An example the hydrogen molecule

The electronic structure of the H2 molecule was considered from an elementary standpoint in Section 1.3, where the ground state was 

There are six possible spin-orbital configurations, the corresponding determinants being (omitting the variables, cf. footnote on p. 57) 

These are in fact spin eigenfunctions, as will become apparent in Chapter 4, with the following quantum numbers (S, M) respectively  

Consequently, the singlet ground state may be approximated by a mixture of the first three terms only, while the remaining functions must describe three distinct and non-mixing excited states. 

To incorporate spatial symmetry requirements, we note that all functions are invariant under spatial rotations about the bond axis, so that we need consider only how they behave under the inversion i, which is the only non-trivial operation of the appropriate symmetry subgroup C, = E, i. The effect of i on the orbitals is simply to interchange a and b, and hence (remembering that interchange of the columns changes the sign of a determinant) we obtain 

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Now let us look at the coupled fermions. Fillaux considers, as an example, the hydrogen molecule and argues that a similar pattern should take place for the oscillators in KHCO3. The vibrational ground-state wave functions are... 

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