Classical and Semiclassical Studies of the Hydrogen Molecule Thomas Sauvaget To cite this version: Thomas Sauvaget. Classical and Semiclassical Studies of the Hydrogen Molecule. Chaotic Dynamics [nlin.CD]. University of Nottingham (UK), 2006. English. tel-02371554 HAL Id: tel-02371554 https://tel.archives-ouvertes.fr/tel-02371554 Submitted on 19 Nov 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution - NonCommercial - NoDerivatives| 4.0 International License . Classical and Semiclassical Studies of the Hydrogen Molecule by Thomas Sauvaget Thesis submitted to the University of Nottingham for the degree of Doctor of Philosophy August 2006 Abstract The aim of this work is to account for the global shape of some Born-Oppenheimer (BO) curves of the hydrogen molecule, in particular the ground state, at the semi- classical level. In the first two chapters we review semiclassical cycle expansion techniques which proved useful in the study of the resonant spectrum of the helium atom, another few-body Coulomb system. Then we first study in detail the geometry of the classical phase space of the hydrogen molecule. This being done, we analyze the dynamics in three of its subspaces and compute various quantities related to their shortest periodic orbits. We then turn to semiclassical studies. One of these subspaces is discarded on instability grounds, while the other two are used as a basis for the construction of semiclassical approximations to the ground state BO curve. We find that each of the two subspaces allow to describe one of the two asymptotic trends of the BO curve. This allows to provide a classical underpinning to bond formation in H2. ❦ i Acknowledgements First I would like to express my gratitude to my advisor Gregor Tanner. He patiently taught me a lot about physics and was always willing to discuss things out. I would also like to thank him and his family for being so kind with me in Bristol and Nottingham. Next I would like to thank Stephen C. Creagh for several very useful discussions in the past years, in particular during the first year when Gregor was in Bristol. From Leicester I thank very much Ben Leimkuhler, whose enthousiastic semi- nars have broaden a lot my knowledge of numerical simulations. Thank you also to Chris C. Sweet and Ruslan Davidchack for showing me some of their C++ codes, where I first learnt about several libraries and tricks. My research was made possible by funding from the EC network MASIE super- vised by Mark Roberts. I would like to thank him, and all those I have met during our many conferences, for such a stimulating atmosphere. The Baines family has been very kind to me in many ways and I would like to thank them very much for their invitations and help. On the french side, Marc Legrand’s enthousiasm and discussion-style lectures have had a great impact on the second half of my undergraduate curriculum, and on my thinking more generally; I am grateful for that influence. Finally I would like to express my most profound gratitude to my dear pa- rents, my fantastic brother, and all my beloved family for their continuing love and encouragements. ii Acknowledgements This work is dedicated to: My Parents & Mr Douglas Whitelaw FRCSEd FRCS, General Surgeon; Mrs Kontouli, Anes- thesist; the theatre crew; the nurses of the High Dependency Unit; the nurses and staff of the Cobham clinic; the physios; and the ambulance crew of the Luton and Dunstable Hospital. “[...] the status of the continuum hypothesis convinced me that Platonist views of mathematical ontology and truth could not be correct [...]” Thimothy W. Gowers iii Contents Abstract .................................... i Acknowledgements.............................. ii ListofFigures................................. viii 1 Introduction 1 1.1 Approximationsinquantummechanics . 2 1.1.1 Basicsetup ........................... 2 1.1.2 Exactsolutions ......................... 3 1.1.3 Directnumericalsolutions . 5 1.1.4 Approximateasymptoticeigenfunctions . 7 1.2 The semiclassical limit of quantum mechanics . 8 1.2.1 Setting.............................. 9 1.2.2 Caseofanintegrablelimit . 10 1.2.3 Caseofanhyperboliclimit. 19 1.3 Organizationofthethesis . 32 2 Few-body Coulomb problems 34 2.1 Motivation................................ 34 2.2 Hydrogenmolecularion . 35 2.2.1 Setting.............................. 35 2.2.2 Solution of the quantum problem . 36 2.2.3 Originofmolecularstability . 40 iv CONTENTS CONTENTS 2.2.4 Semiclassical limit: tunneling geometry, monodromy and bi- furcations ............................ 41 2.2.5 Adiabatic invariant and dynamics beyond the BOA . 45 2.3 Heliumatom: experimentsandquantumaspects . 47 2.3.1 Experimental imput and spectral structure . 47 2.3.2 Quantumapproaches. 49 2.4 Heliumatom:semiclassicalaspects . 54 2.4.1 Setting.............................. 54 2.4.2 Scalingproperty . 56 2.4.3 Collisionsingularities . 56 2.4.4 Symmetries, integrals and reduced space . 63 2.4.5 CollineareZedynamics . 68 2.4.6 CollinearZeedynamics . 72 2.4.7 Planardynamics . 72 2.4.8 Semiclassicalquantization . 76 2.5 Hydrogenmolecule........................... 79 2.5.1 Quantumaspects . 79 2.5.2 Semiclassicalapproches . 81 2.6 Numericalmethods........................... 83 3 Classical dynamics of the hydrogen molecule 85 3.1 Introduction............................... 85 3.2 Setting.................................. 86 3.3 Scalingproperty ............................ 88 3.4 Collisionset............................... 89 3.5 Symmetries,integralsandreduction . 89 3.5.1 Continuoussymmetries . 89 3.5.2 Discretesymmetries . 91 3.6 Dynamics in N(SO(2)) .......................... 92 v CONTENTS CONTENTS 3.6.1 DynamicsoftheeZZeconfiguration . 93 3.7 Dynamics in N{e} ............................ 108 3.7.1 Setting.............................. 108 3.7.2 Flow............................... 110 3.7.3 Fixedpoints........................... 112 3.7.4 Invariantsubspaces. 112 3.7.5 Dynamics in the collinear pendulum configuration . 124 3.7.6 Dynamics in the Z⊥ configuration. 130 4 Semiclassical quantization of the hydrogen molecule 140 4.1 QuantizationoftheeZZeconfiguration . 140 4.2 Quantization of the Z⊥ configuration . 143 4.3 Bondformationmechanism . 145 5 Conclusions 147 5.1 Symmaryofresults........................... 147 5.2 Futurework............................... 148 A Notations for classical mechanics 149 B Group theory and the hydrogen atom 155 B.1 Geometricalgroup ........................... 155 B.2 Degeneracygroup ........................... 157 C Reduction theory for Hamiltonian systems 161 C.1 Setting.................................. 161 C.1.1 Symmetrygroupandintegrals . 162 C.1.2 Isotropytypesandinvariantmanifolds . 163 C.1.3 Freeandnon-freeactions . 164 C.1.4 Energymomentummap . 165 C.2 Regularreduction............................ 165 vi CONTENTS CONTENTS C.3 Singularreduction ........................... 167 Bibliography 168 vii List of Figures 1.1 Some potential V when N = 1 ..................... 10 1.2 The momentum is a multivalued function of the position . 11 + 2.1 Probability density of ground state of H2 (part1). 38 + 2.2 Probability density of ground state of H2 (part2). 39 + 2.3 Probability density of ground state of H2 (part3). 40 + 2.4 The three periodic orbit families of H2 at Lz = 0 .......... 42 + 2.5 EBK and uniform EBK results for low-lying H2 states . 43 + 2.6 Zhilinski´ı diagram of monodromy in H2 ............... 45 2.7 The numerical spectrum of 1Se heliumstates. 48 2.8 HeliumeZeandZeecollinearconfigurations . 66 2.9 Planarheliumconfiguration . 67 2.10 Contour plot of energetically allowed configuration space domain . 67 2.11TypicalorbitofeZe........................... 69 2.12 Poincar´esectionofeZe......................... 69 2.13 Short orbits of the eZe configuration . 71 2.14 One GS orbit in physical, reduced, and MO coordinates . 74 2.15TriplecollisionmanifoldofeZe . 75 2.16 Conveyor belt mechanism between triple collision and double es- capepoints ............................... 75 2.17 Numerical results of semiclassical cycle expansion of eZe ...... 78 1 + 2.18 Potential energy curves of lowest Σg electronic states of H2 .... 80 viii LIST OF FIGURES LIST OF FIGURES 3.1 Classicalhydrogenmolecule. 87 3.2 Fourcollinearhydrogenconfigurations . 91 3.3 Poincar´esectionsofeZZe(part1). 96 3.4 Poincar´esectionsofeZZe(part2). 97 3.5 Evolution of eZZe initial conditions on the PSOS . 100 3.6 R-dependence of the period and action of an eZZe orbit . 100 3.7 TypesoflinearperturbationsofeZZe . 103 3.8 R-dependence of Lyapunov and stability exponents of an eZZe orbit 105 3.9 Effect of parametric triple collision on linear perturbations of eZZe 106 3.10 Gel’fand-Lidskii index of the R 1 orbitatR=3 . 107 − 3.11Equilibriumofthehydogenmolecule . 113 3.12 The Z (4 DOF) and Z⊥ (3DOF)subspaces . 114 3.13 The 2 DOF configurations Wρ and Wz ................ 116 3.14 The 2 DOF configurations S and P .................. 117 3.15 The 2 DOF configuration Wρ,⊥ ..................... 118 3.16 Organization of the various invariant