Self Consistent Fields

Self Consistent Fields
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The total wavefunction of an atom is approximated as the sum of the wavefunctions of the individual atomic orbitals. This is only an approximation as it takes an average value of the electron-electron interactions between electrons in the different orbitals.

The actual wavefunction describing the atom is the solution of Schrodinger's equation:

HΨ_n = E_nΨ_n

H is the Hamiltionian which describes the energy of the system, and is the sum of the kinetic and potential energy terms. The difficulty in many electron atoms is that the expression for the potential energy is complicated, and means that a simple solution of the equation may not be obtained.

The potential energy term for the electrons is:

The second term represents the repulsion between pairs of electrons, and causes the difficulty. Whilst the solution to the Schrodinger equation may not be obtained exactly, the equation may be solved numerically, using the Hartree-Fock method.

The Hartree-Fock method of calculating the wavefunction for an atom is to take an approximation to the wavefunction, and then calculate the energy of the atom with this approximation. Then a different approximation is made, and the energy calculated again, and if the energy is lower then the new approximation is a better one. This procedure if repeated until there is no change in the energy with slight variations in the approximate wavefunction: this lowest energy wavefunction is then taken to be the atomic wavefunction.

The Hartree-Fock method in Sodium

The electronic structure of the Sodium atom can be approximated, using the orbital approximation, as 1s²2s²2p⁶3s¹. If we consider the 3s electron, we can write down the potential energy term (as above) using an average electron-electron repulsion between the 3s electron and all of the other electrons. This average repulsion term depends on the nature of the wavefunctions for each of the other orbitals. Now the Schrodinger equation may be solved (numerically) for the 3s orbital, and this gives an expression for the wavefunction of the 3s orbital.

This expression for the 3s orbital can then be used to calculate a new average electron-electron repulsive potential term, and a new Schrodinger's equation may be constructed with a different orbital, such as the 2p, as the focus of the equation. Solving this equation gives a new, better, expression for the wavefunction of the 2p orbital.

The new expression for the 2p orbital can then be used to calculate new expressions for the 2s and 1s orbitals.

The overall process is then repeated, each time calculating the new electron-electron repulsion term, until each iteration of the method results in only a negligible change in the energies of the atomic orbitals. The overall wavefunction for the atom is then that with the minimum energy with respect to the choice of the original 1s²2s²2p⁶3s¹ orbital approximation.

The orbitals calculated using this procedure are therefore self-consistent with each other, and the atomic orbitals are known as the self-consistent field Hartree-Fock (SCF-HF) orbitals.

The functional forms of the SCF-HF atomic orbitals are like those shown in the radial distribution function section, but the advantage of this procedure is that energies of the orbitals are also calculated.

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