches based on time-scale separation.174 Nonequilibrium systems are characterized by thermal relaxation time scales that are on the order of or even larger than reactive time scales. In this regime, a microcanonical (energy-resolved) description of the systems kinetics is necessary, which is provided by master equation approaches.268 An example for a nonequilibrium system involves the reaction of a borane with an asymmetric alkene.157 While TST predicts a ratio of 99:1 for the Anti-Markovnikov over the Markovnikov product, a master equation ansatz correctly predicted the measured ratio of 9:1. We may recommend the MESMER154 software for routine studies of nonequilibrium systems. Another area building on the master equation is that of stochastic chemical kinetics,153 where the particle number of important intermediates is too small (e.g., in cellular systems) to describe them on the basis of deterministic rate equations, i.e., the assumption of ergodicity is violated. In fact, deterministic rate equations are a consequence of stochastic rate equations for a large number of particles.

The third and last challenge discussed by us concerns the reliable uncertainty estimation for quantum chemical results, which has been initially discussed by Irikura et al.125 While their approach builds on the assumption that an electronic structure model yields similar uncertainties for similar molecules, we are interested in learning a precise covariance function in chemical space based on Gaussian process regression165 or other Bayesian learning methods such that extrapolations into regions far away from the training space will automatically lead to large uncertainties. This way, any user of quantum chemical software could be immediately warned if they are about to apply a density functional to transition metal chemistry that was trained on organic compounds. A topic closely related hereto is the rigorous error control of nonexperimental reference

116 data. If experimental reference data are not available for a given problem, we must rely on the reliability of reference data from electronic structure calculations. We believe that the insights gained from this dissertation can greatly affect developments in high-throughput virtual screening (HTVS), i.e., compound discovery by computational analysis of large virtual structure libraries. On the one hand, HTVS of molecular materials has a long tradition in pharmacological research (drug discovery).269 On the other hand, HTVS of advanced materials with intricate electronic structures is rooted in solid-state research,270 where electronic structure calculations are inevitable. HTVS of advanced molecular materials combines the challenges of systematic and efficient chemical space exploration, high-accuracy first-principles calculations, automation requirements (due to generation, storing, and querying of gigantic data volumes), and distributed computing. Moreover, as HTVS aims for the discovery of novel materials, synthetic feasibility should be a core criterion in systematic chemical space exploration, which may be greatly facilitated by our holistic approach coupling exploration to efficient kinetic modeling. HTVS-based discovery of advanced molecular materials has lead to important progress in fields such as photovoltaics,271–275 electrolytes for flow batter- ies,276,277 and OLED technology.278 For instance, the Aspuru-Guzik research group at Harvard University screened more than 3 million organic molecules for the discovery of efficient solar cells, resulting in 300 million DFT calculations.271,274 The lead candidates revealed unprecedented power conversion efficiencies of up to 11%. HTVS particularly profited from recent advancements in modern computer architectures and novel algorithms, which supported fundamental and development-driven research on electronic properties starting to approach. Parallel and distributed computing allows for a vast number of accurate calculations within affordable time frames, and progress in artificial intelligence research enables the construction of empirical electronic structure models by analyzing quantum chemical data with machine learning techniques. This trend accelerates both the development of evermore sophisticated first-principles models and the data-driven discovery of novel materials with promising properties. Hence, HTVS is no longer a mere search tool building on simplistic structure–property relationships, but an increasingly complex endeavor integrating cutting-edge data analysis and electronic structure models. Moreover, the first-principles character of the screening procedure and the problem-independent formulation of uncertainty assessment will ultimately lead to physically rigorous screenings of advanced molecular materials in general, which demon- strates the potential of this approach for molecular engineering. This process creates a huge amount of high-quality information with acceptable effort, which allows for saving resources (material and staff), and hence, has direct impact on economics and society.

117 Chapter 7 Computational Perspective on Complex Chemical Systems

118 A Kinetic Modeling of Complex Reactions: Technical Details

A.1 Computational Singular Perturbation

A.1.1 Model Simplification Based on a Jacobian Analysis

For the development of our kinetic simulation algorithm applied to the formose model network in Section 2.3.2, we were inspired by two simpliﬁcation approaches, namely Markov state models279,280 and CSP.98,99,*,† Markov state models were developed for molecular dynamics simulations, where the phase space is decomposed into microstates such that a formerly continuous trajectory becomes a jump process, which is no longer Markovian (memoryless) in a locally resolved sense. Since local information is lost in a discrete phase space, the decomposition is chosen such that transitions within microstates are much more likely to occur than transitions between microstates. This way, rapid convergence to local equilibrium can be assumed for these microstates, which recovers Markovianity. As a consequence, a kinetic model can be constructed from these discrete microstates by counting transitions between them. The microstates may in turn form macrostates (kinetic clusters) for which internal transitions are much more likely to occur than transitions between them. These clusters can be determined by studying 281 the eigenvalues λi of the N × N rate matrix K = {Kij}. Its elements Kij are a measure of the rate for a transition from the j-th to the i-th microstate. In the case

*This appendix is reproduced in part from J. Proppe, T. Husch, G. N. Simm, M. Reiher, Faraday Discuss. 2016, 195, 497, with permission from the Royal Society of Chemistry. †In this appendix, we adopt the notation introduced in Chapter 2.

119 Chapter A Kinetic Modeling of Complex Reactions: Technical Details

of linear kinetic models (ﬁrst-order reactions only) as studied in Markov state models,

the rate matrix K is time-invariant and equals the Jacobian J = {Jij} as deﬁned in Eq. (2.14).

The time scale τi corresponding to the process described by the i-th eigenvalue is inversely related to the complex modulus of that eigenvalue,

−1 τi = |λi| , (A.1)

i.e., the larger the complex modulus of an eigenvalue, the faster the corresponding pro- −1 cess. If a predeﬁned gap ϵ ≥ |λi|/|λi+1| (where |λi| ≥ |λi+1|) can be found anywhere in the eigenvalue spectrum, a time scale separation of processes is assumed to be possible such that the rate vector can be decomposed into a fast and a slow part,

g = gfast + gslow . (A.2)

With this decomposition at hand, it is possible to dissipate the fast processes applying a

small time step τfast in the numerical integration until gfast ≈ 0. Subsequently, the slow processes can be modeled from the updated initial conditions applying a much larger

time step τslow. Clearly, the larger the demanded spectral gap, the smaller the error introduced by assuming decoupling of fast and slow processes. Since the Jacobian is time-invariant in the case of linear kinetic models, the time scale separation is also invariant in the course of the global reaction process and needs to be examined only once. However, for nonlinear kinetic models (as discussed in this thesis), the Jacobian is a function of time due to the inclusion of concentrations of reaction partners.90 This poses a challenge to the time scale separation as now a steady examination of the time gap is necessary to ensure valid decoupling of fast and slow processes. One of the most robust approaches in this respect is CSP.282 The basis of CSP is the assumption that the concentration trajectory of a chemical process is rapidly attracted onto a slow invariant manifold Ω,98 which is a (N − M)-dimensional hypersurface in concentration space. N denotes the number of species and also the total

number of time scales, and M denotes the number of fast time scales. Consequently, τM

and τM+1 are the time scales of the slowest of the M fast processes and of the fastest of the (N −M) slow processes, respectively. Two subspaces, the M-dimensional subspace of fast processes and the (N − M)-dimensional subspace of slow processes, are introduced,

which are spanned by MN-dimensional (column) basis vectors ai (i ∈ {1, ··· ,M}) and

(N − M) N-dimensional (column) basis vectors aj (j ∈ {M + 1, ··· ,N}), respectively. Furthermore, a set of N-dimensional dual (row) basis vectors bp (p ∈ {1, ··· ,N}) is p employed, which fulﬁll the condition b aq = δpq, where δpq is the Kronecker delta. The decomposition ansatz for the rate vector reads

120 Computational Singular Perturbation A.1

1 M gfast = [a1, ··· , aM ][b , ··· , b ]g , (A.3)

M+1 N gslow = [aM+1, ··· , aN ][b , ··· , b ]g . (A.4)

p CSP approximates the basis vectors b and aq by an iterative refinement procedure, where each refinement introduces an accuracy increase.89 The refinement procedure requires the time derivatives of the basis vectors. Therefore, computational savings due to the time scale separation may be lost by iteratively determining the basis vectors after each time step.90,174 However, the first refinement does not involve time derivatives and already guarantees numerical stability of the simplified model89,283 at the cost of lower accuracy, which can be controlled by adjusting ϵ.

A.1.2 Kinetic Simulation Coupled with Model Simplification

To determine currently slow and fast processes, we study the eigenvalues of the Jacobian. Given a predeﬁned time gap criterion ϵ (0 < ϵ ≪ 1), we start from the second-smallest complex modulus of the eigenvalues, |λN−1|, and compare it to the next higher complex modulus, |λN−2|. If |λN−1|/|λN−2| ≥ ϵ, we continue by increasing each index by one.

If |λi|/|λi−1| ≥ ϵ for all i ∈ {2, ··· ,N − 1}, our time gap criterion is not fulfilled and we cannot determine a spectral gap. Otherwise, the first eigenvalue pair fulfilling the condition |λi|/|λi−1| < ϵ sets the number of fast time scales, M = i − 1. The left and right eigenvectors of the Jacobian are chosen as an approximation for p the basis vectors b and aq, respectively, according to the CSP formalism. This approximation corresponds to the first refinement of the CSP basis vectors.89 It follows that the eigenvalues of the Jacobian can be obtained from our choice of CSP basis vectors,

i λi = b Jai . (A.5)

To determine which one of the L reaction pairs contributes to the fast processes, we consider the largest complex modulus of eigenvalues of each of the L sub-Jacobians corresponding to the isolated reaction pairs. If a dominant complex modulus is larger than

|λM+1|/ϵ, where λM+1 is associated with the Jacobian of the entire reaction system, the reaction pair connected to it will contribute to the fast processes. The idea behind this approach is that an eigenvalue corresponding to the entire kinetic model is approximately the sum of eigenvalues of sub-Jacobians with similar or smaller moduli.89 With this approach, we introduce the assumption that a reaction pair is either included in or excluded from the fast processes, which certainly is a simpliﬁcation that requires careful investigation.

121 Chapter A Kinetic Modeling of Complex Reactions: Technical Details

Next, we propagate the fast subnetwork (i.e., the subnetwork containing only those edges corresponding to fast reaction pairs) to local equilibrium. The stationary distribution can be determined through a nonlinear optimization algorithm,284 approximately through a so-called radical correction,98 or analytically for simpler networks. Due to the time scale separation, it is assumed that this process occurs immediately, i.e., it is not resolved in the course of the kinetic simulation. The actual simulation starts thereafter. The partially equilibrated concentration

vector ypeq(t) is propagated according to the time scale τ1,slow, which corresponds to the fastest process of the Jacobian of the slow subnetwork (i.e., the subnetwork containing only those edges corresponding to slow reaction pairs). The update of the concentration vector reads

y(t + τ1,slow) = ypeq(t) + g(t)τ1,slow . (A.6)

After that, the Jacobian of the entire network is decomposed again to determine the CSP basis vectors for the next time step. Our kinetic simulation algorithm can be summarized as follows: 1. Determine the number of fast time scales, M, by spectral decomposition of the Jacobian corresponding to the kinetic model under consideration. p 2. Approximate the CSP basis vectors b and aq by the left and right eigenvectors of the Jacobian. 3. Identify a reaction pair as a fast one if the largest complex modulus of eigenvalues of its sub-Jacobian is larger than |λM+1|/ϵ.

4. Propagate the fast subnetwork to local equilibrium, y(t) → ypeq(t).

5. Determine the time step τ1,slow by decomposing the Jacobian of the slow subnetwork. 6. Update the partially equilibrated concentration vector according to Eq. (A.6), ypeq(t) → y(t + τ1,slow). 7. If global equilibrium is not yet reached, repeat steps 1 to 6; otherwise, stop the simulation.

A.2 Construction of Random Covariance Matrices

Covariance matrices are symmetric, positive-semideﬁnite square matrices by deﬁnition, i.e., their eigenvalues are strictly nonnegative. We outline a simple recipe to construct a

random covariance matrix ΣA, which fulﬁlls the condition that its smallest and largest eigenvalues equal zero and σ2 , respectively. Deﬁning σ2 to be the largest eigen- Amax Amax

value ensures that all activation free-energy uncertainties are bound by σAmax , and the introduction of zero as smallest eigenvalue maximizes the variance of the diagonal values

of ΣA.

122 Encoding Chemical Logic into Network Graphs A.3

1. Generate a random (2L × 2L)-dimensional matrix P with elements that are uniformly sampled from [−0.5, +0.5]. L is the number of reversible reactions / bidirectional edges present in the underlying network. 2. The multiplication of P with its transpose leads to a (2L×2L)-dimensional matrix ⊤ 285 Q = PP that already is a covariance matrix with eigenvalues {Wii} and eigenvector matrix V. ( ) ⊤ 3. Generate another covariance matrix R = V W−min{Wii}·I V with eigenvalues Eii, which fulﬁlls the condition min{Eii} = 0. Here, W and I are the eigenvalue matrix of Q and the unit matrix, respectively.

4. The covariance matrix ΣA results from a rescaling of R,

σ2 Amax ΣA = R , max{Eii}

which implies that the largest eigenvalue of Σ equals σ2 . A Amax

Note that for actual CRNs, we expect the user to provide B sets of 2L activation free energies derived from an ensemble of quantum chemical models, instead of sampling from a random (and, hence, problem-unrelated) covariance matrix. However, as it is current practice to generate a single set of activation free energies instead of an ensemble of them, we encourage users to employ these random covariance matrices to develop an intuition for the potential eﬀect of free-energy uncertainty on the kinetics of complex chemical systems.

A.3 Encoding Chemical Logic into Network Graphs

We require AutoNetGen to yield a fully connected network representing isomerization reactions (A ⇌ P), association/dissociation reactions (A + B ⇌ P), and displacement reactions (A + B ⇌ P + Q). Further, we assume a homogeneous and dilute reaction mixture that evolves under isothermal conditions. The dilute solution property supports the assumption that reactive collisions involving more than two species are considered statistically negligible. AutoNetGen requires the speciﬁcation of the number of reactants,

N, their masses, {Mn} with n ∈ {1, ..., N}, the number of network layers to be generated, X, lower and upper bounds to the activation free energy, Amin and Amax, and a maximum standard deviation, σA,max, which applies to all activation free energies. Optionally, it is possible to define reactions of the specified reactants a priori. The following steps are carried out by AutoNetGen for the construction of artificial reaction networks:

1. For the construction of the (x + 1)-th network layer, we first define all potentially reactive intermediate states. At that stage, we register N = N0 + ... + Nx vertices. Since we already explored all possible reactions between the first N − Nx species,

123 Chapter A Kinetic Modeling of Complex Reactions: Technical Details

we will only consider the following potentially reactive intermediate states: Nx unimolecular intermediate states (leading to reactions of type A → P) defined by the Nx species of the x-th network layer; Nx homo-bimolecular intermediate states (type 2A → P); and Nx(Nx − 1)/2 + Nx(N − Nx) hetero-bimolecular intermediate states (type A + B → P). The first Nx(Nx − 1)/2 hetero-bimolecular states represent all possible, nonredundant combinations of the Nx new species among each other, whereas the latter Nx(N − Nx) states represent all possible combinations between the Nx new species and the N − Nx old species. 2. For each of the potentially reactive intermediates states, drawing from an exponential distribution with subsequent rounding determines the number of reaction channels per intermediate state. The specific number of reaction channels determines how many distinct reactive transformations of the corresponding intermediate state will occur and, therefore, how many new product states (potentially reactive intermediate states of the next generation) comprising either one or two molecules will be formed per reactive transformation. For zero reaction channels, the corresponding intermediate state is interpreted to be nonreactive, and no product formation will occur from that state. 3. When the X-th layer has been constructed, the graph structure of the reaction network is considered complete. Subsequently, 2L activation free energies will be calculated according to the following protocol. (a) Absolute free energies of N intermediate states are uniformly sampled from [100·Mn/matom, (100+25)·Mn/matom], which is a rather random choice that, however, respects that the system size (here, represented by the species mass Mn) usually dominates absolute (free) energies over the specific configuration of a given chemical formula. The ratio Mn/matom is defined as a relative species mass with respect to the mass of a single atom, matom = 1 (arbitrary units). (b) Absolute free energies of L transition states, each connecting two intermediate states[ { with absolute} free energies{ }A1 and A2, are uniformly sampled from max A1,A2 + Amin, max A1,A2 + Amax]. (c) 2L (relative) activation free energies are calculated from the free-energy differences of linked intermediate and transition states.

124 B 57Fe Mössbauer Isomer Shift Prediction: Technical Details

B.1 Statistical Calibration Analysis

All statistical results presented in Chapter 5 were produced with our suite of scripts reBoot175 developed in the GNU Octave286 programming language that is mostly compatible with Matlab.*,† The basic functionalities of the calibration methods employed are already described in Chapter 3. Here, we discuss some further details.

B.1.1 Bayesian Linear Regression

Bayesian linear regression119 is an eﬃcient calibration procedure for normally distributed parameters. The best-ﬁt parameter vector, wMAP, is obtained from

⊤ wMAP = βSX y , (B.1) where S is the covariance matrix of the parameters,

S−1 = αI + βX⊤X , (B.2)

*This appendix is reproduced in part with permission from J. Proppe, M. Reiher, J. Chem. Theory Comput. 2017, 13, 3297. Copyright 2017 American Chemical Society. †In this appendix, we adopt the notation introduced in Chapter 3.

125 Chapter B 57Fe Mössbauer Isomer Shift Prediction: Technical Details

I is the (M + 1) × (M + 1) unit matrix (M + 1 is the number of parameters contained in the property model), and α and β are so-called hyperparameters,

γ α = ⊤ , (B.3) wMAPwMAP

−1 N β = MSED , (B.4) N − γ ,wMAP which need to be iteratively reﬁned, a procedure based on the so-called evidence approximation or generalized maximum likelihood.119 For this purpose, the eﬀective number of parameters, γ, is calculated according to

∑M λ β γ = m , (B.5) α + λ β m=0 m

⊤ where the {λm} are eigenvalues of the (X X) matrix of dimension (M + 1) × (M + 1),

⊤ (X X)vm = λmvm , (B.6)

and the vm are the corresponding eigenvectors. For the initialization of α and β, one

can simply choose γ = M + 1 and wMAP = wD. Finally, the locally resolved MPU at

input value x0, s(x0), is estimated as √ −1 ⊤ s(x0) = β + x0 Sx0 . (B.7)

The hyperparameter β represents the inverse noise variance (precision), whereas α represents the belief on the parameter distributions prior to considering the actual data. Choosing α = 0, one assumes prior parameter distributions with inﬁnite variance, which is equivalent to assigning each value on the real line the same probability density. In

that case, Eq. (B.1) reduces to Eq. (3.11) resulting in wMAP = wD, i.e., one simply performs linear least-squares regression and additionally obtains the covariance matrix of the model parameters. For 57Fe Mössbauer isomer shift prediction (cf. Chapter 5), we observe that α optimized with the evidence approximation and α = 0 lead to values for the RMPV (in Eq. (3.19), the RMPV is deﬁned as its squared variant, MPV) deviating

by less than 0.05% (i.e., wD,MAP ≈ wD). Hence, we can directly compare the RMPV with the R632 obtained from bootstrapped linear least-squares regression where α = 0 (the same holds for comparisons between RMPV and RLOO).

The evidence approximation is appealing as it provides, for a given initial model complexity (here, the polynomial degree M), a maximum-transferable set of model

parameters, wMAP, by optimizing the hyperparameters α and β. Hence, the evidence

126 Statistical Calibration Analysis B.1 approximation is an approach suited for model selection.119 With this approach, one also obtains the optimal penalty factor ε = α/β employed in regularized least-squares regression,119 where the regularized MSE is minimized according to ( ) 1 ∑N ( ) regMSE ≡ regMSE = y − f(x , w ) 2 + εw⊤w . (B.8) D,w,ε N n n ε ε ε n=1

Given the normal-population assumption holds, the evidence approximation might yield a good initial value for the penalty factor to be learned in bootstrapped regularized regression. In all cases examined in Chapter 5, the RMPV did not change by more than 0.002% after the ﬁrst iteration of the evidence approximation. This deviation is completely masked by the low experimental resolution of 0.01 mm s−1 for measurements of the 57Fe Mössbauer isomer shift. Therefore, we conclude that a single iteration is suﬃcient to obtain a converged RMPV.

B.1.2 Iteratively Reweighted Linear Least-Squares Regression

In the variant of iteratively reweighted linear least-squares regression141 by Pernot et al.,116 one starts with a weighted linear least-squares regression with respect to a reference data set, D. The elements of the weight matrix U introduced in Eq. (3.13) are updated,   u2 + d2 ··· 0  1   . . .  U =  . .. .  , (B.9) ··· 2 2 0 uN + d where the quantity d2 accounts for the discrepancy between the increased MSE and the average experimental variance, ⟨u⟩2,

2 N 2 d = MSED − ⟨u⟩ , (B.10) N − M − 1 and, hence, is a measure of model inadequacy. Subsequently, one performs another weighted linear least-squares regression with the updated weight matrix. This procedure is repeated until the change in d2 becomes negligible between two iterations. For the problem studied in Chapter 5, we chose a threshold of 0.001%. Note that the approach outlined here is only of limited applicability as the constant discrepancy factor comes at the expense of two critical assumptions: a normal- population distribution and homogeneous residual variance (homoscedasticity). Here, however, both assumptions can be well justiﬁed after removal of the inconsistent data points (#1, #2, #7, #13, #28; Table 5.1 and Fig. 5.1). On the one hand, the boot-

127 Chapter B 57Fe Mössbauer Isomer Shift Prediction: Technical Details

strapped parameter distributions (Fig. 5.2, bottom) resemble normal distributions. On the other hand, we do not observe a trend of residual variance with respect to the input variable. In general, however, one should critically assess these assumptions by bootstrapping prediction intervals.135

B.1.3 Consideration of Experimental Uncertainty in Bootstrap- ping

Bootstrap samples were drawn by pair resampling (cf. Section 3.3.1). Each target value

δexp,n in a bootstrap sample was randomly perturbed according to its experimental uncertainty (assumed to be 0.02 mm s−1 for compounds without reported experimental uncertainty). For instance, if a data point considered refers to the n-th input–target pair in the reference data set, a random value was drawn from a normal distribution

with zero-mean and a standard deviation of un. This random value was rounded to the second decimal place to account for the experimental resolution of 0.01 mm s−1 and

subsequently added to δexp,n.

B.1.4 Statistically Valid Increase of Experimental Resolution

Bootstrap samples were drawn by pair resampling (cf. Section 3.3.1). To increase the experimental resolution from its actual value of 0.01 mm s−1 to 0.001 mm s−1, one must respect that any value of the isomer shift with three decimal places is equiprobable as long as the result rounded to two decimal places equals the reported isomer shift. For this purpose, random values were drawn from a uniform distribution with boundaries of −0.005 mm s−1 and +0.004 mm s−1, and subsequently added to the actual isomer shifts,

δexp,n, assuming the third decimal place equals zero prior to addition. If experimental

uncertainty, un, was explicitly considered in bootstrapping (cf. Section B.1.3), the same

procedure was repeated for un.

B.2 Quantum Chemical Calculations

References to the original crystal structures are listed in Table 5.1. First of all, solute molecules and counterions were removed from all molecular structures. The only excep- tions are structures 22 and 26, where we kept the very small counterions (lithium and sodium, respectively) due to their proximity to the iron nucleus. All molecular structures were fully optimized with the TPSS density functional,287 Ahlrichs’ def2-TZVP basis set on all atoms,253,254 Grimme’s DFT-D3 dispersion, correction288 and the conductor- like screening model (COSMO)289 for electrostatic screening with a dielectric constant of ε = 78 (water). For iodine, the only group-5 element in the reference set of iron compounds, the eﬀective core potential def2-ecp was employed.290 Structure optimiza-

128 Quantum Chemical Calculations B.2 tions were performed with Turbomole 6.4.0291 applying the resolution-of-the-identity approximation, which invokes an auxiliary basis set. The convergence thresholds were set to 10−7 hartree for the electronic energy difference and 10−4 hartree/bohr for the length of the gradient of the electronic energy with respect to the nuclear coordinates. Some molecular structures were further truncated prior to structure optimization (Ta- ble 5.1). For instance, larger alkyl groups at aromatic rings not directly attached to iron were replaced by methyl groups. All subsequent ACED calculations were performed with Orca 3.0.3292 (Table S2 of Ref. 38). For every molecular iron compound contained in our reference set, we determined the ACED with 12 different density functionals; one LDA functional (PWLDA293), four GGA functionals (BP86,294,295 BLYP,295–297 PW91,298 PBE252), two meta-GGA functionals (M06-L,299 TPSS), three hybrid-GGA functionals (B3LYP,296,297,300 B3PW91,252,300 PBE0255), and two meta-hybrid-GGA functionals (M06,301 TPSSh302). Neese’s CP(PPP) basis set128 was employed for iron, whereas Ahlrichs’ def2-TZVP basis set was employed for all other elements. All calculations were based on electrostatic screening (COSMO) with a dielectric constant of ε = 80. The convergence threshold for the electronic energy difference was set to 10−6 hartree (default). In case of error code 16384 (the only type of error that occurred in our ACED calculations), we added either the keyword slowconv to the first line (starting with !) or %scf maxiter end in a separate line ( = 500 always worked in the cases studied). We repeated all ACED calculations with def2-TZVP replaced by def2-SVP253 (Ta- ble S3 of Ref. 38), resulting in a slightly higher MPU for the latter (Tables S6 and S7 of Ref. 38). We did not perform further statistical analyses based on the def2-SVP results, but exploited this smaller basis set to probe the sensitivity of the ACED with respect to the convergence threshold for the electronic energy difference (Table S4 of Ref. 38). We decreased the convergence threshold for the electronic energy difference to 10−8 hartree (B3LYP/def2-SVP). For both thresholds, mean and maximum of the unsigned difference in the ACED are only 0.1% and 3.7% as large as the standard deviation of the ACED itself, respectively. We may conclude that default self-consistent field convergence criteria lead to stable results, especially taking into account the low experimental resolution of 0.01 mm s−1.

129 Chapter B 57Fe Mössbauer Isomer Shift Prediction: Technical Details

130 References

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151 152 Publications

The peer-reviewed publications and preprints that are included in parts or in an extended version in this thesis are denoted by an asterisk.

∗13. J. Proppe, M. Reiher, “Mechanism Deduction from Noisy Chemical Reaction Net- works”, 2018, arXiv:1803.09346. → Chapters 2 and 4. ∗12. T. Hayashi, M. Tinzl, T. Mori, U. Krengel, J. Proppe, J. Soetbeer, D. Klose, G. Jeschke, M. Reiher, D. Hilvert, “Capture and Characterization of a Reactive Haem–Carbenoid Complex in an Artificial Metalloenzyme”, Nat. Catal. 2018, 1, 578. → Chapter 6. 11. T. Weymuth, J. Proppe, M. Reiher, “Statistical Analysis of Semiclassical Disper- sion Corrections”, J. Chem. Theory Comput. 2018, 14, 2480. ∗10. J. Proppe, M. Reiher, “Reliable Estimation of Prediction Uncertainty for Physico- chemical Property Models”, J. Chem. Theory Comput. 2017, 13, 3297. → Chapters 3 and 5, and Appendix B. ∗9. G. N. Simm, J. Proppe, M. Reiher, “Error Assessment of Computational Models in Chemistry”, Chimia 2017, 71, 202. → Chapter 3. 8. G. Angulo, R. D. Astumian, V. Beniwal, P. G. Bolhuis, C. Dellago, J. Ellis, B. Ensing, D. R. Glowacki, S. Hammes-Schiffer, J. Kästner, T. Lelièvre, N. Makri, D. Manolopoulos, G. Menzl, T. F. Miller, A. Mulholland, E. A. Oprzeska-Zingrebe, M. Parrinello, E. Pollak, J. Proppe, M. Reiher, J. Richardson, P. R. Chowdhury, E. Sanz, C. Schütte, D. Shalashilin, R. Szabla, S. Taraphder, A. Tiwari, E. Vanden- Eijnden, A. Vijaykumar, K. Zinovjev, “New Methods: General Discussion”, Fara- day Discuss. 2016, 195, 521. ∗7. J. Proppe, T. Husch, G. N. Simm, M. Reiher, “Uncertainty Quantification for Quantum Chemical Models of Complex Reaction Networks”, Faraday Discuss. 2016, 195, 497. → Chapters 1 and 2, and Appendix A. ∗6. M. Bergeler, G. N. Simm, J. Proppe, M. Reiher, “Heuristics-Guided Exploration of Reaction Mechanisms”, J. Chem. Theory Comput. 2015, 11, 5712. → Chapter 2. 5. J. Proppe, “An Extended Flory Distribution for Kinetically Controlled Step-Growth Polymerizations Perturbed by Intramolecular Reactions”, Macromol. Theory Simul. 2015, 24, 500.

153 4. J. Proppe, C. Herrmann, “Communication Through Molecular Bridges: Different Bridge Orbital Trends Result in Common Property Trends”, J. Comput. Chem. 2015, 36, 201. 3. A. C. Jahnke, J. Proppe, M. Spulber, C. G. Palivan, C. Herrmann, O. S. Wenger, “Charge Delocalization in an Organic Mixed Valent Bithiophene is Greater Than in a Structurally Analogous Biselenophene”, J. Phys. Chem. A 2014, 118, 11293. 2. J. Proppe, G. Luinstra, “A Refined Flory Distribution for Step-Growth Poly- merizations Comprising Cyclic Molecules”, Macromol. Theory Simul. 2013, 22, DOI: 10.1002/mats. 201300117 [retracted, cf. Ref. 21 in Publication 5 of this list]. 1. C. Barreto, J. Proppe, S. Fredriksen, E. Hansen, R. W. Rychwalski, “Graphite Nanoplatelet/Pyromellitic Dianhydride Melt Modified PPC Composites: Prepara- tion and Characterization”, Polymer 2013, 54, 3574.

154 Jonny Proppe

Personal Details

Date of birth August 18, 1987 Place of birth Frankfurt (Oder), Germany Nationality German

Higher Education

07/2014–02/2018 Doctoral Studies in Chemistry (Dr. sc.), ETH Zürich, Switzerland, Supervisor: Prof. Dr. Markus Reiher 10/2008–04/2014 Undergraduate and Graduate Studies in Chemistry (BSc, MSc), University of Hamburg (UHH), Germany

Awards

2017 Early Postdoc.Mobility Fellowship by the Swiss National Science Foundation (project no. 178463) for an 18-month research stay in Prof. Dr. Alán Aspuru-Guzik’s group at Harvard University. 2017 Poster award at the “International Conference on Mathematics in (bio)Chemical Kinetics and Engineering”, Budapest, Hungary. 2017 Chemistry Travel Award by the Swiss Academy of Sciences (SCNAT) and the Swiss Chemical Society (SCS). 2013 Award for the best teaching evaluation of the year, Department of Chemistry, UHH, Germany.

Teaching

2015–2017 Introduction to Computer Science for Chemists (Tutorial and Lec- ture), ETH Zürich, Switzerland 2015–2017 Quantum Chemistry Studies (Practical), ETH Zürich, Switzerland 2014 Introduction to Chemical Reaction Kinetics (Tutorial), ETH Zürich, Switzerland 2010–2014 Mathematics for Chemists (Tutorial), UHH, Germany

155 his thesis was typeset using LATEX, originally developed by TLeslie Lamport and based on Don- ald Knuth’s TEX. The body text is set in 11 point Egenolﬀ-Berner Garamond, a revival of Claude Garamont’s humanist typeface. The above illustration, Science Experiment 02, was created by Ben Schlitter and released under cc by-nc-nd 3.0. A template that can be used to format a PhD dissertation with this look & feel has been released under the permissive agpl license, and can be found online at github.com/suchow/Dissertate or from its lead author, Jordan Suchow, at [email protected].