where the spin singlet is required because the spatial wfn is symmetric under interchange.
The space symmetric state will be the ground state as before.
〈ψ|H|ψ〉= 2EH+
2(RAB)−
e^2
RAB+
〈
ψ∣
∣
∣
∣
e^2
r 12∣
∣
∣
∣ψ〉
From this point, we can do the calculation to obtain
Distance Energy
Calculated 0.85 ̊A -2.68 eV
Actual 0.74 ̊A -4.75 eV.wIth a multiterm wavefunction, we could get good agreement.
27.3 Importance of Unpaired Valence Electrons
Inner (closed shell) electrons stick close to nucleus so they do not get near to other atoms. The
outer (valence) electrons may participate in bonding either by sharing or migrating to the other
atom.Electrons which are paired into spin singlets don’t bond. If we try to share one of
the paired electrons, in a bonding state, with another atom, the electron from the other atom is not
antisymmetric with the (other) paired electron. Therefore only the antibonding (or some excited
state) will work and binding is unlikely. Unpaired electrons don’t have this problem.
↓↑ ↓↑↑... first four don’t bond!
The strongest bonds come from s and p orbitals (not d,f).
27.4 Molecular Orbitals
Even with additional parameters, parity symmetry in diatomic molecules implies we will have sym-
metric and antisymmetric wavefunctions for single electrons. Thesymmetric or bonding state
has a larger probability to be between the two nuclei, sees more positive charge, and is
therefore lower energy. As in our simple model of a molecule, the kinetic energy can be lowered by
sharing an electron.
There is an axis of symmetry fordiatomic molecules. This meansLzcommutes withH and
mℓis a good quantum number. The differentmℓstates, we have seen, have quite different shapes
therefore bond differently. Imagine that a valence electron is in adstate. Themℓ= 0,± 1 ,±2 are
called molecular orbitalsσ,π,δrespectively. Each has a bonding and an antibonding state.
Pictures ofmolecular orbitalsare shown forsandpstates in the following figure. Both bonding
and antibonding orbitals are shown first as atomic states then as molecular. The antibonding states
are denoted by a *.