Interactions

In quantum mechanics, the observable phenomena are interactions, expressed as matrix elementsMatrixelement of operators in a function spaceFunction space. These spaces and operators are like communicating vessels, reality is neither the operation nor the representation, but the interaction. The evaluation of the corresponding matrix elementsMatrixelement requires the coupling of representations and can be factorized into an intrinsic scalar quantity that contains the physics of the interaction, and a tensorial coupling coefficientCouplingcoefficients that contains its symmetry. This factorization is first illustrated for the case ofOverlap integral overlap integralsIntegraloverlap, where the operator is just the unit operator, and then extended to the case of non-trivial operators, such as the HamiltonianHamiltonian, and electric and magnetic dipoleMagneticdipole operators. The Wigner–Eckart theoremWigner-Eckart theorem is introduced, together with the symmetry selection rulesSelection rule, both at the level of representations and subrepresentationsSubrepresentation. The results are applied to chemical reaction theory, and to the theory ofTeller the Jahn–Teller effectJahn−Teller effect. Selection rulesSelection rule are illustrated for linear and circular dichroismCircular dichroism. Finally, the polyhedral Euler theoremEulertheorem is introduced and applied to valence-bond theory for clustersClusters.