Table Of ContentWalterR.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
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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
