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Introduction to Relativistic Quantum Chemistry

University of Helsinki, Spring 2001


Lecturer: Prof Pekka Pyykkö (tel. 09-191 50171, pyykko@chem.helsinki.fi)
Level: Advanced undergraduate and postgraduate
Language: English
Literature: Lecture notes and additional handouts
Credits: 5

Brief Outline

Introductory course to relativistic quantum chemistry.

Lecture Notes

The chapters were revised 21.10.2003: Some misprints were corrected, and the size of the files were reduced. Also, the LaTeX-source file is now available.

The links below point to PDF-files containing the chapter in question. If you discover misprints or other bugs in the material, please report them for example by e-mailing mikael.johansson@helsinki.fi.

Index

  • Preface

  • Table of Contents (pages iiv, 33 kB) version 21.10.2003

  • All-in-One (pages 1–119, 2.24 MB) version 21.10.2003

  • 1 Lorentz Transformations and Lorentz Groups (pages 1–14, 450 kB) version 21.10.2003

    • 1.1 Lorentz transformations
      • 1.1.1 Introduction
      • 1.1.2 Rotations and translations of the four-dimensional space
      • 1.1.3 Lorentz transformations as rotations
      • 1.1.4 Addition of velocities
      • 1.1.5 Perpendicular motion
      • 1.1.6 The relativistic mass transformation
      • 1.1.7 Derivation of E = mc2
      • 1.1.8 Connection between T and p
    • 1.2 Lorentz matrices
    • 1.3 Infinitesimal Lorentz transformations
    • 1.4 The Lorentz group

  • 2 Relativistic Fields (pages 15–20, 61 kB) version 21.10.2003

    • 2.1 Definition
    • 2.2 Scalar fields
    • 2.3 An S = 1 field
    • 2.4 Two-component spinor fields
    • 2.5 Four-component spinor fields

  • 3 Relativistic Wave Equations (pages 21–52, 467 kB) version 21.10.2003

    • 3.1 Klein-Gordon
    • 3.2 Dirac
      • 3.2.1 Why must N >= 4?
      • 3.2.2 Properties of Dirac matrices
    • 3.3 Some properties of the Dirac equation
      • 3.3.1 Solutions with ±E
      • 3.3.2 Inclusion of electromagnetic fields
      • 3.3.3 Free-particle solutions
      • 3.3.4 Probability density
    • 3.4 The Pauli limit
    • 3.5 Central fields
      • 3.5.1 The radial part
      • 3.5.2 Non-relativistic limit
    • 3.6 The Dirac-Coulomb problem
    • 3.7 Virial theorems
      • 3.7.1 Non-relativistic case
      • 3.7.2 Dirac

  • 4 Dirac-Fock (pages 53–70, 234 kB) version 21.10.2003

    • 4.1 The energy expression
    • 4.2 The Dirac-Fock equations
      • 4.2.1 The relativistic Koopmans theorem
      • 4.2.2 Multiconfiguration treatment, a simple example
      • 4.2.3 "Average-of-configuration" treatment
    • 4.3 Numerical solution of the DF equations
      • 4.3.1 Specific features of the DF-OCE method

  • 5 Symmetry (pages 71–86, 572 kB) version 21.10.2003

    • 5.1 Rotation Operators
      • 5.1.1 The Euler angles
      • 5.1.2 Rotation of spherical harmonics
      • 5.1.3 Rotation of |jm> functions
      • 5.1.4 The 2-to-1 homomorphism from SU(2) to SO(3)
    • 5.2 Double groups
      • 5.2.1 Non-relativistic case with spin
      • 5.2.2 Relativistic case
      • 5.2.3 Improper rotations
      • 5.2.4 The Group O(3)
      • 5.2.5 The double group is a symmetry group of the Dirac equation
      • 5.2.6 The Element E
      • 5.2.7 Elements of double groups
      • 5.2.8 Irreducible representations ("irreps")
      • 5.2.9 Classes
      • 5.2.10 Theorem of Opechowski
    • 5.3 Construction of relativistic MO:s
      • 5.3.1 Projection operators
      • 5.3.2 Coupling constant method
    • 5.4 Time reversal
      • 5.4.1 Non-relativistic case
      • 5.4.2 Inclusion of spin
      • 5.4.3 n-electron wave functions
      • 5.4.4 Kramers' theorem
      • 5.4.5 The cases (a), (b) and (c) of Wigner (1932)
      • 5.4.6 Further examples
    • 5.5 Quaternions

  • 6 Molecular Orbital Methods (pages 87–100, 120 kB) version 21.10.2003

    • 6.1 Semi-empirical methods
      • 6.1.1 Extended Hückel methods
      • 6.1.2 Zero Differential Overlap Approximation
      • 6.1.3 Inclusion of Spin-Orbit Splitting
      • 6.1.4 Relativistic Extended Hückel (REX)
    • 6.2 One-electron molecules
      • 6.2.1 The Hamiltonian
      • 6.2.2 Possible coordinate systems
      • 6.2.3 Transformation of the Dirac equation

  • 7 Pseudopotentials (pages 101–104, 61 kB) version 21.10.2003

    • 7.1 Introduction
    • 7.2 A bit of history
    • 7.3 Where to get pseudopotentials

  • 8 On QED (pages 105–112, 600 kB) version 21.10.2003

    • 8.1 Introduction
    • 8.2 Some formulas for vacuum polarization
    • 8.3 Some formulas for self-energy (vacuum fluctuation)

  • 9 On Transformed Hamiltonians (pages 113–119, 96 kB) version 21.10.2003

    • 9.1 General
    • 9.2 The Foldy-Wouthuysen transformation
    • 9.3 The Cowan-Griffin equation
    • 9.4 Douglas-Kroll-Hess
    • 9.5 Zero Order Regular Approximation, ZORA
    • 9.6 Direct Perturbation Theory, DPT
    • 9.7 Further examples
      • 9.7.1 RESC
      • 9.7.2 AMFI

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