The Coulomb Hamiltonian has been an object of study by mathematicians
since the 1930s but the first big advance was made by Kato in 1951 when he
showed that it actually had solutions and he  was
able to determine the differentiablity and analyticity properties of the
solutions. From then on, much was discovered about solutions for neutral
atoms and positive atomic ions and a little about negative atomic ions.
From a mathematical point of view the Coulomb Hamiltonian with clamped
nuclei is much like the atomic problem and so the mathematical properties
of the results of the usual electronic structure calculations are
well-understood. However almost nothing is known about the properties of
its solutions for molecular systems when the nuclei are not treated as
clamped. So from a mathematical point of view it  cannot, as yet, be said
that the full Coulomb Hamiltonian explains the existence of molecules. 

The origin of this rather uncomfortable position will be the subject of
the talk.