Hartree-Fock approximation ↩

aliases: []
shorthands: {"vpi":"\varphi_i", "vp":"\varphi"}
created: 2022-01-09 05:23:10
modified: 2022-01-19 12:39:10

The Hamiltonian operator of an atom with atomic number and electrons:

where is the one-electron Hamiltonian, similar to what we seen for the hydrogen atom:

And the one-electron interaction is:

This Hamiltonian contains only the energy of the electron system and for the core we assumed it is in rest.

This is an electron system of identical particles and electrons are fermions, so the wave function should be in Slater determinant form:

Where is the antisymmetrizing operator:

And the one-electron wavefunctions form an orthonormal set:

Now we will use a functional to describe the energy of the system similar to the case seen in the variational method:

Where is what provides the real valuedness of the energy functional.

The expectation value of the Hamiltonian:

because is Hermitian.¹

The expectation value of for general :

And the expectation value of for general :

Then, bringing it together into the expectation value of the Hamiltonian:

Where, for simplicity, we noted the one electron wavefunctions with , what can not be confused with the electron index, because there is only one and two electron terms in the expression. Note that, in the second and third term of the above equation, the terms eliminate each other leading to the following form of the functional:

Let's calculate the variation of this expression with respect to :

What can be written as:

Where is the functional derivative of :

In the spirit of the variation principle, we require that the functional derivative of vanishes. Obviously, the Slater determinant is invariant under a unitary transformation. Choose the transformation in such way that it diagonalizes the symmetric matrix. If we note the eigenvalues with and we act on the states with the matrix of the unitary transformation, we arrive at the canonical Hartree-Fock equations:

If we consider the one electron wavefunctions as and we utilize that and do not contain any spin operator (do not depend on the spin):

and

(We note that, if , then is allowed.)

If we introduce the Hartree potential:

and the non local exchange potential:

then the Hartree-Fock equations in coordinate space can be written as:

what can be solved by self consistent iterations.

We have a look at the meaning of the Lagrange multipliers. If we multiply the Hartree-Fock equations with and we integrate it over , then we get:

This is the sum of such terms from , what contains the one electron state. This if we remove the one electron state from the electron system and we do not change the other one electron states, then the energy of the system decreases with , so the ionization energy is .

By summing up the one electron energies we end up with the following relation:

where the electrostatic (Hartree) energy coming from the charge density is:

And the exchange energy is:

The energy of the interaction electron system can be calculated if we subtract the interaction and exchange energies calculated with the self consistent solutions (double-counting contributions) from the one electron energies. This result shows high similarity with the mean field approximation used in the spin models, so we can call the Hartree-Fock approximation as the mean field approximation of the interacting electron system.