Wednesday, March 17, 2010

A good place to start a many-body theory

Last week I briefly discussed the notion of quasi-particle weight and recommended P.W. Anderson, Concepts in Solids: Lectures on the Theory of Solids, Chapter 3.

This week we looked at Chapter 4, in Advanced Solid State Physics, The Hartree-Fock approximation.
This is can be viewed as a mean-field theory and/or a variational approximation to the ground state energy. One assumes many-particle ground state wavefunction can be written as a single Slater determinant.

One way to understand the Hartree-Fock equations (4.23) as a mean-field theory is the following. Consider the Coulomb interaction energy between an electron in one orbital (denoted nu) [which has an associated charge density] and the charge density associated with all the other occupied orbitals.
The one electron energies are actually the Lagrange multipliers introduced to care of the normalisation constrain for the one electron orbital wave functions.

The exchange interaction (between electrons in different orbitals) arises because of the fermion statistics. It leads to a splitting of singlet and triplet states. The latter has lower energy because the electrons are forced to occupy different spatial regions by the Pauli exclusion principle.

Hartree-Fock theory is the starting point for almost any attempt at solving a quantum many-body problem.
In Chapter 6, we will see how Anderson used it to solve the problem of local magnetic moments in metals, work recognized in the 1977 Nobel Prize in Physics.

If you want to solve the Hartree-Fock equations for a few simple molecules try WebMO

1 comment:

  1. The Hartree-Fock approach is actually much more general than Slater determinants. See e.g. Tishby and Levine. A Self-Consistent Field Procedure for Stationary States Using An Algebraic Approach and the Maximum Entropy Principle. Chemical Physics Letters (1984) vol. 104 (1) pp. 4-8.

    The Hartree-Fock Hamiltonian is derived here for the state that maximizes the von Neumann entropy subject to constraints on the one-electron observables.

    If the state is constrained to ALSO be an N-electron pure state, then the state is a Slater determinant. If this constraint is NOT required, then the Hartree-Fock Hamiltonian may still apply. This suggests that the Hartree-Fock Hamiltonian is more general than the case of a closed isolated system (i.e. it is more general than Schrodinger equation).

    Either I am raving mad, or this is really important but widely ignored.