Link to HTML version of Power Point Slides for Lecture 2
Reading: Text: Chapter 3.
The background needed for this course is basic quantum mechanics at the level of the first graduate course. Additional background on solid state physics is advantageous for the student: although we will attempt to make the course self-contained including derivations of all the important concepts, there is not time to cover all the background in depth.
This lecture and the material in chapter 3 of the text is meant to be a refresher of basic expressions for the fundamental hamiltonian, rigorous properties of the many-electron system, the Hartree-Fock approximation, perturbation theory, and the simplifications of the expressions in independent particle approximations.
- The fundamental hamiltonian: Eq. (3.1)
- Only one small term - nuclear kinetic energy. All other terms are large and important.
- Atomic units - to simplify the notation - see App. O
- Schrodinger equation and expectation values
- Functional derivatives (see Appendix A) and variational equations
- Important properties of the many-electron system
- The "force theorem" and generalizations
- Crucial aspects of the long-range Coulomb interaction
- Statistical mechanics and the density matrix
- Independent-particle approximation for the electrons
- Non-interacting (Hartree-like) approximation
- Electrons uncorrelated except that they obey exclusion principle
- Assume a modified one-electron hamiltonian that acts on each electron independently - Eq. (3.36)
- N eigenstates of the modified hamiltonian occupied with an electron
- Fermi-Dirac distribution - Eq. (3.38)
- Hartree-Fock Approximation
- Treat the full hamiltonian including electron-electron interactions, but assume electrons are uncorrelated except that they obey exclusion principle
- Slater (Dirac) determinant form of many-electron wavefunction - Eq. (3.43)
- Leads to an orbital-dependent hamiltonian acting on each orbital - the famous Hartree-Fock Eqs. Eq. (3.45)
- Variational approximation - Koopmans' theorem
- Exchange and the exchange hole
- Beyond Hartree-Fock - Correlation
- The correlation function - correlation hole - definition only - discussed more later
- Useful note on perturbation theory - "2n+1" theorem