Who (presumably a chemist) realized first the efficiency of character theory in calculations of orbitals of atoms? In which year?
Introducing groups into quantum theory, by Erhard Scholz (2006):
In quantum chemistry, representations of permutation groups made their first appearance about the same time as they did in spectroscopy. The topic was opened up by a joint publication  of two young physicists, Walter Heitler and Fritz London, who had come to Zürich on Rockefeller grants in 1926, respectively 1927, to work with E. Schrödinger. Group theoretic methods were first applied in two papers by W. Heitler [2,3].
 Heitler, W., London, F. Wechselwirkung neutraler Atome und homöpolare Bindung nach der Quanten-mechanik. Zeitschrift für Physik 44, 455–472 (1927).
 Heitler, W., Zur Gruppentheorie der homöopolaren chemischen Bindung. Zeitschrift für Physik 47, 835–858 (1928).
 Heitler, W., Zur Gruppentheorie der Wechselwirkung von Atomen. Zeitschrift für Physik 51, 805–816 (1928).
I think this realization is an immediate consequence of that of the efficiency of group theory in the quantum theory of angular momentum. The early culprits for the Gruppenpest are Weyl in his 1931 book "Gruppentheorie und quantenmechanik" and B. van der Waerden in his 1932 book "Die gruppentheoretische Methode in der Quantenmechanik". You can probably also find a discussion of this in the more recent book by Biedenharn and Louck.