A.34 Explanation of Hund’s first rule

Hund’s first rule of spin-alignment applies because electrons in atoms prefer to go into spatial states that are antisymmetric with respect to electron exchange. Spin alignment is then an unavoidable consequence of the weird antisymmetrization requirement.

To understand why electrons want to go into antisymmetric spatial states, the interactions between the electrons need to be considered. Sweeping them below the carpet as the discussion of atoms in chapter 5.9 did is not going to cut it.

To keep it as simple as possible, the case of the carbon atom will be considered. As the crude model of chapter 5.9 did correctly deduce, the carbon atom has two 1s electrons locked into a zero-spin singlet state, and similarly two 2s electrons also in a singlet state. Hund’s rule is about the final two electrons that are in 2p states. As far as the simple model of chapter 5.9 was concerned, these electrons can do whatever they want within the 2p subshell.

To go one better than that, the correct interactions between the two 2p electrons will need to be considered. To keep the arguments manageable, it will still be assumed that the effects of the 1s and 2s electrons are independent of where the 2p electrons are.

Call the 2p electrons

where

Now assume that electrons

The two states

The expectation value of energy is

That can be multiplied out and then simplified by noting that in the various inner product integrals involving the single-electron Hamiltonians, the integral over the coordinate unaffected by the Hamiltonian is either zero or one because of orthonormality. Also, the inner product integrals involving

The simplified expectation energy is then:

The first two terms are the single-electron energies of states

The final one of the four terms is the interesting one for
Hund’s rule; it determines how the two electrons occupy the
two states

where

The sign of this inner product can be guesstimated. If

And since this integral is multiplied by

This leaves the philosophical question why for the two electrons of the hydrogen molecule in chapter 5.2 the symmetric state is energetically most favorable, while the antisymmetric state is the one for the 2p electrons. The real difference is in the kinetic energy. In both cases, the antisymmetric combination reduces the Coulomb repulsion energy between the electrons, and in the hydrogen molecule model, it also increases the nuclear attraction energy. But in the hydrogen molecule model, the symmetric state achieves a reduction in kinetic energy that is more than enough to make up for it all. For the 2p electrons, the reduction in kinetic energy is nil. When the positive component wave functions of the hydrogen molecule model are combined into the symmetric state, they allow greater access to fringe areas farther away from the nuclei. Because of the uncertainty principle, less confined electrons tend to have less indeterminacy in momentum, hence less kinetic energy. On the other hand, the 2p states are half positive and half negative, and even their symmetric combination reduces spatial access for the electrons in half the locations.