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Quantum Mechanics of the Diatomic Molecule with Applications Quantum Mechanics of the Diatomic Molecule with Applications Christian G Parigger and James O Hornkohl University of Tennessee Space Institute, Tullahoma, TN, USA IOP Publishing, Bristol, UK ªIOPPublishingLtd2020 Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem ortransmittedinanyformorbyanymeans,electronic,mechanical,photocopying,recording orotherwise,withoutthepriorpermissionofthepublisher,orasexpresslypermittedbylawor undertermsagreedwiththeappropriaterightsorganization.Multiplecopyingispermittedin accordancewiththetermsoflicencesissuedbytheCopyrightLicensingAgency,theCopyright ClearanceCentreandotherreproductionrightsorganizations. PermissiontomakeuseofIOPPublishingcontentotherthanassetoutabovemaybesought [email protected]. ChristianGPariggerandJamesOHornkohlhaveassertedtheirrighttobeidentifiedasthe authorsofthisworkinaccordancewithsections77and78oftheCopyright,DesignsandPatents Act1988. ISBN 978-0-7503-1989-8(ebook) ISBN 978-0-7503-1890-7(print) ISBN 978-0-7503-1941-6(mobi) DOI 10.1088/978-0-7503-1989-8 Version:20191101 IOPebooks BritishLibraryCataloguing-in-PublicationData:Acataloguerecordforthisbookisavailable fromtheBritishLibrary. PublishedbyIOPPublishing,whollyownedbyTheInstituteofPhysics,London IOPPublishing,TempleCircus,TempleWay,Bristol,BS16HG,UK USOffice:IOPPublishing,Inc.,190NorthIndependenceMallWest,Suite601,Philadelphia, PA19106,USA To John, Anna and Melissa To Justin, Jason and Jola Contents Preface xii Acknowledgments xiv Author biographies xv Part I Fundamentals of the diatomic molecule 1 Primer on diatomic spectroscopy 1-1 1.1 Overview 1-1 1.2 Reversed angular momentum 1-2 1.3 Exact diatomic eigenfunction 1-4 1.4 Computation of diatomic spectra 1-5 References 1-6 2 Line strength computations 2-1 2.1 Introduction 2-1 2.2 Idealized computation of spectra 2-4 References 2-5 3 Framework of the Wigner–Witmer eigenfunction (WWE) 3-1 References 3-4 4 Derivation of the Wigner–Witmer eigenfunction 4-1 4.1 Outline of the derivation 4-1 4.2 Time translation symmetry 4-2 4.3 Spatial translation symmetry 4-4 4.4 Two-body symmetry 4-7 4.5 Time and spatial translations together 4-9 4.6 Rotational symmetry 4-9 References 4-10 5 Diatomic formula inferred from the 5-1 Wigner–Witmer eigenfunction References 5-3 vii QuantumMechanicsoftheDiatomicMoleculewithApplications 6 Hund’s cases (a) and (b) 6-1 6.1 Introduction 6-1 6.2 Case (b) basis functions 6-2 6.3 Case (a) eigenfunctions 6-4 References 6-5 7 Basis set for the diatomic molecule 7-1 References 7-3 8 Quantum theory of angular momentum 8-1 8.1 Introduction 8-1 8.2 The standard ∣JM〉 angular momentum representation 8-2 8.3 Rotations 8-3 8.4 Generators of coordinate transformations 8-5 References 8-6 9 Diatomic parity 9-1 9.1 Parity details 9-1 9.1.1 Parity is rotationally invariant 9-2 9.1.2 Spin is immune to the parity operator 9-2 9.1.3 Parity operates on Cartesian coordinates, not angles 9-3 9.1.4 Intrinsic parity and Λ doublets 9-3 9.1.5 Summary of parity details 9-3 9.2 Parity designation 9-3 9.3 The parity operator 9-4 9.4 Parity and angular momentum 9-6 9.5 Diatomic parity 9-7 9.6 Λ doublets 9-8 References 9-8 10 The Condon and Shortley line strength 10-1 Reference 10-2 11 Hönl–London line-strength factors in Hund’s cases (a) and (b) 11-1 11.1 Case (a) basis functions 11-1 11.2 Case (b) basis functions 11-2 viii QuantumMechanicsoftheDiatomicMoleculewithApplications 11.3 Mathematical properties of case (a) and case (b) basis functions 11-2 11.4 Diatomic parity operator 11-3 11.5 Hönl–London line-strength factors 11-4 11.6 Triple integral of three rotation matrix elements 11-5 11.7 Calculation of the Hönl–London line-strength factors 11-5 for cases (a) and (b) 11.8 Hund’s case (b) Hönl–London line-strength factors 11-8 11.9 The electronic–vibrational strength 11-9 Reference 11-10 12 Using the Morse potential in diatomic spectroscopy 12-1 12.1 Introduction 12-1 12.2 Morse eigenfunctions 12-2 12.2.1 Computation of Morse eigenfunctions 12-4 12.3 Morse eigenfunctions as a vibrational basis 12-5 References 12-7 Part II Selected applications of diatomic spectroscopy 13 Introduction to applications of diatomic spectroscopy 13-1 References 13-5 14 Experimental arrangement for laser-plasma diagnosis 14-1 References 14-3 15 Cyanide, CN 15-1 15.1 Analysis of CO laser-plasma 15-1 2 15.2 Analysis of CN in Nd:YAG laser-plasma 15-2 15.3 Spatially and temporally resolved CN spectra 15-4 15.3.1 Laser-beam focusing 15-5 15.3.2 Shadowgraphs 15-6 15.3.3 Raw CN spectra 15-6 15.3.4 Abel-inverted CN spectra 15-7 References 15-12 16 Diatomic carbon, C 16-1 2 16.1 Analysis of C in Nd:YAG laser-plasma 16-1 2 ix