Phys 498CQM Lecture 6

Outline

1. The many-body electron problem
   • General theoretical forms
     • The general many-body hamiltonian for electrons and nuclei
     • The many-body antisymmetric wavefunction
     • Energy as a functional of the wavefunction
     • Variational principle for the ground state
2. General theory of functionals F[f(x)]
   • Functional derivatives
3. Hartree-Fock uncorrelated approximate wavefunction
   • General Slater determinant spin-orbital form
     (Why a "Slater Determinant" instead of a "dirac Determinant"?)
   • Energy = sum of single body terms, direct coulomb, and exchange
   • Solution by minimization using variational theorem and Lagrange multipliers for orthonormality constraints
   • Nonlinear "Schroedinger-like" Equations, with difficult non-local exchange terms
   • Two-electron example (following Koonin, also similar to Thijssen)
   • Restricted Hartree Fock approximations: Open Shell vs. Closed Shell
     • "Spin Restricted": Occupy states just as in a non-interacting theory
       Require equal spin up and spin down orbitals
       Correct solution for cases which are "closed shell"
     • "Orbital Restricted": Allow unequal determinants and orbitals for up and down spins
       Require potential for each state (n,l,m_l,m_s) to depend only on (n,l,m_s); i.e., independent of m_l
       Essential for some cases - like H atom!
       Approximate solution for "Open Shell systems"
     • "Unrestricted": no assumptions on symmetry of potential
4. Numerical methods for solution of equations for an atom
   • Of of the few cases where the H-F equations can be solved directly by integrating differential equations
   • Continuation of discussion of solution of eigenvalue problems related to Homework 2
   • Hartree-Fock solution for atom requires solution of coupled Schroedinger-like and Poisson Eqs.
   • Explicit formulas take advantage of expansions for Coulomb terms
   • Equations for full Unrestricted case prepared by Tim Wilkens based upon formulas from Slater and other sources
   • Ouline for computer program for solution. More later