WalterR.Johnson AtomicStructureTheory Walter R. Johnson Atomic Structure Theory LecturesonAtomicPhysics With21Figuresand45Tables 123 ProfessorDr.WalterR.Johnson UniversityofNotreDame DepartmentofPhysics NieuwlandScienceHall225 46556NotreDame,IN,USA [email protected] LibraryofCongressControlNumber:2006938906 ISBN 978-3-540-68010-9 SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerial isconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broad- casting,reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationof thispublicationorpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLaw ofSeptember9,1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfrom Springer.ViolationsareliableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com ©Springer-VerlagBerlinHeidelberg2007 Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnot imply, even in the absence of a specific statement, thatsuch names are exempt from the relevant protectivelawsandregulationsandthereforefreeforgeneraluse. Typesetting:Camerareadybyauthor Production:LE-TEXJelonek,Schmidt&VöcklerGbR,Leipzig Cover:eStudioCalamarSteinen SPIN11921745 57/3100/YL-543210 Printedonacid-freepaper To Sally for her assistance, encouragement, and patience with this project. Preface This is a set of lecture notes prepared for a course on Atomic Physics for second-year graduate students, first given at the University of Notre Dame during the spring semester of 1994. My aim in this course was to provide op- portunitiesfor“hands-on”practiceinthecalculationofatomicwavefunctions and energies. Thelecturesstartedwithareviewofangularmomentumtheoryincluding the formal rules for manipulating angular momentum operators, a discussion oforbitalandspinangularmomentum,Clebsch-Gordancoefficientsandthree- j symbols. This material should have been familiar to the students from first- year Quantum Mechanics. More advanced material on angular momentum neededinatomicstructurecalculationsfollowed,includinganintroductionto graphical methods, irreducible tensor operators, spherical spinors and vector spherical harmonics. The lectures on angular momentum were followed by an extended discus- sion of the central-field Schro¨dinger equation. The Schro¨dinger equation was reduced to a radial differential equation and analytical solutions for Coulomb wavefunctionswereobtained.Again,thisreductionshouldhavebeenfamiliar from first-year Quantum Mechanics. This preliminary material was followed by an introduction to the finite difference methods used to solve the radial Schro¨dinger equation. A subroutine to find eigenfunctions and eigenvalues of theSchro¨dingerequationwasdeveloped.Thisroutinewasusedtogetherwith parametric potentials to obtain wave functions and energies for alkali atoms. The Thomas-Fermi theory was introduced and used to obtain approximate electron screening potentials. Next, the Dirac equation was considered. The bound-state Dirac equation was reduced to radial form and Dirac-Coulomb wavefunctionsweredeterminedanalytically.Numericalsolutionstotheradial Diracequationwereconsideredandasubroutinetoobtaintheeigenvaluesand eigenfunctions of the radial Dirac equation was developed. Inthethirdpartofthecourse,many-electronwavefunctionswereconsid- eredandthegroundstateofatwo-electronatomwasdeterminedvariationally. This was followed by a discussion of Slater-determinant wave functions and a VIII Preface derivation of the Hartree-Fock (HF) equations for closed-shell atoms. Numer- ical methods for solving the HF equations were described. The HF equations for atoms with one electron beyond closed shells were derived and a code was developed to solve the HF equations for the closed-shell case and for the case of a single valence electron. Finally, the Dirac-Fock equations were derived and discussed. The fourth section of the material began with a discussion of second- quantization. This approach was used to study a number of structure prob- lemsinfirst-orderperturbationtheory,includingexcitedstatesoftwo-electron atoms, excited states of atoms with one or two electrons beyond closed shells and particle-hole states. Relativistic fine-structure effects were considered us- ingthe“no-pair”Hamiltonian.Arathercompletediscussionofthemagnetic- dipole and electric-quadrupole hyperfine structure from the relativistic point of view was given, and nonrelativistic limiting forms were worked out in the Pauli approximation. fortran subroutines to solve the radial Schro´dinger equation and the Hartree-Fockequationswerehandedouttobeusedinconnectionwithweekly homeworkassignments.Someoftheseassignedexercisesrequiredthestudent to write or use fortran codes to determine atomic energy levels or wave functions. Other exercises required the student to write maple or mathe- matica routines to generate formulas for wave functions or matrix elements. Additionally, more standard “pencil and paper” exercises on Atomic Physics were assigned. The second time that this course was taught, material on electromag- netic transitions was included and a chapter on many-body methods was started; the third time through, additional sections on many-body methods were added. The fourth time that this course was taught, material on hyper- finestructurewasmovedtoaseparatechapterandadiscussionoftheisotope shift was included. The chapter on many-body methods was considerably ex- tended and a separate chapter on many-body methods for matrix elements was added. Inordertosqueezeallofthismaterialintoaonesemesterthreecredithour course, it was necessary to skip some of the material on numerical methods. However,Iamconfidentthattheentirebookcouldbecoveredinafourcredit hour course. Notre Dame, Indiana, September 2006 Walter R. Johnson Contents 1 Angular Momentum ....................................... 1 1.1 Orbital Angular Momentum - Spherical Harmonics .......... 1 1.1.1 Quantum Mechanics of Angular Momentum .......... 2 1.1.2 Spherical Coordinates - Spherical Harmonics.......... 4 1.2 Spin Angular Momentum................................. 8 1.2.1 Spin 1/2 and Spinors .............................. 8 1.2.2 Infinitesimal Rotations of Vector Fields .............. 9 1.2.3 Spin 1 and Vectors ................................ 10 1.3 Clebsch-Gordan Coefficients .............................. 12 1.3.1 Wigner 3j Symbols ................................ 16 1.3.2 Irreducible Tensor Operators........................ 17 1.4 Graphical Representation - Basic rules ..................... 19 1.5 Spinor and Vector Spherical Harmonics .................... 22 1.5.1 Spherical Spinors.................................. 22 1.5.2 Vector Spherical Harmonics......................... 24 Problems ................................................... 26 2 Central-Field Schr¨odinger Equation........................ 29 2.1 Radial Schro¨dinger Equation.............................. 29 2.2 Coulomb Wave Functions................................. 31 2.3 Numerical Solution to the Radial Equation ................. 35 2.3.1 Adams Method (adams) ........................... 37 2.3.2 Starting the Outward Integration (outsch) .......... 40 2.3.3 Starting the Inward Integration (insch) ............. 42 2.3.4 Eigenvalue Problem (master) ...................... 43 2.4 Quadrature Rules (rint) ................................. 46 2.5 Potential Models ........................................ 48 2.5.1 Parametric Potentials.............................. 49 2.5.2 Thomas-Fermi Potential............................ 51 2.6 Separation of Variables for Dirac Equation.................. 55 2.7 Radial Dirac Equation for a Coulomb Field ................. 56 X Contents 2.8 Numerical Solution to Dirac Equation...................... 60 2.8.1 Outward and Inward Integrations (adams, outdir, indir) ........................................... 61 2.8.2 Eigenvalue Problem for Dirac Equation (master) ..... 64 2.8.3 Examples using Parametric Potentials................ 65 Problems ................................................... 66 3 Self-Consistent Fields...................................... 71 3.1 Two-Electron Systems ................................... 71 3.2 HF Equations for Closed-Shell Atoms ...................... 77 3.3 Numerical Solution to the HF Equations ................... 88 3.3.1 Starting Approximation (hart) ..................... 88 3.3.2 Refining the Solution (nrhf) ....................... 90 3.4 Atoms with One Valence Electron ......................... 93 3.5 Dirac-Fock Equations .................................... 97 Problems ...................................................104 4 Atomic Multiplets ........................................107 4.1 Second-Quantization.....................................107 4.2 6-j Symbols.............................................111 4.3 Two-Electron Atoms.....................................114 4.4 Atoms with One or Two Valence Electrons .................118 4.5 Particle-Hole Excited States ..............................123 4.6 9-j Symbols.............................................126 4.7 Relativity and Fine Structure .............................128 4.7.1 He-like Ions.......................................128 4.7.2 Atoms with Two Valence Electrons ..................132 4.7.3 Particle-Hole States................................133 Problems ...................................................134 5 Hyperfine Interaction & Isotope Shift .....................137 5.1 Hyperfine Structure......................................137 5.2 Atoms with One Valence Electron .........................142 5.2.1 Pauli Approximation...............................144 5.3 Isotope Shift............................................146 5.3.1 Normal and Specific Mass Shifts.....................148 5.4 Calculations of the SMS..................................149 5.4.1 Angular Decomposition ............................149 5.4.2 Application to One-Electron Atoms..................151 5.5 Field Shift..............................................152 Problems ...................................................155 Contents XI 6 Radiative Transitions ......................................157 6.1 Review of Classical Electromagnetism......................157 6.1.1 Electromagnetic Potentials .........................157 6.1.2 Electromagnetic Plane Waves .......................159 6.2 Quantized Electromagnetic Field ..........................160 6.2.1 Eigenstates of N ..................................161 i 6.2.2 Interaction Hamiltonian............................162 6.2.3 Time-Dependent Perturbation Theory ...............163 6.2.4 Transition Matrix Elements.........................164 6.2.5 Gauge Invariance .................................168 6.2.6 Electric-Dipole Transitions .........................169 6.2.7 Magnetic-Dipole and Electric-Quadrupole Transitions..175 6.2.8 Nonrelativistic Many-Body Amplitudes ..............182 6.3 Theory of Multipole Transitions ...........................185 Problems ...................................................192 7 Introduction to MBPT ....................................195 7.1 Closed-Shell Atoms ......................................197 7.1.1 Angular Momentum Reduction .....................199 7.1.2 Example: Second-Order Energy in Helium ............202 7.2 B-Spline Basis Sets ......................................203 7.2.1 Hartree-Fock Equation and B-splines ................206 7.2.2 B-spline Basis for the Dirac Equation ................207 7.2.3 Application: Helium Correlation Energy..............208 7.3 Atoms with One Valence Electron .........................209 7.3.1 Second-Order Energy ..............................210 7.3.2 Angular Momentum Decomposition..................211 7.3.3 Quasi-Particle Equation and Brueckner Orbitals.......212 7.3.4 Monovalent Negative Ions ..........................214 7.4 Relativistic Calculations..................................216 7.4.1 Breit Interaction ..................................217 7.4.2 Angular Reduction of the Breit Interaction ...........218 7.4.3 Coulomb-Breit Many-Electron Hamiltonian...........221 7.4.4 Closed-Shell Energies ..............................221 7.4.5 One Valence Electron ..............................223 7.5 CI Calculations .........................................224 7.5.1 Relativistic CI Calculations.........................226 7.6 MBPT for Divalent Atoms and Ions .......................227 7.6.1 Two-Particle Model Spaces .........................227 7.6.2 First-Order Perturbation Theory ....................230 7.7 Second-Order Perturbation Theory ........................231 7.7.1 Angular Momentum Reduction......................233 Problems ...................................................234