Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

Appendix D
Two-electron atoms
Let us assume that the spin and the coordinates are separable variables (this
is surely true if the Hamiltonian does not contain spin-dependent terms): i.e.,
one can write
ψ(1, 2) = Φ(r 1 , r 2 )χ(σ 1 , σ 2 )
(D.1)
where Φ is a function of coordinates r alone, χ of the spins σ alone. ψ(1, 2)
is always antisymmetric because electrons are fermions. It is however clear
that this can be achieved by an antisymmetric Φ and a symmetric χ, or by a
symmetric Φ and an antisymmetric χ. The spin eigenfunctions of the single
electron have two possible values that we indicate by v + and v − . We can build
three symmetric functions for the spin:
χ 1,1 = v + (σ 1 )v + (σ 2 ) (D.2)
χ 1,0 = √ 1 [v + (σ 1 )v − (σ 2 ) + v − (σ 1 )v + (σ 2 )] (D.3)
2
χ 1,−1 = v − (σ 1 )v − (σ 2 ) (D.4)
and an antisymmetric one:
χ 0,0 = 1 √
2
[v + (σ 1 )v − (σ 2 ) − v − (σ 1 )v + (σ 2 )]
(D.5)
The symmetric functions constitute a triplet and correspond to a state of
the two-electron system with total spin equal to 1 and three possible values:
−1, 0, +1, for the projection of spin along z. The antisymmetric function constitutes
a singlet and corresponds to a state with total spin equal to 0.
The value of total spin thus determines the symmetry of the spin part, and
as of consequence, of the coordinate part of the wave function. An antisymmetric
coordinate part tends to “push apart” the two electrons, since Φ(r, r) = 0,
i.e. the wave function tends to zero when the two electrons move into the
same position. The presence of electrostatic repulsion lowers the energy for
the antisymmetric coordinate wave function with respect to the corresponding
symmetric case, in which there is a finite probability that electrons are close
together (no reason for Φ(r, r) to vanish). This is why excited states in He are
labeled as ortho-helium (triplet state with symmetric spin part and antisymmetric
coordinate part) and para-helium (singlet state with antisymmetric spin
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