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Phys 498CQM Lecture # 9

Monday, February 19, 2001
Lecturer: Richard Martin

Reading:
Thijssen Ch. 5

Outline

Continue discussion of methods to treat many-electron atoms:
Hartree-Fock Approx. and the Kohn-Sham Approach in Density Functional Theory
  • A postscript file with lecture notes on Density Functional Theory to accompany lectures 9 and 10 is available.
  • See Atomic Reference Data for Electronic Structure Calculations at NIST for descriptions of the equations and many results on atoms
    1. Recall key aspects of the Hartree Fock Approx.
      • Exchange interaction
        • Non-local (orbital-dependent) potential operator
        • Cancels unphysical self-interaction term exactly
        • Solve equations by a self consistent calculation
      • Aspects of programs for calculations for atoms in the Hartree Fock Approx.
    2. Density Functional Theory
      • The Hohenberg-Kohn Theorem
        • The ground state electron density uniquely determines the external potential acting on the electrons
        • Applies to the exact solution of the full interacting many-body system
        • Therefore all properties must be functionals of the ground state electron density
        • All properties are determined by one scalar function, the density n(r), instead of the complex many-body wavefunction
        • But no one has ever constructed an explicit functional, except for a single particle! Even two non-interacting electrons of same spin has not been done
      • The Kohn-Sham Ansatz
        • Replace the original (intractable) many-body problem with a different problem
        • Kohn-Sham proposed one consider a non-interacting electron system with the same density n(r) as the true interacting system
          • This system is exactly solvable (at least numerically)
          • The Hohenberg-Kohn theorems apply to this system
        • Define energy of many body system to be sum of kinetic energy of independent-particle wavefunctions + Hartree energy (functional of density) + exchange-correlation Exc[n]
        • Leads to self-consistent sets of independent-particle Schroedinger-like Eqs.
        • In principle solve many-body problem by single body methods!
        • Note that this applies only to the ground state density and energy, NOT to other properties
        • One ,ust extend the Kohn-Sham idea for other properties
        • In practice use various approximations for Exc[n]
      • Results using the Kohn-Sham Ansatz and simple approximations for Exc

    Last Modified Feb. 20
    Email question/comments/corrections to rmartin@uiuc.edu .