Extension of a VCI program for the calculation of rovibrational intensities Von der Fakultät Chemie der Universität Stuttgart zur Erlangung der Würde eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung. Vorgelegt von Martin Tschöpe aus Quierschied Hauptberichter: apl. Prof. Dr. Guntram Rauhut Mitberichter: Prof. Dr. Ronny Nawrodt Prüfungsausschussvorsitzender: Prof. Dr. Johannes Kästner Tag der mündlichen Prüfung: 22. Juni 2023 Institut für Theoretische Chemie Universität Stuttgart 2023 ii Abstract The identification of molecules in the interstellar medium (ISM), circumstellar spheres and low-temperature exoplanet atmospheres is a major challenge in astrophysics and is mainly based on highly accurate rotational and rovibrational infrared reference spectra. One way to determine these reference spectra is through ab initio calculations, since they allow for an efficient simulation of a wide range of conditions, including extremely low pressure and tem- perature. In this thesis, a new realization of rovibrational configuration interaction (RVCI) theory for the calculation of rovibrational infrared spectra via configuration interaction theory has been developed and implemented to allow for the calculation of these reference spectra. The approach is based on a multi-mode expansion of the multi-dimensional potential energy sur- face (PES) and dipole moment surface (DMS), vibrational self-consistent field (VSCF) and vibrational configuration interaction (VCI) theory. A direct product between vibrational ba- sis functions (VCI wave functions) and rotational basis functions is used. Thus, in contrast to the previously introduced rotational configuration interaction (RCI), the interaction be- tween rotational and vibrational bands is taken into account. This is done with high accuracy by including the higher order term of the inverse effective moment of inertia tensor µ for the rotational term and the Coriolis coupling term in theWatsonHamiltonian. Moreover, a new rotational basis called molecule specific rotational basis (MSRB) is introduced. The convergence behavior of several different series expansionswithinRVCI theory showed very individual effects for the five parameters investigated. If the maximum total angular mo- mentum quantum number Jmax or the vibrational basis set is not sufficiently converged, large artifacts occur. Efficient ways to detect and avoid these issues are presented. The CI space in the VCI calculations is another crucial parameter in terms of quantitative convergence. It was found that the best indicator for convergence of the coupling strength is the spectral separation between the vibrational bands. For the two quasi-degenerate vibrational modes of H2CS the 0th order Coriolis coupling is significant, while the 1st order terms show only small changes. Compared to the Coriolis coupling terms, the rotational terms require a one order higher µ- tensor expansion for the same accuracy. The 1st order introduces at most energy shifts for iii entire progressions of 5 cm−1. The changes induced by the 2nd order terms are more than one order of magnitude smaller. Since the absence of higher order coupling terms does not cause artifacts in the spectrum, an insufficient convergence is hard to detect in the resulting spectra. The calculation for the first paper in this thesis relied on a number of approximations, that could be removed in the further course of this thesis. Most of these approximations did not drastically affect the spectra of ketenimine, as there are onlyminor changes in the spectrumup to 2900 cm−1. Above that, however, the ν1 band and the coupling between ν8 + ν12 and ν11 show that the quality of the quantum number assignment, the consideration of the coupling and the consistency of the intensities improved significantly over the last three years. The new calculations also revealed an interesting turnaround progression in this region. The line broadening study using propynal as a benchmark molecule gives evidence for the assumption that for molecules with 6− 10 atoms there is no need to consider sophisticated beyond Voigt profiles, as they are used for small molecules (N2, H2O, CH4, NH3, etc). The reason for this is that the higher mass and the larger moment of inertia tensor lead to a higher rovibrational state density. Therefore, the variation between different broadening profiles is negligible. At the end of this thesis, several runtime optimizations are analyzed. The parallelization shows an almost perfect scaling in the number of CPU cores for the precalculations and the intensity calculation. In addition, the precalculations of the vibrational integrals save about a factor of 8 in total computation time. The contraction of the MSRB coefficients and the RVCI coefficients results in a total reduction of computational time of 50% for H2CS and 97% for ketenimine. The current implementation of the RVCI theory in MOLPRO is capable of calculating rovibrational infrared and Raman spectra for up to 10 atoms, up to room temperature and over a broad spectral range. However, combining all these features together requires large computational resources. A list of optimizations to increase the computational efficiency is presented in theoutlook. Moreover, a numberofpossible additional functionalities andmeth- ods to increase the robustness of the code are provided. iv Zusammenfassung Die Identifizierung von Molekülen im interstellaren Medium, in zirkumstellaren Scheiben und in den Atmosphären kalter Exoplaneten ist eine große Herausforderung in der Astro- physik und basiert hauptsächlich auf hochgenauen Rotations- und Rotationsschwingungs- Referenzspektren. Eine Möglichkeit, diese Referenzspektren zu bestimmen, sind ab initio- Berechnungen, da sie eine effiziente Simulation eines breiten Bereichs von Bedingungen (ein- schließlich extrem niedriger Drücke und Temperaturen) ermöglichen. In dieser Arbeit wurde eine neue und besonders effiziente Implementierung der Rota- tionsschwingungskonfigurationswechselwirkungstheorie für die Berechnung von Infrarot- Rotationschwingungsspektren entwickelt, umdie Berechnung dieserReferenzspektren zu er- möglichen. Der Ansatz basiert auf Normalkoordinaten und einer Mehrmodenentwicklung der mehrdimensionalen Potential- und Dipolmomentflächen sowie Schwingungs-Selbst- konsistentes-Feld-Verfahren und Schwingungskonfigurationswechselwirkungstheorie. Dabei wird ein direktes Produkt zwischen Schwingungsbasisfunktionen und Rotationsbasisfunk- tionen verwendet. So kann im Gegensatz zu der zuvor eingeführten Rotationskonfigura- tionswechselwirkungstheorie die Wechselwirkung zwischen Rotations- und Vibrationsban- den berücksichtigt werden. Dies geschieht mit hoher Genauigkeit, indem die Terme höherer Ordnung des inversen effektiven Trägheitsmomenttensors µ für den Rotationsterm und die Coriolis-Kopplung imWatson Hamiltonian berücksichtigt werden. Darüber hinaus werden eine neue Rotationsbasis namens Molekülspezifische Rotationsbasis (MSRB) und eine neue Art der Zuweisung von Rotationsschwingungsquantenzahlen eingeführt. Das Konvergenzverhalten verschiedener Entwicklungen für die Rotationsschwingungs- konfigurationswechselwirkungstheorie (RVCI) zeigte sehr individuelle Effekte für die fünf untersuchten Parameter. Wenn die maximale Gesamtdrehimpulsquantenzahl Jmax oder die Größe der Schwingungsbasis nicht ausreichend konvergiert, treten besonders große Arte- fakte auf. Es werden effiziente Methoden zur Erkennung und Vermeidung dieser Pro- bleme vorgestellt. Auch die Größe des Schwingungsbasissatzes ist ein entscheidender Para- meter für die Konvergenz des Spektrums. Der beste Indikator für die Konvergenz bezüglich dieses Parameters und für die Stärke der Kopplung ist der spektrale Abstand zwischen den v Schwingungsbanden. Für die beiden quasi-entarteten Schwingungsmoden von H2CS ist die Coriolis-Kopplung nullter Ordnung sehr entscheidend, während die Terme erster Ordnung nur geringe Änderungen verursachen. Im Vergleich zu den Coriolis-Kopplungstermen er- fordern die Rotationsterme eine um eine Ordnung höhere µ-Tensorentwicklung für die glei- cheGenauigkeit. Die ersteOrdnung führt für ganze Progressionen zuEnergieverschiebungen vonhöchstens 5 cm−1. Die durchdieTermeder zweitenOrdnunghervorgerufenenÄnderun- gen sind um mehr als eine Größenordnung geringer. Da das Fehlen von Kopplungstermen höherer Ordnung keine Artefakte im Spektrum verursacht, ist eine unzureichende Konver- genz in den resultierenden Spektren sehr schwierig zu erkennen. Die Berechnungen für die erste Veröffentlichung in dieser Dissertation beruhten auf einer Reihe von Näherungen, die im weiteren Verlauf dieser Arbeit entfernt werden konnten. Die meisten dieser Näherungen hatten kaum Auswirkungen auf die Spektren von Ketenimin, da sie bis 2900 cm−1 nur zu geringfügigen Änderungen des Spektrums führten. Oberhalb dieser Grenze zeigen jedoch die ν1-Bande und die Kopplung zwischen ν8 + ν12 und ν11, dass sich die Qualität der Quantenzahlzuordnung und die Konsistenz der Intensitäten in den letzten drei Jahren deutlich verbessert haben. Die neuen Berechnungen zeigen auch eine interessante turnaroundProgression in diesemBereich. Die Studie zur Linienverbreiterung unterVerwen- dung von Propynal als Anwendungsmolekül bestätigte die Annahme, dass für Moleküle mit 6 − 10 Atomen keine Notwendigkeit besteht, beyond Voigt-Profile zu verwenden, wie sie für kleine Moleküle (N2, H2O, CH4, NH3, etc.) benutzt werden. Der Grund dafür ist, dass die höhere Masse und der größere Trägheitstensor zu einer hohen Schwingungszustandsdichte führen, wodurch die genaue Form des Verbreiterungsprofil weniger relevant wird. Am Ende dieser Arbeit werden verschiedene Laufzeitoptimierungen analysiert. Die Pa- rallelisierung zeigt eine nahezu perfekte Skalierung in der Anzahl der CPU-Kerne für die Vorberechnungen der Schwingungsintegrale und für die Intensitätsberechnung. Darüber hinaus sparen die Vorberechnungen der Schwingungsintegrale etwa einen Faktor von 8 an Gesamtrechenzeit ein. Die Kontraktion der MSRB-Koeffizienten mit den RVCI- Koeffizienten führt zu einer Gesamtrechenzeitreduktion von 50% für H2CS und 97% für Ketenimin. Die derzeitige Implementierung der RVCI-Theorie inMOLPRO ist in der Lage, Infrarot- und Raman-Spektren für bis zu 10 Atome, von T = 0K bis zu Raumtemperatur und über einen weiten Spektralbereich zu berechnen. Die Kombination all dieser Eigenschaften er- fordert jedoch große Rechenressourcen. Im Ausblick wird daher eine Liste von Optimierun- gen zur Steigerung der Recheneffizienz vorgestellt. Darüber hinaus wird eine Reihe von möglichen zusätzlichen Funktionalitäten und Methoden zur Erhöhung der Robustheit des Programms aufgelistet. vi Danksagung In den letzten vier Jahren haben mich viele Menschen in vielfältiger Weise unterstützt und dazu beigetragen, diese Arbeit zu ermöglichen. Ihnenmöchte ich nun an dieser Stelle meinen Dank aussprechen. Als Erstes möchte ich mich bei Prof. Dr. Guntram Rauhut bedanken. Er hat mir nicht nur diese Promotion ermöglicht, sondern mich auch fachlich und beim Schreiben der zahlreichen Berichte und Paper viel gelehrt. Besonders froh bin ich darüber, dass er mir die Freiheiten gelassen hat, unsere Forschung in Richtung astrophysikalischer An- wendungen voranzutreiben und er mir meine Reise in die USA ermöglichte. Mein Dank gilt auch Prof. Dr. Ronny Nawrodt als Mitberichter und Prof. Dr. Jo- hannes Kästner als Prüfungsausschussvorsitzender, die beide ohne Zögern und auf sehr un- kompliziert Weise bereit waren, diese Aufgaben zu übernommen. Darüber hinaus danke ich Prof. Dr. Laura Kreidberg, Dr. Iouli Gordan und Prof. Dr. Heather A. Knutson für Ihre Gastfreundschaft und dafür, dass ichmeine Ergebnisse in Ihren Arbeitsgruppen präsentieren durfte. Der Studienstiftung des deutschen Volkes danke ich für die finanzielle Unterstützung in den letzten drei Jahren und für die Möglichkeit, in die USA zu reisen. Auch einige Arbeitskollegen möchte ich nicht unerwähnt lassen. So habe ich die enge Zusammenarbeit mit Sebastian Erfort sehr genossen. Von ihm habe ich nicht nur viel über Rotationen und Schwingungen gelernt, sondern auch somanches beim Bouldern. Ein weite- rer, ehemaliger Zimmerkollege, mit dem ich viele gesellige Stunden verbracht habe, istMoritz Schneider. Ich kennewenig andereMenschen,mit denen ich so gut über Themen diskutieren kann, egal ob wir einer Meinung waren oder nicht. Von Dr. Benjamin Ziegler habe ich nicht nur die Phrase “UmGotteswillen! Wie konnte das jemals funktionieren?” zu schätzen und zu fürchten gelernt, sondern auch viele Kniffe in Fortran. Mit Dr. Taras Petrenko verbinde ich viele, sehr gesellige Stunden, auch wenn er meistens eine Spur aus Kekskrümeln hinterlassen hatte. Unserem ehemaligen Post Doc Dr. Benjamin Schröder möchte ich dafür danken, dass er mich an seinem allumfassenden Wissen zu Rotationsschwingungsspektren hat teilhaben lassen undDr. TinaMathea für die schöne Zeit und die angenehmeArbeitsatmosphäre. Den Abschluss dieser Runde aus ehemaligen Arbeitskollegen darf Dennis Dinu bereiten. Auch vii wenn die Zeit mit ihm in Stuttgart nur sehr kurz war, so war sie doch sehr intensiv und bleibt mir überaus positiv in Erinnerung. Ebenfalls in meinem Universitätsalltag, wenn auch nicht in Form von Arbeitskollegen, haben mich einige weitere Personen begleitet. Als Erstes möchte ich Juliane Heitkämper und Tizian Wenzel für manch mächtig munteres Mensa-Mahl merci mitteilen. Meinem ehemali- gen Betreuer während der Bachelorarbeit, Dr. Nicolai Lang, möchte ich besonders herzlich dafür danken, dass er meine Frage “Hast du kurz Zeit?” stets bejahte, obschon er wusste, dass ich damit seine Abendplanung über denHaufen geworfen hatte. Ich schätze an dir nicht nur, dass du mir geduldig, die abstraktesten physikalischen Zusammenhänge wieder und wieder erklärt hast, sondern auch, dass du mich dazu ermutigt hast, die Forschungsreise in die USA anzutreten. Für viele musikalische und kulinarische Stunden, so wie einige der wichtigsten Unterhaltungen inmeinem Leben kurz vor Beginn der Promotion danke ich Dr. Daniel Diz- darevic. Darüber hinaus möchte ich Dr. Johannes Reiff danken für 9 Jahre in denen wir uns durch Praktikumsprotokolle, Übungsblätter und die Masterarbeit gekämpft haben. Nun möchte ich drei ganz besonderen Männern danken, die mich in den letzten fünf Jahren, weniger in meiner fachlichen als vielmehr in meiner persönlichen Entwicklung un- terstützt haben. Beginnen möchte ich mit Mario Zinßer. Er ist ein schier unerschöpflicher Quell positiver Energie und schafft es immer, in allem das Gute zu sehen. Dies hat mir in vie- len schwierigen Phasenmeiner Promotion, aber auch darüber hinaus sehr geholfen. Auch du hast mich dazu ermutigt, die Reise in die USA anzutreten und vielleicht warst du mit deiner Reise nachHawaii sogar einweiteresMal eineQuelle der Inspiration. Wirwaren in den letzten Jahrennicht nur gemeinsamaufderWildspitze, inHawaii (mit einem Jeep), beimKletternund Ski fahren, sondern haben auch so manche persönliche Herausforderung gemeistert. Trotz- dem glaube ich, dass unser mit Abstand größtes Abenteuer noch vor uns liegt. Auch wenn das zeitliche Zusammenfallenmit meiner Promotion ein Zufall war, so hat mir Dr. Manfred Schrode in den letzten fünf Jahren maßgeblich mit meiner Persönlichkeitsent- wicklung geholfen. Fast noch wichtiger ist aber, dass er mir das Handwerkszeug gegeben hat, um diesen Prozess in Zukunft alleine weiterführen zu können und dafür möchte ich mich an dieser Stelle in aller Form bedanken. Auch wenn mein nächster Dank über die Maßen exzentrisch wirken mag, geht dieser an Elon Musk. Die Entwicklung der Falcon 9 und Star- shipRaketen und das zugehörige Raumfahrtprogramm hatte auch den Zweck, dieMenschen zu motivieren und ihnen etwas zu geben, worauf sie sich freuen können, weil es im Leben nicht nur darum gehen kann, Probleme zu lösen. Das größte Abenteuer in der Geschichte der Menschheit hat dies definitiv bei mir bewirkt. Zu guter Letzt möchte ich mich bei meiner Familie bedanken. Gabi Tschöpe undMichael Tschöpe, ihr habtmich in den 13 Jahrenmeiner akademischenAusbildung auf demWeg vom mittelmäßigen Realschüler zum Dr. rer. nat. immer bedingungslos unterstützt und dafür viii bin ich euch unendlich dankbar. Meinem Bruder Matthias Tschöpe bin ich nicht nur dafür dankbar, dass er mich an seinem Wissen über theoretische Informatik und künstliche Intel- ligenz hat teilhaben lassen. Ich bin dir auch dafür dankbar, dass du den Mut hattest, mit mir zahlreiche Projekte in Angriff zu nehmen, sei es handwerklicher Art (unsere Jahrhundert- Betonage imStickoder die selbst-geschweißtenGartentore) oder automobilerArt (die Fahrten an der Nordschleife). Zuletzt möchte ichDr. Andreas Tschöpe danken. Du bist für mich seit langem ein Vorbild, weil du der erste in unserer Familie warst, der promoviert hat, du trotz- dem einer der bodenständigstenMenschen geblieben bist, den ichmir vorstellen kann und du mich damals ermutigt hast, diese Promotion hier anzutreten. ix x Peer-reviewed publications This cumulative dissertation summarizes results that have been published in (I) Martin Tschöpe, Benjamin Schröder, Sebastian Erfort and Guntram Rauhut High- Level Rovibrational Calculations on Ketenimine. Frontiers in Chemistry, Section As- trochemistry, 8, 623641 (2021) Copyright: Reprinted from Ref. [1], Copyright 2021, CC-BY 4.0. Contributions: M.T.: Implementation (RVCI), Simulation, Data curation, Visualiza- tion, Formal analysis (partially), Project administration (partially), Writing – original draft sectionsAbstract, Theory andComputationalDetails, Results, Discussion andSum- mary (partially), Writing – review & editing (partially) B.S.: Formal analysis (partially), Writing – original draft sections Introduction, Discus- sion and Summary (partially), Writing – review & editing (partially) S.E.: Conceptualization, Methodology, Implementation (RCI), Writing – review & editing (partially) G.R.: Project administration (partially), Funding acquisition, Resources, Writing – re- view & editing (partially) DOI: https://doi.org/10.3389/fchem.2020.623641 (II) Martin Tschöpe andGuntramRauhutConvergence of series expansions in rovibrational configuration interaction (RVCI) calculations. The Journal of Chemical Physics, 157, 234105 (2022) Copyright: Reprinted from [2], with the permission of AIP Publishing, Contributions: M.T.: Methodology, Implementation, Simulation, Data curation, Vi- sualization, Writing – original draft sections Theory (RVCI), Computational Details, Results, Discussion and Summary andWriting – review & editing (partially) G.R.: Project administration, Funding acquisition,Writing–original draft sectionsAb- stract, Introduction, Theory (PES and VCI) andWriting – review & editing (partially) DOI: https://doi.org/10.1063/5.0129828 xi https://doi.org/10.3389/fchem.2020.623641 https://doi.org/10.1063/5.0129828 (III) Martin Tschöpe and Guntram Rauhut A theoretical study of propynal under interstel- lar conditions and beyond, covering low-frequency infrared spectra, spectroscopic constants, and hot bands. Monthly Notices of the Royal Astronomical Society, 520, 3345–3354 (2023), Issue 3 Copyright:Reprintedwith the permission fromRef. [3]. Copyright 2023,OxfordUni- versity Press. Contributions: M.T.: Methodology, Implementation, Simulation, Data curation, Vi- sualization, Writing – original draft andWriting – review & editing (partially) G.R.: Project administration, Funding acquisition, Resources,Writing – review& edit- ing (partially) DOI: https://doi.org/10.1093/mnras/stad251 (IV) Martin Tschöpe and Guntram Rauhut Spectroscopic Characterization of Diazophos- phane – A Candidate for Astrophysical Observations. The Astrophysical Journal, 949, 1 (2023) Copyright: Reprinted from Ref. [4], Copyright 2023, CC-BY 4.0. Contributions: M.T.: Methodology, Implementation, Simulation, Data curation, Vi- sualization, Project administration,Writing–original draft andWriting– review&edit- ing (partially) G.R.: Funding acquisition, Resources, Writing – review & editing (partially) DOI: https://doi.org/10.3847/1538-4357/acc9ad xii https://doi.org/10.1093/mnras/stad251 https://doi.org/10.3847/1538-4357/acc9ad Other publications by the author, not included in this thesis: (V) Erfort, S., Tschöpe, M., & Rauhut, G. Toward a fully automated calculation of rovi- brational infrared intensities for semi-rigid polyatomic molecules. J. Chem. Phys. 152, 244104 (2020) DOI: https://doi.org/10.1063/5.0011832 (VI) Erfort, S., Tschöpe, M., Rauhut, G., Zeng, X., & Tew, D. P. Ab initio calculation of rovibrational states for non-degenerate double-well potentials: cis–trans isomerization of HOPO. J. Chem. Phys. 152, 174306 (2020) DOI: https://doi.org/10.1063/5.0005497 (VII) Erfort, S.,Tschöpe, M., & Rauhut, G. Efficient and automated quantum chemical cal- culation of rovibrational nonresonant Raman spectra. J. Chem. Phys. 156, 124102 (2022) DOI: https://doi.org/10.1063/5.0087359 (VIII) Dinu, D. F., Tschöpe, M., Schröder, B., Liedl, K. R., & Rauhut, G. Determination of spectroscopic constants from rovibrational configuration interaction calculations. J. Chem. Phys. 157, 154107 (2022) DOI: https://doi.org/10.1063/5.0116018 xiii https://doi.org/10.1063/5.0011832 https://doi.org/10.1063/5.0005497 https://doi.org/10.1063/5.0087359 https://doi.org/10.1063/5.0116018 The results of this thesis have also been presented in the following talks and conferences: (I) Martin Tschöpe, Benjamin Schröder, Sebastian Erfort, Guntram Rauhut,High-Level Rovibrational Calculations on Ketenimine. Online presentation at 75th International Symposium onMolecular Spectroscopy virtual, June 2021 (II) Martin Tschöpe, Adaptable High-Level Rovibrational Calculations. Seminar presentation, invited by Prof. Laura Kreidberg Max-Planck-Institute, APEx, Heidelberg, Germany March 2022 (III) Martin Tschöpe, Adaptable Infrared Line List Calculation for Medium Sized Molecules Seminar presentation, invited by Dr. Iouli E. Gordon Harvard-Smithsonian Center for Astrophysics, Boston, Massachusetts, USA April 2022 (IV) Martin Tschöpe, Guntram Rauhut, Efficient and Automated Ab Initio Calculation of Infrared Spectra forMedium SizedMolecules. Poster presentation at Exoplanet IV conference Las Vegas, Nevada, USA May 2022 (V) Martin Tschöpe, Adaptable Infrared Line List Calculation for Medium Sized Molecules. Serveral presentations in different groups, invited by Prof. Heather A. Knutson California Institute of Technology, Pasadena, California, USA May 2022 (VI) Martin Tschöpe, Adaptable Infrared Line List Calculation for Medium Sized Molecules. Online seminar presentation, invited by Prof. Sara Seager Massachusetts Institute of Technology, virtual July 2022 (VII) Martin Tschöpe, Sebastian Erfort, GuntramRauhut, Efficient and Automated Ab Ini- tio Simulation of Rovibrational Infrared Spectra forMedium-SizedMolecules. Poster presentation at 58th Symposium on Theoretical Chemistry Heidelberg, Germany, September 2022 xiv Contents Abstract iii Zusammenfassung v Danksagung vii Peer-reviewed publications xi Contents xv 1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Scope of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Delimitation to other program parts in Molpro . . . . . . . . . . . . 4 1.2.2 Delimitation to rovibrational theory literature . . . . . . . . . . . . . 7 1.2.3 Delimitation to other rovibrational software . . . . . . . . . . . . . . 7 2 Theory 11 2.1 Watson Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.1 Vibrational term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.2 Rotational term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.3 Coriolis coupling term . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.1.4 RVCI Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2 Infrared Intensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3 Raman Intensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4 Line Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.4.1 Natural linewidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.4.2 Doppler broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.4.3 Pressure broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 xv 2.4.4 Voigt broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3 Results 47 3.1 Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.1.1 Total angular momentum . . . . . . . . . . . . . . . . . . . . . . . . 50 3.1.2 VCI space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.1.3 Vibrational basis set size . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.1.4 Coriolis coupling term order . . . . . . . . . . . . . . . . . . . . . . 54 3.1.5 Rotational term order . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.1.6 Influence of NSSWs . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2 Ketenimine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3 Line broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.4 Runtime optimizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.4.1 Parallelization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.4.2 Precalculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.4.3 Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.4.4 RVCI Coefficient Threshold . . . . . . . . . . . . . . . . . . . . . . 77 4 Summary and Conclusion 81 5 Outlook 85 5.1 Runtime and memory savings . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.2 Additional functionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.3 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Abbreviations 93 List of Figures 95 List of Tables 97 Bibliography 99 Publication 1: High-Level Rovibrational Calculations on Ketenimine 111 Publication 2: Convergence of series expansions in rovibrational configu- ration interaction (RVCI) calculations 127 xvi Publication 3: A theoretical study of propynal under interstellar con- ditions and beyond, covering low-frequency infrared spectra, spectro- scopic constants, and hot bands 139 Publication 4: Spectroscopic characterization of diazophosphane - a candi- date for astrophysical observations 151 Declaration of Authorship 161 xvii xviii 1 Introduction Choosing a subject for a PhD is a difficult decision and above all it depends on personal pref- erences. Section 1.1 describes, from my subjective perspective, why my topic is so important and motivating these days. The Section 1.2 addresses the scope of the thesis. Since a PhD thesis is build upon previous research, it is important to precisely define the border between the previous status quo and the own new research. This will be given in Subsec. 1.2.1 and 1.2.2, along with a short overview what distinguishes the approach in this thesis from other rovibrational research groups in Subsec. 1.2.3. 1.1 Motivation The question of whether places outside the solar system are habitable has long preoccupied mankind [5–9]. Themost promising extraterrestrial places are planets orbiting other stars and their moons (exoplanet (EP) and exomoon (EM))[9–13]. Over the last three decades more than 3500 systems with altogethermore than 5000 confirmed EPs and about 3000 candidates were found [14]. In contrast to that, so far no EM has been confirmed, although there were several promising candidates [15–17], therefore the subsequent discussion focuses on EPs. After the planets are discovered, the usual procedure is to study them in more detail. Param- eters like the radius of the planet, the mass, the mean distance to the star (semi-major axis) and the calculated average temperature on the planet are determined [14, 18, 19]. The latter 1 2 1.1. Motivation depends on the one hand on the distance to the star. On the other hand it depends on the existence and the composition of the atmosphere. More atmospheric greenhouse gases cause higher temperatures, as can be seen onVenus in a very extremeway. This is one of the reasons, why the investigation of exoplanet atmosphere (EPA) is very important for the habitability of planets beyond earth. Other reasons are, that the atmosphere protects the surface of a planet against radiation and meteorites. Hence, EP research relies heavily on the spectroscopy inves- tigation of the atmospheres and therefore also accurate reference spectra. Due to the tempera- ture of EPA, microwave and infrared (IR) spectroscopy are widely used. For very hot planets, there is a significant number of electronic excitations that also allow spectroscopy in the visible or ultraviolet wavelengths. There are many different methods to detect EPs, but not all of them allow for a subsequent analysis of the atmosphere [20]. Only 3% of the planets are detected by the direct imaging method (DIM) [14, 21], meaning that the planet is far enough away from the star and the re- solving power of the telescope is high enough to distinguish the star and the planet on images. With this method an analysis of the atmosphere is in principle possible. In this case emission spectroscopy can be used. There are also indirectmethods to detect exoplanets, where the planet and the star are too close to spatially resolve them, but only the influence of the planet on the host star can be detected [20]. One indirect detection method, that also allows for an analysis of the atmosphere is for example the transit method (TM). It can be applied, when coinci- dentally the star, the EP and the observer are in a straight line. In that case there is a primary eclipse (earth - planet - star) that can be used for detection via absorption spectroscopy. The opposite case is called secondary eclipse (earth - star - planet), which can be used for emission spectroscopy [22]. Another detection method that contributed to the majority of the detec- tions in the early days of EP discovery (between 1990 and 2010) is the radial velocity method (RVM) [14, 21]. It uses the fact that the star and planet orbit around their common center of gravity. This causes a red and blue shift of the star, that can be detected and yields for example the mass of the EP. However, it does not provide any information about the EPA. The most promising methods for the investigation of EPA are therefore the DIM and the TM. In general, a larger planet yields a higher signal-to-noise ratio (SNR) of the atmospheric signals due to the larger volume causing scattering or emission (depending on the detection method). For this reason, atmospheres have been detected in most cases for Jupiter-like and Neptune-like planets, some have been found for super-Earth planets [23], but no EPA detec- tion has been successful for Earth-sized EPs [24–27], like for example rocky planets. However, this is mainly due to two experimental limitations. The first is that for Earth-like planets, the majority of the shading effect during the eclipse is due to the core shadow and not due to the atmosphere. The core of the planet simply reduces all the light from the star (called a flat trans- mission spectrum) [28–30], without anymeasurable spectral signature. In contrast, an atmo- Chapter 1. Introduction 3 sphere shows a frequency dependent dimming, i.e. spectral features, due to the absorption lines of the molecules in the atmosphere. Therefore measuring the atmosphere of a rocky, icy or watery planet simply requires a better telescope. The second experimental limitation, that prevents the detection of Earth-like EPA is a systematic observational bias in theTM,meaning that the SNR is higher when the planet is closer to the star [22]. Therefore the vast majority of planets observed by TM are closer to their star, than the distance betweenMercury and the Sun [14]. Hence, the stellar pressure is strong and in most cases stronger than the gravity of small planets and therefore these planets can not hold their atmosphere. This effect is known as photoevaporation-driven atmosphere loss/mass loss and is related to the explanation of the small planet radius gap [31, 32]. Since, better telescopes allow to apply the TM to EP further away from their host star, these planets should be able to protect their atmosphere against the stellar pressure, even if these planets are relatively light. This means that the reason, why the majority of the EPAs discovered so far are around gas giants is that they are the easiest targets to probe newmethods. However, the future EP and EPA research strives to smaller and colder planets, as can be seen in the following: In 2001 the first detection of a chemical substance (sodium) in an EPA was achieved for the hot Jupiter HD 209458 b, which has a diameter of about 1.35 times that of Jupiter and it orbits its star on a very narrow orbit (about 1/8 of the distance Mercury- Sun). As a result, a year on the planet lasts only 3.5 Earth days and the surface temperature is about 1000K [33]. Many organic compounds are destroyed at such high temperatures. After that, a number of other molecules such as, water, carbon monoxide, carbon dioxide, methane, ozone were also detected on other hot Jupiters [34, 35]. In addition, the limit for the lightest planet where an atmosphere could be detected was lowered from the hot Jupiter HD 209458 b with 220 times the mass of Earth to Neptune-sized planets (HAT-P-11b, 23 times the mass of Earth) [36] to super-Earths (55 Cancri e, 8 Earth masses) [37]. Another super-Earth with a detected atmosphere is K2-18b with 8.6 Earth masses [38]. This planet is also in the habitable zone, making its atmospheric detection even more groundbreaking [38], since organic compounds can exist there. The results mentioned above show that enormous progress has been made in this field in the last two decades. Within the community it seems to be common sense that the next step is the investigation of these atmospheres in respect to biological relevant molecules [39–41]. There are different definitions and an ongoing discussion about how to properly define these gases. Deciding which molecules are suitable for this task is a very difficult interdisciplinary challenge. The contribution that spectroscopists canmake is primarily to provide highly accu- rate reference spectra. To summarize this task a bit more precisely: We need to provide highly accurate reference spectra for temperatures up to 370K (upper end of the habitable zone) for a 4 1.2. Scope of the Thesis large number of relatively small molecules (typically 13 atoms at most [40]) with a potentially high abundance in the atmosphere. However, the detection of these gases is still in the distant future because it requires a com- pletely new type of space-based telescopes [42, 43], using either a starshade/coronagraph (e.g. NewWorlds Mission, LUVOIR or Terrestrial Planet Finder-C) or an optical interferometer (e.g. Large Interferometer For Exoplanets (LIFE) orTerrestrial Planet Finder-I) [44–47]. This is also the reason why it is a relatively new and small field of research and therefore there is not yet a consensus on which molecules to study [39–41]. Fortunately, there is a related field of research where biologically relevant molecules have already been found, albeit under very different conditions. In the interstellar medium (ISM) and in circumstellar shells more than 200molecules have been detected [48–50]. These regions represent a wide chemical diversity, from rather stable to highly reactive species, from simple diatomics (e.g. CO, N2 andOH), to simple organicmolecules such asmethanol (CH3OH; [51]) and up to even larger compounds such as polycyclic aromatic hydrocarbons (PAHs; [52]) and fullerenes (C60; [53]). For this reason, the ISM and circumstellar shells are a very interesting test case for the detection of bi- ologically relevant molecules in EPs. In this context, complex organic molecules (COMs; [54]) are of particular interest. These are molecules with 6 or more atoms, including at least one carbon molecule. Such compounds are thought to be important building blocks for biologi- cally relevant molecules [55–57]. Two examples of such COMs are ketenimine and propynal. Both of them were studied in the course of this PhD [1, 3]. 1.2 Scope of the Thesis Since a dissertation builds on previous research, it is important to clearly define the boundary between the previous state of the art and the own new research. This will be done in Sub- sec. 1.2.1 and 1.2.2, along with a brief overview of what distinguishes the approach in this thesis from other rovibrational research groups in Subsec. 1.2.3. 1.2.1 Delimitation to other program parts inMolpro The rovibrational configuration interaction (RVCI) theory developed in the context of this thesis and the according algorithms is at the very end of a long sequence of theoretical meth- ods. Hence, the question appears which of these methods should and need to be explained in this thesis. The overview of the program structure in Fig. 1.1 allows for a rough under- standing of the different subprograms. The black arrows guide the main information flow. However, the illustration is by no means complete, as there are many more minor dependen- cies. The main information within this graphic is encoded in the color of the boxes. The algorithms in the red boxes are not explained at all, but only some computational details are Chapter 1. Introduction 5 Analytical representation Surface calculation Vib. basis optimization (VSCF) Rovib. state determination (RVCI) Vib. state determination (VCI) Rot. basis optimization (RCI) Electronic structure method Geometry optimization Harm. frequency & normal coordinate determination Figure 1.1: Overview of the program structure with the level of detail for the explanations within this thesis. Algorithms in red boxes are not explained, but sometimes computational details are given. The yellow boxes denote algorithms that are briefly discussed, program parts in the green boxes will be explained in more detail and the algorithm in the blue box in the bottom represents the main content of the thesis. 6 1.2. Scope of the Thesis mentioned. For example the specific electronic structure theory method and the correspond- ing basis set, will be given for some of the calculations. The details behind it are not necessary for the understanding of the rest of the thesis. The same holds true for the geometry optimiza- tion algorithms. The harmonic frequency calculation is to a certain extent important for the understanding because it yields the coordinate system. However, this will be explained along with the surface calculation. The fitting of the potential energy surface is a non-trivial and crucial part in the code. However, to understand the RVCI theory, it is sufficient to know that the grid representation is replaced by analytical functions, to save computational time. For the algorithms in the yellow boxes, the basic ideas will be explained. This is because the design decisionmade there affects the latter calculations, namely the choice of the coordinates and multi-mode expansion in the surface calculation and the basis set for the vibrational con- figuration interaction (VCI) program depends on the vibrational self-consistent field (VSCF) program. The VCI theory and rotational configuration interaction (RCI) theory in green re- quire a more detailed explanation. They provide the basis functions for the RVCI theory (in blue), which is the main topic of this thesis. VCI theory is briefly described in this thesis, but since the vibrational theory is the primary research object of the Rauhut Group for the last decades, there aremany theses from previous PhD students, that are more suited to give a deeper understanding. Another reason why I want to keep the VCI theory very short. For the RVCI theory, I derived all equations again, if they were identical to the RCI theory (such as for the partition functions, rotational integrals or in parts for the intensity calculation) or the equations were derived by me for the first time. For most of theMOLPRO program it is easy to drawing a clear line betweenmy ownwork and the previous status quo. However, the transition between theRCI implementation of Er- fort andmyRVCI program ismore involved andwill be explained in the following: The calcu- lation of the partition functions and theRCI are both implemented byErfort. In contrast, the calculation and diagonalization of the RVCI matrix, including all higher order µ-tensor cou- pling terms (Coriolis-coupling and rotational terms), is mywork. The rigid rotor basis (RRB) and the Wang basis (WB) were introduced decades ago in the literature and implemented in MOLPRO by Erfort. The molecule specific rotational basis (MSRB) was introduced and im- plemented by me in the course of this PhD. The assignment of the k and ν quantum number only occurred in the RVCI program and the two algorithms (leading coefficient method and partial trace method) were implemented by me. The same holds true for the two assignment algorithms for the rovibrational irrep (based on the rovibrational wave function and based on a partial trace on the rovibrational basis functions). The infrared intensity calculations for RCI were the work of Erfort. The necessary ad- justments for the RVCI intensities and the introduction of the intensity calculations with the MSRBwere done within this thesis. The different broadening profiles were introduced byme Chapter 1. Introduction 7 and later combined by Erfort in the DAT2GRAPH program. For the hot bands, Erfort in- troduced the vibrational transition moment integrals and I did the rest of the programming. The majority of runtime optimization was implemented by me (see Section 3.4), besides, a symmetry optimization for the rotational integrals. 1.2.2 Delimitation to rovibrational theory literature There are a number of books and publications on the subject of rovibrational spectroscopy. Usually several assumptions aremade at the beginning (choice of coordinate origin, space fixed or molecule fixed coordinate system, internal or normal coordinates, grid based calculations or the use of fit functions, Taylor series expansion ormulti-mode expansion, RRBorWB etc). Only if many of these assumptions match with the own implementation and additionally the derivation is as close as possible to equations, which can be programmed later, there is a sub- stantial benefit of following this literature. For the MOLPRO software, a comparison with the book of Bunker and Jensen [58] is most suitable. It also uses the Eckard conditions, the transition between space-fixed and molecule-fixed coordinates is also described by means of Wigner rotation matrix elements, and normal coordinates and equivalent rotations are used to determine the rotational irreps and thus the nuclear spin statistical weights (NSSW). In ad- dition, intensities are addressed fairly detailed. However, neither the multi-mode expansion and nor a specific vibrational basis set is selected. Thus, the expressions for the vibrational integrals cannot be simplified at an early point in the analytic calculation, and thus the deriva- tion is also terminated early. Moreover, the rotational integrals are not even computed in the RRB. The Wigner 3-J symbols are introduced as a general concept, but the calculation of in- dividual elements is not included. Hence, they were also self-derivated and then compared with the tabulated results in the appendix of [59]. There is various literature about partition functions, but little about the actual and efficient implementation. The same is true for the assignment of quantum numbers and rovibrational irreps. Most literature either refers to un- coupled systems, or uses the Hose-Taylor theorem [60], which is a rather academic but not very practical approach. (If the leading coefficient is smaller than 0.5, then an unambiguous assignment is not possible.) The issues regarding runtime optimization are either analogous to other places in the code, whichmakes similar methods possible (parallelization, thresholds) or so specialized that there is also no literature about them. 1.2.3 Delimitation to other rovibrational software Over the last decades a number of quantum chemical programs describing nuclear motions appeared. Of course, it is not possible to present all the groupsworking on this topic, but some of them will be introduced. For example Bowman and Carter et al. did some early work on 8 1.2. Scope of the Thesis the VSCF and VCI theory and developed theMULTIMODE software [61, 62]. However, since they focus on a different accuracy regime and usually do not calculate complete rovibra- tional spectra, but some selected lines for smallJ it is not really comparable toMOLPRO [63]. Carrington et al. developed a lot of different, out of the box approaches [64–66]. However, a comparison toMOLPRO is also difficult, since it is not a software package, but more a set of independent algorithms. Furthermore, the focus is more on innovative, academic prob- lem solving than on broad applicability. In contrast, the groups of Mátyus and Császár et al. developed the software packagesGENUISH andMARVEL [67–70]. The former is an ab ini- tio program using internal coordinates and it also allows for the calculation of rovibrational spectra. The latter is an empirical software, that relies on high quality experimental data. In addition to that there are some very successful programsmaking use of fitting amodel Hamil- tonian. One example is the SPFIT/SPCAT program introduced by Pickett et al. [71, 72]. An- other example is PGOPHER, which was invented by Western [73]. Moreover, the methods of Stanton and Franke et al. [74] rely on VPT2 using a large effective model Hamiltonian. A group that has so far specialized in a low number of small molecules (e.g. water, methane and ozone), but with a very high accuracy demand is Tyuterev et al. [75–77]. Over decades they successively improve the quality of their potential energy surface (PES) with more and more experimental results. Since MOLPRO is a pure ab initio program, allowing for the simulation of molecules for which no or only scarce experimental data exist, this is again not really a suitable comparison. However, one of the members of the latter group devel- oped its own more generalized software [78]. Similarities to MOLPRO are the use of the Born-Oppenheimer approximation and therefore the use of a PES and dipolemoment surface (DMS). Moreover, it is in principle an ab initio method relying on a non-empirical effective Hamiltonian, that exploits symmetry properties. The basic idea is to combine the Van Vleck perturbation theory (also known as contact transformation) with the polyad scheme. Both the contact transformation and the new method of Ref. [78] apply a series of unitary transfor- mations on the nuclear motion Hamiltonian. However, the former results in an expansion of the Hamiltonian in a power series and the latter uses an expansion in the polyad scheme. Furthermore, the approach of Ref. [78] does also allows for an empirical corrections, if this is requested. The program TROVE [79] was developed by Yurchenko, Thiel and Jensen and is since then extensively used by Yurchenko, Yachmenev and Tennyson [80–83] for calculations for their EXOMOL [84] database. Similarities to MOLPRO are again the use of the Born- Oppenheimer approximation and hence of an electronic PES and DMS. Moreover, they use a variationally computed eigenvector matrix. In principle they use an ab initio approach that can be refined using semi-empirical PES, an empirical basis set correction [85] or similarmeth- ods [86]. However, there are much more differences: For example, in TROVE, the rovibra- Chapter 1. Introduction 9 tional Hamiltonian is represented by a Taylor series expansion in terms of linearized coordi- nates around the non-rigid reference configuration that is treated explicitly on a grid. In con- trast,MOLPRO relies on an analytic respresentation of the PES and DMS in terms of poly- nomials or B-splines, represented in normal coordinates, a multi mode expansion and VCI theory. A particular good comparison is possible for H2CS, which was calculated with both ap- proaches [81, 87]. In TROVE, the kinetic energy operator was terminated after the 6th order and the PES after the 8th order. However, this cannot be directly compared with our values because the 8th order Taylor series contains different terms than our 4th ordermulti-mode ex- pansion. Inboth cases there are termswhich arenot included in the othermethod. Yachmenev et al. control the size of the vibrational basis set by the polyad number P and its maximum value Pmax. Then they define a function that defines how important the different vibrational modes are for the convergence. For example, for H2CS they define P = nCS + 2(nCH1 + nCH2) + nH1CS + nH2CS + nτ < Pmax (1.1) This means for example that the level of excitation nCH1 of the C−H1 stretching mode is weighted twice as much as the level of excitation nCS for the C− Smode, due to the prefactor 2. In contrast, in MOLPRO the selection of the vibrational basis functions in the rovibra- tional code is either set manually via a user input or giving by an upper energetic bound. This shows that although there are a large number of methods for calculating rovibrational infrared spectra, they are so different that the theory has to be recalculated from scratch each time, besides the basics already developed decades ago and summarized for example in [58, 59, 88, 89], as mentioned above. 10 1.2. Scope of the Thesis 2 Theory The goal of this thesis is to calculate rovibrational spectra. Therefore we need to determine two quantities: 1. The frequencies νi,f (i for initial state, f for final state) and 2. the intensi- ties Ii,f of each transition. This means that in a first step, the rovibrational state energies and wave functions need to be determined. This is done by using the Schrödinger equation, de- termining a Hamiltonian that describes the rovibrational problem, choosing a rovibrational basis set, building up the corresponding matrix and diagonalizing it. The second quantity that is needed for the spectrum is a measurement for the strength of the transition. This is the intensity for IR spectra and the differential cross-sections for Raman spectra. As mentioned in Subsec. 1.2.1, there is a long series of programs and algorithms in the MOLPRO software that is required before the rovibrational calculations can be done. How- ever, for the understanding of the RVCI theory that is developed in the course of this thesis, it is sufficient to explain the basics about the surface calculation (PES, DMS and polarisability surface (PS)), VCI and RCI theory. This is done while introducing theWatsonHamiltonian. 2.1 WatsonHamiltonian There are many different ways to describe the nuclear motion of molecules. Many of them are using the Born-Oppenheimer approximation to separate the nuclear motion and the elec- tronic motion, making use of the different time scale of these processes and yielding a PES 11 12 2.1. Watson Hamiltonian term in the Hamiltonian. Determining the Euler angles and applying the Eckart conditions allows to separate the rotational and the vibrational motion as much as possible. The vibra- tional motion of the cores corresponds to a physical angular momentum and is not due to the choice of the coordinate system. Therefore, a complete decoupling is not possible. The Coriolis coupling term describes this effect. In a second step, the different vibrational modes can now also be separated as far as possible. For this purpose, the potential is approximated quadratically in the vicinity of the minimum. This harmonic approximation allows then to set up and diagonalize the Hessian matrix. The resulting eigenvalues are proportional to the harmonic frequencies and the eigenvectors correspond to the so-called normal coordinates qi, which are used in the following. Combining these steps leads to theWatsonHamiltonian [90] HW = 1 2 ∑ αβ (Jα − πα)µαβ(Jβ − πβ)− 1 8 ∑ α µαα − 1 2 3N−6∑ i=1 ∂2 ∂q2i + V (q1, . . . , q3N−6). (2.1) Here µ denotes the inverse of the effective moment of inertia tensor, Jα the total angular momentum, the indices α and β correspond to the Cartesian coordinates {x, y, z}, πα de- scribes the vibrational angular momentum and N is the number of atoms. Hence, there are M = 3N − 6 vibrational modes, since we only consider non-linear molecules. This results inM normal coordinates, which span the potential energy surface (PES) V (q1, . . . , qM ). The Watson Hamiltonian can be split in the following three parts 1. Vibrational term (subsection 2.1.1) Hvib = 1 2 ∑ αβ∈{x,y,z} παµαβπβ − 1 8 ∑ α∈{x,y,z} µαα − 1 2 M∑ i=1 ∂2 ∂q2i + V (q1, . . . , qM ) 2. Rotational term (subsection 2.1.2) Hrot = 1 2 ∑ αβ∈{x,y,z} JαµαβJβ 3. Coriolis coupling term (subsection 2.1.3) Hcc = −1 2 ∑ αβ∈{x,y,z} ( Jαµαβπβ + παµαβJβ ) HW = Hvib +Hrot +Hcc. Chapter 2. Theory 13 2.1.1 Vibrational term TheWatson Hamiltonian for non-rotating molecules has the form [90] Hvib = 1 2 ∑ αβ∈{x,y,z} παµαβπβ − 1 8 ∑ α∈{x,y,z} µαα − 1 2 M∑ i=1 ∂2 ∂q2i + V (q1, . . . , qM ). (2.2) For didactic reasons it is most useful to start with the PES term V (q1, . . . , qM ). (2.3) This is due to the fact that it relies on a multi-mode expansion, which is also used for another part of the Watson Hamiltonian. Since the calculation of an equidistant grid is not compu- tationally feasible for large molecules, an expansion that is truncated after a certain order is an intuitive solution for this issue. In addition, it is advantageous to use difference surfaces, since this leads to decreasing contributions for the higher order terms, which is essential for the convergence of the expansion. ATaylor expansion and amulti-mode expansion are closely related in that sense, that they consist of the same terms, but their assignment to the specific expansion orders is differently. The PES described by a multi-mode expansion is given by V (q1, ..., qM ) = M∑ i Vi(qi) + M∑ i