Set of supplementary notes.

9.2. MAGNETISM 137
Here, E Coul looks like Coulomb repulsion between charge densities, and E ex resembles E Coul ,
but the electrons have traded places (⇒ exchange term). For short range interactions, such as
V = δ(r 1 − r 2 ), E Coul → E ex .
We now find that the energy of the singlet state is
E singlet = 1 )
(〈ab +
2
ba|Ĥ|ab + ba〉 (9.20)
= E 0 + E ex (9.21)
The energy of the triplet state, however, is lower:
E triplet = 1 )
(〈ab −
2
ba|Ĥ|ab − ba〉 (9.22)
= E 0 − E ex (9.23)
There is, therefore, a spin dependent effective interaction in this simple model system. Note
that this interaction arises, because the electrons have been constrained to single occupancy of
the two orbitals, leaving only spin flips as the remaining degrees of freedom.
This simple example reflects a general phenomenon: the spin triplet state is symmetric under
particle exchange and must therefore be multiplied by an antisymmetric spatial wavefunction.
An antisymmetric spatial wavefunction must have nodes whenever two spatial coordinates are
equal: ψ(...., r i = r, ...r j = r, ...) = 0. So it is then clear that the particles stay farther apart
in an antisymmetrised spatial state than in a symmetric state. This reduces the effect of the
repulsive Coulomb interaction. Therefore it is because of the combination of Pauli principle and
Coulomb repulsion that states with antisymmetric spatial wavefunction (which will generally
have high spin) have lower energy.
When the orbitals concerned are orthogonal, E ex is positive in sign, i.e. the lowest energy
state is a triplet. However, if the overlapping orbitals are not orthogonal – as will happen
between two orbitals between neighbouring atoms – the interaction may be of a negative sign,
so the lowest energy is a singlet.
Heisenberg Hamiltonian
We can express the spin-dependent interaction between the electrons, which has arisen from
the direct exchange term E ex , in terms of the spin states of the two electrons, which are probed
by the spin operators Ŝ1 for electron 1 and Ŝ2 for electron 2. Because triplet and singlet states
differ in the expectation value of the magnitude of the total spin Ŝ = Ŝ1 + Ŝ2, we can use this
to distinguish between the singlet and triplet states:
Ŝ 2 = (Ŝ1 + Ŝ2) 2 = 3 2 + 2Ŝ1 · Ŝ2
This leads to the definition of a new operator Ĥspin
Ĥ spin = 1 4 (E singlet + 3E triplet ) − (E singlet − E triplet )Ŝ1 · Ŝ2