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Contents
Contents
Index
Lectures, The University of Manchester 2005
Quantum Mechanics of Atoms and Molecules
Dr. T. Brandes
Contents
Short Historical Introduction
Atoms and Molecules as a Concept
Greek Philosophy
Chemistry
Thermodynamics, Statistical Mechanics
Opponents to Atoms and Molecules
Discovery of Atoms
`Splitting of the Atom'
Theory of Atoms: Quantum Mechanics
`Old Quantum Mechanics'
Modern Quantum Mechanics
Some Revision, Fine-Structure of Atomic Spectra
Hydrogen Atom (non-relativistic)
Non-relativistic Single Particle Quantum Mechanics
Coulomb Potential
Orbital Angular Momentum
Radial Solutions
A `Mini-Molecule': Perturbation Theory vs Non-Perturbative Bonding
Example: Two-Level System
Exact solution
Second Order Perturbation Theory
Hydrogen Atom: Fine Structure
Kinetic Energy and Darwin Term
Kinetic Energy Correction
Darwin term
Spin-Orbit Coupling
Spin-Orbit Coupling in Atoms
Spin-Orbit Coupling in Solids
Perturbation Theory for Fine Structure
Degenerate Perturbation Theory
Degenerate Perturbation Theory for Spin-Orbit Coupling
Putting everything together
Introduction into Many-Particle Systems
Indistinguishable Particles
Permutations
Basis vectors for Fermi and Bose systems
Single Particle
-particle system
Permutations
-Boson systems
-Fermion systems
2-Fermion Systems
Two Electrons
Properties of Spin-Singlets and Triplets
Total Spin
Entanglement
The Exchange Interaction
Spin-independent Hamiltonian
Perturbation Theory
Direct and Exchange Term: Discussion
Two-electron Atoms and Ions
Perturbation theory in
Ground state
Excited states
The Hartree-Fock Method
The Hartree Equations, Atoms, and the Periodic Table
Effective Average Potential
Angular Average, Shells, and Periodic Table
Periodic Table
Hamiltonian for
Fermions
Expectation value of
Expectation value of
Spin independent symmetric
Hartree-Fock Equations
The Variational Principle
Functional Derivates
Lagrange Multiplier
The Variational Principle for Many-Electron Systems
Functional Derivative
`Direct' and `Exchange' Operators
Hartree-Fock Equations
Direct Term
Exchange Term
Example:
, `closed shell'
Ground State Energy
Molecules
Introduction
Model Hamiltonian
The Born-Oppenheimer Approximation
Derivation
Unsuccessful Attempt
More Successful Attempt
Discussion of the Born-Oppenheimer Approximation
Adiabaticity and Geometric Phases
Breakdown of the Born-Oppenheimer Approximation
The Hydrogen Molecule Ion
Hamiltonian for
The Rayleigh-Ritz Variational Method
Bonding and Antibonding
Rayleigh-Ritz Results
Explicit Calculation of
,
,
Symmetries of MOs in LCAO
Molecular Potential Energy
Hartree-Fock for Molecules
Roothan Equations
Time-Dependent Fields
Time-Dependence in Quantum Mechanics
Time-evolution with time-independent
Example: Two-Level System
Power Series
Eigenvectors
Quantum Oscillations in Two-Level Systems
Time-dependent Hamiltonians
Spin
in Magnetic Field
Constant
Rotating Field
Landau-Zener-Rosen problem
Time-Dependent Perturbation Theory
Model Hamiltonian
The Interaction Picture
First Order Perturbation Theory
Time-Independent Hamiltonian
Higher Order Perturbation Theory
States
Successive Interation
Interaction with Light
Electromagnetic Fields and Maxwells Equations
Longitudinal and transversal parts
Longitudinal
and
are `trivial'
Potentials
Gauge Transformations
Coulomb Gauge
Gauge invariance in single-particle non-relativistic QM
Local Gauge Transformation
Example: spatially constant electric field, zero magnetic field
Gauge invariance for many charges in non-relativistic QM
Charge and current densities, polarization and magnetization
The Hamiltonian
Coulomb Gauge
Power-Zienau-Woolley Transformation
Some Remarks on Fields
Rotations and Vibrations of Molecules
Vibrations and Rotations in Diatomic Molecules
Hamiltonian
Angular Momentum of Two Particles
Born-Oppenheimer Approximation
Angular Momentum
Spin
Radial SE
Harmonic Approximation
The Energy Spectrum
Spin
Beyond the Harmonic Approximation
Selection Rules
Dipole Approximation
Pure Rotation
Pure Vibration
Recap of the Harmonic Oscillator
Pure Vibrational Dipole Transitions
Vibration-Rotation Spectra
Electronic Transitions
The Franck-Condon Principle
Interaction between Molecules
From microscopic to macroscopic
Introduction
General considerations
Effective Potentials
Electrostatics: multipole expansions
Remark on Dipole-Dipole Interaction
Effective Interaction between Molecules
From Classical to Quantum
The
-Matrix
Two molecules
First oder term: static dipole-dipole interaction
Second oder term: (London) dispersion forces (van-der-Waals forces)
Physical Picture
Derivation from Second Order Term
Matrix Elements
Examples
Two-Level System
.
Hydrogen Atom
.
.
Bosons and Fermions
.
.
.
.
Hartree-Fock
.
.
Molecules
.
.
Time-Dependence
.
.
Bibliography
Index
About this document ...
Tobias Brandes 2005-04-26
Fermions
, `closed shell'
,
,
in Magnetic Field
and
are `trivial'
-Matrix