Helium Atom - Hartree-Fock Method

Hartree-Fock Method

The Hartree-Fock method is used for a variety of atomic systems. However it is just an approximation, and there are more accurate and efficient methods used today to solve atomic systems. The "many-body problem" for helium and other few electron systems can be solved quite accurately. For example the ground state of helium is known to fifteen digits. In Hartree-Fock theory, the electrons are assumed to move in a potential created by the nucleus and the other electrons. The Hamiltonian for helium with 2 electrons can be written as a sum of the Hamiltonians for each electron:

where the zero-order unperturbed Hamiltonian is

while the perturbation term:

is the electron-electron interaction. H₀ is just the sum of the two hydrogenic Hamiltonians:

where

E_n1, the energy eigenvalues and, the corresponding eigenfunctions of the hydrogenic Hamiltonian will denote the normalized energy eigenvalues and the normalized eigenfunctions. So:

where

Neglecting the electron-electron repulsion term, the Schrödinger equation for the spatial part of the two-electron wave function will reduce to the 'zero-order' equation

This equation is separable and the eigenfunctions can be written in the form of single products of hydrogenic wave functions:

The corresponding energies are (in a.u.):

Note that the wave function

An exchange of electron labels corresponds to the same energy . This particular case of degeneracy with respect to exchange of electron labels is called exchange degeneracy. The exact spatial wave functions of two-electron atoms must either be symmetric or antisymmetric with respect to the interchange of the coordinates and of the two electrons. The proper wave function then must be composed of the symmetric (+) and antisymmetric(-) linear combinations:

This comes from Slater determinants.

The factor normalizes . In order to get this wave function into a single product of one-particle wave functions, we use the fact that this is in the ground state. So . So the will vanish, in agreement with the original formulation of the Pauli exclusion principle, in which two electrons cannot be in the same state. Therefore the wave function for helium can be written as

Where and use the wave functions for the hydrogen Hamiltonian. For helium, Z = 2 from

where E a.u. which is approximately -108.8 eV, which corresponds to an ionization potential V a.u. ( eV). The experimental values are E a.u. ( eV) and V a.u. ( eV).

The energy that we obtained is too low because the repulsion term between the electrons was ignored, whose effect is to raise the energy levels. As Z gets bigger, our approach should yield better results, since the electron-electron repulsion term will get smaller.

So far a very crude independent-particle approximation has been used, in which the electron-electron repulsion term is completely omitted. Splitting the Hamiltonian showed below will improve the results:

where

and

V(r) is a central potential which is chosen so that the effect of the perturbation is small. The net effect of each electron on the motion of the other one is to screen somewhat the charge of the nucleus, so a simple guess for V(r) is

where S is a screening constant and the quantity Z_e is the effective charge. The potential is a Coulomb interaction, so the corresponding individual electron energies are given (in a.u.) by

and the corresponding wave function is given by

If Z_e was 1.70, that would make the expression above for the ground state energy agree with the experimental value E₀ = -2.903 a.u. of the ground state energy of helium. Since Z = 2 in this case, the screening constant is S = .30. For the ground state of helium, for the average shielding approximation, the screening effect of each electron on the other one is equivalent to about of the electronic charge.