Indiana University-Purdue University Indianapolis (IUPUI) / In this thesis, we present novel computational methods and frameworks to address the challenges associated with the determination of free energy profiles for condensed-phase chemical reactions using combined quantum mechanical and molecular mechanical (QM/MM) approaches. We focus on overcoming issues related to force matching, molecular polarizability, and convergence of free energy profiles. First, we introduce a method called Reaction Path-Force Matching in Collective Variables (RP-FM-CV) that efficiently carries out ab initio QM/MM free energy simulations through mean force fitting. This method provides accurate and robust simulations of solution-phase chemical reactions by significantly reducing deviations on the collective variables forces, thereby bringing simulated free energy profiles closer to experimental and benchmark AI/MM results. Second, we explore the role of pairwise repulsive correcting potentials in generating converged free energy profiles for chemical reactions using QM/MM simulations. We develop a free energy correcting model that sheds light on the behavior of repulsive pairwise potentials with large force deviations in collective variables. Our findings contribute to a deeper understanding of force matching models, paving the way for more accurate predictions of free energy profiles in chemical reactions. Next, we address the underpolarization problem in semiempirical (SE) molecular orbital methods by introducing a hybrid framework called doubly polarized QM/MM (dp-QM/MM). This framework improves the response property of SE/MM methods through high-level molecular polarizability fitting using machine learning (ML)-derived corrective polarizabilities, referred to as chaperone polarizabilities. We demonstrate the effectiveness of the dp-QM/MM method in simulating the Menshutkin reaction in water, showing that ML chaperones significantly reduce the error in solute molecular polarizability, bringing simulated free energy profiles closer to experimental results. In summary, this thesis presents a series of novel methods and frameworks that improve the accuracy and reliability of free energy profile estimations in condensed-phase chemical reactions using QM/MM simulations. By addressing the challenges of force matching, molecular polarizability, and convergence, these advancements have the potential to impact various fields, including computational chemistry, materials science, and drug design.

Die vorliegende Arbeit behandelt die kollektive Dynamik identischer Klasse-I-anregbarer Elemente. Diese können im Rahmen der nichtlinearen Dynamik als Systeme nahe einer Sattel-Knoten-Bifurkation auf einem invarianten Kreis beschrieben werden. Der Fokus der Arbeit liegt auf dem Studium aktiver Rotatoren als Prototypen solcher Elemente. In Teil eins der Arbeit besprechen wir das klassische Modell abstoßend gekoppelter aktiver Rotatoren von Shinomoto und Kuramoto und generalisieren es indem wir höhere Fourier-Moden in der internen Dynamik der Rotatoren berücksichtigen. Wir besprechen außerdem die mathematischen Methoden die wir zur Untersuchung des Aktive-Rotatoren-Modells verwenden. In Teil zwei untersuchen wir Existenz und Stabilität periodischer Zwei-Cluster-Lösungen für generalisierte aktive Rotatoren und beweisen anschließend die Existenz eines Kontinuums periodischer Lösungen für eine Klasse Watanabe-Strogatz-integrabler Systeme zu denen insbesondere das klassische Aktive-Rotatoren-Modell gehört und zeigen dass (i) das Kontinuum eine normal-anziehende invariante Mannigfaltigkeit bildet und (ii) eine der auftretenden periodischen Lösungen Splay-State-Dynamik besitzt. Danach entwickeln wir mit Hilfe der Averaging-Methode eine Störungstheorie für solche Systeme. Mit dieser können wir Rückschlüsse auf die asymptotische Dynamik des generalisierten Aktive-Rotatoren-Modells ziehen. Als Hauptergebnis stellen wir fest dass sowohl periodische Zwei-Cluster-Lösungen als auch Splay States robuste Lösungen für das Aktive-Rotatoren-Modell darstellen. Wir untersuchen außerdem einen "Stabilitätstransfer" zwischen diesen Lösungen durch sogenannte Broken-Symmetry States. In Teil drei untersuchen wir Ensembles gekoppelter Morris-Lecar-Neuronen und stellen fest, dass deren asymptotische Dynamik der der aktiven Rotatoren vergleichbar ist was nahelegt dass die Ergebnisse aus Teil zwei ein qualitatives Bild für solch kompliziertere und realistischere Neuronenmodelle liefern. / We study the collective dynamics of class I excitable elements, which can be described within the theory of nonlinear dynamics as systems close to a saddle-node bifurcation on an invariant circle. The focus of the thesis lies on the study of active rotators as a prototype for such elements. In part one of the thesis, we motivate the classic model of repulsively coupled active rotators by Shinomoto and Kuramoto and generalize it by considering higher-order Fourier modes in the on-site dynamics of the rotators. We also discuss the mathematical methods which our work relies on, in particular the concept of Watanabe-Strogatz (WS) integrability which allows to describe systems of identical angular variables in terms of Möbius transformations. In part two, we investigate the existence and stability of periodic two-cluster states for generalized active rotators and prove the existence of a continuum of periodic orbits for a class of WS-integrable systems which includes, in particular, the classic active rotator model. We show that (i) this continuum constitutes a normally attracting invariant manifold and that (ii) one of the solutions yields splay state dynamics. We then develop a perturbation theory for such systems, based on the averaging method. By this approach, we can deduce the asymptotic dynamics of the generalized active rotator model. As a main result, we find that periodic two-cluster states and splay states are robust periodic solutions for systems of identical active rotators. We also investigate a 'transfer of stability' between these solutions by means of so-called broken-symmetry states. In part three, we study ensembles of higher-dimensional class I excitable elements in the form of Morris-Lecar neurons and find the asymptotic dynamics of such systems to be similar to those of active rotators, which suggests that our results from part two yield a suitable qualitative description for more complicated and realistic neural models.

The topic of this work is the structural characterization and theoretical modeling of organic single and heterolayers. The growth of sub-monolayers and monolayers (ML) of the two polycyclic aromatic hydrocarbons quaterrylene (QT) and hexa-peri-hexabenzocoronene (HBC) on Ag(111) and Au(111) was investigated. A transition from a disordered, isotropic phase to an ordered phase with increasing coverage was found. The lattice of the ordered phase turned out to be coverage dependent. The intermolecular potential was modeled including Coulomb and van der Waals interaction by a force-field approach. The postulated repulsive character of the potential could be connected to the non-uniform intramolecular charge distribution and to a screening of the van der Waals forces. Furthermore, the influence of the variable lattice constant on the epitaxial growth of HBC was studied. The second part of this work deals with a ML of 3,4,9,10-perylenetetracarboxylic dianhydride (PTCDA) on a ML of HBC. In dependency on the initial lattice constant of HBC, a total of three line-on-line (LOL) and point-on-line coincident phases of PTCDA (with respect to HBC) was found. Following an analysis of the general properties of LOL coincident systems via force-field calculations, a new method to predict the structure of such systems is introduced. / Thema dieser Arbeit ist die strukturelle Charakterisierung von organischen Einfach- und Heterolagen sowie deren theoretische Beschreibung und Modellierung. Es wurden Submonolagen und Monolagen (ML) der polyzyklischen Kohlenwasserstoffe Quaterrylen (QT) und Hexa-peri-hexabenzocoronen (HBC) auf Ag(111) und Au(111) Einkristallen untersucht und ein Übergang von einer ungeordneten, isotropen Phase zu einer geordneten Phase mit steigender Bedeckung beobachtet. Die geordnete Phase wies dabei bedeckungsabhängige Gitterkonstanten auf. Das intermolekulare Potential wurde unter Berücksichtigung von Coulomb und van der Waals Anteilen mittels Kraftfeldmethoden modelliert. Der postulierte repulsive Charakter des Potentials konnte auf die Ladungsverteilung im Molekül und eine Abschwächung des van der Waals Potentials zurückgeführt werden. Weiterhin wurde der Einfluss der variablen HBC Gitterkonstante auf die epitaktische Relation des Gitters zum Metallsubstrat untersucht. Der zweite Teil der Arbeit widmet sich der Untersuchung einer ML 3,4,9,10-Perylenetetracarboxylic dianhydrid (PTCDA) auf einer ML HBC. Dabei wurden, in Abhängigkeit von der HBC Gitterkonstante, insgesamt drei verschiedene Typen von line-on-line bzw. point-on-line Epitaxie nachgewiesen. Im Anschluss an eine Analyse der generellen Eigenschaften solcher epitaktischer Lagen mittels Kraftfeldrechnungen wird eine neue Methode zur Vorhersage der Struktur konkreter Systeme vorgestellt.

The topic of this work is the structural characterization and theoretical modeling of organic single and heterolayers. The growth of sub-monolayers and monolayers (ML) of the two polycyclic aromatic hydrocarbons quaterrylene (QT) and hexa-peri-hexabenzocoronene (HBC) on Ag(111) and Au(111) was investigated. A transition from a disordered, isotropic phase to an ordered phase with increasing coverage was found. The lattice of the ordered phase turned out to be coverage dependent. The intermolecular potential was modeled including Coulomb and van der Waals interaction by a force-field approach. The postulated repulsive character of the potential could be connected to the non-uniform intramolecular charge distribution and to a screening of the van der Waals forces. Furthermore, the influence of the variable lattice constant on the epitaxial growth of HBC was studied. The second part of this work deals with a ML of 3,4,9,10-perylenetetracarboxylic dianhydride (PTCDA) on a ML of HBC. In dependency on the initial lattice constant of HBC, a total of three line-on-line (LOL) and point-on-line coincident phases of PTCDA (with respect to HBC) was found. Following an analysis of the general properties of LOL coincident systems via force-field calculations, a new method to predict the structure of such systems is introduced.:1 Introduction 2 Experimental Methods 2.1 Organic molecular beam epitaxy 2.2 Scanning tunneling microscopy (STM) 2.3 Low-energy electron diffraction (LEED) 2.4 Molecules and substrates: Basic properties and literature review 2.4.1 3,4,9,10-Perylenetetracarboxylic dianhydride 2.4.2 Hexa-peri-hexabenzocoronene 2.4.3 Quaterrylene 2.4.4 Metal substrates: Au(111) and Ag(111) 3 Theory and Modeling 3.1 Reciprocal space and LEED 3.1.1 Fourier transform and geometrical LEED theory 3.1.2 Kinematic and dynamic LEED theory 3.1.3 Further applications of the Fourier transform 3.2 Computational chemistry 3.2.1 Calculating molecular properties 3.2.2 The atomic force-field method 3.2.3 Potential energy calculations in extended systems 4 Epitaxy in terms of potential energy 5 Interaction of QT and HBC at Sub-ML and ML Coverage 5.1 Experimental results 5.2 Modeling technique 5.3 Results of the model calculation 5.4 Discussion of results 5.5 Conclusion 6 The Ordered Phases of HBC on Ag(111) and Au(111 6.1 Geometrical analysis of epitaxy 6.2 Energetic gain of epitaxial structures 6.3 Comparison to experiment 6.4 Influence of the Au(111) surface reconstruction 6.5 Conclusion 7 Organic Heterosystems of PTCDA and HBC on Au(111) 7.1 PTCDA on Au(111) revisited 7.2 LEED and STM on PTCDA/HBC/Au(111) samples 7.2.1 A “compact” HBC layer substrate 7.2.2 A “loosely packed” HBC layer substrate 7.2.3 Summary of LEED results 7.2.4 STM results 7.3 Epitaxial relations in the system PTCDA/HBC/Au(111) 7.3.1 Geometrical analysis of epitaxy 7.3.2 Energetic gain of epitaxial structures 7.3.3 Mutual alignment of lattices 7.4 Heterosystems of PTCDA and HBC with inverted stacking sequence 8 General Properties of POL and LOL Epitaxy 8.1 A new coordinate system 8.2 Specific properties of the substrate-adsorbate potential 8.3 The “natural order” of the lattice lines 8.4 Prediction of epitaxial growth - a “LOL predictor” 8.4.1 Method 8.4.2 Results 9 General Conclusions and Future Perspectives 9.1 Conclusion 9.2 Outlook Appendix A.1 Conductance in a STM: The 1D WKB model A.2 Extraction of the DOS from STS measurements by means of the 1D WKB model A.3 Practical application of the 1D WKB model A.4 The normalized differential conductivity A.5 A new normalization method / Thema dieser Arbeit ist die strukturelle Charakterisierung von organischen Einfach- und Heterolagen sowie deren theoretische Beschreibung und Modellierung. Es wurden Submonolagen und Monolagen (ML) der polyzyklischen Kohlenwasserstoffe Quaterrylen (QT) und Hexa-peri-hexabenzocoronen (HBC) auf Ag(111) und Au(111) Einkristallen untersucht und ein Übergang von einer ungeordneten, isotropen Phase zu einer geordneten Phase mit steigender Bedeckung beobachtet. Die geordnete Phase wies dabei bedeckungsabhängige Gitterkonstanten auf. Das intermolekulare Potential wurde unter Berücksichtigung von Coulomb und van der Waals Anteilen mittels Kraftfeldmethoden modelliert. Der postulierte repulsive Charakter des Potentials konnte auf die Ladungsverteilung im Molekül und eine Abschwächung des van der Waals Potentials zurückgeführt werden. Weiterhin wurde der Einfluss der variablen HBC Gitterkonstante auf die epitaktische Relation des Gitters zum Metallsubstrat untersucht. Der zweite Teil der Arbeit widmet sich der Untersuchung einer ML 3,4,9,10-Perylenetetracarboxylic dianhydrid (PTCDA) auf einer ML HBC. Dabei wurden, in Abhängigkeit von der HBC Gitterkonstante, insgesamt drei verschiedene Typen von line-on-line bzw. point-on-line Epitaxie nachgewiesen. Im Anschluss an eine Analyse der generellen Eigenschaften solcher epitaktischer Lagen mittels Kraftfeldrechnungen wird eine neue Methode zur Vorhersage der Struktur konkreter Systeme vorgestellt.:1 Introduction 2 Experimental Methods 2.1 Organic molecular beam epitaxy 2.2 Scanning tunneling microscopy (STM) 2.3 Low-energy electron diffraction (LEED) 2.4 Molecules and substrates: Basic properties and literature review 2.4.1 3,4,9,10-Perylenetetracarboxylic dianhydride 2.4.2 Hexa-peri-hexabenzocoronene 2.4.3 Quaterrylene 2.4.4 Metal substrates: Au(111) and Ag(111) 3 Theory and Modeling 3.1 Reciprocal space and LEED 3.1.1 Fourier transform and geometrical LEED theory 3.1.2 Kinematic and dynamic LEED theory 3.1.3 Further applications of the Fourier transform 3.2 Computational chemistry 3.2.1 Calculating molecular properties 3.2.2 The atomic force-field method 3.2.3 Potential energy calculations in extended systems 4 Epitaxy in terms of potential energy 5 Interaction of QT and HBC at Sub-ML and ML Coverage 5.1 Experimental results 5.2 Modeling technique 5.3 Results of the model calculation 5.4 Discussion of results 5.5 Conclusion 6 The Ordered Phases of HBC on Ag(111) and Au(111 6.1 Geometrical analysis of epitaxy 6.2 Energetic gain of epitaxial structures 6.3 Comparison to experiment 6.4 Influence of the Au(111) surface reconstruction 6.5 Conclusion 7 Organic Heterosystems of PTCDA and HBC on Au(111) 7.1 PTCDA on Au(111) revisited 7.2 LEED and STM on PTCDA/HBC/Au(111) samples 7.2.1 A “compact” HBC layer substrate 7.2.2 A “loosely packed” HBC layer substrate 7.2.3 Summary of LEED results 7.2.4 STM results 7.3 Epitaxial relations in the system PTCDA/HBC/Au(111) 7.3.1 Geometrical analysis of epitaxy 7.3.2 Energetic gain of epitaxial structures 7.3.3 Mutual alignment of lattices 7.4 Heterosystems of PTCDA and HBC with inverted stacking sequence 8 General Properties of POL and LOL Epitaxy 8.1 A new coordinate system 8.2 Specific properties of the substrate-adsorbate potential 8.3 The “natural order” of the lattice lines 8.4 Prediction of epitaxial growth - a “LOL predictor” 8.4.1 Method 8.4.2 Results 9 General Conclusions and Future Perspectives 9.1 Conclusion 9.2 Outlook Appendix A.1 Conductance in a STM: The 1D WKB model A.2 Extraction of the DOS from STS measurements by means of the 1D WKB model A.3 Practical application of the 1D WKB model A.4 The normalized differential conductivity A.5 A new normalization method