Molecular term symbol

In molecular physics, the molecular term symbol is a shorthand expression of the group representation and angular momenta that characterize the state of a molecule, i.e. its electronic quantum state which is an eigenstate of the electronic molecular Hamiltonian. It is the equivalent of the term symbol for the atomic case. However, the following presentation is restricted to the case of homonuclear diatomic molecules, or other symmetric molecules with an inversion centre. For heteronuclear diatomic molecules, the u/g symbol does not correspond to any exact symmetry of the electronic molecular Hamiltonian. In the case of less symmetric molecules the molecular term symbol contains the symbol of the group representation to which the molecular electronic state belongs.

It has the general form:

where

For atoms, we use S, L, J and M_J to characterize a given state. In linear molecules, however, the lack of spherical symmetry destroys the relationship ${\displaystyle [{\hat {\mathbf {L} }}^{2},{\hat {H}}]=0}$, so L ceases to be a good quantum number. A new set of operators have to be used instead: ${\displaystyle \{{\hat {\mathbf {S} }}^{2},{\hat {\mathbf {S} }}_{z},{\hat {\mathbf {L} }}_{z},{\hat {\mathbf {J} }}_{z}={\hat {\mathbf {S} }}_{z}+{\hat {\mathbf {L} }}_{z}\}}$, where the z-axis is defined along the internuclear axis of the molecule. Since these operators commute with each other and with the Hamiltonian on the limit of negligible spin-orbit coupling, their eigenvalues may be used to describe a molecule state through the quantum numbers S, M_S, M_L and M_J.

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