The Hartree-Fock Method

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The Herman-Skillman method is based on Slater's simplified version of the Hartree-Fock equations. We start with the Hartree formalism, approximating the many-electron Hamiltonian for the atomic system as a sum of one-electron Hamiltonians

where N is the number of electrons in the system. Each electron is treated as interacting individually with the atomic nucleus and can be solved independent of the other electrons. Within the potential energy portion of the Hamiltonian, we take the Born-Oppenheimer approximation and ignore nuclear-nuclear interactions as being constant. In the simplest form, we are also assuming the electrons are non-interacting.

For the simplest form of Happrox. with no electron-electron interaction, the exact eigenfunction will be a product function

where the subscripts 1, 2, 3, ..., N specify which unique quantum state the single-electron wave function represents. Any permutation of the single-electron functions, which we can call spin-orbitals, will also lead to an eigenfunction of Happrox..

In the full Hartree method, electron-electron interactions are included in the Hamiltonian by an additional potential energy term. This extra term is like a perturbation of the simple solution for the non-interacting electron situation. This more complete Hamiltonian is also the starting point for the Hartree-Fock method.

The Hartree-Fock method introduces full quantum mechanical considerations into the analysis of the many-electron problem. Symmetry constraints are applied to the eigenfunction. As electrons are fermions, the total wave function must be antisymmetric with respect to the interchange of any two of the electrons. This is equivalent to claiming the electrons must satisfy the Pauli exclusion principle. We can form an eigenfunction with this property by antisymmetrizing the product function above. This can be represented in matrix notation as

and this is the Slater Determinant representation of the total wave function. If you consider matrix properties, the exclusion principle can be seen to be enforced as the determinant will be zero if any orbital i = orbital j.

As a simple example consider a helium atom with its two electrons in their ground state configuration. The antisymmetrized wave function will then be

where the generic spin-orbital subscripts have been replaced with the definite quantum levels of the electrons. Note, in principle the set of all possible Slater determinants of order N forms a complete set for describing a system. For the helium case, we could also have contributions from the singly excited states with 1s-3s or 2s-3s; with diminishing contribution from more highly excited situations.

If we want the total wave function to consist of one Slater determinant, how do we choose the "best" one? The expectation value of the total energy for a state represented by a Slater determinant is given by the expectation value of the total Hamiltonian. The "best" determinant for the ground state should be the one which minimizes the expectation value of H; shown here in detail (and dropping the subscript approx.):

We minimize the energy with respect to changes in the Slater determinant eigenfunction through a variational derivative with respect to the spin-orbitals using a lagrange multiplier which turns out to be the energy eigenvalue for the spin-orbital. This leads to the Hartree-Fock equations.

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Created: November 12, 1998 --- Last Updated: November 28, 2001
By Mark D. Pauli