Skript / lecture notes - Universität Paderborn

Prof. Dr. Wolf Gero Schmidt
Universität Paderborn, Lehrstuhl für Theoretische Physik
The approach to solving this problem numerically is essentially:
• Guess a set of single particle states, ϕ i .
• Compute the Hartree potential, V H , for each of the particles using the states guessed
above.
• Solve the single-particle (Hartree) equations for the states using V H .
• If the states obtained are the same as the states entering V H , you are done.
• If the states obtained are different than the states entering V H , use these as a new
guess at the states.
• Repeat until convergence (solutions = guesses).
There are some subtleties in the process. The most important is that after solving for
the single particle states, the new guess at the states is a mixture of the old guess from
and the new solutions. If one uses the new states as the guesses, wild oscillations in the
numerics often occur. Of course, the solution to the Hartree equation is not exact. The
approximation used is that the solution to the problem can be written as a product of
single particle states. This is the usual variational simplification, where the region of
Hilbert space considered is limited by some guess as to the form of the states. Better
solutions to the N−particle problem involve linear combinations of these Hartree-like
basis states. This will be discussed in the next paragraph.
4.4 Hartree-Fock-Methode
Hartree-Fock Method
Hartree-Verfahren: Hartree 〈 method 〉
Minimieren Minimize Ψ H |Ĥ|Ψ H bez. with respect to |Ψ H 〉
Problem: Raum der Funktion |Ψ H 〉, die zur Minimierung benutzt wurde, enthält die
”wahre” WF gar nicht! However, the region of the Hilbert space that is considered in the
minimization does necessarily not contain the ”true” solution!
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